An Introduction to Categorical Data Analysis
Alan Agresti, 3rd Edition

Chapter 1: Introduction

1.1 Categorical Response Data

Binary variable: a variable taking on one of two mutually exclusive categories
Nominal variable: a categorical variable having unordered/arbitrarily ordered scales
Ordinal variable: a variable having naturally ordered categories

1.2 Probability Distributions for Categorical Data

Parametric inferential statistical analysis requires assumptions about the probability distributions of categorical response variables

Key distributions for categorical response:

Binomial
Multinomial

(1.2.1) Binomial Distribution

Fixed number n of independent & identical trials with two possible outcomes for each; Identical means that P[Success] is the same for each trial; Independent means the response outcomes are independent random variables (the outcome of one trial has no effect on the outcome of another, AKA Bernoulli Trials)
\(\pi\) = probability of success for each trial
\(Y\) = the number of success out of the n trials
Under the assumption of n independent, identical trials Y has the binomial distribution with index n and parameter \(\pi\)
Probability of a particular outcome y for Y equals:
\[P[y] = \frac{n!}{y!(n-y)!}\pi^{y}(1-\pi)^{n-y}, y = 0,1,2,...,n\]
\[E[Y] = \mu = n\pi; \sigma = \sqrt{n\pi(1-\pi)}\]

When \(\pi = 0.5\), the binomial distribution is symmetric

For fixed n, the distribution becomes more bell-shaped as \(\pi\) approaches 0.5
For fixed \(\pi\), the distribution becomes more bell-shaped as n increases

Normal Approximation to Binomial Distribution:

When n is large, \(\mu = n\pi; \sigma = \sqrt{n\pi(1-\pi)}\)
General guideline: the expected number of each outcome should be at least 5
That is, the closer \(\pi\) is to 0 or 1, more observations are required before a symmetric bell-shape distribution occurs

(1.2.2) Multinomial Distribution

When the observations of nominal/ordinal categorical variables with more than two possible outcomes are independent with the same categorical probabilities for each category, the probability distribution of counts in the response categories is said to be multinomial
c is the number of possible outcome categories
\((\pi_{1}, \pi_{2}, ..., \pi_{c})\) where \(\Sigma_{j} = \pi_{j} = 1\) denotes the probability of each category
For n independent observations, the multinomial probability that \(y_{1}\) fall in category 1, \(y_{2}\) fall in category 2, …, \(y_{c}\) fall in category c, where \(\Sigma_{j} y_{j} = n\), equals:
\[P[y_{1}, y_{2}, ..., y_{c}] = \frac{n!}{y_{1}!y_{2}!...y_{c}!}\pi_{1}^{y_{1}}\pi_{2}^{y_{2}}...\pi_{c}^{y_{c}}\]
The binomial distribution is a special case of the multinomial with c = 2 categories
Additionally, we will not need this formula because our focus is on inference methods that use sampling distributions of statistics computed from multinomial counts, and those sampling distributions are approximately normal
Explore the binomial distribution here

1.3 Statistical Inference for a Proportion

(1.3.1) Likelihood & Maximum Likelihood Estimation

The probability of the observed data expressed as a function of the parameter, is called the likelihood function
The MLE for the binomial parameter \(\pi\) is: \[\hat{\pi} = y/n\]
Results that apply to sample means with random sampling also apply to sample proportions
- Central Limit Theorem: the sampling distribution of the sampling proportion \(\hat{\pi}\) is approximately normal for large n
- Law of Large Numbers: \(\hat{\pi}\) converges to the population proportion \(\pi\) as n increases

(1.3.2) Significance Tests about a binomial parameter

For the binomial distribution, the ML estimator of \(\pi\) is the sample proportion, \(\hat{\pi}\)
\(\hat{\pi}\) has a sampling distribution characterized by:
- \(E[\hat{\pi}] = \frac{y}{n} = \pi\)
- \(\sigma(\hat{\pi}) = \sqrt{\frac{\hat{\pi}(1-\hat{\pi})}{n}} = \sqrt{\frac{\pi(1-\pi)}{n}}\)
Assuming \(H_{0}: \pi = \pi_{0}\) is true, the SE of \(\hat{\pi}\) is:
- \(SE_{0} = \sqrt{\frac{\pi_{0}(1-\pi_{0})}{n}}\)
- referred to as the null standard error
Test statistic:
- \(z = \frac{\hat{\pi}-\pi_{0}}{SE_{0}} = \frac{\hat{\pi}-\pi_{0}}{\sqrt{\frac{\pi_{0}(1-\pi_{0})}{n}}}\)
z measures the number of SE’s that \(\hat{\pi}\) falls from the \(H_{0}\) value

Ch. 1 Code

# obtain and plot Binomial(n=10, pi=1/4) probabilities
options(scipen = 999)
y <- 0:10
prob <- round(dbinom(y, 10, 0.25), 6)
prob

 [1] 0.056314 0.187712 0.281568 0.250282 0.145998 0.058399 0.016222
 [8] 0.003090 0.000386 0.000029 0.000001

plot(y, prob, pch = 19, xlab = "Number of Diamonds in 10 Draws", ylab = "Probability")

# Poisson orange hair probs, lambda = 8
orangehair <- round(dpois(0:20, 8), 4)
orangehair

 [1] 0.0003 0.0027 0.0107 0.0286 0.0573 0.0916 0.1221 0.1396 0.1396 0.1241
[11] 0.0993 0.0722 0.0481 0.0296 0.0169 0.0090 0.0045 0.0021 0.0009 0.0004
[21] 0.0002

plot(0:20, orangehair, pch = 19, xlab = "Number of Orange-haired Students", 
    ylab = "Probability")

# Exact prob of at most 10 orange-haired students
ppois(10, 8)

[1] 0.8158858

dpois(0, 8) + dpois(1, 8) + dpois(2, 8) + dpois(3, 8) + dpois(4, 8) + dpois(5, 
    8) + dpois(6, 8) + dpois(7, 8) + dpois(8, 8) + dpois(9, 8) + dpois(10, 8)

[1] 0.8158858

# Normal approximation to above
zstar <- (10.5 - 8)/sqrt(8)
pnorm(zstar)

[1] 0.8116204

# Multinomial sample of size 4, 4 categories, probs = 0.4 0.3 0.2 0.1 prob
# of getting one of each
dmultinom(c(1, 1, 1, 1), size = 4, prob = c(0.4, 0.3, 0.2, 0.1), log = FALSE)

[1] 0.0576

24 * (0.4^1) * (0.3^1) * (0.2^1) * (0.1^1)

[1] 0.0576

# Multinomial calculations Shafer Lecture 3
dmultinom(c(3, 2, 7), size = 12, prob = c(0.2, 0.15, 0.65), log = FALSE)

[1] 0.06988616

dmultinom(c(4, 0, 8), size = 12, prob = c(0.2, 0.15, 0.65), log = FALSE)

[1] 0.02523667

dbinom(0, 12, 0.2) + dbinom(1, 12, 0.2)

[1] 0.2748779

pbinom(1, 12, 0.2)

[1] 0.2748779

# Plot Likelihood for Binomial (n = 5, y = 2)
L.binomial <- function(pi, n, y) choose(n, y) * (pi^y) * ((1 - pi)^(n - y))
pix <- seq(0, 1, length = 1000)
like <- L.binomial(pix, 5, 2)
plot(pix, like, type = "l", lwd = 3, xlab = "Success Probability", ylab = "Likelihood", 
    col = "orange")

# Wald confidence interval for binomial proportion
0.4 + c(-1, 1) * qnorm(0.975) * sqrt(0.4 * 0.6/5)

[1] -0.02940659  0.82940659

y <- 2
n <- 5
pi.hat <- y/n
pi.hat + c(-1, 1) * qnorm(0.975) * sqrt(pi.hat * (1 - pi.hat)/5)

[1] -0.02940659  0.82940659

binom.confint(y, n, conf.level = 0.95, method = "asymptotic")

      method x n mean       lower     upper
1 asymptotic 2 5  0.4 -0.02940659 0.8294066

# Binomial LL plot to show effect of increasing sample size
LL.binomial <- function(pi, n, y) y * log(pi) + (n - y) * log(1 - pi)
pi1 <- seq(0, 1, length = 1000)
z1 <- LL.binomial(pi1, 4, 1) - (1 * log(1) + 3 * log(3) - 4 * log(4))
z2 <- LL.binomial(pi1, 40, 10) - (10 * log(10) + 30 * log(30) - 40 * log(40))
z3 <- LL.binomial(pi1, 400, 100) - (100 * log(100) + 300 * log(300) - 400 * 
    log(400))
yrange <- c(-4, 0)
plot(pi1, z1, type = "l", lwd = 3, xlab = "Success Probability", ylab = "Log-Likelihood (shifted up to zero)", 
    ylim = yrange, col = "black")
par(new = T)
plot(pi1, z2, type = "l", lwd = 3, xlab = "", ylab = "", ylim = yrange, col = "blue")
par(new = T)
plot(pi1, z3, type = "l", lwd = 3, xlab = "", ylab = "", ylim = yrange, col = "purple")

# Wald confidence interval for binomial proportion
0.25 + c(-1, 1) * qnorm(0.975) * sqrt(0.25 * 0.75/4)

[1] -0.1743447  0.6743447

y <- 1
n <- 4
pi.hat <- y/n
pi.hat + c(-1, 1) * qnorm(0.975) * sqrt(pi.hat * (1 - pi.hat)/4)

[1] -0.1743447  0.6743447

binom.confint(y, n, conf.level = 0.95, method = "asymptotic")

      method x n mean      lower     upper
1 asymptotic 1 4 0.25 -0.1743447 0.6743447

# Finding LRCI for binomial proportion
LBCI.endpoint <- function(pi) {
    y <- 1
    n <- 4
    y * log(pi) + (n - y) * log(1 - pi) - (y * log(y) + (n - y) * log(n - y) - 
        n * log(n) - 0.5 * qchisq(0.95, 1))
}
left <- uniroot(LBCI.endpoint, c(0, 0.25))
left$root

[1] 0.01623382

right <- uniroot(LBCI.endpoint, c(0.25, 1))
right$root

[1] 0.7224464

y <- 1
n <- 4
y * log(y) + (n - y) * log(n - y) - n * log(n) - 0.5 * qchisq(0.95, 1)

[1] -4.17007

binom.confint(y, n, conf.level = 0.95, methods = "lrt")

  method x n mean      lower     upper
1    lrt 1 4 0.25 0.01623238 0.7224088

# Now plot the LL function and cut line and LRCI points on the plot
pi <- seq(1e-04, 0.8, length = 1000)
LL <- log(pi) + 3 * log(1 - pi)
LLmax <- log(0.25) + 3 * log(0.75)
LLmax

[1] -2.249341

LLmax - 0.5 * qchisq(0.95, 1)

[1] -4.17007

yrange <- range(-5, -2.249341)
xrange <- range(pi)
plot(pi, LL, type = "l", lwd = 3, xlab = "pi", ylab = "Log Likelihood", xlim = xrange, 
    ylim = yrange)
cutline <- rep(-4.17007, length(pi))
par(new = T)
plot(pi, cutline, type = "l", lwd = 3, lty = 3, xlab = "", ylab = "", xlim = xrange, 
    ylim = yrange, col = "purple")
points(0.01623238, -4.17007, pch = 19, cex = 1.5, col = "darkgreen")
points(0.7224088, -4.17007, pch = 19, cex = 1.5, col = "darkgreen")

# Now Shafer-type (Lecture 2, p.14) Modified Wald confidence interval
y <- 1
n <- 4
p.tilde <- (y + 0.5)/(n + 1)
p.tilde + c(-1, 1) * qnorm(0.975) * sqrt(p.tilde * (1 - p.tilde)/(n + 1))

[1] -0.1016731  0.7016731

# Working on the logit scale: CI for logit of pi
log(1/3) + c(-1, 1) * qnorm(0.975)/sqrt(0.75)

[1] -3.361784  1.164559

hi <- log(1/3) + c(-1, 1) * qnorm(0.975)/sqrt(0.75)
exp(hi)/(1 + exp(hi))

[1] 0.0335114 0.7621602

binom.confint(y, n, conf.level = 0.95, methods = "logit")

  method x n mean     lower     upper
1  logit 1 4 0.25 0.0335114 0.7621602

# Goodness-of-Fit test - testing 12:3:1 Genetic Theory
1 - pchisq(6.9106, 2)

[1] 0.03157783

obs.data <- c(1550, 400, 100)
gof.test <- chisq.test(obs.data, p = c(12/16, 3/16, 1/16))
gof.test


    Chi-squared test for given probabilities

data:  obs.data
X-squared = 6.9106, df = 2, p-value = 0.03158

# GOF test for primary/secondard infection example Pearson chi-square
# calculation and p-value; Agresti, p.37
xsq <- ((30 - 38.06590216)^2/38.06590216 + (63 - 38.9943388)^2/38.9943388 + 
    (63 - 78.93975904)^2/78.93975904)
xsq

[1] 19.70606

1 - pchisq(xsq, 1)

[1] 0.000009031458

# Likelihood chi-square calculation and p-value; Agresti, p.38
gsq <- 2 * (30 * log(30/38.06590216) + 63 * log(63/38.9943388) + 63 * log(63/78.93975904))
gsq

[1] 17.73787

1 - pchisq(gsq, 1)

[1] 0.00002535289

# Agresti Intro CDA book, Chapter 1 work Agresti p.7ff HT and CI binomial
# proportion
y <- 837
n <- 1810
prop.test(y, n, p = 0.5, alternative = "two.sided", correct = F)


    1-sample proportions test without continuity correction

data:  y out of n, null probability 0.5
X-squared = 10.219, df = 1, p-value = 0.00139
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.4395653 0.4854557
sample estimates:
        p 
0.4624309

binom.confint(y, n, conf.level = 0.95, method = "asymptotic")

      method   x    n      mean     lower     upper
1 asymptotic 837 1810 0.4624309 0.4394616 0.4854003

binom.confint(y, n, conf.level = 0.95, method = "wilson")

  method   x    n      mean     lower     upper
1 wilson 837 1810 0.4624309 0.4395653 0.4854557

# So the CI method in the prop.test function is the Wilson/Score method
# Agresti p.9 Score CI
y <- 9
n <- 10
binom.confint(y, n, conf.level = 0.95, method = "wilson")

  method x  n mean   lower     upper
1 wilson 9 10  0.9 0.59585 0.9821238

binom.confint(y, n, conf.level = 0.95, method = "asymptotic")

      method x  n mean     lower    upper
1 asymptotic 9 10  0.9 0.7140615 1.085939

binom.confint(y, n, conf.level = 0.95, method = "lrt")

  method x  n mean     lower     upper
1    lrt 9 10  0.9 0.6283633 0.9940051

# Agresti p.9 Witmer Modified Wald also called Agresti-Coull
y <- 10
n <- 10
pi.tilde2 <- (y + 2)/(n + 4)
pi.tilde2

[1] 0.8571429

pi.tilde2 + c(-1, 1) * qnorm(0.975) * sqrt(pi.tilde2 * (1 - pi.tilde2)/14)

[1] 0.6738432 1.0404425

binom.confint(y, n, conf.level = 0.95, method = "wilson")

  method  x  n mean     lower upper
1 wilson 10 10    1 0.7224672     1

# Agresti p.12 Binomial example
y <- 9
n <- 10
pi.hat <- y/n
se <- sqrt(pi.hat * (1 - pi.hat)/10)
WTS <- (pi.hat - 0.5)/se
CSTS <- WTS^2
CSTS

[1] 17.77778

Wpvalue <- 1 - pchisq(CSTS, 1)
Wpvalue

[1] 0.00002482661

prop.test(y, n, p = 0.5, alternative = "two.sided", correct = F)


    1-sample proportions test without continuity correction

data:  y out of n, null probability 0.5
X-squared = 6.4, df = 1, p-value = 0.01141
alternative hypothesis: true p is not equal to 0.5
95 percent confidence interval:
 0.5958500 0.9821238
sample estimates:
  p 
0.9

# Agresti p.19 Binomial example
y <- 9
n <- 10
binom.confint(y, n, conf.level = 0.95, method = "asymptotic")

      method x  n mean     lower    upper
1 asymptotic 9 10  0.9 0.7140615 1.085939

binom.confint(y, n, conf.level = 0.95, method = "wilson")

  method x  n mean   lower     upper
1 wilson 9 10  0.9 0.59585 0.9821238

binom.confint(y, n, conf.level = 0.95, method = "agresti-coull")

         method x  n mean     lower    upper
1 agresti-coull 9 10  0.9 0.5740323 1.003941

binom.confint(y, n, conf.level = 0.95, method = "lrt")

  method x  n mean     lower     upper
1    lrt 9 10  0.9 0.6283633 0.9940051

binom.test(y, n, 0.5, alternative = "two.sided")


    Exact binomial test

data:  y and n
number of successes = 9, number of trials = 10, p-value = 0.02148
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval:
 0.5549839 0.9974714
sample estimates:
probability of success 
                   0.9

binom.test(y, n, 0.5, alternative = "greater")


    Exact binomial test

data:  y and n
number of successes = 9, number of trials = 10, p-value = 0.01074
alternative hypothesis: true probability of success is greater than 0.5
95 percent confidence interval:
 0.6058367 1.0000000
sample estimates:
probability of success 
                   0.9

exactci::binom.exact(y, n, 0.5, alternative = "greater", midp = T)


    Exact one-sided binomial test, mid-p version

data:  y and n
number of successes = 9, number of trials = 10, p-value = 0.005859
alternative hypothesis: true probability of success is greater than 0.5
95 percent confidence interval:
 0.6504873 1.0000000
sample estimates:
probability of success 
                   0.9

exactci::binom.exact(y, n, 0.5, alternative = "two.sided", midp = T)


    Exact two-sided binomial test (central method), mid-p version

data:  y and n
number of successes = 9, number of trials = 10, p-value = 0.01172
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval:
 0.5965206 0.9950264
sample estimates:
probability of success 
                   0.9

midPci(y, n, 0.95)




data:  

95 percent confidence interval:
 0.5966 0.9946

Chapter 2: Analyzing Contingency Tables

Ch. 2 Code

# p.30 Table 2.2

# Find Wald CI for difference of binomial proportions

n11 <- 189
n1 <- 11034
pi1.hat <- n11/n1
n21 <- 104
n2 <- 11037
pi2.hat <- n21/n2
diff <- pi1.hat - pi2.hat
se.diff <- sqrt(pi1.hat * (1 - pi1.hat)/n1 + pi2.hat * (1 - pi2.hat)/n2)
se.diff

[1] 0.001539964

WCI.diff <- diff + c(-1, 1) * qnorm(0.975) * se.diff
WCI.diff

[1] 0.004687751 0.010724297

prop.test(c(n11, n21), c(n1, n2), conf.level = 0.95, correct = F)


    2-sample test for equality of proportions without continuity
    correction

data:  c(n11, n21) out of c(n1, n2)
X-squared = 25.014, df = 1, p-value = 0.0000005692
alternative hypothesis: two.sided
95 percent confidence interval:
 0.004687751 0.010724297
sample estimates:
    prop 1     prop 2 
0.01712887 0.00942285

# Find Agresti-Caffo adjusted CI for difference of binomial proportions -
# adds 1 to each cell install.packages('PropCIs') library(PropCIs)

wald2ci(n11, n1, n21, n2, conf.level = 0.95, adjust = "AC")




data:  

95 percent confidence interval:
 0.004676768 0.010732536
sample estimates:
[1] 0.007704652

# Find SCORE CI for difference of binomial proportions
# install.packages('PropCIs') library(PropCIs)

diffscoreci(n11, n1, n21, n2, conf.level = 0.95)  # Produces SCORE CI for diff




data:  

95 percent confidence interval:
 0.004716821 0.010788501

# Relative risk (RR)

# Find Modified Wald CI for RR

phi.hat <- log(pi1.hat/pi2.hat)
se.phi.hat <- sqrt((1 - pi1.hat)/n11 + (1 - pi2.hat)/n21)
WCI.trans.RR <- phi.hat + c(-1, 1) * qnorm(0.975) * se.phi.hat
WCI.RR <- exp(WCI.trans.RR)
WCI.RR

[1] 1.433031 2.305884

RR.hat <- pi1.hat/pi2.hat
aa <- exp(qnorm(0.975) * se.phi.hat)
RR.hat

[1] 1.817802

aa

[1] 1.268501

(RR.hat/aa)

[1] 1.433031

(aa * RR.hat)  # So this modified WCI is of the form ( (1/a)*RR , a*RR )

[1] 2.305884

# Find SCORE CI for RR

riskscoreci(n11, n1, n21, n2, conf.level = 0.95)




data:  

95 percent confidence interval:
 1.433904 2.304713

# Odds Ratio (OR)

# Find Modified Wald CI for OR

n11

[1] 189

n1

[1] 11034

n21

[1] 104

n2

[1] 11037

theta.hat <- (n11 * (n2 - n21))/(n21 * (n1 - n11))
Ltheta.hat <- log(theta.hat)
se.Ltheta.hat <- sqrt(1/n11 + 1/(n2 - n21) + 1/n21 + 1/(n1 - n11))
WCI.LOR <- Ltheta.hat + c(-1, 1) * qnorm(0.975) * se.Ltheta.hat
WCI.OR <- exp(WCI.LOR)
WCI.OR

[1] 1.440042 2.330780

bb <- exp(qnorm(0.975) * se.Ltheta.hat)
theta.hat

[1] 1.832054

bb

[1] 1.272222

(theta.hat/bb)

[1] 1.440042

(bb * theta.hat)  # So this modified WCI is of the form ( (1/b)*RR , b*RR )

[1] 2.33078

epitools::oddsratio(c(n11, (n1 - n11), n21, (n2 - n21)), method = "wald", conf = 0.95, 
    correct = F)

$data
          Outcome
Predictor  Disease1 Disease2 Total
  Exposed1      189    10845 11034
  Exposed2      104    10933 11037
  Total         293    21778 22071

$measure
          odds ratio with 95% C.I.
Predictor  estimate    lower   upper
  Exposed1 1.000000       NA      NA
  Exposed2 1.832054 1.440042 2.33078

$p.value
          two-sided
Predictor       midp.exact    fisher.exact      chi.square
  Exposed1              NA              NA              NA
  Exposed2 0.0000004989646 0.0000005032836 0.0000005691897

$correction
[1] FALSE

attr(,"method")
[1] "Unconditional MLE & normal approximation (Wald) CI"

# Find SCORE CI for OR

orscoreci(n11, n1, n21, n2, conf.level = 0.95)




data:  

95 percent confidence interval:
 1.440802 2.329551

# Chi-square Test of Indendence 2 by 3 Table 2.4 on p.39

## PROBLEMATIC BECAUSE LINK WILL NOT WORK & DATA CANNOT BE IMPORTED##

Political <- read.table("/Users/Patrick/Dropbox/data/agresti/Political.dat", 
    header = T)
head(Political)

  person gender party
1      1 female   Dem
2      2 female   Dem
3      3 female   Dem
4      4 female   Dem
5      5 female   Dem
6      6 female   Dem

tail(Political)

     person gender party
2445   2445   male   Ind
2446   2446   male   Ind
2447   2447   male   Ind
2448   2448   male   Ind
2449   2449   male   Ind
2450   2450   male   Ind

summary(Political)

     person          gender     party     
 Min.   :   1.0   female:1357   Dem: 825  
 1st Qu.: 613.2   male  :1093   Ind:1088  
 Median :1225.5                 Rep: 537  
 Mean   :1225.5                           
 3rd Qu.:1837.8                           
 Max.   :2450.0

str(Political)

'data.frame':   2450 obs. of  3 variables:
 $ person: int  1 2 3 4 5 6 7 8 9 10 ...
 $ gender: Factor w/ 2 levels "female","male": 1 1 1 1 1 1 1 1 1 1 ...
 $ party : Factor w/ 3 levels "Dem","Ind","Rep": 1 1 1 1 1 1 1 1 1 1 ...

Party <- factor(Political$party, levels = c("Dem", "Rep", "Ind"))
GenderGap <- xtabs(~gender + Party, data = Political)
GenderGap

        Party
gender   Dem Rep Ind
  female 495 272 590
  male   330 265 498

GenderGap2 <- matrix(c(495, 272, 590, 330, 265, 498), ncol = 3, byrow = T)
GenderGap2

     [,1] [,2] [,3]
[1,]  495  272  590
[2,]  330  265  498

chisq.test(GenderGap)


    Pearson's Chi-squared test

data:  GenderGap
X-squared = 12.569, df = 2, p-value = 0.001865

y11 <- 495
y12 <- 272
y13 <- 590
n1p <- y11 + y12 + y13
y21 <- 330
y22 <- 265
y23 <- 498
n2p <- y21 + y22 + y23
npp <- n1p + n2p
np1 <- y11 + y21
np2 <- y12 + y22
np3 <- y13 + y23
e11 <- np1 * n1p/npp
e12 <- np2 * n1p/npp
e13 <- np3 * n1p/npp
e21 <- np1 * n2p/npp
e22 <- np2 * n2p/npp
e23 <- np3 * n2p/npp
X2 <- (y11 - e11)^2/e11 + (y12 - e12)^2/e12 + (y13 - e13)^2/e13 + (y21 - e21)^2/e21 + 
    (y22 - e22)^2/e22 + (y23 - e23)^2/e23
X2

[1] 12.56926

(1 - pchisq(X2, 2))

[1] 0.00186475

G2 <- 2 * (y11 * log(y11/e11) + y12 * log(y12/e12) + y13 * log(y13/e13) + y21 * 
    log(y21/e21) + y22 * log(y22/e22) + y23 * log(y23/e23))
G2

[1] 12.6009

(1 - pchisq(G2, 2))

[1] 0.001835477

# Residuals on p.40 and Mosaic Plot on p.41

stdres <- chisq.test(GenderGap)$stdres
stdres

        Party
gender         Dem       Rep       Ind
  female  3.272365 -2.498557 -1.032199
  male   -3.272365  2.498557  1.032199

mosaic(GenderGap, gp = shading_Friendly, residuals = stdres, residuals_type = "Std\nresiduals", 
    labeling = labeling_residuals)

# Now let's partition the above G2 TS into two orthogonal pieces; see
# pp.41-42

# First piece

y11 <- 495
y12 <- 272
n1p <- y11 + y12
y21 <- 330
y22 <- 265
n2p <- y21 + y22
npp <- n1p + n2p
np1 <- y11 + y21
np2 <- y12 + y22
e11 <- np1 * n1p/npp
e12 <- np2 * n1p/npp
e21 <- np1 * n2p/npp
e22 <- np2 * n2p/npp
G2 <- 2 * (y11 * log(y11/e11) + y12 * log(y12/e12) + y21 * log(y21/e21) + y22 * 
    log(y22/e22))
G2

[1] 11.53574

(1 - pchisq(G2, 1))

[1] 0.0006827076

# Second piece

y11 <- 767
y12 <- 590
n1p <- y11 + y12
y21 <- 595
y22 <- 498
n2p <- y21 + y22
npp <- n1p + n2p
np1 <- y11 + y21
np2 <- y12 + y22
e11 <- np1 * n1p/npp
e12 <- np2 * n1p/npp
e21 <- np1 * n2p/npp
e22 <- np2 * n2p/npp
G2 <- 2 * (y11 * log(y11/e11) + y12 * log(y12/e12) + y21 * log(y21/e21) + y22 * 
    log(y22/e22))
G2

[1] 1.065158

(1 - pchisq(G2, 1))

[1] 0.3020418

# p.44 Table 2.6

Malform <- matrix(c(17066, 14464, 788, 126, 37, 48, 38, 5, 1, 1), ncol = 2)
Malform

      [,1] [,2]
[1,] 17066   48
[2,] 14464   38
[3,]   788    5
[4,]   126    1
[5,]    37    1

CMHtest(Malform, rscores = c(0, 0.5, 1.5, 4, 7))

Cochran-Mantel-Haenszel Statistics 

                 AltHypothesis   Chisq Df     Prob
cor        Nonzero correlation  6.5699  1 0.010372
rmeans  Row mean scores differ 12.0817  4 0.016754
cmeans  Col mean scores differ  6.5699  1 0.010372
general    General association 12.0817  4 0.016754

M <- sqrt(6.5699)
M

[1] 2.563182

1 - pnorm(M)

[1] 0.005185889

R <- sqrt(6.5699/32574)
R

[1] 0.01420181

# FET: Fisher's Exact Test

# In-class Example

Madeuptea <- matrix(c(5, 2, 1, 7), ncol = 2)
Madeuptea

     [,1] [,2]
[1,]    5    1
[2,]    2    7

fisher.test(Madeuptea, alternative = "greater")


    Fisher's Exact Test for Count Data

data:  Madeuptea
p-value = 0.03497
alternative hypothesis: true odds ratio is greater than 1
95 percent confidence interval:
 1.179718      Inf
sample estimates:
odds ratio 
  13.59412

choose(6, 5) * choose(9, 2)/choose(15, 7) + choose(6, 6) * choose(9, 1)/choose(15, 
    7)

[1] 0.03496503

choose(7, 5) * choose(8, 1)/choose(15, 6) + choose(7, 6) * choose(8, 0)/choose(15, 
    6)

[1] 0.03496503

f0 <- choose(6, 0) * choose(9, 7)/choose(15, 7)
f1 <- choose(6, 1) * choose(9, 6)/choose(15, 7)
f2 <- choose(6, 2) * choose(9, 5)/choose(15, 7)
f3 <- choose(6, 3) * choose(9, 4)/choose(15, 7)
f4 <- choose(6, 4) * choose(9, 3)/choose(15, 7)
f5 <- choose(6, 5) * choose(9, 2)/choose(15, 7)
f6 <- choose(6, 6) * choose(9, 1)/choose(15, 7)
cbind(0:6, c(f0, f1, f2, f3, f4, f5, f6))

     [,1]        [,2]
[1,]    0 0.005594406
[2,]    1 0.078321678
[3,]    2 0.293706294
[4,]    3 0.391608392
[5,]    4 0.195804196
[6,]    5 0.033566434
[7,]    6 0.001398601

fisher.test(Madeuptea, alternative = "two.sided")


    Fisher's Exact Test for Count Data

data:  Madeuptea
p-value = 0.04056
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
   0.8646648 934.0087368
sample estimates:
odds ratio 
  13.59412

choose(6, 5) * choose(9, 2)/choose(15, 7) + choose(6, 6) * choose(9, 1)/choose(15, 
    7) + choose(6, 0) * choose(9, 7)/choose(15, 7)

[1] 0.04055944

ormidp.test(5, 2, 1, 7, or = 1)

   one.sided  two.sided
1 0.01818182 0.03636364

or.midp(c(5, 2, 1, 7), conf.level = 0.95)$conf.int

[1]   1.154754 465.760830

# Textbook Example on p.47

Fisherstea <- matrix(c(3, 1, 1, 3), ncol = 2)
Fisherstea

     [,1] [,2]
[1,]    3    1
[2,]    1    3

fisher.test(Fisherstea)


    Fisher's Exact Test for Count Data

data:  Fisherstea
p-value = 0.4857
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
   0.2117329 621.9337505
sample estimates:
odds ratio 
  6.408309

fisher.test(Fisherstea, alternative = "greater")


    Fisher's Exact Test for Count Data

data:  Fisherstea
p-value = 0.2429
alternative hypothesis: true odds ratio is greater than 1
95 percent confidence interval:
 0.3135693       Inf
sample estimates:
odds ratio 
  6.408309

ormidp.test(3, 1, 1, 3, or = 1)

  one.sided two.sided
1 0.1285714 0.2571429

or.midp(c(3, 1, 1, 3), conf.level = 0.95)$conf.int

[1]   0.3100508 306.6338538

Chapter 3: Generalized Linear Models

Examples in R

Data

# Snoring and Heart Disease pp.69-72
df.heart <- read.table("/Users/Patrick/Dropbox/data/agresti/Heart.dat", header = T)
df.heart <- df.heart %>% mutate(x = dplyr::recode(snoring, never = 0, occasional = 2, 
    nearly_every_night = 4, every_night = 5), n = yes + no, PctYes = yes/n) %>% 
    dplyr::select(snoring, x, yes, no, n, PctYes)
kable(df.heart, caption = "Snoring and Heart Disease pp.69-72")

Snoring and Heart Disease pp.69-72
snoring	x	yes	no	n	PctYes
never	0	24	1355	1379	0.0174039
occasional	2	35	603	638	0.0548589
nearly_every_night	4	21	192	213	0.0985915
every_night	5	30	224	254	0.1181102

Models

fit1 <- glm(yes/n ~ x, family = binomial(link = logit), weights = n, data = df.heart)
summary(fit1)


Call:
glm(formula = yes/n ~ x, family = binomial(link = logit), data = df.heart, 
    weights = n)

Deviance Residuals: 
      1        2        3        4  
-0.8346   1.2521   0.2758  -0.6845  

Coefficients:
            Estimate Std. Error z value             Pr(>|z|)    
(Intercept) -3.86625    0.16621 -23.261 < 0.0000000000000002 ***
x            0.39734    0.05001   7.945  0.00000000000000194 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 65.9045  on 3  degrees of freedom
Residual deviance:  2.8089  on 2  degrees of freedom
AIC: 27.061

Number of Fisher Scoring iterations: 4

fitted(fit1)

         1          2          3          4 
0.02050742 0.04429511 0.09305411 0.13243885

fit2 <- glm(yes/n ~ x, family = quasi(link = identity, variance = "mu(1-mu)"), 
    weights = n, data = df.heart)
summary(fit2, dispersion = 1)


Call:
glm(formula = yes/n ~ x, family = quasi(link = identity, variance = "mu(1-mu)"), 
    data = df.heart, weights = n)

Deviance Residuals: 
       1         2         3         4  
 0.04478  -0.21322   0.11010   0.09798  

Coefficients:
            Estimate Std. Error z value         Pr(>|z|)    
(Intercept) 0.017247   0.003451   4.998 0.00000058029057 ***
x           0.019778   0.002805   7.051 0.00000000000177 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for quasi family taken to be 1)

    Null deviance: 65.904481  on 3  degrees of freedom
Residual deviance:  0.069191  on 2  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 3

Plots

binomial.logit.fit <- function(x, a, b) exp(a + b * x)/(1 + exp(a + b * x))
binomial.identity.fit <- function(x, a, b) a + b * x
xx <- seq(0.0001, 5, length = 1000)
yy1 <- binomial.logit.fit(xx, coefficients(fit1)[1], coefficients(fit1)[2])
yy2 <- binomial.identity.fit(xx, coefficients(fit2)[1], coefficients(fit2)[2])
xrange <- range(xx)
yrange <- range(0, 0.2)

plot(xx, yy1, type = "l", lwd = 3, xlim = xrange, ylim = yrange, xlab = "", 
    ylab = "")

par(new = T)
plot(xx, yy2, type = "l", lty = 4, lwd = 3, xlim = xrange, ylim = yrange, xlab = "x (snoring level)", 
    ylab = "probability", col = "purple")
par(new = T)
plot(df.heart$x, df.heart$pctyes, cex = 2, pch = 19, xlim = xrange, ylim = yrange, 
    xlab = "", ylab = "", col = "darkgreen")
legend("topleft", legend = c("Logistic Fit", "Linear Fit"), lty = c(1, 4), lwd = 3, 
    col = c("black", "purple"))

Data 2

df.heart2 <- read.table("/Users/Patrick/Dropbox/data/agresti/Heart2.dat", header = T)
head(df.heart2)

  subject snoring x y
1       1   never 0 1
2       2   never 0 1
3       3   never 0 1
4       4   never 0 1
5       5   never 0 1
6       6   never 0 1

# Dobson and Barnett Poisson data analysis

x <- c(rep(-1, 2), rep(0, 4), rep(1, 3))
y <- c(2, 3, 6, 7, 8, 9, 10, 12, 15)

DB <- data.frame(x, y)

kable(DB)

x	y
-1	2
-1	3
0	6
0	7
0	8
0	9
1	10
1	12
1	15

Models & Predictions

fit1 <- glm(y ~ x, family = poisson(link = identity), data = DB)
summary(fit1)


Call:
glm(formula = y ~ x, family = poisson(link = identity), data = DB)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-0.7019  -0.3377  -0.1105   0.2958   0.7184  

Coefficients:
            Estimate Std. Error z value             Pr(>|z|)    
(Intercept)   7.4516     0.8841   8.428 < 0.0000000000000002 ***
x             4.9353     1.0892   4.531           0.00000586 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 18.4206  on 8  degrees of freedom
Residual deviance:  1.8947  on 7  degrees of freedom
AIC: 40.008

Number of Fisher Scoring iterations: 3

# predict response at x = -2 using ID link
coefficients(fit1)[1] + coefficients(fit1)[2] * (-2)

(Intercept) 
  -2.418969

fit2 <- glm(y ~ x, family = poisson(link = log), data = DB)
summary(fit2)


Call:
glm(formula = y ~ x, family = poisson(link = log), data = DB)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-0.8472  -0.2601  -0.2137   0.5214   0.8788  

Coefficients:
            Estimate Std. Error z value             Pr(>|z|)    
(Intercept)   1.8893     0.1421  13.294 < 0.0000000000000002 ***
x             0.6698     0.1787   3.748             0.000178 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 18.4206  on 8  degrees of freedom
Residual deviance:  2.9387  on 7  degrees of freedom
AIC: 41.052

Number of Fisher Scoring iterations: 4

# predict response at x = -2 using LOG link
exp(coefficients(fit2)[1] + coefficients(fit2)[2] * (-2))

(Intercept) 
   1.732734

Other Examples

# Agresti horseshoe crabs pp.73-76
df.horsecrab <- read.table("/Users/Patrick/Dropbox/data/agresti/Crabs.dat", 
    header = T)
summary(df.horsecrab)

      crab          sat               y              weight     
 Min.   :  1   Min.   : 0.000   Min.   :0.0000   Min.   :1.200  
 1st Qu.: 44   1st Qu.: 0.000   1st Qu.:0.0000   1st Qu.:2.000  
 Median : 87   Median : 2.000   Median :1.0000   Median :2.350  
 Mean   : 87   Mean   : 2.919   Mean   :0.6416   Mean   :2.437  
 3rd Qu.:130   3rd Qu.: 5.000   3rd Qu.:1.0000   3rd Qu.:2.850  
 Max.   :173   Max.   :15.000   Max.   :1.0000   Max.   :5.200  
     width          color           spine      
 Min.   :21.0   Min.   :1.000   Min.   :1.000  
 1st Qu.:24.9   1st Qu.:2.000   1st Qu.:2.000  
 Median :26.1   Median :2.000   Median :3.000  
 Mean   :26.3   Mean   :2.439   Mean   :2.486  
 3rd Qu.:27.7   3rd Qu.:3.000   3rd Qu.:3.000  
 Max.   :33.5   Max.   :4.000   Max.   :3.000

table(df.horsecrab$sat, df.horsecrab$y, dnn = c("Satellite Count", "Success"))

               Success
Satellite Count  0  1
             0  62  0
             1   0 16
             2   0  9
             3   0 19
             4   0 19
             5   0 15
             6   0 13
             7   0  4
             8   0  6
             9   0  3
             10  0  3
             11  0  1
             12  0  1
             14  0  1
             15  0  1

xx <- seq(21, 33.5, length = 1000)
xrange <- range(xx)
yrange <- range(df.horsecrab$sat)

plot(sat ~ width, xlab = "Width", ylab = "Number of satellites", xlim = xrange, 
    ylim = yrange, col = "darkgreen", cex = 1.3, data = df.horsecrab)

fit <- glm(sat ~ width, family = poisson(link = log), data = df.horsecrab)
summary(fit)


Call:
glm(formula = sat ~ width, family = poisson(link = log), data = df.horsecrab)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.8526  -1.9884  -0.4933   1.0970   4.9221  

Coefficients:
            Estimate Std. Error z value             Pr(>|z|)    
(Intercept) -3.30476    0.54224  -6.095         0.0000000011 ***
width        0.16405    0.01997   8.216 < 0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 632.79  on 172  degrees of freedom
Residual deviance: 567.88  on 171  degrees of freedom
AIC: 927.18

Number of Fisher Scoring iterations: 6

yy <- exp(coefficients(fit)[1] + coefficients(fit)[2] * xx)
par(new = T)
plot(xx, yy, type = "l", lwd = 3, xlab = "", ylab = "", xlim = xrange, ylim = yrange, 
    col = "lightblue")

gam.fit <- gam(sat ~ s(width), family = poisson, data = df.horsecrab)
# curve(predict(gam.fit, data.frame(width = x), type = 'resp'),col =
# 'purple', lty = 4, lwd = 3, add = T) legend('topleft', legend =
# c('Loglinear Fit','GAM Fit'), lty = c(1, 4), lwd = 3, col = c('lightblue',
# 'purple'))

# Political Ideology and Evolution pp.78-79

EVO <- read.table("/Users/Patrick/Dropbox/data/agresti/Evolution.dat", header = T)
attach(EVO)
EVO

  ideology true false
1        1   11    37
2        2   46   104
3        3   70    72
4        4  241   214
5        5   78    36
6        6   89    24
7        7   36     6

EVO$n <- EVO$true + EVO$false
fit <- glm(true/n ~ ideology, family = binomial, weights = n, data = EVO)
summary(fit)


Call:
glm(formula = true/n ~ ideology, family = binomial, data = EVO, 
    weights = n)

Deviance Residuals: 
      1        2        3        4        5        6        7  
 0.1430  -0.2697   1.4614  -1.0791   0.2922   0.4471   0.2035  

Coefficients:
            Estimate Std. Error z value            Pr(>|z|)    
(Intercept) -1.75658    0.20500  -8.569 <0.0000000000000002 ***
ideology     0.49422    0.05092   9.706 <0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 113.20  on 6  degrees of freedom
Residual deviance:   3.72  on 5  degrees of freedom
AIC: 42.332

Number of Fisher Scoring iterations: 3

confint(fit)[2, ]

    2.5 %    97.5 % 
0.3961660 0.5959414

Anova(fit)

Analysis of Deviance Table (Type II tests)

Response: true/n
         LR Chisq Df            Pr(>Chisq)    
ideology   109.48  1 < 0.00000000000000022 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

fit0 <- glm(true/n ~ 1, family = binomial, weights = n, data = EVO)

glm.scoretest(fit0, EVO$ideology)^2

[1] 104.101

cbind(ideology, true, false, EVO$n, true/EVO$n, fitted(fit), rstandard(fit, 
    type = "pearson"))

  ideology true false                                   
1        1   11    37  48 0.2291667 0.2205679  0.1611162
2        2   46   104 150 0.3066667 0.3168813 -0.3515386
3        3   70    72 142 0.4929577 0.4319445  1.6480176
4        4  241   214 455 0.5296703 0.5548525 -1.4995488
5        5   78    36 114 0.6842105 0.6713982  0.3248519
6        6   89    24 113 0.7876106 0.7700750  0.5413625
7        7   36     6  42 0.8571429 0.8459201  0.2206605

Chapter 4: Logistic Regression

Basics of LRM (Logistic Regression Modelling)

General

Most popular model for binary data is logistic regression
- Logistic regression model introduced in section 3.2.2 as a GLM (generalized linear model) for a binomial random component with the logit link function
- How does one interpret the model, conduct statistical inference for its parameters, and summarize effects/predictive power?
Models success probability, P[Y = 1]
- Estimated success probability is denoted by \(\pi(x)\) to emphasize that it depends on the value of the predictor variable, x
Assumptions:
- Observations are independent binomial variates with parameter \(\pi(x)\), which itself varies according to the value of x

Model

Section 4.1.1: Logistic Regression Model (4.1)
\[logit[\pi(x)] = log\frac{\pi(x)}{1 - \pi(x)} = \alpha + \beta x\]
LRM using the exponential function (4.2):
\[\pi(x) = \frac{e^{\alpha + \beta x}}{1 + e^{\alpha + \beta x}}\]
\(\beta\): effect parameter determining the rate of increase or decrease of the S-shaped curve for \(\pi(x)\)
- sign of \(\beta\) indicates an ascending curve (+) or a descending curve (-) & magnitude of \(|\beta|\) indicates the rate of change
- when \(\beta = 0\), the curve flattens to a horizontal line indicating that the binary response is independent of the explanatory variable

Interpretation

Model 4.1 indicates that the logit increases by \(\beta\) for every 1 unit increase in x
Exponentiating model 4.1 provides a more intuitive interpretation using odds ratios
\[\frac{\pi(x)}{1 - \pi(x)} = exp(\alpha + \beta x) = e^{\alpha}(e^{\beta})^{x}\]
The odds of success multiply by \(e^{\beta}\) for every 1 unit increase in x; that is, the odds of success at level x + 1 equal the odds at x multiplied by \(e^{\beta}\), so when \(\beta = 0\), \(e^{\beta} = e^{0} = 1\) so odds do not change as x changes
Using a tangent line on the curve at a particular value of x describes the rate of change in predicted probability for that point
- For Logistic Regression parameter \(\beta\), the tangent line has a slope equal to \(\beta \pi(x) [1 - \pi(x)]\)
  - For instance, the line tangent to the curve at x for which \(\pi(x) = 0.5\) has slope \(\beta(0.5)(0.5) = 0.25\beta\); by contrast when \(\pi(x) = 0.9\), the tangent line has slope \(0.09\beta\)
  - The steepest slope occurs at x when \(\pi(x) = 0.5\)
  - That x value, called the median effective level, relates to the logistic regression parameter by \(x = -\alpha/\beta\) & represents the point in which each outcome is equally likely

Horshoe Crab Example

Data

# Horseshoe crabs: original/ungrouped data
df.horsecrab <- read.table("/Users/Patrick/Dropbox/data/agresti/Crabs.dat", 
    header = TRUE)
table(df.horsecrab$sat, df.horsecrab$y, dnn = c("Sattelite Count", "Success"))

               Success
Sattelite Count  0  1
             0  62  0
             1   0 16
             2   0  9
             3   0 19
             4   0 19
             5   0 15
             6   0 13
             7   0  4
             8   0  6
             9   0  3
             10  0  3
             11  0  1
             12  0  1
             14  0  1
             15  0  1

Model

# Binary/Binomial (log-link) logistic fit p.92
fit <- glm(y ~ width, family = binomial, data = df.horsecrab)
summary(fit)


Call:
glm(formula = y ~ width, family = binomial, data = df.horsecrab)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.0281  -1.0458   0.5480   0.9066   1.6942  

Coefficients:
            Estimate Std. Error z value   Pr(>|z|)    
(Intercept) -12.3508     2.6287  -4.698 0.00000262 ***
width         0.4972     0.1017   4.887 0.00000102 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 225.76  on 172  degrees of freedom
Residual deviance: 194.45  on 171  degrees of freedom
AIC: 198.45

Number of Fisher Scoring iterations: 4

Plot

# Plots pg. 92
plot(jitter(y, 0.08) ~ width, pch = 19, col = "purple", xlab = "Width in cm", 
    ylab = "At least one satellite", data = df.horsecrab)
gam.fit <- gam(y ~ (width), family = binomial, data = df.horsecrab)

curve(predict(gam.fit, data.frame(width = x), type = "resp"), col = "orange", 
    lty = 4, lwd = 3, add = TRUE)

curve(predict(fit, data.frame(width = x), type = "resp"), col = "darkgreen", 
    lty = 2, lwd = 3, add = TRUE)

legend("bottomright", legend = c("Logistic Fit", "GAM Fit"), lty = c(2, 4), 
    lwd = 3, col = c("darkgreen", "orange"))

Predictions

# Predictions p.92 Estimated Probability of Satellite at width = 21.0
predict(fit, data.frame(width = 21), type = "response")

       1 
0.129096

# Estimated Probability of Satellite at width = 33.5
predict(fit, data.frame(width = 33.5), type = "response")

        1 
0.9866974

# Estimated Probability of Satellite at mean width
pimean <- predict(fit, data.frame(width = mean(df.horsecrab$width)), type = "response")
pimean

        1 
0.6738768

# Estimated Probability of Satellite at mean width + unit increase
pimean1 <- predict(fit, data.frame(width = mean(df.horsecrab$width) + 1), type = "response")
pimean1

        1 
0.7725915

Odds/OR’s

# Demonstrating OR p.93
oddsmean <- pimean/(1 - pimean)
oddsmean

       1 
2.066325

oddsmean1 <- pimean1/(1 - pimean1)
oddsmean1

       1 
3.397373

oddsmean1/oddsmean

       1 
1.644162

exp(coefficients(fit)[2])

   width 
1.644162

LD50/MEL

# LD50 or Point estimate of Median Effective Level (MEL = LD50/ED50/EC50)
MEL <- (-coefficients(fit)[1]/coefficients(fit)[2])
MEL

(Intercept) 
   24.83922

Retrospective Studies

Section 4.1.4: Logistic regression with Retrospective Studies

Section 4.2: Statistical Inference for LRM

General

For fixed sample size (n), SE’s are relatively large when estimated probabilities (\(\hat{\pi_{i}}\)) are closer to 0 or 1; when \(\hat{\pi_{i}}\)’s are closer to their limits [0,1] (such as occurence of a rare disease), effects of explanatory variables are more difficult to estimate accurately compared to a more evenly distributed binary condition.

Testing & CI’s

LRT vs Wald vs Score

# Testing p.96
Wald.chisq <- (coefficients(fit)[2]/sqrt(diag(vcov(fit)))[2])^2
Wald.chisq

   width 
23.88748

Wald.pvalue <- 1 - pchisq(Wald.chisq, 1)
Wald.pvalue

        width 
0.00000102134

Anova(fit)

Analysis of Deviance Table (Type II tests)

Response: y
      LR Chisq Df    Pr(>Chisq)    
width   31.306  1 0.00000002204 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

CI: \(\beta\)

# WCI and PLCI for beta p.95
coefficients(fit)[2] + c(-1, 1) * qnorm(0.975) * se.coef(fit)[2]

[1] 0.2978326 0.6966286

confint(fit)[2, ]

    2.5 %    97.5 % 
0.3083806 0.7090167

CI: OR

# WCI and PLCI for OR p.95
exp(coefficients(fit)[2] + c(-1, 1) * qnorm(0.975) * sqrt(diag(vcov(fit)))[2])

[1] 1.346936 2.006975

exp(coefficients(fit)[2] + c(-1, 1) * qnorm(0.975) * se.coef(fit)[2])

[1] 1.346936 2.006975

exp(confint(fit)[2, ])

   2.5 %   97.5 % 
1.361219 2.031992

Predictions

# Predicted probabilities for fitted values pp.96-98
pred.prob <- fitted(fit)
Linpred <- predict(fit, se.fit = T)
Linpred.CIL <- Linpred$fit - qnorm(0.975) * Linpred$se.fit
Linpred.CIU <- Linpred$fit + qnorm(0.975) * Linpred$se.fit
pred.prob.CIL <- exp(Linpred.CIL)/(1 + exp(Linpred.CIL))
pred.prob.CIU <- exp(Linpred.CIU)/(1 + exp(Linpred.CIU))
crabs.probs <- cbind(df.horsecrab$width, pred.prob, pred.prob.CIL, pred.prob.CIU)
head(crabs.probs, 7)

       pred.prob pred.prob.CIL pred.prob.CIU
1 28.3 0.8482329     0.7528521     0.9111481
2 22.5 0.2380991     0.1299857     0.3952793
3 26.0 0.6404177     0.5580894     0.7152343
4 24.8 0.4951254     0.3982566     0.5923614
5 26.0 0.6404177     0.5580894     0.7152343
6 23.8 0.3736172     0.2613945     0.5013184
7 26.5 0.6954646     0.6120544     0.7677464

Plot

# Plot jittered data, fitted logistic, with confidence bands p.97
plot(jitter(y, 0.1) ~ width, pch = 19, col = "purple", xlim = c(18, 34), xlab = "Width in cm", 
    ylab = "At least one satellite", data = df.horsecrab)
data.plot <- data.frame(width = (18:34))
Linpred2 <- predict(fit, newdata = data.plot, se.fit = T)
pred.prob2 <- (exp(Linpred2$fit)/(1 + exp(Linpred2$fit)))

LB <- Linpred2$fit - qnorm(0.975) * Linpred2$se.fit
UB <- Linpred2$fit + qnorm(0.975) * Linpred2$se.fit

LB.p <- exp(LB)/(1 + exp(LB))
UB.p <- exp(UB)/(1 + exp(UB))

lines(18:34, pred.prob2, lwd = 3, col = "darkgreen")
lines(18:34, LB.p, lwd = 3, lty = 4, col = "orange")
lines(18:34, UB.p, lwd = 3, lty = 4, col = "orange")

LRM with “real” x, categorical variables, & both

Multiple Logistic Regression

General

Model

Interpretations

MLR Horseshoe Crab Example

Data

# HSCrabs with nominal Color and quant Width as predictors pp. 102-104 Color
# is nominal with 1 = medium light, 2 = medium, 3 = medium dark, 4 = dark
table(df.horsecrab$color)


 1  2  3  4 
12 95 44 22

Model Fit

fit2 <- glm(y ~ width + factor(color), family = binomial, data = df.horsecrab)
summary(fit2)


Call:
glm(formula = y ~ width + factor(color), family = binomial, data = df.horsecrab)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.1124  -0.9848   0.5243   0.8513   2.1413  

Coefficients:
                Estimate Std. Error z value   Pr(>|z|)    
(Intercept)    -11.38519    2.87346  -3.962 0.00007426 ***
width            0.46796    0.10554   4.434 0.00000926 ***
factor(color)2   0.07242    0.73989   0.098      0.922    
factor(color)3  -0.22380    0.77708  -0.288      0.773    
factor(color)4  -1.32992    0.85252  -1.560      0.119    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 225.76  on 172  degrees of freedom
Residual deviance: 187.46  on 168  degrees of freedom
AIC: 197.46

Number of Fisher Scoring iterations: 4

OR

bhat <- coefficients(fit2)[2]
a1hat <- coefficients(fit2)[1]
a2hat <- a1hat + coefficients(fit2)[3]
a3hat <- a1hat + coefficients(fit2)[4]
a4hat <- a1hat + coefficients(fit2)[5]
ORwidth <- exp(bhat)
ORwidth

   width 
1.596727

ORcolor1to4 <- exp(a1hat - a4hat)
ORcolor1to4

(Intercept) 
   3.780738

Plot

# Plot on p.103
xx <- seq(18, 34, length = 1000)
y1 <- binomial.logit.fit(xx, a1hat, bhat)
y2 <- binomial.logit.fit(xx, a2hat, bhat)
y3 <- binomial.logit.fit(xx, a3hat, bhat)
y4 <- binomial.logit.fit(xx, a4hat, bhat)
xrange <- range(xx)
yrange <- range(0, 1)
plot(xx, y1, type = "l", lwd = 3, xlim = xrange, ylim = yrange, col = "orange", 
    xlab = "Width in cm", ylab = "Predicted Probability")
par(new = T)
plot(xx, y2, type = "l", lwd = 3, xlim = xrange, ylim = yrange, lty = 2, col = "darkgreen", 
    xlab = "", ylab = "")
par(new = T)
plot(xx, y3, type = "l", lwd = 3, xlim = xrange, ylim = yrange, lty = 3, col = "darkblue", 
    xlab = "", ylab = "")
par(new = T)
plot(xx, y4, type = "l", lwd = 3, xlim = xrange, ylim = yrange, lty = 4, xlab = "", 
    ylab = "")
title("Horseshoe Crabs: Logistic Predicted Curves by Color")

Test: Color

# Testing to drop Nominal Color variable p.104 Recall the following:
fit <- glm(y ~ width, family = binomial, data = df.horsecrab)
summary(fit)


Call:
glm(formula = y ~ width, family = binomial, data = df.horsecrab)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.0281  -1.0458   0.5480   0.9066   1.6942  

Coefficients:
            Estimate Std. Error z value   Pr(>|z|)    
(Intercept) -12.3508     2.6287  -4.698 0.00000262 ***
width         0.4972     0.1017   4.887 0.00000102 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 225.76  on 172  degrees of freedom
Residual deviance: 194.45  on 171  degrees of freedom
AIC: 198.45

Number of Fisher Scoring iterations: 4

Anova(fit2)

Analysis of Deviance Table (Type II tests)

Response: y
              LR Chisq Df   Pr(>Chisq)    
width          24.6038  1 0.0000007041 ***
factor(color)   6.9956  3      0.07204 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# For Width, Full Resid Dev = 187.46, Reduced Resid Dev = 194.45, Diff =
# 6.99 Treating Color as Quantitative, p.105
fit3 <- glm(y ~ width + color, family = binomial, data = df.horsecrab)
summary(fit3)


Call:
glm(formula = y ~ width + color, family = binomial, data = df.horsecrab)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.1692  -0.9889   0.5429   0.8700   1.9742  

Coefficients:
            Estimate Std. Error z value  Pr(>|z|)    
(Intercept) -10.0708     2.8068  -3.588  0.000333 ***
width         0.4583     0.1040   4.406 0.0000105 ***
color        -0.5090     0.2237  -2.276  0.022860 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 225.76  on 172  degrees of freedom
Residual deviance: 189.12  on 170  degrees of freedom
AIC: 195.12

Number of Fisher Scoring iterations: 4

ORcdelta1 <- exp(coefficients(fit3)[3])
ORcdelta1

    color 
0.6010683

ORcdelta3 <- exp(3 * coefficients(fit3)[3])
ORcdelta3  # this is dark vs medium-light

    color 
0.2171558

anova(fit3, fit2, test = "LRT")

Analysis of Deviance Table

Model 1: y ~ width + color
Model 2: y ~ width + factor(color)
  Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1       170     189.12                     
2       168     187.46  2   1.6641   0.4351

# Discretizing Color with Dummy for Dark color, pp.105-6
df.horsecrab$c4 <- ifelse(df.horsecrab$color == 4, 1, 0)
table(df.horsecrab$c4)


  0   1 
151  22

fit4 <- glm(y ~ width + c4, family = binomial, data = df.horsecrab)
summary(fit4)


Call:
glm(formula = y ~ width + c4, family = binomial, data = df.horsecrab)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.0821  -0.9932   0.5274   0.8606   2.1553  

Coefficients:
            Estimate Std. Error z value   Pr(>|z|)    
(Intercept) -11.6790     2.6925  -4.338 0.00001440 ***
width         0.4782     0.1041   4.592 0.00000439 ***
c4           -1.3005     0.5259  -2.473     0.0134 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 225.76  on 172  degrees of freedom
Residual deviance: 187.96  on 170  degrees of freedom
AIC: 193.96

Number of Fisher Scoring iterations: 4

ORcdarkvsrest <- exp(coefficients(fit4)[3])
ORcdarkvsrest

       c4 
0.2723923

anova(fit4, fit2, test = "LRT")

Analysis of Deviance Table

Model 1: y ~ width + c4
Model 2: y ~ width + factor(color)
  Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1       170     187.96                     
2       168     187.46  2  0.50085   0.7785

# Notice Agresti doesn't compare fit4 and fit3, which are not nested

Chapter 5: Building & Applying Logistic Regression Models

Model Selection

Strategies

Whether or not a variable should be included in the model can be assessed using confirmatory analyses on restricted sets of models
- How does leaving a variable out of certain models affect the model?
- Is a better fit worth inducing bias through overfitting or does a more simple model provide easier interpretation without sacrificing a decent fit?
How many variables can a model handle?
- Guideline: dataset should contain at least 10 outcomes of each type for every explanatory variable
- ex: if y = 1 for 30 observations from a sample of 1000, the model should not have more than p = 3 predictors despite a large sample size
- This rule is conservative & might be too simple
Including correlated explanatory variables could lead to a poor fit due to multicollinearity
Overall LR-tests indicate if any of the explanatory variables may be significant even if individial LR tests on \(\beta\) estimates may not show any significance

Stepwise/Canned Procedures

Stepwise procedures involve algoritms for selecting/deleting explanatory variables for a model
Backward elimination starts with a saturated, over complex model and removes variables according to some established criteria
Forward selection starts with the null model and sequentially includes predictors which improve the model fit the most until including any other additional predictor will not improve model fit
With categorical predictors, stepwise procedures should consider the entire variable rather than just a dummy variable at a time
- Otherwise results will depend on how the baseline level was selected
Stepwise methods do not necessarily yield a meaningful model and some terms may seem important by chance, so be skeptical when using these methods for variable selection
- There is a subtle difference between statistical significance and practical significance

Purposeful Selection

Purposeful variable selection account for:
- Goals of the Study
- Relative Statistical Significance
- Multicollinearity
- Potential Confounding
Steps of Purposeful Selection (Hosmer, 2013):
1. Construct an initial main effects model containing known important variables & any other variable showing any evidence of being relevant when used as sole predictors (p-vale < 0.2)
2. Conduct backward elimination, keeping a variable if it is significant at a more stringent level OR if it shows evidence of being a relevant confounder (the estimated effect of a key variable changes substantially when it is removed)
3. Add any variables that did not clear step 1 but are significant when adjusting for the variables in the model after step 2, since a variable may not be significantly associated with y but may make an important contribution in the presence of other variables
4. Check for plausible interactions among variables in the model after step 3 using significance tests at conventional levels (\(\alpha = 0.05\))
5. Conduct follow up diagnostic investigations

AIC

Bias-Variance Tradeoff & Overfitting

Model Checking

Residuals

GoF

Diagnostics

Chapter 6: Multicategory Logit Models

Nominal vs Ordinal Response Variable, number of levels

BCL Modeling

General

Predictions

Testing/GoF

Interpretations

Cumulative logit modelling

PO Model

NPO Model

Predictions

Testing/GoF

Interpretations

Latent Variables

Other Ch 6 Topics

Cumulative link models and Paired-Category Logit Models (ACL, CRA and CRB)

Calculating Probabilities & Interpreting Odds Ratios in All Models

Ch. 6 Code

# BCL: Alligators pp.159 - 164

gators <- read.table("/Users/Patrick/Dropbox/data/agresti/Alligators.dat", header = T)
summary(gators)

       x         y     
 Min.   :1.240   F:31  
 1st Qu.:1.575   I:20  
 Median :1.850   O: 8  
 Mean   :2.130         
 3rd Qu.:2.450         
 Max.   :3.890

head(gators)

     x y
1 1.24 I
2 1.30 I
3 1.30 I
4 1.32 F
5 1.32 F
6 1.40 F

hist(gators$x)

sm.density.compare(gators$x, gators$y, lwd = 3, xlab = "Alligator length in meters")
legend("topright", levels(as.factor(gators$y)), fill = c(2:4))

fit <- vglm(y ~ x, family = multinomial, data = gators)
summary(fit)


Call:
vglm(formula = y ~ x, family = multinomial, data = gators)

Pearson residuals:
                      Min      1Q  Median     3Q   Max
log(mu[,1]/mu[,3]) -2.330 -0.5075  0.5538 0.6836 1.452
log(mu[,2]/mu[,3]) -2.687 -0.4821 -0.1653 0.7093 3.439

Coefficients: 
              Estimate Std. Error z value Pr(>|z|)   
(Intercept):1   1.6177     1.3073   1.237  0.21591   
(Intercept):2   5.6974     1.7937   3.176  0.00149 **
x:1            -0.1101     0.5171  -0.213  0.83137   
x:2            -2.4654     0.8996      NA       NA   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])

Residual deviance: 98.3412 on 114 degrees of freedom

Log-likelihood: -49.1706 on 114 degrees of freedom

Number of Fisher scoring iterations: 5 

Warning: Hauck-Donner effect detected in the following estimate(s):
'x:2'


Reference group is level  3  of the response

coefficients(fit)[1] - coefficients(fit)[2]

(Intercept):1 
    -4.079713

coefficients(fit)[3] - coefficients(fit)[4]

     x:1 
2.355337

fit2 <- vglm(y ~ x, family = multinomial(refLevel = "I"), data = gators)
summary(fit2)


Call:
vglm(formula = y ~ x, family = multinomial(refLevel = "I"), data = gators)

Pearson residuals:
                      Min      1Q  Median      3Q   Max
log(mu[,1]/mu[,2]) -2.882 -0.8133  0.4313  0.6837 1.691
log(mu[,3]/mu[,2]) -2.163 -0.3755 -0.2311 -0.1262 3.530

Coefficients: 
              Estimate Std. Error z value Pr(>|z|)   
(Intercept):1  -4.0797     1.4686  -2.778  0.00547 **
(Intercept):2  -5.6974     1.7937  -3.176  0.00149 **
x:1             2.3553     0.8032      NA       NA   
x:2             2.4654     0.8996   2.741  0.00613 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: log(mu[,1]/mu[,2]), log(mu[,3]/mu[,2])

Residual deviance: 98.3412 on 114 degrees of freedom

Log-likelihood: -49.1706 on 114 degrees of freedom

Number of Fisher scoring iterations: 5 

Warning: Hauck-Donner effect detected in the following estimate(s):
'x:1'


Reference group is level  2  of the response

fit0 <- vglm(y ~ 1, family = multinomial, data = gators)
deviance(fit0)

[1] 115.1419

lrtest(fit0, fit)

Likelihood ratio test

Model 1: y ~ 1
Model 2: y ~ x
  #Df  LogLik Df  Chisq Pr(>Chisq)    
1 116 -57.571                         
2 114 -49.171 -2 16.801  0.0002248 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

confint(fit2, method = "profile")

                   2.5 %    97.5 %
(Intercept):1 -7.3475058 -1.518699
(Intercept):2 -9.5913317 -2.485991
x:1            1.0111800  4.199073
x:2            0.8775186  4.463608

fits <- fitted(fit)
head(fits)

          F         I          O
1 0.2265307 0.7219640 0.05150528
2 0.2502564 0.6924668 0.05727683
3 0.2502564 0.6924668 0.05727683
4 0.2584591 0.6822562 0.05928463
5 0.2584591 0.6822562 0.05928463
6 0.2925882 0.6397049 0.06770685

tail(fits)

           F           I         O
54 0.7650896 0.010326420 0.2245840
55 0.7650743 0.009851059 0.2250746
56 0.7648452 0.008156804 0.2269980
57 0.7647495 0.007780500 0.2274700
58 0.7645799 0.007248095 0.2281720
59 0.7630060 0.004733748 0.2322603

# to make plot
BCL.pi1 <- function(x, a1, b1, a2, b2) {
    ex1 <- exp(a1 + b1 * x)
    ex2 <- exp(a2 + b2 * x)
    den <- 1 + ex1 + ex2
    ex1/den
}

BCL.pi2 <- function(x, a1, b1, a2, b2) {
    ex1 <- exp(a1 + b1 * x)
    ex2 <- exp(a2 + b2 * x)
    den <- 1 + ex1 + ex2
    ex2/den
}

BCL.pi3 <- function(x, a1, b1, a2, b2) {
    ex1 <- exp(a1 + b1 * x)
    ex2 <- exp(a2 + b2 * x)
    den <- 1 + ex1 + ex2
    1/den
}

xx <- seq(0, 5, length = 1000)
a1hat <- coefficients(fit)[1]
a2hat <- coefficients(fit)[2]
b1hat <- coefficients(fit)[3]
b2hat <- coefficients(fit)[4]
pi1 <- BCL.pi1(xx, a1hat, b1hat, a2hat, b2hat)
pi2 <- BCL.pi2(xx, a1hat, b1hat, a2hat, b2hat)
pi3 <- BCL.pi3(xx, a1hat, b1hat, a2hat, b2hat)
xrange <- range(xx)
yrange <- range(0, 1)

plot(xx, pi1, type = "l", lwd = 3, xlim = xrange, ylim = yrange, xlab = "Alligator length", 
    ylab = "Probability")
par(new = T)
plot(xx, pi2, type = "l", lwd = 3, lty = 3, col = "purple", xlim = xrange, ylim = yrange, 
    xlab = "", ylab = "")
par(new = T)
plot(xx, pi3, type = "l", lwd = 3, lty = 4, col = "orange", xlim = xrange, ylim = yrange, 
    xlab = "", ylab = "")
legend("topright", legend = c("Fish", "Invertebrates", "Other"), lty = c(1, 
    3, 4), lwd = 3, col = c("black", "purple", "orange"))
title("BCL Fit to Alligator Data")

# BCL: Belief in the Afterlife, pp.165 - 166
after <- read.table("/Users/Patrick/Dropbox/data/agresti/Afterlife.dat", header = T)
after$n <- after$yes + after$undecided + after$no

fit <- vglm(cbind(yes, undecided, no) ~ gender + race, family = multinomial, 
    data = after)
summary(fit)


Call:
vglm(formula = cbind(yes, undecided, no) ~ gender + race, family = multinomial, 
    data = after)

Pearson residuals:
  log(mu[,1]/mu[,3]) log(mu[,2]/mu[,3])
1            -0.2190            -0.1143
2             0.2284             0.1115
3             0.4709             0.2303
4            -0.6184            -0.2795

Coefficients: 
              Estimate Std. Error z value     Pr(>|z|)    
(Intercept):1   1.3016     0.2265   5.747 0.0000000091 ***
(Intercept):2  -0.6529     0.3405  -1.918       0.0551 .  
gendermale:1   -0.4186     0.1713  -2.444       0.0145 *  
gendermale:2   -0.1051     0.2465  -0.426       0.6700    
racewhite:1     0.3418     0.2370   1.442       0.1493    
racewhite:2     0.2710     0.3541   0.765       0.4442    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])

Residual deviance: 0.8539 on 2 degrees of freedom

Log-likelihood: -19.7324 on 2 degrees of freedom

Number of Fisher scoring iterations: 3 

No Hauck-Donner effect found in any of the estimates


Reference group is level  3  of the response

fitR <- vglm(cbind(yes, undecided, no) ~ race, family = multinomial, data = after)
summary(fitR)


Call:
vglm(formula = cbind(yes, undecided, no) ~ race, family = multinomial, 
    data = after)

Pearson residuals:
  log(mu[,1]/mu[,3]) log(mu[,2]/mu[,3])
1              1.418            -0.2420
2             -1.648             0.2811
3              1.024             0.1910
4             -1.465            -0.2733

Coefficients: 
              Estimate Std. Error z value     Pr(>|z|)    
(Intercept):1   1.1564     0.2167   5.337 0.0000000945 ***
(Intercept):2  -0.6931     0.3273  -2.118       0.0342 *  
racewhite:1     0.2982     0.2355   1.266       0.2055    
racewhite:2     0.2597     0.3531   0.736       0.4620    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])

Residual deviance: 8.0465 on 4 degrees of freedom

Log-likelihood: -23.3287 on 4 degrees of freedom

Number of Fisher scoring iterations: 4 

No Hauck-Donner effect found in any of the estimates


Reference group is level  3  of the response

lrtest(fit, fitR)

Likelihood ratio test

Model 1: cbind(yes, undecided, no) ~ gender + race
Model 2: cbind(yes, undecided, no) ~ race
  #Df  LogLik Df  Chisq Pr(>Chisq)  
1   2 -19.732                       
2   4 -23.329  2 7.1926    0.02742 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

fitG <- vglm(cbind(yes, undecided, no) ~ gender, family = multinomial, data = after)
summary(fitG)


Call:
vglm(formula = cbind(yes, undecided, no) ~ gender, family = multinomial, 
    data = after)

Pearson residuals:
  log(mu[,1]/mu[,3]) log(mu[,2]/mu[,3])
1             0.1991            0.03849
2             0.4982            0.21597
3            -0.4716           -0.09120
4            -1.4535           -0.63008

Coefficients: 
              Estimate Std. Error z value            Pr(>|z|)    
(Intercept):1   1.5867     0.1163  13.639 <0.0000000000000002 ***
(Intercept):2  -0.4282     0.1688  -2.537              0.0112 *  
gendermale:1   -0.4008     0.1705  -2.350              0.0188 *  
gendermale:2   -0.0906     0.2457  -0.369              0.7123    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])

Residual deviance: 2.8481 on 4 degrees of freedom

Log-likelihood: -20.7295 on 4 degrees of freedom

Number of Fisher scoring iterations: 3 

No Hauck-Donner effect found in any of the estimates


Reference group is level  3  of the response

lrtest(fit, fitG)

Likelihood ratio test

Model 1: cbind(yes, undecided, no) ~ gender + race
Model 2: cbind(yes, undecided, no) ~ gender
  #Df  LogLik Df  Chisq Pr(>Chisq)
1   2 -19.732                     
2   4 -20.730  2 1.9942     0.3689

# p.166 table of predicted probabilities
data.frame(after$race, after$gender, fitted(fit))

  after.race after.gender       yes  undecided        no
1      white       female 0.7545608 0.09956287 0.1458763
2      white         male 0.6782703 0.12244794 0.1992817
3      black       female 0.7073517 0.10018119 0.1924671
4      black         male 0.6221640 0.12055943 0.2572766

data.frame(after$race, after$gender, after$yes, after$undecided, after$no, after$n * 
    fitted(fit))

  after.race after.gender after.yes after.undecided after.no       yes
1      white       female       371              49       74 372.75305
2      white         male       250              45       71 248.24695
3      black       female        64               9       15  62.24695
4      black         male        25               5       13  26.75305
  undecided       no
1 49.184055 72.06289
2 44.815945 72.93711
3  8.815945 16.93711
4  5.184055 11.06289

# PO: Political Ideology pp.170-171
politics <- read.table("/Users/Patrick/Dropbox/data/agresti/Polviews.dat", header = T)
politics$n <- politics$y1 + politics$y2 + politics$y3 + politics$y4 + politics$y5

fit <- vglm(cbind(y1, y2, y3, y4, y5) ~ party + gender, family = cumulative(parallel = T), 
    data = politics)
summary(fit)


Call:
vglm(formula = cbind(y1, y2, y3, y4, y5) ~ party + gender, family = cumulative(parallel = T), 
    data = politics)

Pearson residuals:
  logitlink(P[Y<=1]) logitlink(P[Y<=2]) logitlink(P[Y<=3])
1            -0.2505            -0.6258            0.63464
2            -0.7075             0.6437           -0.08234
3             0.5157             0.9772           -0.52995
4            -0.5193            -1.3318           -0.32096
  logitlink(P[Y<=4])
1            -1.1078
2             0.7512
3            -1.3012
4            -0.1741

Coefficients: 
              Estimate Std. Error z value            Pr(>|z|)    
(Intercept):1 -2.12233    0.16875 -12.577 <0.0000000000000002 ***
(Intercept):2  0.16892    0.11481   1.471               0.141    
(Intercept):3  1.85716    0.15103  12.297 <0.0000000000000002 ***
(Intercept):4  4.65005    0.23496  19.791 <0.0000000000000002 ***
partyrepub    -3.63366    0.21785 -16.680 <0.0000000000000002 ***
gendermale     0.04731    0.14955   0.316               0.752    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: logitlink(P[Y<=1]), logitlink(P[Y<=2]), 
logitlink(P[Y<=3]), logitlink(P[Y<=4])

Residual deviance: 9.8072 on 10 degrees of freedom

Log-likelihood: -35.2032 on 10 degrees of freedom

Number of Fisher scoring iterations: 4 

No Hauck-Donner effect found in any of the estimates


Exponentiated coefficients:
partyrepub gendermale 
0.02641936 1.04844945

attach(politics)
data.frame(gender, party, fitted(fit))

  gender party          y1         y2        y3        y4          y5
1 female   dem 0.106945695 0.43518307 0.3228363 0.1255644 0.009470552
2 female repub 0.003153813 0.02717842 0.1144033 0.5895341 0.265730342
3   male   dem 0.111549168 0.44229817 0.3165490 0.1205668 0.009036867
4   male repub 0.003306108 0.02844904 0.1189361 0.5927070 0.256601745

data.frame(gender, party, y1, y2, y3, y4, y5, n * fitted(fit))

     gender party         y1         y2         y3         y4 y5      y1.1
1    female   dem 0.04917311 0.05267166 0.03970426 0.01349427  4 65.455334
2    female repub 0.04952472 0.05304689 0.03999101 0.01359440 32  1.930268
3      male   dem 0.04987871 0.05342464 0.04027974 0.01369527  3 68.272856
4      male repub 0.05023509 0.05380493 0.04057047 0.01379688 32  2.023480
5    female   dem 0.05059389 0.05418776 0.04086321 0.01389923  4 65.455334
6    female repub 0.05095511 0.05457317 0.04115797 0.01400233 32  1.930268
7      male   dem 0.05131877 0.05496116 0.04145476 0.01410618  3 68.272856
8      male repub 0.05168488 0.05535174 0.04175360 0.01421080 32  2.023480
9    female   dem 0.05205347 0.05574493 0.04205450 0.01431617  4 65.455334
10   female repub 0.05242453 0.05614076 0.04235748 0.01442232 32  1.930268
11     male   dem 0.05279810 0.05653922 0.04266254 0.01452924  3 68.272856
12     male repub 0.05317417 0.05694034 0.04296969 0.01463694 32  2.023480
13   female   dem 0.05355278 0.05734414 0.04327896 0.01474543  4 65.455334
14   female repub 0.05393392 0.05775062 0.04359035 0.01485471 32  1.930268
15     male   dem 0.05431763 0.05815981 0.04390389 0.01496479  3 68.272856
16     male repub 0.05470390 0.05857172 0.04421957 0.01507567 32  2.023480
17   female   dem 0.05509276 0.05898636 0.04453741 0.01518737  4 65.455334
18   female repub 0.05548423 0.05940375 0.04485744 0.01529987 32  1.930268
19     male   dem 0.05587831 0.05982390 0.04517965 0.01541320  3 68.272856
20     male repub 0.05627502 0.06024684 0.04550407 0.01552735 32  2.023480
21   female   dem 0.05667438 0.06067258 0.04583070 0.01564233  4 65.455334
22   female repub 0.05707641 0.06110113 0.04615957 0.01575816 32  1.930268
23     male   dem 0.05748111 0.06153250 0.04649068 0.01587482  3 68.272856
24     male repub 0.05788850 0.06196672 0.04682405 0.01599234 32  2.023480
25   female   dem 0.05829861 0.06240380 0.04715969 0.01611071  4 65.455334
26   female repub 0.05871143 0.06284376 0.04749762 0.01622994 32  1.930268
27     male   dem 0.05912700 0.06328661 0.04783785 0.01635004  3 68.272856
28     male repub 0.05954532 0.06373237 0.04818039 0.01647102 32  2.023480
29   female   dem 0.05996642 0.06418105 0.04852526 0.01659287  4 65.455334
30   female repub 0.06039030 0.06463268 0.04887248 0.01671561 32  1.930268
31     male   dem 0.06081698 0.06508726 0.04922204 0.01683925  3 68.272856
32     male repub 0.06124648 0.06554481 0.04957398 0.01696378 32  2.023480
33   female   dem 0.06167882 0.06600536 0.04992830 0.01708921  4 65.455334
34   female repub 0.06211400 0.06646891 0.05028502 0.01721556 32  1.930268
35     male   dem 0.06255205 0.06693548 0.05064416 0.01734283  3 68.272856
36     male repub 0.06299299 0.06740509 0.05100572 0.01747102 32  2.023480
37   female   dem 0.06343682 0.06787775 0.05136972 0.01760014  4 65.455334
38   female repub 0.06388356 0.06835349 0.05173618 0.01773019 32  1.930268
39     male   dem 0.06433324 0.06883231 0.05210511 0.01786120  3 68.272856
40     male repub 0.06478586 0.06931424 0.05247652 0.01799315 32  2.023480
41   female   dem 0.06524144 0.06979929 0.05285044 0.01812605  4 65.455334
42   female repub 0.06570000 0.07028748 0.05322686 0.01825993 32  1.930268
43     male   dem 0.06616156 0.07077883 0.05360582 0.01839477  3 68.272856
44     male repub 0.06662613 0.07127334 0.05398732 0.01853059 32  2.023480
45   female   dem 0.06709372 0.07177104 0.05437138 0.01866739  4 65.455334
46   female repub 0.06756436 0.07227195 0.05475802 0.01880518 32  1.930268
47     male   dem 0.06803806 0.07277608 0.05514724 0.01894397  3 68.272856
48     male repub 0.06851484 0.07328345 0.05553907 0.01908377 32  2.023480
49   female   dem 0.06899471 0.07379407 0.05593352 0.01922457  4 65.455334
50   female repub 0.06947769 0.07430797 0.05633060 0.01936640 32  1.930268
51     male   dem 0.06996380 0.07482516 0.05673033 0.01950925  3 68.272856
52     male repub 0.07045305 0.07534565 0.05713273 0.01965313 32  2.023480
53   female   dem 0.07094547 0.07586947 0.05753781 0.01979805  4 65.455334
54   female repub 0.07144106 0.07639663 0.05794558 0.01994402 32  1.930268
55     male   dem 0.07193984 0.07692714 0.05835606 0.02009104  3 68.272856
56     male repub 0.07244183 0.07746103 0.05876927 0.02023913 32  2.023480
57   female   dem 0.07294706 0.07799831 0.05918522 0.02038828  4 65.455334
58   female repub 0.07345552 0.07853901 0.05960393 0.02053851 32  1.930268
59     male   dem 0.07396725 0.07908313 0.06002541 0.02068983  3 68.272856
60     male repub 0.07448226 0.07963069 0.06044968 0.02084223 32  2.023480
61   female   dem 0.07500056 0.08018171 0.06087676 0.02099573  4 65.455334
62   female repub 0.07552217 0.08073622 0.06130665 0.02115034 32  1.930268
63     male   dem 0.07604711 0.08129421 0.06173939 0.02130607  3 68.272856
64     male repub 0.07657540 0.08185572 0.06217497 0.02146291 32  2.023480
65   female   dem 0.07710706 0.08242077 0.06261342 0.02162088  4 65.455334
66   female repub 0.07764209 0.08298936 0.06305476 0.02177999 32  1.930268
67     male   dem 0.07818052 0.08356151 0.06349899 0.02194025  3 68.272856
68     male repub 0.07872237 0.08413725 0.06394615 0.02210166 32  2.023480
69   female   dem 0.07926765 0.08471659 0.06439623 0.02226422  4 65.455334
70   female repub 0.07981638 0.08529954 0.06484927 0.02242796 32  1.930268
71     male   dem 0.08036858 0.08588613 0.06530526 0.02259287  3 68.272856
72     male repub 0.08092426 0.08647638 0.06576424 0.02275897 32  2.023480
73   female   dem 0.08148344 0.08707029 0.06622622 0.02292626  4 65.455334
74   female repub 0.08204614 0.08766789 0.06669121 0.02309475 32  1.930268
75     male   dem 0.08261238 0.08826919 0.06715923 0.02326445  3 68.272856
76     male repub 0.08318217 0.08887422 0.06763029 0.02343536 32  2.023480
77   female   dem 0.08375553 0.08948299 0.06810442 0.02360750  4 65.455334
78   female repub 0.08433249 0.09009552 0.06858163 0.02378088 32  1.930268
79     male   dem 0.08491304 0.09071182 0.06906194 0.02395549  3 68.272856
80     male repub 0.08549722 0.09133191 0.06954535 0.02413136 32  2.023480
81   female   dem 0.08608505 0.09195581 0.07003190 0.02430849  4 65.455334
82   female repub 0.08667653 0.09258355 0.07052159 0.02448688 32  1.930268
83     male   dem 0.08727168 0.09321512 0.07101445 0.02466655  3 68.272856
84     male repub 0.08787053 0.09385056 0.07151048 0.02484750 32  2.023480
85   female   dem 0.08847309 0.09448988 0.07200971 0.02502975  4 65.455334
86   female repub 0.08907938 0.09513310 0.07251215 0.02521330 32  1.930268
87     male   dem 0.08968942 0.09578023 0.07301783 0.02539816  3 68.272856
88     male repub 0.09030322 0.09643130 0.07352675 0.02558434 32  2.023480
89   female   dem 0.09092080 0.09708632 0.07403893 0.02577185  4 65.455334
90   female repub 0.09154217 0.09774530 0.07455440 0.02596070 32  1.930268
91     male   dem 0.09216737 0.09840827 0.07507316 0.02615089  3 68.272856
92     male repub 0.09279640 0.09907524 0.07559524 0.02634244 32  2.023480
93   female   dem 0.09342928 0.09974624 0.07612065 0.02653536  4 65.455334
94   female repub 0.09406602 0.10042127 0.07664940 0.02672965 32  1.930268
95     male   dem 0.09470666 0.10110035 0.07718153 0.02692532  3 68.272856
96     male repub 0.09535120 0.10178351 0.07771704 0.02712238 32  2.023480
97   female   dem 0.09599965 0.10247076 0.07825594 0.02732085  4 65.455334
98   female repub 0.09665205 0.10316212 0.07879827 0.02752072 32  1.930268
99     male   dem 0.09730841 0.10385760 0.07934403 0.02772202  3 68.272856
100    male repub 0.09796874 0.10455722 0.07989324 0.02792475 32  2.023480
101  female   dem 0.09863306 0.10526100 0.08044592 0.02812892  4 65.455334
102  female repub 0.09930138 0.10596896 0.08100209 0.02833453 32  1.930268
103    male   dem 0.09997374 0.10668111 0.08156177 0.02854161  3 68.272856
104    male repub 0.10065014 0.10739748 0.08212496 0.02875015 32  2.023480
105  female   dem 0.10133060 0.10811807 0.08269169 0.02896017  4 65.455334
106  female repub 0.10201513 0.10884291 0.08326198 0.02917168 32  1.930268
107    male   dem 0.10270377 0.10957201 0.08383585 0.02938469  3 68.272856
108    male repub 0.10339652 0.11030539 0.08441330 0.02959921 32  2.023480
109  female   dem 0.10409339 0.11104306 0.08499436 0.02981524  4 65.455334
110  female repub 0.10479442 0.11178505 0.08557905 0.03003280 32  1.930268
111    male   dem 0.10549961 0.11253137 0.08616738 0.03025190  3 68.272856
112    male repub 0.10620898 0.11328204 0.08675937 0.03047255 32  2.023480
113  female   dem 0.10692256 0.11403707 0.08735504 0.03069476  4 65.455334
114  female repub 0.10764035 0.11479648 0.08795441 0.03091854 32  1.930268
115    male   dem 0.10836237 0.11556029 0.08855749 0.03114389  3 68.272856
116    male repub 0.10908864 0.11632851 0.08916430 0.03137083 32  2.023480
117  female   dem 0.10981919 0.11710117 0.08977486 0.03159938  4 65.455334
118  female repub 0.11055402 0.11787827 0.09038919 0.03182953 32  1.930268
119    male   dem 0.11129315 0.11865983 0.09100729 0.03206131  3 68.272856
120    male repub 0.11203660 0.11944588 0.09162920 0.03229471 32  2.023480
121  female   dem 0.11278438 0.12023642 0.09225493 0.03252976  4 65.455334
122  female repub 0.11353652 0.12103148 0.09288450 0.03276646 32  1.930268
123    male   dem 0.11429303 0.12183106 0.09351791 0.03300483  3 68.272856
124    male repub 0.11505392 0.12263519 0.09415520 0.03324486 32  2.023480
125  female   dem 0.11581922 0.12344388 0.09479638 0.03348659  4 65.455334
126  female repub 0.11658894 0.12425715 0.09544147 0.03373001 32  1.930268
127    male   dem 0.11736309 0.12507501 0.09609047 0.03397514  3 68.272856
128    male repub 0.11814170 0.12589748 0.09674342 0.03422198 32  2.023480
129  female   dem 0.11892478 0.12672458 0.09740033 0.03447056  4 65.455334
130  female repub 0.11971234 0.12755632 0.09806122 0.03472088 32  1.930268
131    male   dem 0.12050440 0.12839271 0.09872609 0.03497294  3 68.272856
132    male repub 0.12130099 0.12923378 0.09939498 0.03522678 32  2.023480
133  female   dem 0.12210210 0.13007953 0.10006790 0.03548238  4 65.455334
134  female repub 0.12290777 0.13092999 0.10074487 0.03573977 32  1.930268
135    male   dem 0.12371801 0.13178516 0.10142589 0.03599896  3 68.272856
136    male repub 0.12453283 0.13264507 0.10211100 0.03625996 32  2.023480
137  female   dem 0.12535224 0.13350972 0.10280021 0.03652278  4 65.455334
138  female repub 0.12617627 0.13437914 0.10349353 0.03678743 32  1.930268
139    male   dem 0.12700494 0.13525334 0.10419099 0.03705393  3 68.272856
140    male repub 0.12783825 0.13613232 0.10489259 0.03732228 32  2.023480
141  female   dem 0.12867621 0.13701612 0.10559837 0.03759250  4 65.455334
142  female repub 0.12951886 0.13790474 0.10630833 0.03786459 32  1.930268
143    male   dem 0.13036620 0.13879819 0.10702249 0.03813858  3 68.272856
144    male repub 0.13121825 0.13969650 0.10774087 0.03841448 32  2.023480
145  female   dem 0.13207502 0.14059967 0.10846348 0.03869229  4 65.455334
146  female repub 0.13293653 0.14150771 0.10919035 0.03897202 32  1.930268
147    male   dem 0.13380279 0.14242065 0.10992149 0.03925370  3 68.272856
148    male repub 0.13467382 0.14333850 0.11065692 0.03953733 32  2.023480
149  female   dem 0.13554963 0.14426127 0.11139665 0.03982292  4 65.455334
150  female repub 0.13643024 0.14518897 0.11214070 0.04011049 32  1.930268
151    male   dem 0.13731566 0.14612162 0.11288910 0.04040004  3 68.272856
152    male repub 0.13820591 0.14705922 0.11364184 0.04069161 32  2.023480
153  female   dem 0.13910100 0.14800181 0.11439896 0.04098518  4 65.455334
154  female repub 0.14000095 0.14894937 0.11516047 0.04128078 32  1.930268
155    male   dem 0.14090576 0.14990194 0.11592639 0.04157842  3 68.272856
156    male repub 0.14181546 0.15085952 0.11669672 0.04187811 32  2.023480
157  female   dem 0.14273006 0.15182212 0.11747150 0.04217987  4 65.455334
158  female repub 0.14364957 0.15278977 0.11825073 0.04248371 32  1.930268
159    male   dem 0.14457400 0.15376246 0.11903443 0.04278964  3 68.272856
160    male repub 0.14550337 0.15474021 0.11982262 0.04309767 32  2.023480
161  female   dem 0.14643769 0.15572303 0.12061531 0.04340781  4 65.455334
162  female repub 0.14737697 0.15671094 0.12141253 0.04372009 32  1.930268
163    male   dem 0.14832124 0.15770395 0.12221428 0.04403451  3 68.272856
164    male repub 0.14927049 0.15870206 0.12302058 0.04435109 32  2.023480
165  female   dem 0.15022475 0.15970530 0.12383146 0.04466984  4 65.455334
166  female repub 0.15118403 0.16071366 0.12464692 0.04499077 32  1.930268
167    male   dem 0.15214833 0.16172717 0.12546698 0.04531389  3 68.272856
168    male repub 0.15311767 0.16274582 0.12629165 0.04563923 32  2.023480
169  female   dem 0.15409207 0.16376964 0.12712096 0.04596679  4 65.455334
170  female repub 0.15507154 0.16479864 0.12795492 0.04629658 32  1.930268
171    male   dem 0.15605608 0.16583281 0.12879354 0.04662863  3 68.272856
172    male repub 0.15704571 0.16687218 0.12963684 0.04696294 32  2.023480
173  female   dem 0.15804044 0.16791676 0.13048484 0.04729953  4 65.455334
174  female repub 0.15904028 0.16896654 0.13133755 0.04763841 32  1.930268
175    male   dem 0.16004524 0.17002155 0.13219498 0.04797960  3 68.272856
176    male repub 0.16105534 0.17108179 0.13305715 0.04832310 32  2.023480
177  female   dem 0.16207059 0.17214727 0.13392407 0.04866894  4 65.455334
178  female repub 0.16309099 0.17321800 0.13479577 0.04901713 32  1.930268
179    male   dem 0.16411655 0.17429398 0.13567225 0.04936768  3 68.272856
180    male repub 0.16514730 0.17537523 0.13655353 0.04972060 32  2.023480
181  female   dem 0.16618323 0.17646176 0.13743963 0.05007592  4 65.455334
182  female repub 0.16722435 0.17755357 0.13833055 0.05043364 32  1.930268
183    male   dem 0.16827068 0.17865066 0.13922632 0.05079377  3 68.272856
184    male repub 0.16932223 0.17975306 0.14012694 0.05115635 32  2.023480
185  female   dem 0.17037900 0.18086076 0.14103244 0.05152136  4 65.455334
186  female repub 0.17144101 0.18197377 0.14194282 0.05188884 32  1.930268
187    male   dem 0.17250826 0.18309210 0.14285810 0.05225880  3 68.272856
188    male repub 0.17358076 0.18421575 0.14377829 0.05263125 32  2.023480
189  female   dem 0.17465853 0.18534474 0.14470341 0.05300621  4 65.455334
190  female repub 0.17574156 0.18647906 0.14563347 0.05338368 32  1.930268
191    male   dem 0.17682987 0.18761873 0.14656849 0.05376369  3 68.272856
192    male repub 0.17792346 0.18876375 0.14750847 0.05414625 32  2.023480
193  female   dem 0.17902235 0.18991412 0.14845343 0.05453138  4 65.455334
194  female repub 0.18012653 0.19106985 0.14940338 0.05491909 32  1.930268
195    male   dem 0.18123602 0.19223094 0.15035834 0.05530939  3 68.272856
196    male repub 0.18235083 0.19339740 0.15131831 0.05570230 32  2.023480
197  female   dem 0.18347096 0.19456924 0.15228332 0.05609784  4 65.455334
198  female repub 0.18459641 0.19574645 0.15325337 0.05649602 32  1.930268
199    male   dem 0.18572720 0.19692905 0.15422847 0.05689685  3 68.272856
200    male repub 0.18686333 0.19811703 0.15520864 0.05730036 32  2.023480
201  female   dem 0.18800480 0.19931039 0.15619389 0.05770655  4 65.455334
202  female repub 0.18915162 0.20050915 0.15718424 0.05811544 32  1.930268
203    male   dem 0.19030380 0.20171330 0.15817968 0.05852706  3 68.272856
204    male repub 0.19146134 0.20292285 0.15918023 0.05894140 32  2.023480
205  female   dem 0.19262425 0.20413780 0.16018592 0.05935849  4 65.455334
206  female repub 0.19379253 0.20535814 0.16119673 0.05977835 32  1.930268
207    male   dem 0.19496618 0.20658389 0.16221270 0.06020099  3 68.272856
208    male repub 0.19614521 0.20781504 0.16323382 0.06062642 32  2.023480
209  female   dem 0.19732962 0.20905159 0.16426010 0.06105466  4 65.455334
210  female repub 0.19851942 0.21029355 0.16529157 0.06148573 32  1.930268
211    male   dem 0.19971461 0.21154091 0.16632822 0.06191964  3 68.272856
212    male repub 0.20091519 0.21279368 0.16737008 0.06235641 32  2.023480
213  female   dem 0.20212116 0.21405186 0.16841714 0.06279606  4 65.455334
214  female repub 0.20333253 0.21531544 0.16946941 0.06323860 32  1.930268
215    male   dem 0.20454930 0.21658442 0.17052692 0.06368404  3 68.272856
216    male repub 0.20577147 0.21785880 0.17158965 0.06413241 32  2.023480
217  female   dem 0.20699904 0.21913859 0.17265764 0.06458371  4 65.455334
218  female repub 0.20823201 0.22042377 0.17373087 0.06503797 32  1.930268
219    male   dem 0.20947039 0.22171435 0.17480937 0.06549520  3 68.272856
220    male repub 0.21071418 0.22301033 0.17589314 0.06595542 32  2.023480
221  female   dem 0.21196337 0.22431170 0.17698218 0.06641865  4 65.455334
222  female repub 0.21321796 0.22561845 0.17807652 0.06688489 32  1.930268
223    male   dem 0.21447796 0.22693060 0.17917614 0.06735417  3 68.272856
224    male repub 0.21574336 0.22824812 0.18028107 0.06782651 32  2.023480
225  female   dem 0.21701417 0.22957102 0.18139130 0.06830191  4 65.455334
226  female repub 0.21829038 0.23089930 0.18250685 0.06878040 32  1.930268
227    male   dem 0.21957199 0.23223294 0.18362772 0.06926199  3 68.272856
228    male repub 0.22085899 0.23357195 0.18475392 0.06974671 32  2.023480
229  female   dem 0.22215140 0.23491632 0.18588545 0.07023455  4 65.455334
230  female repub 0.22344919 0.23626604 0.18702232 0.07072556 32  1.930268
231    male   dem 0.22475238 0.23762110 0.18816454 0.07121973  3 68.272856
232    male repub 0.22606096 0.23898151 0.18931211 0.07171708 32  2.023480
233  female   dem 0.22737492 0.24034725 0.19046504 0.07221765  4 65.455334
234  female repub 0.22869426 0.24171831 0.19162333 0.07272143 32  1.930268
235    male   dem 0.23001898 0.24309469 0.19278699 0.07322844  3 68.272856
236    male repub 0.23134907 0.24447638 0.19395601 0.07373872 32  2.023480
237  female   dem 0.23268452 0.24586338 0.19513042 0.07425226  4 65.455334
238  female repub 0.23402534 0.24725567 0.19631020 0.07476909 32  1.930268
239    male   dem 0.23537152 0.24865324 0.19749536 0.07528922  3 68.272856
240    male repub 0.23672304 0.25005609 0.19868591 0.07581268 32  2.023480
241  female   dem 0.23807991 0.25146421 0.19988185 0.07633947  4 65.455334
242  female repub 0.23944212 0.25287758 0.20108318 0.07686963 32  1.930268
243    male   dem 0.24080965 0.25429619 0.20228991 0.07740315  3 68.272856
244    male repub 0.24218252 0.25572004 0.20350203 0.07794007 32  2.023480
245  female   dem 0.24356069 0.25714911 0.20471955 0.07848039  4 65.455334
246  female repub 0.24494418 0.25858340 0.20594248 0.07902414 32  1.930268
247    male   dem 0.24633296 0.26002288 0.20717080 0.07957133  3 68.272856
248    male repub 0.24772703 0.26146755 0.20840453 0.08012198 32  2.023480
249  female   dem 0.24912639 0.26291740 0.20964367 0.08067610  4 65.455334
250  female repub 0.25053102 0.26437241 0.21088820 0.08123372 32  1.930268
251    male   dem 0.25194091 0.26583256 0.21213815 0.08179485  3 68.272856
252    male repub 0.25335605 0.26729785 0.21339350 0.08235951 32  2.023480
253  female   dem 0.25477644 0.26876826 0.21465425 0.08292772  4 65.455334
254  female repub 0.25620205 0.27024378 0.21592041 0.08349949 32  1.930268
255    male   dem 0.25763289 0.27172439 0.21719197 0.08407483  3 68.272856
256    male repub 0.25906893 0.27321007 0.21846893 0.08465378 32  2.023480
257  female   dem 0.26051016 0.27470081 0.21975129 0.08523635  4 65.455334
258  female repub 0.26195658 0.27619659 0.22103905 0.08582254 32  1.930268
259    male   dem 0.26340818 0.27769740 0.22233221 0.08641239  3 68.272856
260    male repub 0.26486492 0.27920322 0.22363076 0.08700591 32  2.023480
261  female   dem 0.26632681 0.28071403 0.22493470 0.08760311  4 65.455334
262  female repub 0.26779384 0.28222981 0.22624403 0.08820401 32  1.930268
263    male   dem 0.26926597 0.28375055 0.22755874 0.08880864  3 68.272856
264    male repub 0.27074321 0.28527623 0.22887883 0.08941700 32  2.023480
265  female   dem 0.27222553 0.28680683 0.23020430 0.09002912  4 65.455334
266  female repub 0.27371292 0.28834232 0.23153514 0.09064501 32  1.930268
267    male   dem 0.27520536 0.28988270 0.23287134 0.09126469  3 68.272856
268    male repub 0.27670285 0.29142793 0.23421291 0.09188818 32  2.023480
269  female   dem 0.27820535 0.29297801 0.23555983 0.09251550  4 65.455334
270  female repub 0.27971286 0.29453290 0.23691210 0.09314666 32  1.930268
271    male   dem 0.28122535 0.29609258 0.23826972 0.09378168  3 68.272856
272    male repub 0.28274281 0.29765704 0.23963267 0.09442058 32  2.023480
273  female   dem 0.28426522 0.29922626 0.24100095 0.09506337  4 65.455334
274  female repub 0.28579256 0.30080020 0.24237455 0.09571008 32  1.930268
275    male   dem 0.28732481 0.30237885 0.24375347 0.09636072  3 68.272856
276    male repub 0.28886196 0.30396218 0.24513769 0.09701531 32  2.023480
277  female   dem 0.29040398 0.30555018 0.24652722 0.09767387  4 65.455334
278  female repub 0.29195085 0.30714280 0.24792203 0.09833641 32  1.930268
279    male   dem 0.29350255 0.30874004 0.24932212 0.09900294  3 68.272856
280    male repub 0.29505906 0.31034187 0.25072748 0.09967350 32  2.023480
281  female   dem 0.29662036 0.31194825 0.25213811 0.10034810  4 65.455334
282  female repub 0.29818643 0.31355917 0.25355398 0.10102674 32  1.930268
283    male   dem 0.29975725 0.31517460 0.25497510 0.10170946  3 68.272856
284    male repub 0.30133278 0.31679450 0.25640144 0.10239627 32  2.023480
285  female   dem 0.30291301 0.31841887 0.25783300 0.10308718  4 65.455334
286  female repub 0.30449792 0.32004766 0.25926977 0.10378221 32  1.930268
287    male   dem 0.30608748 0.32168085 0.26071173 0.10448139  3 68.272856
288    male repub 0.30768167 0.32331841 0.26215888 0.10518472 32  2.023480
289  female   dem 0.30928046 0.32496031 0.26361119 0.10589222  4 65.455334
290  female repub 0.31088383 0.32660653 0.26506865 0.10660392 32  1.930268
291    male   dem 0.31249175 0.32825703 0.26653126 0.10731983  3 68.272856
292    male repub 0.31410419 0.32991179 0.26799900 0.10803997 32  2.023480
293  female   dem 0.31572113 0.33157077 0.26947185 0.10876434  4 65.455334
294  female repub 0.31734255 0.33323394 0.27094979 0.10949298 32  1.930268
295    male   dem 0.31896841 0.33490128 0.27243282 0.11022590  3 68.272856
296    male repub 0.32059869 0.33657275 0.27392092 0.11096311 32  2.023480
297  female   dem 0.32223336 0.33824832 0.27541407 0.11170463  4 65.455334
298  female repub 0.32387239 0.33992795 0.27691225 0.11245048 32  1.930268
299    male   dem 0.32551575 0.34161162 0.27841545 0.11320068  3 68.272856
300    male repub 0.32716342 0.34329929 0.27992366 0.11395523 32  2.023480
301  female   dem 0.32881536 0.34499093 0.28143684 0.11471417  4 65.455334
302  female repub 0.33047155 0.34668651 0.28295500 0.11547750 32  1.930268
303    male   dem 0.33213195 0.34838599 0.28447810 0.11624525  3 68.272856
304    male repub 0.33379653 0.35008933 0.28600612 0.11701742 32  2.023480
305  female   dem 0.33546527 0.35179651 0.28753906 0.11779404  4 65.455334
306  female repub 0.33713813 0.35350748 0.28907689 0.11857512 32  1.930268
307    male   dem 0.33881507 0.35522221 0.29061959 0.11936069  3 68.272856
308    male repub 0.34049607 0.35694067 0.29216713 0.12015074 32  2.023480
309  female   dem 0.34218110 0.35866282 0.29371951 0.12094531  4 65.455334
310  female repub 0.34387011 0.36038862 0.29527669 0.12174440 32  1.930268
311    male   dem 0.34556309 0.36211804 0.29683866 0.12254804  3 68.272856
312    male repub 0.34725998 0.36385103 0.29840539 0.12335624 32  2.023480
313  female   dem 0.34896077 0.36558756 0.29997686 0.12416901  4 65.455334
314  female repub 0.35066541 0.36732760 0.30155305 0.12498637 32  1.930268
315    male   dem 0.35237387 0.36907110 0.30313394 0.12580835  3 68.272856
316    male repub 0.35408612 0.37081803 0.30471950 0.12663494 32  2.023480
317  female   dem 0.35580212 0.37256834 0.30630971 0.12746618  4 65.455334
318  female repub 0.35752183 0.37432199 0.30790454 0.12830207 32  1.930268
319    male   dem 0.35924521 0.37607896 0.30950397 0.12914263  3 68.272856
320    male repub 0.36097224 0.37783919 0.31110797 0.12998787 32  2.023480
321  female   dem 0.36270286 0.37960265 0.31271653 0.13083782  4 65.455334
322  female repub 0.36443706 0.38136929 0.31432960 0.13169248 32  1.930268
323    male   dem 0.36617478 0.38313908 0.31594717 0.13255188  3 68.272856
324    male repub 0.36791599 0.38491197 0.31756921 0.13341602 32  2.023480
325  female   dem 0.36966065 0.38668792 0.31919569 0.13428492  4 65.455334
326  female repub 0.37140872 0.38846689 0.32082658 0.13515860 32  1.930268
327    male   dem 0.37316017 0.39024883 0.32246186 0.13603707  3 68.272856
328    male repub 0.37491495 0.39203371 0.32410150 0.13692034 32  2.023480
329  female   dem 0.37667302 0.39382148 0.32574546 0.13780844  4 65.455334
330  female repub 0.37843435 0.39561210 0.32739373 0.13870137 32  1.930268
331    male   dem 0.38019888 0.39740553 0.32904627 0.13959915  3 68.272856
332    male repub 0.38196659 0.39920171 0.33070304 0.14050179 32  2.023480
333  female   dem 0.38373743 0.40100061 0.33236403 0.14140931  4 65.455334
334  female repub 0.38551136 0.40280218 0.33402919 0.14232172 32  1.930268
335    male   dem 0.38728833 0.40460638 0.33569851 0.14323904  3 68.272856
336    male repub 0.38906831 0.40641315 0.33737193 0.14416127 32  2.023480
337  female   dem 0.39085126 0.40822247 0.33904945 0.14508844  4 65.455334
338  female repub 0.39263712 0.41003428 0.34073101 0.14602055 32  1.930268
339    male   dem 0.39442585 0.41184853 0.34241660 0.14695762  3 68.272856
340    male repub 0.39621743 0.41366518 0.34410617 0.14789967 32  2.023480
341  female   dem 0.39801179 0.41548418 0.34579969 0.14884669  4 65.455334
342  female repub 0.39980889 0.41730549 0.34749713 0.14979872 32  1.930268
343    male   dem 0.40160870 0.41912905 0.34919846 0.15075576  3 68.272856
344    male repub 0.40341116 0.42095483 0.35090364 0.15171782 32  2.023480
345  female   dem 0.40521623 0.42278278 0.35261263 0.15268491  4 65.455334
346  female repub 0.40702387 0.42461284 0.35432540 0.15365706 32  1.930268
347    male   dem 0.40883403 0.42644497 0.35604192 0.15463426  3 68.272856
348    male repub 0.41064667 0.42827912 0.35776214 0.15561654 32  2.023480
349  female   dem 0.41246174 0.43011524 0.35948604 0.15660390  4 65.455334
350  female repub 0.41427919 0.43195329 0.36121357 0.15759635 32  1.930268
351    male   dem 0.41609897 0.43379321 0.36294470 0.15859391  3 68.272856
352    male repub 0.41792105 0.43563496 0.36467938 0.15959659 32  2.023480
353  female   dem 0.41974537 0.43747849 0.36641759 0.16060440  4 65.455334
354  female repub 0.42157188 0.43932375 0.36815929 0.16161735 32  1.930268
355    male   dem 0.42340055 0.44117068 0.36990443 0.16263545  3 68.272856
356    male repub 0.42523131 0.44301925 0.37165297 0.16365871 32  2.023480
357  female   dem 0.42706413 0.44486940 0.37340489 0.16468715  4 65.455334
358  female repub 0.42889895 0.44672107 0.37516013 0.16572076 32  1.930268
359    male   dem 0.43073573 0.44857422 0.37691866 0.16675957  3 68.272856
360    male repub 0.43257442 0.45042881 0.37868043 0.16780358 32  2.023480
361  female   dem 0.43441496 0.45228477 0.38044542 0.16885280  4 65.455334
362  female repub 0.43625732 0.45414206 0.38221356 0.16990724 32  1.930268
363    male   dem 0.43810144 0.45600062 0.38398484 0.17096692  3 68.272856
364    male repub 0.43994727 0.45786042 0.38575919 0.17203183 32  2.023480
365  female   dem 0.44179476 0.45972138 0.38753659 0.17310199  4 65.455334
366  female repub 0.44364386 0.46158348 0.38931698 0.17417741 32  1.930268
367    male   dem 0.44549453 0.46344664 0.39110034 0.17525809  3 68.272856
368    male repub 0.44734671 0.46531083 0.39288660 0.17634404 32  2.023480
369  female   dem 0.44920035 0.46717598 0.39467574 0.17743528  4 65.455334
370  female repub 0.45105541 0.46904205 0.39646770 0.17853180 32  1.930268
371    male   dem 0.45291182 0.47090899 0.39826245 0.17963363  3 68.272856
372    male repub 0.45476955 0.47277675 0.40005993 0.18074075 32  2.023480
373  female   dem 0.45662854 0.47464526 0.40186011 0.18185319  4 65.455334
374  female repub 0.45848873 0.47651449 0.40366294 0.18297094 32  1.930268
375    male   dem 0.46035008 0.47838437 0.40546838 0.18409402  3 68.272856
376    male repub 0.46221254 0.48025486 0.40727637 0.18522243 32  2.023480
377  female   dem 0.46407606 0.48212591 0.40908688 0.18635618  4 65.455334
378  female repub 0.46594058 0.48399745 0.41089986 0.18749527 32  1.930268
379    male   dem 0.46780605 0.48586945 0.41271526 0.18863971  3 68.272856
380    male repub 0.46967242 0.48774184 0.41453304 0.18978950 32  2.023480
381  female   dem 0.47153964 0.48961457 0.41635315 0.19094465  4 65.455334
382  female repub 0.47340766 0.49148760 0.41817554 0.19210516 32  1.930268
383    male   dem 0.47527642 0.49336086 0.42000017 0.19327104  3 68.272856
384    male repub 0.47714587 0.49523431 0.42182699 0.19444230 32  2.023480
385  female   dem 0.47901597 0.49710790 0.42365595 0.19561893  4 65.455334
386  female repub 0.48088665 0.49898157 0.42548700 0.19680094 32  1.930268
387    male   dem 0.48275787 0.50085526 0.42732010 0.19798834  3 68.272856
388    male repub 0.48462957 0.50272893 0.42915520 0.19918112 32  2.023480
389  female   dem 0.48650170 0.50460253 0.43099225 0.20037930  4 65.455334
390  female repub 0.48837422 0.50647599 0.43283120 0.20158287 32  1.930268
391    male   dem 0.49024706 0.50834928 0.43467200 0.20279183  3 68.272856
392    male repub 0.49212017 0.51022233 0.43651460 0.20400619 32  2.023480
393  female   dem 0.49399350 0.51209509 0.43835896 0.20522595  4 65.455334
394  female repub 0.49586701 0.51396751 0.44020503 0.20645112 32  1.930268
395    male   dem 0.49774062 0.51583954 0.44205275 0.20768168  3 68.272856
396    male repub 0.49961431 0.51771113 0.44390207 0.20891765 32  2.023480
397  female   dem 0.50148800 0.51958222 0.44575295 0.21015903  4 65.455334
398  female repub 0.50336165 0.52145276 0.44760534 0.21140581 32  1.930268
399    male   dem 0.50523521 0.52332270 0.44945918 0.21265799  3 68.272856
400    male repub 0.50710862 0.52519198 0.45131443 0.21391558 32  2.023480
401  female   dem 0.50898183 0.52706056 0.45317103 0.21517857  4 65.455334
402  female repub 0.51085479 0.52892838 0.45502894 0.21644697 32  1.930268
403    male   dem 0.51272744 0.53079540 0.45688809 0.21772077  3 68.272856
404    male repub 0.51459974 0.53266155 0.45874845 0.21899997 32  2.023480
405  female   dem 0.51647162 0.53452678 0.46060997 0.22028457  4 65.455334
406  female repub 0.51834305 0.53639106 0.46247258 0.22157457 32  1.930268
407    male   dem 0.52021396 0.53825431 0.46433623 0.22286997  3 68.272856
408    male repub 0.52208430 0.54011650 0.46620089 0.22417075 32  2.023480
409  female   dem 0.52395402 0.54197757 0.46806649 0.22547693  4 65.455334
410  female repub 0.52582308 0.54383746 0.46993298 0.22678849 32  1.930268
411    male   dem 0.52769141 0.54569614 0.47180031 0.22810543  3 68.272856
412    male repub 0.52955896 0.54755354 0.47366844 0.22942775 32  2.023480
413  female   dem 0.53142569 0.54940962 0.47553730 0.23075545  4 65.455334
414  female repub 0.53329154 0.55126433 0.47740685 0.23208851 32  1.930268
415    male   dem 0.53515645 0.55311761 0.47927703 0.23342694  3 68.272856
416    male repub 0.53702039 0.55496941 0.48114779 0.23477073 32  2.023480
417  female   dem 0.53888329 0.55681969 0.48301908 0.23611987  4 65.455334
418  female repub 0.54074510 0.55866840 0.48489084 0.23747435 32  1.930268
419    male   dem 0.54260578 0.56051548 0.48676303 0.23883418  3 68.272856
420    male repub 0.54446527 0.56236088 0.48863559 0.24019934 32  2.023480
421  female   dem 0.54632352 0.56420456 0.49050847 0.24156983  4 65.455334
422  female repub 0.54818048 0.56604647 0.49238162 0.24294564 32  1.930268
423    male   dem 0.55003610 0.56788655 0.49425498 0.24432676  3 68.272856
424    male repub 0.55189033 0.56972477 0.49612850 0.24571318 32  2.023480
425  female   dem 0.55374312 0.57156106 0.49800214 0.24710490  4 65.455334
426  female repub 0.55559442 0.57339539 0.49987582 0.24850190 32  1.930268
427    male   dem 0.55744417 0.57522769 0.50174951 0.24990418  3 68.272856
428    male repub 0.55929233 0.57705794 0.50362315 0.25131173 32  2.023480
429  female   dem 0.56113885 0.57888607 0.50549669 0.25272453  4 65.455334
430  female repub 0.56298368 0.58071203 0.50737008 0.25414258 32  1.930268
431    male   dem 0.56482677 0.58253580 0.50924326 0.25556586  3 68.272856
432    male repub 0.56666807 0.58435730 0.51111618 0.25699437 32  2.023480
433  female   dem 0.56850753 0.58617651 0.51298878 0.25842809  4 65.455334
434  female repub 0.57034510 0.58799336 0.51486102 0.25986701 32  1.930268
435    male   dem 0.57218073 0.58980783 0.51673285 0.26131113  3 68.272856
436    male repub 0.57401438 0.59161985 0.51860421 0.26276041 32  2.023480
437  female   dem 0.57584599 0.59342938 0.52047504 0.26421486  4 65.455334
438  female repub 0.57767553 0.59523638 0.52234530 0.26567446 32  1.930268
439    male   dem 0.57950293 0.59704081 0.52421493 0.26713920  3 68.272856
440    male repub 0.58132816 0.59884261 0.52608389 0.26860906 32  2.023480
441  female   dem 0.58315117 0.60064174 0.52795211 0.27008402  4 65.455334
442  female repub 0.58497190 0.60243816 0.52981955 0.27156408 32  1.930268
443    male   dem 0.58679032 0.60423183 0.53168616 0.27304921  3 68.272856
444    male repub 0.58860637 0.60602269 0.53355188 0.27453940 32  2.023480
445  female   dem 0.59042001 0.60781071 0.53541667 0.27603464  4 65.455334
446  female repub 0.59223120 0.60959585 0.53728046 0.27753491 32  1.930268
447    male   dem 0.59403988 0.61137805 0.53914321 0.27904019  3 68.272856
448    male repub 0.59584602 0.61315728 0.54100487 0.28055046 32  2.023480
449  female   dem 0.59764956 0.61493350 0.54286539 0.28206571  4 65.455334
450  female repub 0.59945047 0.61670665 0.54472471 0.28358591 32  1.930268
451    male   dem 0.60124869 0.61847671 0.54658278 0.28511106  3 68.272856
452    male repub 0.60304419 0.62024363 0.54843956 0.28664112 32  2.023480
453  female   dem 0.60483691 0.62200737 0.55029499 0.28817609  4 65.455334
454  female repub 0.60662682 0.62376788 0.55214902 0.28971594 32  1.930268
455    male   dem 0.60841387 0.62552513 0.55400160 0.29126065  3 68.272856
456    male repub 0.61019802 0.62727908 0.55585268 0.29281020 32  2.023480
457  female   dem 0.61197922 0.62902968 0.55770222 0.29436457  4 65.455334
458  female repub 0.61375744 0.63077690 0.55955015 0.29592374 32  1.930268
459    male   dem 0.61553263 0.63252070 0.56139644 0.29748769  3 68.272856
460    male repub 0.61730475 0.63426104 0.56324103 0.29905639 32  2.023480
461  female   dem 0.61907375 0.63599788 0.56508387 0.30062982  4 65.455334
462  female repub 0.62083960 0.63773119 0.56692491 0.30220796 32  1.930268
463    male   dem 0.62260225 0.63946092 0.56876411 0.30379079  3 68.272856
464    male repub 0.62436166 0.64118703 0.57060142 0.30537828 32  2.023480
465  female   dem 0.62611780 0.64290950 0.57243677 0.30697041  4 65.455334
466  female repub 0.62787063 0.64462828 0.57427014 0.30856715 32  1.930268
467    male   dem 0.62962009 0.64634334 0.57610147 0.31016848  3 68.272856
468    male repub 0.63136616 0.64805464 0.57793071 0.31177437 32  2.023480
469  female   dem 0.63310880 0.64976215 0.57975782 0.31338480  4 65.455334
470  female repub 0.63484796 0.65146583 0.58158274 0.31499974 32  1.930268
471    male   dem 0.63658361 0.65316564 0.58340543 0.31661917  3 68.272856
472    male repub 0.63831571 0.65486156 0.58522584 0.31824305 32  2.023480
473  female   dem 0.64004423 0.65655355 0.58704393 0.31987136  4 65.455334
474  female repub 0.64176911 0.65824157 0.58885965 0.32150408 32  1.930268
475    male   dem 0.64349034 0.65992559 0.59067295 0.32314117  3 68.272856
476    male repub 0.64520787 0.66160557 0.59248379 0.32478260 32  2.023480
477  female   dem 0.64692166 0.66328150 0.59429212 0.32642836  4 65.455334
478  female repub 0.64863169 0.66495332 0.59609790 0.32807840 32  1.930268
479    male   dem 0.65033791 0.66662102 0.59790108 0.32973269  3 68.272856
480    male repub 0.65204028 0.66828456 0.59970161 0.33139122 32  2.023480
481  female   dem 0.65373879 0.66994391 0.60149946 0.33305394  4 65.455334
482  female repub 0.65543338 0.67159903 0.60329457 0.33472083 32  1.930268
483    male   dem 0.65712403 0.67324990 0.60508690 0.33639185  3 68.272856
484    male repub 0.65881070 0.67489650 0.60687641 0.33806698 32  2.023480
485  female   dem 0.66049336 0.67653878 0.60866306 0.33974618  4 65.455334
486  female repub 0.66217198 0.67817672 0.61044680 0.34142941 32  1.930268
487    male   dem 0.66384653 0.67981029 0.61222759 0.34311665  3 68.272856
488    male repub 0.66551696 0.68143946 0.61400539 0.34480787 32  2.023480
489  female   dem 0.66718326 0.68306421 0.61578015 0.34650302  4 65.455334
490  female repub 0.66884539 0.68468451 0.61755184 0.34820208 32  1.930268
491    male   dem 0.67050331 0.68630033 0.61932040 0.34990500  3 68.272856
492    male repub 0.67215700 0.68791164 0.62108581 0.35161177 32  2.023480
493  female   dem 0.67380643 0.68951842 0.62284801 0.35332233  4 65.455334
494  female repub 0.67545157 0.69112065 0.62460697 0.35503666 32  1.930268
495    male   dem 0.67709239 0.69271828 0.62636265 0.35675472  3 68.272856
496    male repub 0.67872886 0.69431131 0.62811501 0.35847647 32  2.023480
497  female   dem 0.68036095 0.69589971 0.62986400 0.36020188  4 65.455334
498  female repub 0.68198863 0.69748345 0.63160959 0.36193090 32  1.930268
499    male   dem 0.68361188 0.69906251 0.63335175 0.36366351  3 68.272856
500    male repub 0.68523067 0.70063686 0.63509042 0.36539966 32  2.023480
501  female   dem 0.68684497 0.70220649 0.63682558 0.36713932  4 65.455334
502  female repub 0.68845476 0.70377136 0.63855718 0.36888245 32  1.930268
503    male   dem 0.69006000 0.70533147 0.64028519 0.37062901  3 68.272856
504    male repub 0.69166068 0.70688678 0.64200957 0.37237895 32  2.023480
505  female   dem 0.69325676 0.70843727 0.64373028 0.37413225  4 65.455334
506  female repub 0.69484823 0.70998293 0.64544729 0.37588886 32  1.930268
507    male   dem 0.69643506 0.71152373 0.64716056 0.37764874  3 68.272856
508    male repub 0.69801722 0.71305965 0.64887006 0.37941185 32  2.023480
509  female   dem 0.69959469 0.71459067 0.65057574 0.38117815  4 65.455334
510  female repub 0.70116745 0.71611678 0.65227758 0.38294760 32  1.930268
511    male   dem 0.70273547 0.71763795 0.65397554 0.38472016  3 68.272856
512    male repub 0.70429874 0.71915417 0.65566958 0.38649578 32  2.023480
513  female   dem 0.70585723 0.72066541 0.65735968 0.38827442  4 65.455334
514  female repub 0.70741091 0.72217167 0.65904580 0.39005605 32  1.930268
515    male   dem 0.70895978 0.72367291 0.66072790 0.39184062  3 68.272856
516    male repub 0.71050379 0.72516913 0.66240595 0.39362808 32  2.023480
517  female   dem 0.71204295 0.72666031 0.66407992 0.39541839  4 65.455334
518  female repub 0.71357722 0.72814643 0.66574978 0.39721151 32  1.930268
519    male   dem 0.71510658 0.72962748 0.66741550 0.39900740  3 68.272856
520    male repub 0.71663103 0.73110344 0.66907704 0.40080601 32  2.023480
521  female   dem 0.71815053 0.73257430 0.67073437 0.40260729  4 65.455334
522  female repub 0.71966507 0.73404003 0.67238747 0.40441120 32  1.930268
523    male   dem 0.72117463 0.73550064 0.67403631 0.40621771  3 68.272856
524    male repub 0.72267919 0.73695609 0.67568084 0.40802675 32  2.023480
525  female   dem 0.72417874 0.73840639 0.67732106 0.40983829  4 65.455334
526  female repub 0.72567326 0.73985151 0.67895692 0.41165228 32  1.930268
527    male   dem 0.72716274 0.74129145 0.68058839 0.41346867  3 68.272856
528    male repub 0.72864715 0.74272619 0.68221546 0.41528742 32  2.023480
529  female   dem 0.73012648 0.74415571 0.68383809 0.41710848  4 65.455334
530  female repub 0.73160072 0.74558002 0.68545625 0.41893181 32  1.930268
531    male   dem 0.73306985 0.74699909 0.68706992 0.42075735  3 68.272856
532    male repub 0.73453385 0.74841292 0.68867908 0.42258506 32  2.023480
533  female   dem 0.73599272 0.74982149 0.69028368 0.42441489  4 65.455334
534  female repub 0.73744643 0.75122480 0.69188372 0.42624680 32  1.930268
535    male   dem 0.73889498 0.75262283 0.69347916 0.42808074  3 68.272856
536    male repub 0.74033836 0.75401558 0.69506999 0.42991665 32  2.023480
537  female   dem 0.74177654 0.75540304 0.69665616 0.43175449  4 65.455334
538  female repub 0.74320952 0.75678519 0.69823767 0.43359422 32  1.930268
539    male   dem 0.74463728 0.75816204 0.69981449 0.43543577  3 68.272856
540    male repub 0.74605982 0.75953357 0.70138659 0.43727911 32  2.023480
541  female   dem 0.74747712 0.76089977 0.70295395 0.43912418  4 65.455334
542  female repub 0.74888918 0.76226064 0.70451655 0.44097094 32  1.930268
543    male   dem 0.75029597 0.76361617 0.70607437 0.44281933  3 68.272856
544    male repub 0.75169750 0.76496635 0.70762738 0.44466931 32  2.023480
545  female   dem 0.75309375 0.76631119 0.70917557 0.44652082  4 65.455334
546  female repub 0.75448471 0.76765066 0.71071891 0.44837382 32  1.930268
547    male   dem 0.75587038 0.76898477 0.71225738 0.45022825  3 68.272856
548    male repub 0.75725074 0.77031352 0.71379097 0.45208406 32  2.023480
549  female   dem 0.75862579 0.77163689 0.71531965 0.45394121  4 65.455334
550  female repub 0.75999552 0.77295488 0.71684340 0.45579964 32  1.930268
551    male   dem 0.76135993 0.77426749 0.71836221 0.45765930  3 68.272856
552    male repub 0.76271899 0.77557472 0.71987606 0.45952015 32  2.023480
553  female   dem 0.76407272 0.77687656 0.72138492 0.46138212  4 65.455334
554  female repub 0.76542110 0.77817300 0.72288879 0.46324517 32  1.930268
555    male   dem 0.76676412 0.77946405 0.72438764 0.46510925  3 68.272856
556    male repub 0.76810179 0.78074971 0.72588146 0.46697430 32  2.023480
557  female   dem 0.76943409 0.78202996 0.72737022 0.46884028  4 65.455334
558  female repub 0.77076102 0.78330482 0.72885393 0.47070712 32  1.930268
559    male   dem 0.77208257 0.78457427 0.73033255 0.47257479  3 68.272856
560    male repub 0.77339875 0.78583832 0.73180607 0.47444323 32  2.023480
561  female   dem 0.77470954 0.78709696 0.73327449 0.47631238  4 65.455334
562  female repub 0.77601495 0.78835020 0.73473778 0.47818220 32  1.930268
563    male   dem 0.77731496 0.78959804 0.73619592 0.48005262  3 68.272856
564    male repub 0.77860959 0.79084047 0.73764892 0.48192361 32  2.023480
565  female   dem 0.77989882 0.79207749 0.73909675 0.48379510  4 65.455334
566  female repub 0.78118265 0.79330911 0.74053940 0.48566705 32  1.930268
567    male   dem 0.78246108 0.79453533 0.74197685 0.48753940  3 68.272856
568    male repub 0.78373411 0.79575615 0.74340911 0.48941210 32  2.023480
569  female   dem 0.78500174 0.79697156 0.74483614 0.49128510  4 65.455334
570  female repub 0.78626396 0.79818158 0.74625795 0.49315834 32  1.930268
571    male   dem 0.78752078 0.79938620 0.74767452 0.49503177  3 68.272856
572    male repub 0.78877220 0.80058543 0.74908584 0.49690535 32  2.023480
573  female   dem 0.79001821 0.80177926 0.75049190 0.49877901  4 65.455334
574  female repub 0.79125881 0.80296771 0.75189270 0.50065270 32  1.930268
575    male   dem 0.79249401 0.80415077 0.75328821 0.50252638  3 68.272856
576    male repub 0.79372381 0.80532846 0.75467843 0.50439999 32  2.023480
577  female   dem 0.79494820 0.80650076 0.75606336 0.50627347  4 65.455334
578  female repub 0.79616719 0.80766769 0.75744298 0.50814677 32  1.930268
579    male   dem 0.79738078 0.80882925 0.75881729 0.51001985  3 68.272856
580    male repub 0.79858898 0.80998545 0.76018628 0.51189265 32  2.023480
581  female   dem 0.79979178 0.81113629 0.76154993 0.51376511  4 65.455334
582  female repub 0.80098918 0.81228177 0.76290826 0.51563718 32  1.930268
583    male   dem 0.80218120 0.81342190 0.76426124 0.51750882  3 68.272856
584    male repub 0.80336783 0.81455669 0.76560887 0.51937997 32  2.023480
585  female   dem 0.80454908 0.81568614 0.76695114 0.52125057  4 65.455334
586  female repub 0.80572494 0.81681026 0.76828806 0.52312058 32  1.930268
587    male   dem 0.80689543 0.81792905 0.76961961 0.52498993  3 68.272856
588    male repub 0.80806055 0.81904253 0.77094579 0.52685859 32  2.023480
589  female   dem 0.80922030 0.82015069 0.77226659 0.52872650  4 65.455334
590  female repub 0.81037468 0.82125354 0.77358202 0.53059360 32  1.930268
591    male   dem 0.81152371 0.82235110 0.77489206 0.53245985  3 68.272856
592    male repub 0.81266738 0.82344337 0.77619672 0.53432518 32  2.023480
593  female   dem 0.81380571 0.82453036 0.77749598 0.53618956  4 65.455334
594  female repub 0.81493870 0.82561207 0.77878985 0.53805293 32  1.930268
595    male   dem 0.81606635 0.82668851 0.78007833 0.53991524  3 68.272856
596    male repub 0.81718867 0.82775970 0.78136141 0.54177643 32  2.023480
597  female   dem 0.81830566 0.82882563 0.78263908 0.54363645  4 65.455334
598  female repub 0.81941734 0.82988633 0.78391136 0.54549526 32  1.930268
599    male   dem 0.82052371 0.83094179 0.78517823 0.54735281  3 68.272856
600    male repub 0.82162478 0.83199203 0.78643970 0.54920903 32  2.023480
601  female   dem 0.82272055 0.83303706 0.78769577 0.55106389  4 65.455334
602  female repub 0.82381103 0.83407688 0.78894643 0.55291732 32  1.930268
603    male   dem 0.82489624 0.83511151 0.79019168 0.55476929  3 68.272856
604    male repub 0.82597617 0.83614095 0.79143153 0.55661974 32  2.023480
605  female   dem 0.82705083 0.83716522 0.79266598 0.55846862  4 65.455334
606  female repub 0.82812025 0.83818433 0.79389502 0.56031587 32  1.930268
607    male   dem 0.82918441 0.83919828 0.79511866 0.56216146  3 68.272856
608    male repub 0.83024334 0.84020709 0.79633690 0.56400533 32  2.023480
609  female   dem 0.83129704 0.84121077 0.79754974 0.56584743  4 65.455334
610  female repub 0.83234551 0.84220933 0.79875718 0.56768772 32  1.930268
611    male   dem 0.83338878 0.84320279 0.79995923 0.56952614  3 68.272856
612    male repub 0.83442685 0.84419114 0.80115588 0.57136264 32  2.023480
613  female   dem 0.83545973 0.84517441 0.80234714 0.57319718  4 65.455334
614  female repub 0.83648742 0.84615260 0.80353302 0.57502971 32  1.930268
615    male   dem 0.83750995 0.84712573 0.80471352 0.57686018  3 68.272856
616    male repub 0.83852731 0.84809381 0.80588863 0.57868854 32  2.023480
617  female   dem 0.83953953 0.84905685 0.80705837 0.58051474  4 65.455334
618  female repub 0.84054661 0.85001487 0.80822274 0.58233874 32  1.930268
619    male   dem 0.84154856 0.85096787 0.80938174 0.58416050  3 68.272856
620    male repub 0.84254539 0.85191587 0.81053538 0.58597995 32  2.023480
621  female   dem 0.84353712 0.85285889 0.81168366 0.58779706  4 65.455334
622  female repub 0.84452375 0.85379693 0.81282658 0.58961179 32  1.930268
623    male   dem 0.84550530 0.85473001 0.81396417 0.59142407  3 68.272856
624    male repub 0.84648179 0.85565814 0.81509641 0.59323388 32  2.023480
625  female   dem 0.84745321 0.85658134 0.81622331 0.59504116  4 65.455334
626  female repub 0.84841959 0.85749961 0.81734489 0.59684586 32  1.930268
627    male   dem 0.84938093 0.85841298 0.81846114 0.59864795  3 68.272856
628    male repub 0.85033725 0.85932145 0.81957208 0.60044737 32  2.023480
629  female   dem 0.85128857 0.86022504 0.82067771 0.60224409  4 65.455334
630  female repub 0.85223488 0.86112377 0.82177803 0.60403805 32  1.930268
631    male   dem 0.85317622 0.86201765 0.82287307 0.60582922  3 68.272856
632    male repub 0.85411258 0.86290669 0.82396281 0.60761755 32  2.023480
633  female   dem 0.85504399 0.86379090 0.82504728 0.60940300  4 65.455334
634  female repub 0.85597045 0.86467031 0.82612648 0.61118552 32  1.930268
635    male   dem 0.85689198 0.86554492 0.82720041 0.61296508  3 68.272856
636    male repub 0.85780860 0.86641475 0.82826909 0.61474162 32  2.023480
637  female   dem 0.85872031 0.86727982 0.82933252 0.61651511  4 65.455334
638  female repub 0.85962714 0.86814014 0.83039072 0.61828551 32  1.930268
639    male   dem 0.86052909 0.86899573 0.83144369 0.62005277  3 68.272856
640    male repub 0.86142617 0.86984659 0.83249144 0.62181685 32  2.023480
641  female   dem 0.86231841 0.87069276 0.83353398 0.62357771  4 65.455334
642  female repub 0.86320582 0.87153423 0.83457132 0.62533532 32  1.930268
643    male   dem 0.86408841 0.87237103 0.83560347 0.62708962  3 68.272856
644    male repub 0.86496620 0.87320317 0.83663045 0.62884059 32  2.023480
645  female   dem 0.86583919 0.87403067 0.83765225 0.63058818  4 65.455334
646  female repub 0.86670742 0.87485354 0.83866890 0.63233235 32  1.930268
647    male   dem 0.86757088 0.87567181 0.83968040 0.63407307  3 68.272856
648    male repub 0.86842960 0.87648548 0.84068676 0.63581029 32  2.023480
649  female   dem 0.86928359 0.87729457 0.84168799 0.63754398  4 65.455334
650  female repub 0.87013286 0.87809909 0.84268411 0.63927409 32  1.930268
651    male   dem 0.87097744 0.87889907 0.84367513 0.64100060  3 68.272856
652    male repub 0.87181734 0.87969452 0.84466105 0.64272347 32  2.023480
653  female   dem 0.87265256 0.88048546 0.84564190 0.64444265  4 65.455334
654  female repub 0.87348313 0.88127189 0.84661767 0.64615811 32  1.930268
655    male   dem 0.87430907 0.88205385 0.84758839 0.64786982  3 68.272856
656    male repub 0.87513038 0.88283134 0.84855407 0.64957774 32  2.023480
657  female   dem 0.87594709 0.88360438 0.84951471 0.65128184  4 65.455334
658  female repub 0.87675921 0.88437299 0.85047033 0.65298207 32  1.930268
659    male   dem 0.87756676 0.88513718 0.85142095 0.65467841  3 68.272856
660    male repub 0.87836975 0.88589697 0.85236657 0.65637082 32  2.023480
661  female   dem 0.87916820 0.88665238 0.85330721 0.65805927  4 65.455334
662  female repub 0.87996212 0.88740343 0.85424288 0.65974373 32  1.930268
663    male   dem 0.88075154 0.88815013 0.85517359 0.66142415  3 68.272856
664    male repub 0.88153646 0.88889249 0.85609937 0.66310052 32  2.023480
665  female   dem 0.88231690 0.88963054 0.85702021 0.66477279  4 65.455334
666  female repub 0.88309289 0.89036430 0.85793614 0.66644093 32  1.930268
667    male   dem 0.88386443 0.89109377 0.85884717 0.66810492  3 68.272856
668    male repub 0.88463154 0.89181898 0.85975332 0.66976472 32  2.023480
669  female   dem 0.88539425 0.89253994 0.86065458 0.67142031  4 65.455334
670  female repub 0.88615256 0.89325667 0.86155099 0.67307164 32  1.930268
671    male   dem 0.88690650 0.89396919 0.86244256 0.67471870  3 68.272856
672    male repub 0.88765607 0.89467751 0.86332929 0.67636144 32  2.023480
673  female   dem 0.88840131 0.89538166 0.86421121 0.67799986  4 65.455334
674  female repub 0.88914221 0.89608164 0.86508833 0.67963390 32  1.930268
675    male   dem 0.88987881 0.89677748 0.86596066 0.68126355  3 68.272856
676    male repub 0.89061111 0.89746920 0.86682822 0.68288878 32  2.023480
677  female   dem 0.89133914 0.89815680 0.86769102 0.68450956  4 65.455334
678  female repub 0.89206292 0.89884032 0.86854908 0.68612587 32  1.930268
679    male   dem 0.89278245 0.89951976 0.86940241 0.68773767  3 68.272856
680    male repub 0.89349775 0.90019514 0.87025103 0.68934494 32  2.023480
681  female   dem 0.89420885 0.90086648 0.87109495 0.69094765  4 65.455334
682  female repub 0.89491576 0.90153381 0.87193419 0.69254579 32  1.930268
683    male   dem 0.89561850 0.90219712 0.87276876 0.69413932  3 68.272856
684    male repub 0.89631709 0.90285645 0.87359869 0.69572822 32  2.023480
685  female   dem 0.89701153 0.90351182 0.87442398 0.69731246  4 65.455334
686  female repub 0.89770186 0.90416323 0.87524465 0.69889203 32  1.930268
687    male   dem 0.89838808 0.90481070 0.87606072 0.70046689  3 68.272856
688    male repub 0.89907022 0.90545426 0.87687220 0.70203703 32  2.023480
689  female   dem 0.89974828 0.90609392 0.87767911 0.70360242  4 65.455334
690  female repub 0.90042230 0.90672969 0.87848146 0.70516304 32  1.930268
691    male   dem 0.90109229 0.90736161 0.87927928 0.70671887  3 68.272856
692    male repub 0.90175825 0.90798967 0.88007257 0.70826988 32  2.023480
693  female   dem 0.90242022 0.90861391 0.88086136 0.70981606  4 65.455334
694  female repub 0.90307821 0.90923433 0.88164565 0.71135739 32  1.930268
695    male   dem 0.90373224 0.90985096 0.88242547 0.71289384  3 68.272856
696    male repub 0.90438231 0.91046382 0.88320084 0.71442539 32  2.023480
697  female   dem 0.90502847 0.91107291 0.88397176 0.71595203  4 65.455334
698  female repub 0.90567070 0.91167826 0.88473826 0.71747374 32  1.930268
699    male   dem 0.90630905 0.91227989 0.88550035 0.71899049  3 68.272856
700    male repub 0.90694352 0.91287782 0.88625805 0.72050228 32  2.023480
701  female   dem 0.90757413 0.91347205 0.88701138 0.72200907  4 65.455334
702  female repub 0.90820090 0.91406261 0.88776035 0.72351086 32  1.930268
703    male   dem 0.90882385 0.91464952 0.88850497 0.72500762  3 68.272856
704    male repub 0.90944299 0.91523279 0.88924528 0.72649935 32  2.023480
705  female   dem 0.91005834 0.91581244 0.88998128 0.72798602  4 65.455334
706  female repub 0.91066993 0.91638849 0.89071298 0.72946762 32  1.930268
707    male   dem 0.91127775 0.91696096 0.89144042 0.73094413  3 68.272856
708    male repub 0.91188185 0.91752986 0.89216360 0.73241553 32  2.023480
709  female   dem 0.91248222 0.91809521 0.89288254 0.73388183  4 65.455334
710  female repub 0.91307890 0.91865703 0.89359726 0.73534298 32  1.930268
711    male   dem 0.91367189 0.91921533 0.89430777 0.73679900  3 68.272856
712    male repub 0.91426122 0.91977014 0.89501410 0.73824985 32  2.023480
713  female   dem 0.91484689 0.92032146 0.89571625 0.73969554  4 65.455334
714  female repub 0.91542894 0.92086933 0.89641426 0.74113603 32  1.930268
715    male   dem 0.91600737 0.92141374 0.89710813 0.74257133  3 68.272856
716    male repub 0.91658221 0.92195473 0.89779788 0.74400143 32  2.023480
717  female   dem 0.91715347 0.92249231 0.89848353 0.74542630  4 65.455334
718  female repub 0.91772117 0.92302650 0.89916510 0.74684593 32  1.930268
719    male   dem 0.91828533 0.92355731 0.89984260 0.74826033  3 68.272856
720    male repub 0.91884596 0.92408476 0.90051605 0.74966947 32  2.023480
721  female   dem 0.91940308 0.92460886 0.90118548 0.75107335  4 65.455334
722  female repub 0.91995671 0.92512965 0.90185089 0.75247195 32  1.930268
723    male   dem 0.92050686 0.92564712 0.90251230 0.75386527  3 68.272856
724    male repub 0.92105356 0.92616130 0.90316973 0.75525330 32  2.023480
725  female   dem 0.92159682 0.92667221 0.90382321 0.75663603  4 65.455334
726  female repub 0.92213665 0.92717986 0.90447273 0.75801345 32  1.930268
727    male   dem 0.92267308 0.92768427 0.90511834 0.75938555  3 68.272856
728    male repub 0.92320613 0.92818546 0.90576003 0.76075233 32  2.023480
729  female   dem 0.92373580 0.92868344 0.90639784 0.76211378  4 65.455334
730  female repub 0.92426212 0.92917824 0.90703177 0.76346988 32  1.930268
731    male   dem 0.92478510 0.92966985 0.90766184 0.76482065  3 68.272856
732    male repub 0.92530476 0.93015832 0.90828808 0.76616606 32  2.023480
733  female   dem 0.92582113 0.93064364 0.90891049 0.76750611  4 65.455334
734  female repub 0.92633420 0.93112584 0.90952910 0.76884080 32  1.930268
735    male   dem 0.92684401 0.93160493 0.91014393 0.77017013  3 68.272856
736    male repub 0.92735056 0.93208094 0.91075499 0.77149408 32  2.023480
737  female   dem 0.92785388 0.93255387 0.91136229 0.77281266  4 65.455334
738  female repub 0.92835399 0.93302375 0.91196587 0.77412585 32  1.930268
739    male   dem 0.92885089 0.93349058 0.91256572 0.77543366  3 68.272856
740    male repub 0.92934461 0.93395439 0.91316188 0.77673608 32  2.023480
741  female   dem 0.92983516 0.93441520 0.91375436 0.77803311  4 65.455334
742  female repub 0.93032256 0.93487301 0.91434318 0.77932474 32  1.930268
743    male   dem 0.93080683 0.93532785 0.91492835 0.78061098  3 68.272856
744    male repub 0.93128798 0.93577973 0.91550989 0.78189182 32  2.023480
745  female   dem 0.93176603 0.93622867 0.91608782 0.78316726  4 65.455334
746  female repub 0.93224099 0.93667468 0.91666216 0.78443729 32  1.930268
747    male   dem 0.93271289 0.93711779 0.91723292 0.78570193  3 68.272856
748    male repub 0.93318174 0.93755800 0.91780012 0.78696116 32  2.023480
749  female   dem 0.93364755 0.93799533 0.91836379 0.78821498  4 65.455334
750  female repub 0.93411034 0.93842980 0.91892392 0.78946340 32  1.930268
751    male   dem 0.93457014 0.93886142 0.91948056 0.79070641  3 68.272856
752    male repub 0.93502694 0.93929021 0.92003370 0.79194402 32  2.023480
753  female   dem 0.93548078 0.93971619 0.92058337 0.79317623  4 65.455334
754  female repub 0.93593167 0.94013937 0.92112959 0.79440303 32  1.930268
755    male   dem 0.93637962 0.94055977 0.92167237 0.79562443  3 68.272856
756    male repub 0.93682464 0.94097740 0.92221173 0.79684043 32  2.023480
757  female   dem 0.93726677 0.94139228 0.92274769 0.79805103  4 65.455334
758  female repub 0.93770600 0.94180443 0.92328026 0.79925623 32  1.930268
759    male   dem 0.93814237 0.94221385 0.92380946 0.80045604  3 68.272856
760    male repub 0.93857587 0.94262057 0.92433531 0.80165046 32  2.023480
761  female   dem 0.93900654 0.94302460 0.92485783 0.80283949  4 65.455334
762  female repub 0.93943438 0.94342595 0.92537703 0.80402314 32  1.930268
763    male   dem 0.93985941 0.94382465 0.92589293 0.80520140  3 68.272856
764    male repub 0.94028165 0.94422070 0.92640555 0.80637429 32  2.023480
765  female   dem 0.94070111 0.94461412 0.92691490 0.80754180  4 65.455334
766  female repub 0.94111781 0.94500493 0.92742101 0.80870394 32  1.930268
767    male   dem 0.94153177 0.94539315 0.92792388 0.80986071  3 68.272856
768    male repub 0.94194299 0.94577877 0.92842353 0.81101213 32  2.023480
769  female   dem 0.94235150 0.94616184 0.92891999 0.81215819  4 65.455334
770  female repub 0.94275731 0.94654234 0.92941326 0.81329890 32  1.930268
771    male   dem 0.94316043 0.94692031 0.92990337 0.81443427  3 68.272856
772    male repub 0.94356089 0.94729576 0.93039034 0.81556430 32  2.023480
773  female   dem 0.94395869 0.94766869 0.93087417 0.81668899  4 65.455334
774  female repub 0.94435385 0.94803914 0.93135488 0.81780836 32  1.930268
775    male   dem 0.94474639 0.94840710 0.93183250 0.81892241  3 68.272856
776    male repub 0.94513632 0.94877260 0.93230704 0.82003114 32  2.023480
777  female   dem 0.94552366 0.94913565 0.93277851 0.82113457  4 65.455334
778  female repub 0.94590842 0.94949626 0.93324693 0.82223270 32  1.930268
779    male   dem 0.94629062 0.94985445 0.93371232 0.82332554  3 68.272856
780    male repub 0.94667027 0.95021023 0.93417470 0.82441310 32  2.023480
781  female   dem 0.94704738 0.95056362 0.93463407 0.82549538  4 65.455334
782  female repub 0.94742198 0.95091463 0.93509046 0.82657239 32  1.930268
783    male   dem 0.94779407 0.95126328 0.93554389 0.82764415  3 68.272856
784    male repub 0.94816367 0.95160958 0.93599436 0.82871065 32  2.023480
785  female   dem 0.94853080 0.95195354 0.93644190 0.82977191  4 65.455334
786  female repub 0.94889547 0.95229517 0.93688653 0.83082794 32  1.930268
787    male   dem 0.94925770 0.95263450 0.93732825 0.83187874  3 68.272856
788    male repub 0.94961749 0.95297154 0.93776708 0.83292433 32  2.023480
789  female   dem 0.94997486 0.95330629 0.93820304 0.83396472  4 65.455334
790  female repub 0.95032984 0.95363878 0.93863615 0.83499991 32  1.930268
791    male   dem 0.95068242 0.95396901 0.93906642 0.83602991  3 68.272856
792    male repub 0.95103263 0.95429701 0.93949387 0.83705474 32  2.023480
793  female   dem 0.95138048 0.95462278 0.93991851 0.83807440  4 65.455334
794  female repub 0.95172599 0.95494633 0.94034036 0.83908892 32  1.930268
795    male   dem 0.95206917 0.95526769 0.94075944 0.84009828  3 68.272856
796    male repub 0.95241002 0.95558686 0.94117575 0.84110252 32  2.023480
797  female   dem 0.95274858 0.95590386 0.94158933 0.84210163  4 65.455334
798  female repub 0.95308484 0.95621870 0.94200017 0.84309563 32  1.930268
799    male   dem 0.95341883 0.95653140 0.94240830 0.84408454  3 68.272856
800    male repub 0.95375055 0.95684196 0.94281373 0.84506835 32  2.023480
801  female   dem 0.95408003 0.95715040 0.94321648 0.84604709  4 65.455334
802  female repub 0.95440727 0.95745674 0.94361657 0.84702077 32  1.930268
803    male   dem 0.95473229 0.95776098 0.94401400 0.84798940  3 68.272856
804    male repub 0.95505510 0.95806315 0.94440880 0.84895298 32  2.023480
805  female   dem 0.95537572 0.95836324 0.94480097 0.84991154  4 65.455334
806  female repub 0.95569416 0.95866128 0.94519054 0.85086509 32  1.930268
807    male   dem 0.95601042 0.95895728 0.94557752 0.85181363  3 68.272856
808    male repub 0.95632454 0.95925125 0.94596192 0.85275718 32  2.023480
809  female   dem 0.95663651 0.95954320 0.94634376 0.85369576  4 65.455334
810  female repub 0.95694636 0.95983315 0.94672305 0.85462938 32  1.930268
811    male   dem 0.95725409 0.96012111 0.94709981 0.85555804  3 68.272856
812    male repub 0.95755972 0.96040708 0.94747406 0.85648177 32  2.023480
813  female   dem 0.95786326 0.96069109 0.94784581 0.85740058  4 65.455334
814  female repub 0.95816472 0.96097315 0.94821506 0.85831447 32  1.930268
815    male   dem 0.95846412 0.96125326 0.94858185 0.85922347  3 68.272856
816    male repub 0.95876147 0.96153145 0.94894617 0.86012759 32  2.023480
817  female   dem 0.95905678 0.96180771 0.94930806 0.86102685  4 65.455334
818  female repub 0.95935007 0.96208207 0.94966751 0.86192125 32  1.930268
819    male   dem 0.95964134 0.96235454 0.95002455 0.86281081  3 68.272856
820    male repub 0.95993061 0.96262512 0.95037919 0.86369554 32  2.023480
821  female   dem 0.96021790 0.96289384 0.95073144 0.86457547  4 65.455334
822  female repub 0.96050321 0.96316070 0.95108132 0.86545060 32  1.930268
823    male   dem 0.96078656 0.96342571 0.95142885 0.86632095  3 68.272856
824    male repub 0.96106796 0.96368888 0.95177403 0.86718653 32  2.023480
825  female   dem 0.96134742 0.96395023 0.95211688 0.86804736  4 65.455334
826  female repub 0.96162496 0.96420978 0.95245741 0.86890346 32  1.930268
827    male   dem 0.96190058 0.96446752 0.95279565 0.86975484  3 68.272856
828    male repub 0.96217430 0.96472347 0.95313159 0.87060151 32  2.023480
829  female   dem 0.96244613 0.96497765 0.95346526 0.87144349  4 65.455334
830  female repub 0.96271608 0.96523006 0.95379667 0.87228079 32  1.930268
831    male   dem 0.96298416 0.96548072 0.95412583 0.87311343  3 68.272856
832    male repub 0.96325039 0.96572963 0.95445277 0.87394144 32  2.023480
833  female   dem 0.96351478 0.96597682 0.95477748 0.87476481  4 65.455334
834  female repub 0.96377734 0.96622228 0.95509998 0.87558357 32  1.930268
835    male   dem 0.96403808 0.96646603 0.95542029 0.87639773  3 68.272856
836    male repub 0.96429702 0.96670809 0.95573843 0.87720732 32  2.023480
837  female   dem 0.96455415 0.96694845 0.95605440 0.87801234  4 65.455334
838  female repub 0.96480950 0.96718714 0.95636821 0.87881281 32  1.930268
839    male   dem 0.96506308 0.96742417 0.95667989 0.87960875  3 68.272856
840    male repub 0.96531490 0.96765954 0.95698944 0.88040017 32  2.023480
841  female   dem 0.96556497 0.96789326 0.95729687 0.88118709  4 65.455334
842  female repub 0.96581330 0.96812536 0.95760221 0.88196953 32  1.930268
843    male   dem 0.96605990 0.96835583 0.95790546 0.88274750  3 68.272856
844    male repub 0.96630478 0.96858468 0.95820663 0.88352102 32  2.023480
845  female   dem 0.96654796 0.96881194 0.95850574 0.88429011  4 65.455334
846  female repub 0.96678944 0.96903760 0.95880281 0.88505478 32  1.930268
847    male   dem 0.96702924 0.96926168 0.95909784 0.88581505  3 68.272856
848    male repub 0.96726737 0.96948420 0.95939084 0.88657093 32  2.023480
849  female   dem 0.96750384 0.96970515 0.95968183 0.88732245  4 65.455334
850  female repub 0.96773865 0.96992455 0.95997083 0.88806961 32  1.930268
851    male   dem 0.96797182 0.97014241 0.96025784 0.88881245  3 68.272856
852    male repub 0.96820337 0.97035874 0.96054288 0.88955096 32  2.023480
853  female   dem 0.96843329 0.97057356 0.96082595 0.89028518  4 65.455334
854  female repub 0.96866160 0.97078686 0.96110708 0.89101511 32  1.930268
855    male   dem 0.96888832 0.97099866 0.96138627 0.89174078  3 68.272856
856    male repub 0.96911345 0.97120897 0.96166354 0.89246220 32  2.023480
857  female   dem 0.96933700 0.97141780 0.96193889 0.89317939  4 65.455334
858  female repub 0.96955899 0.97162516 0.96221235 0.89389236 32  1.930268
859    male   dem 0.96977941 0.97183106 0.96248392 0.89460114  3 68.272856
860    male repub 0.96999829 0.97203551 0.96275361 0.89530573 32  2.023480
861  female   dem 0.97021564 0.97223852 0.96302143 0.89600617  4 65.455334
862  female repub 0.97043145 0.97244010 0.96328741 0.89670246 32  1.930268
863    male   dem 0.97064576 0.97264025 0.96355154 0.89739462  3 68.272856
864    male repub 0.97085855 0.97283899 0.96381384 0.89808266 32  2.023480
865  female   dem 0.97106985 0.97303633 0.96407433 0.89876662  4 65.455334
866  female repub 0.97127966 0.97323227 0.96433301 0.89944650 32  1.930268
867    male   dem 0.97148799 0.97342683 0.96458990 0.90012232  3 68.272856
868    male repub 0.97169486 0.97362001 0.96484500 0.90079410 32  2.023480
869  female   dem 0.97190027 0.97381182 0.96509834 0.90146186  4 65.455334
870  female repub 0.97210423 0.97400228 0.96534991 0.90212561 32  1.930268
871    male   dem 0.97230675 0.97419139 0.96559973 0.90278537  3 68.272856
872    male repub 0.97250785 0.97437916 0.96584782 0.90344116 32  2.023480
873  female   dem 0.97270752 0.97456560 0.96609418 0.90409299  4 65.455334
874  female repub 0.97290579 0.97475072 0.96633883 0.90474089 32  1.930268
875    male   dem 0.97310265 0.97493452 0.96658177 0.90538487  3 68.272856
876    male repub 0.97329813 0.97511702 0.96682301 0.90602495 32  2.023480
877  female   dem 0.97349222 0.97529823 0.96706258 0.90666115  4 65.455334
878  female repub 0.97368494 0.97547815 0.96730047 0.90729348 32  1.930268
879    male   dem 0.97387629 0.97565679 0.96753671 0.90792196  3 68.272856
880    male repub 0.97406629 0.97583416 0.96777129 0.90854661 32  2.023480
881  female   dem 0.97425495 0.97601028 0.96800424 0.90916744  4 65.455334
882  female repub 0.97444227 0.97618514 0.96823555 0.90978448 32  1.930268
883    male   dem 0.97462826 0.97635875 0.96846525 0.91039774  3 68.272856
884    male repub 0.97481293 0.97653113 0.96869334 0.91100724 32  2.023480
885  female   dem 0.97499630 0.97670229 0.96891984 0.91161300  4 65.455334
886  female repub 0.97517836 0.97687222 0.96914475 0.91221503 32  1.930268
887    male   dem 0.97535913 0.97704095 0.96936808 0.91281335  3 68.272856
888    male repub 0.97553862 0.97720847 0.96958984 0.91340799 32  2.023480
889  female   dem 0.97571683 0.97737480 0.96981006 0.91399894  4 65.455334
890  female repub 0.97589378 0.97753994 0.97002872 0.91458625 32  1.930268
891    male   dem 0.97606947 0.97770391 0.97024585 0.91516991  3 68.272856
892    male repub 0.97624391 0.97786670 0.97046146 0.91574995 32  2.023480
893  female   dem 0.97641710 0.97802834 0.97067555 0.91632639  4 65.455334
894  female repub 0.97658907 0.97818882 0.97088813 0.91689924 32  1.930268
895    male   dem 0.97675981 0.97834815 0.97109922 0.91746853  3 68.272856
896    male repub 0.97692934 0.97850634 0.97130882 0.91803426 32  2.023480
897  female   dem 0.97709765 0.97866341 0.97151695 0.91859646  4 65.455334
898  female repub 0.97726477 0.97881935 0.97172362 0.91915514 32  1.930268
899    male   dem 0.97743070 0.97897417 0.97192882 0.91971033  3 68.272856
900    male repub 0.97759545 0.97912789 0.97213258 0.92026203 32  2.023480
901  female   dem 0.97775901 0.97928051 0.97233491 0.92081027  4 65.455334
902  female repub 0.97792142 0.97943204 0.97253580 0.92135505 32  1.930268
903    male   dem 0.97808266 0.97958248 0.97273528 0.92189641  3 68.272856
904    male repub 0.97824275 0.97973184 0.97293335 0.92243436 32  2.023480
905  female   dem 0.97840170 0.97988013 0.97313002 0.92296891  4 65.455334
906  female repub 0.97855951 0.98002736 0.97332530 0.92350009 32  1.930268
907    male   dem 0.97871619 0.98017354 0.97351920 0.92402790  3 68.272856
908    male repub 0.97887176 0.98031866 0.97371173 0.92455237 32  2.023480
909  female   dem 0.97902621 0.98046275 0.97390289 0.92507351  4 65.455334
910  female repub 0.97917955 0.98060580 0.97409271 0.92559134 32  1.930268
911    male   dem 0.97933180 0.98074782 0.97428117 0.92610587  3 68.272856
912    male repub 0.97948296 0.98088883 0.97446831 0.92661714 32  2.023480
913  female   dem 0.97963304 0.98102882 0.97465412 0.92712514  4 65.455334
914  female repub 0.97978204 0.98116781 0.97483861 0.92762990 32  1.930268
915    male   dem 0.97992997 0.98130579 0.97502179 0.92813144  3 68.272856
916    male repub 0.98007684 0.98144279 0.97520367 0.92862976 32  2.023480
917  female   dem 0.98022266 0.98157880 0.97538426 0.92912490  4 65.455334
918  female repub 0.98036744 0.98171383 0.97556357 0.92961686 32  1.930268
919    male   dem 0.98051117 0.98184789 0.97574160 0.93010566  3 68.272856
920    male repub 0.98065388 0.98198099 0.97591837 0.93059132 32  2.023480
921  female   dem 0.98079555 0.98211312 0.97609389 0.93107386  4 65.455334
922  female repub 0.98093622 0.98224431 0.97626815 0.93155329 32  1.930268
923    male   dem 0.98107587 0.98237455 0.97644118 0.93202963  3 68.272856
924    male repub 0.98121452 0.98250385 0.97661297 0.93250289 32  2.023480
925  female   dem 0.98135217 0.98263222 0.97678354 0.93297310  4 65.455334
926  female repub 0.98148883 0.98275967 0.97695290 0.93344026 32  1.930268
927    male   dem 0.98162451 0.98288620 0.97712105 0.93390440  3 68.272856
928    male repub 0.98175921 0.98301181 0.97728800 0.93436553 32  2.023480
929  female   dem 0.98189294 0.98313652 0.97745377 0.93482366  4 65.455334
930  female repub 0.98202572 0.98326033 0.97761835 0.93527882 32  1.930268
931    male   dem 0.98215753 0.98338324 0.97778175 0.93573102  3 68.272856
932    male repub 0.98228840 0.98350527 0.97794399 0.93618028 32  2.023480
933  female   dem 0.98241832 0.98362641 0.97810507 0.93662661  4 65.455334
934  female repub 0.98254731 0.98374668 0.97826500 0.93707003 32  1.930268
935    male   dem 0.98267536 0.98386609 0.97842379 0.93751055  3 68.272856
936    male repub 0.98280250 0.98398462 0.97858145 0.93794819 32  2.023480
937  female   dem 0.98292872 0.98410231 0.97873797 0.93838297  4 65.455334
938  female repub 0.98305402 0.98421914 0.97889338 0.93881490 32  1.930268
939    male   dem 0.98317843 0.98433512 0.97904768 0.93924399  3 68.272856
940    male repub 0.98330193 0.98445027 0.97920087 0.93967028 32  2.023480
941  female   dem 0.98342455 0.98456459 0.97935297 0.94009376  4 65.455334
942  female repub 0.98354628 0.98467807 0.97950397 0.94051446 32  1.930268
943    male   dem 0.98366713 0.98479074 0.97965390 0.94093239  3 68.272856
944    male repub 0.98378710 0.98490259 0.97980275 0.94134756 32  2.023480
945  female   dem 0.98390621 0.98501363 0.97995053 0.94176000  4 65.455334
946  female repub 0.98402446 0.98512386 0.98009726 0.94216972 32  1.930268
947    male   dem 0.98414186 0.98523330 0.98024293 0.94257673  3 68.272856
948    male repub 0.98425840 0.98534194 0.98038756 0.94298105 32  2.023480
949  female   dem 0.98437410 0.98544980 0.98053115 0.94338269  4 65.455334
950  female repub 0.98448897 0.98555687 0.98067371 0.94378167 32  1.930268
951    male   dem 0.98460300 0.98566317 0.98081525 0.94417801  3 68.272856
952    male repub 0.98471621 0.98576870 0.98095577 0.94457172 32  2.023480
953  female   dem 0.98482860 0.98587346 0.98109528 0.94496281  4 65.455334
954  female repub 0.98494018 0.98597746 0.98123379 0.94535130 32  1.930268
955    male   dem 0.98505094 0.98608070 0.98137130 0.94573721  3 68.272856
956    male repub 0.98516091 0.98618320 0.98150782 0.94612055 32  2.023480
957  female   dem 0.98527008 0.98628495 0.98164337 0.94650133  4 65.455334
958  female repub 0.98537845 0.98638596 0.98177793 0.94687957 32  1.930268
959    male   dem 0.98548604 0.98648624 0.98191153 0.94725529  3 68.272856
960    male repub 0.98559286 0.98658579 0.98204417 0.94762850 32  2.023480
961  female   dem 0.98569889 0.98668462 0.98217585 0.94799921  4 65.455334
962  female repub 0.98580416 0.98678273 0.98230659 0.94836744 32  1.930268
963    male   dem 0.98590866 0.98688013 0.98243638 0.94873320  3 68.272856
964    male repub 0.98601241 0.98697681 0.98256524 0.94909652 32  2.023480
965  female   dem 0.98611540 0.98707280 0.98269316 0.94945739  4 65.455334
966  female repub 0.98621764 0.98716808 0.98282017 0.94981584 32  1.930268
967    male   dem 0.98631915 0.98726268 0.98294626 0.95017188  3 68.272856
968    male repub 0.98641991 0.98735658 0.98307144 0.95052553 32  2.023480
969  female   dem 0.98651994 0.98744980 0.98319572 0.95087680  4 65.455334
970  female repub 0.98661925 0.98754234 0.98331910 0.95122570 32  1.930268
971    male   dem 0.98671783 0.98763421 0.98344159 0.95157225  3 68.272856
972    male repub 0.98681570 0.98772541 0.98356320 0.95191646 32  2.023480
973  female   dem 0.98691286 0.98781595 0.98368392 0.95225835  4 65.455334
974  female repub 0.98700931 0.98790582 0.98380378 0.95259793 32  1.930268
975    male   dem 0.98710505 0.98799504 0.98392277 0.95293521  3 68.272856
976    male repub 0.98720010 0.98808361 0.98404090 0.95327021 32  2.023480
977  female   dem 0.98729446 0.98817154 0.98415817 0.95360295  4 65.455334
978  female repub 0.98738814 0.98825882 0.98427460 0.95393342 32  1.930268
979    male   dem 0.98748113 0.98834547 0.98439018 0.95426166  3 68.272856
980    male repub 0.98757344 0.98843148 0.98450493 0.95458767 32  2.023480
981  female   dem 0.98766508 0.98851687 0.98461885 0.95491146  4 65.455334
982  female repub 0.98775606 0.98860163 0.98473195 0.95523306 32  1.930268
983    male   dem 0.98784637 0.98868578 0.98484422 0.95555246  3 68.272856
984    male repub 0.98793602 0.98876931 0.98495568 0.95586970 32  2.023480
985  female   dem 0.98802502 0.98885223 0.98506634 0.95618477  4 65.455334
986  female repub 0.98811338 0.98893455 0.98517619 0.95649770 32  1.930268
987    male   dem 0.98820108 0.98901627 0.98528525 0.95680849  3 68.272856
988    male repub 0.98828815 0.98909739 0.98539352 0.95711716 32  2.023480
989  female   dem 0.98837458 0.98917791 0.98550100 0.95742373  4 65.455334
990  female repub 0.98846039 0.98925785 0.98560770 0.95772819 32  1.930268
991    male   dem 0.98854556 0.98933720 0.98571363 0.95803058  3 68.272856
992    male repub 0.98863012 0.98941598 0.98581879 0.95833090 32  2.023480
993  female   dem 0.98871406 0.98949418 0.98592319 0.95862916  4 65.455334
994  female repub 0.98879738 0.98957180 0.98602683 0.95892538 32  1.930268
995    male   dem 0.98888010 0.98964886 0.98612971 0.95921957  3 68.272856
996    male repub 0.98896221 0.98972536 0.98623186 0.95951174 32  2.023480
997  female   dem 0.98904373 0.98980129 0.98633325 0.95980190  4 65.455334
998  female repub 0.98912465 0.98987667 0.98643392 0.96009007 32  1.930268
999    male   dem 0.98920497 0.98995150 0.98653385 0.96037626  3 68.272856
1000   male repub 0.98928471 0.99002578 0.98663305 0.96066049 32  2.023480
          y2.1      y3.1      y4.1       y5.1
1    266.35063 197.58961  76.85076   5.796383
2     16.63435  70.01972 360.82006 162.638320
3    270.70537 193.74153  73.79200   5.530949
4     17.41203  72.79395 362.76203 157.051229
5    266.35063 197.58961  76.85076   5.796383
6     16.63435  70.01972 360.82006 162.638320
7    270.70537 193.74153  73.79200   5.530949
8     17.41203  72.79395 362.76203 157.051229
9    266.35063 197.58961  76.85076   5.796383
10    16.63435  70.01972 360.82006 162.638320
11   270.70537 193.74153  73.79200   5.530949
12    17.41203  72.79395 362.76203 157.051229
13   266.35063 197.58961  76.85076   5.796383
14    16.63435  70.01972 360.82006 162.638320
15   270.70537 193.74153  73.79200   5.530949
16    17.41203  72.79395 362.76203 157.051229
17   266.35063 197.58961  76.85076   5.796383
18    16.63435  70.01972 360.82006 162.638320
19   270.70537 193.74153  73.79200   5.530949
20    17.41203  72.79395 362.76203 157.051229
21   266.35063 197.58961  76.85076   5.796383
22    16.63435  70.01972 360.82006 162.638320
23   270.70537 193.74153  73.79200   5.530949
24    17.41203  72.79395 362.76203 157.051229
25   266.35063 197.58961  76.85076   5.796383
26    16.63435  70.01972 360.82006 162.638320
27   270.70537 193.74153  73.79200   5.530949
28    17.41203  72.79395 362.76203 157.051229
29   266.35063 197.58961  76.85076   5.796383
30    16.63435  70.01972 360.82006 162.638320
31   270.70537 193.74153  73.79200   5.530949
32    17.41203  72.79395 362.76203 157.051229
33   266.35063 197.58961  76.85076   5.796383
34    16.63435  70.01972 360.82006 162.638320
35   270.70537 193.74153  73.79200   5.530949
36    17.41203  72.79395 362.76203 157.051229
37   266.35063 197.58961  76.85076   5.796383
38    16.63435  70.01972 360.82006 162.638320
39   270.70537 193.74153  73.79200   5.530949
40    17.41203  72.79395 362.76203 157.051229
41   266.35063 197.58961  76.85076   5.796383
42    16.63435  70.01972 360.82006 162.638320
43   270.70537 193.74153  73.79200   5.530949
44    17.41203  72.79395 362.76203 157.051229
45   266.35063 197.58961  76.85076   5.796383
46    16.63435  70.01972 360.82006 162.638320
47   270.70537 193.74153  73.79200   5.530949
48    17.41203  72.79395 362.76203 157.051229
49   266.35063 197.58961  76.85076   5.796383
50    16.63435  70.01972 360.82006 162.638320
51   270.70537 193.74153  73.79200   5.530949
52    17.41203  72.79395 362.76203 157.051229
53   266.35063 197.58961  76.85076   5.796383
54    16.63435  70.01972 360.82006 162.638320
55   270.70537 193.74153  73.79200   5.530949
56    17.41203  72.79395 362.76203 157.051229
57   266.35063 197.58961  76.85076   5.796383
58    16.63435  70.01972 360.82006 162.638320
59   270.70537 193.74153  73.79200   5.530949
60    17.41203  72.79395 362.76203 157.051229
61   266.35063 197.58961  76.85076   5.796383
62    16.63435  70.01972 360.82006 162.638320
63   270.70537 193.74153  73.79200   5.530949
64    17.41203  72.79395 362.76203 157.051229
65   266.35063 197.58961  76.85076   5.796383
66    16.63435  70.01972 360.82006 162.638320
67   270.70537 193.74153  73.79200   5.530949
68    17.41203  72.79395 362.76203 157.051229
69   266.35063 197.58961  76.85076   5.796383
70    16.63435  70.01972 360.82006 162.638320
71   270.70537 193.74153  73.79200   5.530949
72    17.41203  72.79395 362.76203 157.051229
73   266.35063 197.58961  76.85076   5.796383
74    16.63435  70.01972 360.82006 162.638320
75   270.70537 193.74153  73.79200   5.530949
76    17.41203  72.79395 362.76203 157.051229
77   266.35063 197.58961  76.85076   5.796383
78    16.63435  70.01972 360.82006 162.638320
79   270.70537 193.74153  73.79200   5.530949
80    17.41203  72.79395 362.76203 157.051229
81   266.35063 197.58961  76.85076   5.796383
82    16.63435  70.01972 360.82006 162.638320
83   270.70537 193.74153  73.79200   5.530949
84    17.41203  72.79395 362.76203 157.051229
85   266.35063 197.58961  76.85076   5.796383
86    16.63435  70.01972 360.82006 162.638320
87   270.70537 193.74153  73.79200   5.530949
88    17.41203  72.79395 362.76203 157.051229
89   266.35063 197.58961  76.85076   5.796383
90    16.63435  70.01972 360.82006 162.638320
91   270.70537 193.74153  73.79200   5.530949
92    17.41203  72.79395 362.76203 157.051229
93   266.35063 197.58961  76.85076   5.796383
94    16.63435  70.01972 360.82006 162.638320
95   270.70537 193.74153  73.79200   5.530949
96    17.41203  72.79395 362.76203 157.051229
97   266.35063 197.58961  76.85076   5.796383
98    16.63435  70.01972 360.82006 162.638320
99   270.70537 193.74153  73.79200   5.530949
100   17.41203  72.79395 362.76203 157.051229
101  266.35063 197.58961  76.85076   5.796383
102   16.63435  70.01972 360.82006 162.638320
103  270.70537 193.74153  73.79200   5.530949
104   17.41203  72.79395 362.76203 157.051229
105  266.35063 197.58961  76.85076   5.796383
106   16.63435  70.01972 360.82006 162.638320
107  270.70537 193.74153  73.79200   5.530949
108   17.41203  72.79395 362.76203 157.051229
109  266.35063 197.58961  76.85076   5.796383
110   16.63435  70.01972 360.82006 162.638320
111  270.70537 193.74153  73.79200   5.530949
112   17.41203  72.79395 362.76203 157.051229
113  266.35063 197.58961  76.85076   5.796383
114   16.63435  70.01972 360.82006 162.638320
115  270.70537 193.74153  73.79200   5.530949
116   17.41203  72.79395 362.76203 157.051229
117  266.35063 197.58961  76.85076   5.796383
118   16.63435  70.01972 360.82006 162.638320
119  270.70537 193.74153  73.79200   5.530949
120   17.41203  72.79395 362.76203 157.051229
121  266.35063 197.58961  76.85076   5.796383
122   16.63435  70.01972 360.82006 162.638320
123  270.70537 193.74153  73.79200   5.530949
124   17.41203  72.79395 362.76203 157.051229
125  266.35063 197.58961  76.85076   5.796383
126   16.63435  70.01972 360.82006 162.638320
127  270.70537 193.74153  73.79200   5.530949
128   17.41203  72.79395 362.76203 157.051229
129  266.35063 197.58961  76.85076   5.796383
130   16.63435  70.01972 360.82006 162.638320
131  270.70537 193.74153  73.79200   5.530949
132   17.41203  72.79395 362.76203 157.051229
133  266.35063 197.58961  76.85076   5.796383
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153  266.35063 197.58961  76.85076   5.796383
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213  266.35063 197.58961  76.85076   5.796383
214   16.63435  70.01972 360.82006 162.638320
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217  266.35063 197.58961  76.85076   5.796383
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221  266.35063 197.58961  76.85076   5.796383
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225  266.35063 197.58961  76.85076   5.796383
226   16.63435  70.01972 360.82006 162.638320
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229  266.35063 197.58961  76.85076   5.796383
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233  266.35063 197.58961  76.85076   5.796383
234   16.63435  70.01972 360.82006 162.638320
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237  266.35063 197.58961  76.85076   5.796383
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241  266.35063 197.58961  76.85076   5.796383
242   16.63435  70.01972 360.82006 162.638320
243  270.70537 193.74153  73.79200   5.530949
244   17.41203  72.79395 362.76203 157.051229
245  266.35063 197.58961  76.85076   5.796383
246   16.63435  70.01972 360.82006 162.638320
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248   17.41203  72.79395 362.76203 157.051229
249  266.35063 197.58961  76.85076   5.796383
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251  270.70537 193.74153  73.79200   5.530949
252   17.41203  72.79395 362.76203 157.051229
253  266.35063 197.58961  76.85076   5.796383
254   16.63435  70.01972 360.82006 162.638320
255  270.70537 193.74153  73.79200   5.530949
256   17.41203  72.79395 362.76203 157.051229
257  266.35063 197.58961  76.85076   5.796383
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261  266.35063 197.58961  76.85076   5.796383
262   16.63435  70.01972 360.82006 162.638320
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264   17.41203  72.79395 362.76203 157.051229
265  266.35063 197.58961  76.85076   5.796383
266   16.63435  70.01972 360.82006 162.638320
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269  266.35063 197.58961  76.85076   5.796383
270   16.63435  70.01972 360.82006 162.638320
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272   17.41203  72.79395 362.76203 157.051229
273  266.35063 197.58961  76.85076   5.796383
274   16.63435  70.01972 360.82006 162.638320
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276   17.41203  72.79395 362.76203 157.051229
277  266.35063 197.58961  76.85076   5.796383
278   16.63435  70.01972 360.82006 162.638320
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281  266.35063 197.58961  76.85076   5.796383
282   16.63435  70.01972 360.82006 162.638320
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284   17.41203  72.79395 362.76203 157.051229
285  266.35063 197.58961  76.85076   5.796383
286   16.63435  70.01972 360.82006 162.638320
287  270.70537 193.74153  73.79200   5.530949
288   17.41203  72.79395 362.76203 157.051229
289  266.35063 197.58961  76.85076   5.796383
290   16.63435  70.01972 360.82006 162.638320
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292   17.41203  72.79395 362.76203 157.051229
293  266.35063 197.58961  76.85076   5.796383
294   16.63435  70.01972 360.82006 162.638320
295  270.70537 193.74153  73.79200   5.530949
296   17.41203  72.79395 362.76203 157.051229
297  266.35063 197.58961  76.85076   5.796383
298   16.63435  70.01972 360.82006 162.638320
299  270.70537 193.74153  73.79200   5.530949
300   17.41203  72.79395 362.76203 157.051229
301  266.35063 197.58961  76.85076   5.796383
302   16.63435  70.01972 360.82006 162.638320
303  270.70537 193.74153  73.79200   5.530949
304   17.41203  72.79395 362.76203 157.051229
305  266.35063 197.58961  76.85076   5.796383
306   16.63435  70.01972 360.82006 162.638320
307  270.70537 193.74153  73.79200   5.530949
308   17.41203  72.79395 362.76203 157.051229
309  266.35063 197.58961  76.85076   5.796383
310   16.63435  70.01972 360.82006 162.638320
311  270.70537 193.74153  73.79200   5.530949
312   17.41203  72.79395 362.76203 157.051229
313  266.35063 197.58961  76.85076   5.796383
314   16.63435  70.01972 360.82006 162.638320
315  270.70537 193.74153  73.79200   5.530949
316   17.41203  72.79395 362.76203 157.051229
317  266.35063 197.58961  76.85076   5.796383
318   16.63435  70.01972 360.82006 162.638320
319  270.70537 193.74153  73.79200   5.530949
320   17.41203  72.79395 362.76203 157.051229
321  266.35063 197.58961  76.85076   5.796383
322   16.63435  70.01972 360.82006 162.638320
323  270.70537 193.74153  73.79200   5.530949
324   17.41203  72.79395 362.76203 157.051229
325  266.35063 197.58961  76.85076   5.796383
326   16.63435  70.01972 360.82006 162.638320
327  270.70537 193.74153  73.79200   5.530949
328   17.41203  72.79395 362.76203 157.051229
329  266.35063 197.58961  76.85076   5.796383
330   16.63435  70.01972 360.82006 162.638320
331  270.70537 193.74153  73.79200   5.530949
332   17.41203  72.79395 362.76203 157.051229
333  266.35063 197.58961  76.85076   5.796383
334   16.63435  70.01972 360.82006 162.638320
335  270.70537 193.74153  73.79200   5.530949
336   17.41203  72.79395 362.76203 157.051229
337  266.35063 197.58961  76.85076   5.796383
338   16.63435  70.01972 360.82006 162.638320
339  270.70537 193.74153  73.79200   5.530949
340   17.41203  72.79395 362.76203 157.051229
341  266.35063 197.58961  76.85076   5.796383
342   16.63435  70.01972 360.82006 162.638320
343  270.70537 193.74153  73.79200   5.530949
344   17.41203  72.79395 362.76203 157.051229
345  266.35063 197.58961  76.85076   5.796383
346   16.63435  70.01972 360.82006 162.638320
347  270.70537 193.74153  73.79200   5.530949
348   17.41203  72.79395 362.76203 157.051229
349  266.35063 197.58961  76.85076   5.796383
350   16.63435  70.01972 360.82006 162.638320
351  270.70537 193.74153  73.79200   5.530949
352   17.41203  72.79395 362.76203 157.051229
353  266.35063 197.58961  76.85076   5.796383
354   16.63435  70.01972 360.82006 162.638320
355  270.70537 193.74153  73.79200   5.530949
356   17.41203  72.79395 362.76203 157.051229
357  266.35063 197.58961  76.85076   5.796383
358   16.63435  70.01972 360.82006 162.638320
359  270.70537 193.74153  73.79200   5.530949
360   17.41203  72.79395 362.76203 157.051229
361  266.35063 197.58961  76.85076   5.796383
362   16.63435  70.01972 360.82006 162.638320
363  270.70537 193.74153  73.79200   5.530949
364   17.41203  72.79395 362.76203 157.051229
365  266.35063 197.58961  76.85076   5.796383
366   16.63435  70.01972 360.82006 162.638320
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368   17.41203  72.79395 362.76203 157.051229
369  266.35063 197.58961  76.85076   5.796383
370   16.63435  70.01972 360.82006 162.638320
371  270.70537 193.74153  73.79200   5.530949
372   17.41203  72.79395 362.76203 157.051229
373  266.35063 197.58961  76.85076   5.796383
374   16.63435  70.01972 360.82006 162.638320
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376   17.41203  72.79395 362.76203 157.051229
377  266.35063 197.58961  76.85076   5.796383
378   16.63435  70.01972 360.82006 162.638320
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380   17.41203  72.79395 362.76203 157.051229
381  266.35063 197.58961  76.85076   5.796383
382   16.63435  70.01972 360.82006 162.638320
383  270.70537 193.74153  73.79200   5.530949
384   17.41203  72.79395 362.76203 157.051229
385  266.35063 197.58961  76.85076   5.796383
386   16.63435  70.01972 360.82006 162.638320
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388   17.41203  72.79395 362.76203 157.051229
389  266.35063 197.58961  76.85076   5.796383
390   16.63435  70.01972 360.82006 162.638320
391  270.70537 193.74153  73.79200   5.530949
392   17.41203  72.79395 362.76203 157.051229
393  266.35063 197.58961  76.85076   5.796383
394   16.63435  70.01972 360.82006 162.638320
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396   17.41203  72.79395 362.76203 157.051229
397  266.35063 197.58961  76.85076   5.796383
398   16.63435  70.01972 360.82006 162.638320
399  270.70537 193.74153  73.79200   5.530949
400   17.41203  72.79395 362.76203 157.051229
401  266.35063 197.58961  76.85076   5.796383
402   16.63435  70.01972 360.82006 162.638320
403  270.70537 193.74153  73.79200   5.530949
404   17.41203  72.79395 362.76203 157.051229
405  266.35063 197.58961  76.85076   5.796383
406   16.63435  70.01972 360.82006 162.638320
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408   17.41203  72.79395 362.76203 157.051229
409  266.35063 197.58961  76.85076   5.796383
410   16.63435  70.01972 360.82006 162.638320
411  270.70537 193.74153  73.79200   5.530949
412   17.41203  72.79395 362.76203 157.051229
413  266.35063 197.58961  76.85076   5.796383
414   16.63435  70.01972 360.82006 162.638320
415  270.70537 193.74153  73.79200   5.530949
416   17.41203  72.79395 362.76203 157.051229
417  266.35063 197.58961  76.85076   5.796383
418   16.63435  70.01972 360.82006 162.638320
419  270.70537 193.74153  73.79200   5.530949
420   17.41203  72.79395 362.76203 157.051229
421  266.35063 197.58961  76.85076   5.796383
422   16.63435  70.01972 360.82006 162.638320
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424   17.41203  72.79395 362.76203 157.051229
425  266.35063 197.58961  76.85076   5.796383
426   16.63435  70.01972 360.82006 162.638320
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428   17.41203  72.79395 362.76203 157.051229
429  266.35063 197.58961  76.85076   5.796383
430   16.63435  70.01972 360.82006 162.638320
431  270.70537 193.74153  73.79200   5.530949
432   17.41203  72.79395 362.76203 157.051229
433  266.35063 197.58961  76.85076   5.796383
434   16.63435  70.01972 360.82006 162.638320
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436   17.41203  72.79395 362.76203 157.051229
437  266.35063 197.58961  76.85076   5.796383
438   16.63435  70.01972 360.82006 162.638320
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440   17.41203  72.79395 362.76203 157.051229
441  266.35063 197.58961  76.85076   5.796383
442   16.63435  70.01972 360.82006 162.638320
443  270.70537 193.74153  73.79200   5.530949
444   17.41203  72.79395 362.76203 157.051229
445  266.35063 197.58961  76.85076   5.796383
446   16.63435  70.01972 360.82006 162.638320
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448   17.41203  72.79395 362.76203 157.051229
449  266.35063 197.58961  76.85076   5.796383
450   16.63435  70.01972 360.82006 162.638320
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452   17.41203  72.79395 362.76203 157.051229
453  266.35063 197.58961  76.85076   5.796383
454   16.63435  70.01972 360.82006 162.638320
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456   17.41203  72.79395 362.76203 157.051229
457  266.35063 197.58961  76.85076   5.796383
458   16.63435  70.01972 360.82006 162.638320
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460   17.41203  72.79395 362.76203 157.051229
461  266.35063 197.58961  76.85076   5.796383
462   16.63435  70.01972 360.82006 162.638320
463  270.70537 193.74153  73.79200   5.530949
464   17.41203  72.79395 362.76203 157.051229
465  266.35063 197.58961  76.85076   5.796383
466   16.63435  70.01972 360.82006 162.638320
467  270.70537 193.74153  73.79200   5.530949
468   17.41203  72.79395 362.76203 157.051229
469  266.35063 197.58961  76.85076   5.796383
470   16.63435  70.01972 360.82006 162.638320
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472   17.41203  72.79395 362.76203 157.051229
473  266.35063 197.58961  76.85076   5.796383
474   16.63435  70.01972 360.82006 162.638320
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476   17.41203  72.79395 362.76203 157.051229
477  266.35063 197.58961  76.85076   5.796383
478   16.63435  70.01972 360.82006 162.638320
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480   17.41203  72.79395 362.76203 157.051229
481  266.35063 197.58961  76.85076   5.796383
482   16.63435  70.01972 360.82006 162.638320
483  270.70537 193.74153  73.79200   5.530949
484   17.41203  72.79395 362.76203 157.051229
485  266.35063 197.58961  76.85076   5.796383
486   16.63435  70.01972 360.82006 162.638320
487  270.70537 193.74153  73.79200   5.530949
488   17.41203  72.79395 362.76203 157.051229
489  266.35063 197.58961  76.85076   5.796383
490   16.63435  70.01972 360.82006 162.638320
491  270.70537 193.74153  73.79200   5.530949
492   17.41203  72.79395 362.76203 157.051229
493  266.35063 197.58961  76.85076   5.796383
494   16.63435  70.01972 360.82006 162.638320
495  270.70537 193.74153  73.79200   5.530949
496   17.41203  72.79395 362.76203 157.051229
497  266.35063 197.58961  76.85076   5.796383
498   16.63435  70.01972 360.82006 162.638320
499  270.70537 193.74153  73.79200   5.530949
500   17.41203  72.79395 362.76203 157.051229
501  266.35063 197.58961  76.85076   5.796383
502   16.63435  70.01972 360.82006 162.638320
503  270.70537 193.74153  73.79200   5.530949
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505  266.35063 197.58961  76.85076   5.796383
506   16.63435  70.01972 360.82006 162.638320
507  270.70537 193.74153  73.79200   5.530949
508   17.41203  72.79395 362.76203 157.051229
509  266.35063 197.58961  76.85076   5.796383
510   16.63435  70.01972 360.82006 162.638320
511  270.70537 193.74153  73.79200   5.530949
512   17.41203  72.79395 362.76203 157.051229
513  266.35063 197.58961  76.85076   5.796383
514   16.63435  70.01972 360.82006 162.638320
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516   17.41203  72.79395 362.76203 157.051229
517  266.35063 197.58961  76.85076   5.796383
518   16.63435  70.01972 360.82006 162.638320
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521  266.35063 197.58961  76.85076   5.796383
522   16.63435  70.01972 360.82006 162.638320
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525  266.35063 197.58961  76.85076   5.796383
526   16.63435  70.01972 360.82006 162.638320
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528   17.41203  72.79395 362.76203 157.051229
529  266.35063 197.58961  76.85076   5.796383
530   16.63435  70.01972 360.82006 162.638320
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532   17.41203  72.79395 362.76203 157.051229
533  266.35063 197.58961  76.85076   5.796383
534   16.63435  70.01972 360.82006 162.638320
535  270.70537 193.74153  73.79200   5.530949
536   17.41203  72.79395 362.76203 157.051229
537  266.35063 197.58961  76.85076   5.796383
538   16.63435  70.01972 360.82006 162.638320
539  270.70537 193.74153  73.79200   5.530949
540   17.41203  72.79395 362.76203 157.051229
541  266.35063 197.58961  76.85076   5.796383
542   16.63435  70.01972 360.82006 162.638320
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545  266.35063 197.58961  76.85076   5.796383
546   16.63435  70.01972 360.82006 162.638320
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549  266.35063 197.58961  76.85076   5.796383
550   16.63435  70.01972 360.82006 162.638320
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552   17.41203  72.79395 362.76203 157.051229
553  266.35063 197.58961  76.85076   5.796383
554   16.63435  70.01972 360.82006 162.638320
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556   17.41203  72.79395 362.76203 157.051229
557  266.35063 197.58961  76.85076   5.796383
558   16.63435  70.01972 360.82006 162.638320
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561  266.35063 197.58961  76.85076   5.796383
562   16.63435  70.01972 360.82006 162.638320
563  270.70537 193.74153  73.79200   5.530949
564   17.41203  72.79395 362.76203 157.051229
565  266.35063 197.58961  76.85076   5.796383
566   16.63435  70.01972 360.82006 162.638320
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568   17.41203  72.79395 362.76203 157.051229
569  266.35063 197.58961  76.85076   5.796383
570   16.63435  70.01972 360.82006 162.638320
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572   17.41203  72.79395 362.76203 157.051229
573  266.35063 197.58961  76.85076   5.796383
574   16.63435  70.01972 360.82006 162.638320
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576   17.41203  72.79395 362.76203 157.051229
577  266.35063 197.58961  76.85076   5.796383
578   16.63435  70.01972 360.82006 162.638320
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581  266.35063 197.58961  76.85076   5.796383
582   16.63435  70.01972 360.82006 162.638320
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585  266.35063 197.58961  76.85076   5.796383
586   16.63435  70.01972 360.82006 162.638320
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588   17.41203  72.79395 362.76203 157.051229
589  266.35063 197.58961  76.85076   5.796383
590   16.63435  70.01972 360.82006 162.638320
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592   17.41203  72.79395 362.76203 157.051229
593  266.35063 197.58961  76.85076   5.796383
594   16.63435  70.01972 360.82006 162.638320
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596   17.41203  72.79395 362.76203 157.051229
597  266.35063 197.58961  76.85076   5.796383
598   16.63435  70.01972 360.82006 162.638320
599  270.70537 193.74153  73.79200   5.530949
600   17.41203  72.79395 362.76203 157.051229
601  266.35063 197.58961  76.85076   5.796383
602   16.63435  70.01972 360.82006 162.638320
603  270.70537 193.74153  73.79200   5.530949
604   17.41203  72.79395 362.76203 157.051229
605  266.35063 197.58961  76.85076   5.796383
606   16.63435  70.01972 360.82006 162.638320
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608   17.41203  72.79395 362.76203 157.051229
609  266.35063 197.58961  76.85076   5.796383
610   16.63435  70.01972 360.82006 162.638320
611  270.70537 193.74153  73.79200   5.530949
612   17.41203  72.79395 362.76203 157.051229
613  266.35063 197.58961  76.85076   5.796383
614   16.63435  70.01972 360.82006 162.638320
615  270.70537 193.74153  73.79200   5.530949
616   17.41203  72.79395 362.76203 157.051229
617  266.35063 197.58961  76.85076   5.796383
618   16.63435  70.01972 360.82006 162.638320
619  270.70537 193.74153  73.79200   5.530949
620   17.41203  72.79395 362.76203 157.051229
621  266.35063 197.58961  76.85076   5.796383
622   16.63435  70.01972 360.82006 162.638320
623  270.70537 193.74153  73.79200   5.530949
624   17.41203  72.79395 362.76203 157.051229
625  266.35063 197.58961  76.85076   5.796383
626   16.63435  70.01972 360.82006 162.638320
627  270.70537 193.74153  73.79200   5.530949
628   17.41203  72.79395 362.76203 157.051229
629  266.35063 197.58961  76.85076   5.796383
630   16.63435  70.01972 360.82006 162.638320
631  270.70537 193.74153  73.79200   5.530949
632   17.41203  72.79395 362.76203 157.051229
633  266.35063 197.58961  76.85076   5.796383
634   16.63435  70.01972 360.82006 162.638320
635  270.70537 193.74153  73.79200   5.530949
636   17.41203  72.79395 362.76203 157.051229
637  266.35063 197.58961  76.85076   5.796383
638   16.63435  70.01972 360.82006 162.638320
639  270.70537 193.74153  73.79200   5.530949
640   17.41203  72.79395 362.76203 157.051229
641  266.35063 197.58961  76.85076   5.796383
642   16.63435  70.01972 360.82006 162.638320
643  270.70537 193.74153  73.79200   5.530949
644   17.41203  72.79395 362.76203 157.051229
645  266.35063 197.58961  76.85076   5.796383
646   16.63435  70.01972 360.82006 162.638320
647  270.70537 193.74153  73.79200   5.530949
648   17.41203  72.79395 362.76203 157.051229
649  266.35063 197.58961  76.85076   5.796383
650   16.63435  70.01972 360.82006 162.638320
651  270.70537 193.74153  73.79200   5.530949
652   17.41203  72.79395 362.76203 157.051229
653  266.35063 197.58961  76.85076   5.796383
654   16.63435  70.01972 360.82006 162.638320
655  270.70537 193.74153  73.79200   5.530949
656   17.41203  72.79395 362.76203 157.051229
657  266.35063 197.58961  76.85076   5.796383
658   16.63435  70.01972 360.82006 162.638320
659  270.70537 193.74153  73.79200   5.530949
660   17.41203  72.79395 362.76203 157.051229
661  266.35063 197.58961  76.85076   5.796383
662   16.63435  70.01972 360.82006 162.638320
663  270.70537 193.74153  73.79200   5.530949
664   17.41203  72.79395 362.76203 157.051229
665  266.35063 197.58961  76.85076   5.796383
666   16.63435  70.01972 360.82006 162.638320
667  270.70537 193.74153  73.79200   5.530949
668   17.41203  72.79395 362.76203 157.051229
669  266.35063 197.58961  76.85076   5.796383
670   16.63435  70.01972 360.82006 162.638320
671  270.70537 193.74153  73.79200   5.530949
672   17.41203  72.79395 362.76203 157.051229
673  266.35063 197.58961  76.85076   5.796383
674   16.63435  70.01972 360.82006 162.638320
675  270.70537 193.74153  73.79200   5.530949
676   17.41203  72.79395 362.76203 157.051229
677  266.35063 197.58961  76.85076   5.796383
678   16.63435  70.01972 360.82006 162.638320
679  270.70537 193.74153  73.79200   5.530949
680   17.41203  72.79395 362.76203 157.051229
681  266.35063 197.58961  76.85076   5.796383
682   16.63435  70.01972 360.82006 162.638320
683  270.70537 193.74153  73.79200   5.530949
684   17.41203  72.79395 362.76203 157.051229
685  266.35063 197.58961  76.85076   5.796383
686   16.63435  70.01972 360.82006 162.638320
687  270.70537 193.74153  73.79200   5.530949
688   17.41203  72.79395 362.76203 157.051229
689  266.35063 197.58961  76.85076   5.796383
690   16.63435  70.01972 360.82006 162.638320
691  270.70537 193.74153  73.79200   5.530949
692   17.41203  72.79395 362.76203 157.051229
693  266.35063 197.58961  76.85076   5.796383
694   16.63435  70.01972 360.82006 162.638320
695  270.70537 193.74153  73.79200   5.530949
696   17.41203  72.79395 362.76203 157.051229
697  266.35063 197.58961  76.85076   5.796383
698   16.63435  70.01972 360.82006 162.638320
699  270.70537 193.74153  73.79200   5.530949
700   17.41203  72.79395 362.76203 157.051229
701  266.35063 197.58961  76.85076   5.796383
702   16.63435  70.01972 360.82006 162.638320
703  270.70537 193.74153  73.79200   5.530949
704   17.41203  72.79395 362.76203 157.051229
705  266.35063 197.58961  76.85076   5.796383
706   16.63435  70.01972 360.82006 162.638320
707  270.70537 193.74153  73.79200   5.530949
708   17.41203  72.79395 362.76203 157.051229
709  266.35063 197.58961  76.85076   5.796383
710   16.63435  70.01972 360.82006 162.638320
711  270.70537 193.74153  73.79200   5.530949
712   17.41203  72.79395 362.76203 157.051229
713  266.35063 197.58961  76.85076   5.796383
714   16.63435  70.01972 360.82006 162.638320
715  270.70537 193.74153  73.79200   5.530949
716   17.41203  72.79395 362.76203 157.051229
717  266.35063 197.58961  76.85076   5.796383
718   16.63435  70.01972 360.82006 162.638320
719  270.70537 193.74153  73.79200   5.530949
720   17.41203  72.79395 362.76203 157.051229
721  266.35063 197.58961  76.85076   5.796383
722   16.63435  70.01972 360.82006 162.638320
723  270.70537 193.74153  73.79200   5.530949
724   17.41203  72.79395 362.76203 157.051229
725  266.35063 197.58961  76.85076   5.796383
726   16.63435  70.01972 360.82006 162.638320
727  270.70537 193.74153  73.79200   5.530949
728   17.41203  72.79395 362.76203 157.051229
729  266.35063 197.58961  76.85076   5.796383
730   16.63435  70.01972 360.82006 162.638320
731  270.70537 193.74153  73.79200   5.530949
732   17.41203  72.79395 362.76203 157.051229
733  266.35063 197.58961  76.85076   5.796383
734   16.63435  70.01972 360.82006 162.638320
735  270.70537 193.74153  73.79200   5.530949
736   17.41203  72.79395 362.76203 157.051229
737  266.35063 197.58961  76.85076   5.796383
738   16.63435  70.01972 360.82006 162.638320
739  270.70537 193.74153  73.79200   5.530949
740   17.41203  72.79395 362.76203 157.051229
741  266.35063 197.58961  76.85076   5.796383
742   16.63435  70.01972 360.82006 162.638320
743  270.70537 193.74153  73.79200   5.530949
744   17.41203  72.79395 362.76203 157.051229
745  266.35063 197.58961  76.85076   5.796383
746   16.63435  70.01972 360.82006 162.638320
747  270.70537 193.74153  73.79200   5.530949
748   17.41203  72.79395 362.76203 157.051229
749  266.35063 197.58961  76.85076   5.796383
750   16.63435  70.01972 360.82006 162.638320
751  270.70537 193.74153  73.79200   5.530949
752   17.41203  72.79395 362.76203 157.051229
753  266.35063 197.58961  76.85076   5.796383
754   16.63435  70.01972 360.82006 162.638320
755  270.70537 193.74153  73.79200   5.530949
756   17.41203  72.79395 362.76203 157.051229
757  266.35063 197.58961  76.85076   5.796383
758   16.63435  70.01972 360.82006 162.638320
759  270.70537 193.74153  73.79200   5.530949
760   17.41203  72.79395 362.76203 157.051229
761  266.35063 197.58961  76.85076   5.796383
762   16.63435  70.01972 360.82006 162.638320
763  270.70537 193.74153  73.79200   5.530949
764   17.41203  72.79395 362.76203 157.051229
765  266.35063 197.58961  76.85076   5.796383
766   16.63435  70.01972 360.82006 162.638320
767  270.70537 193.74153  73.79200   5.530949
768   17.41203  72.79395 362.76203 157.051229
769  266.35063 197.58961  76.85076   5.796383
770   16.63435  70.01972 360.82006 162.638320
771  270.70537 193.74153  73.79200   5.530949
772   17.41203  72.79395 362.76203 157.051229
773  266.35063 197.58961  76.85076   5.796383
774   16.63435  70.01972 360.82006 162.638320
775  270.70537 193.74153  73.79200   5.530949
776   17.41203  72.79395 362.76203 157.051229
777  266.35063 197.58961  76.85076   5.796383
778   16.63435  70.01972 360.82006 162.638320
779  270.70537 193.74153  73.79200   5.530949
780   17.41203  72.79395 362.76203 157.051229
781  266.35063 197.58961  76.85076   5.796383
782   16.63435  70.01972 360.82006 162.638320
783  270.70537 193.74153  73.79200   5.530949
784   17.41203  72.79395 362.76203 157.051229
785  266.35063 197.58961  76.85076   5.796383
786   16.63435  70.01972 360.82006 162.638320
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788   17.41203  72.79395 362.76203 157.051229
789  266.35063 197.58961  76.85076   5.796383
790   16.63435  70.01972 360.82006 162.638320
791  270.70537 193.74153  73.79200   5.530949
792   17.41203  72.79395 362.76203 157.051229
793  266.35063 197.58961  76.85076   5.796383
794   16.63435  70.01972 360.82006 162.638320
795  270.70537 193.74153  73.79200   5.530949
796   17.41203  72.79395 362.76203 157.051229
797  266.35063 197.58961  76.85076   5.796383
798   16.63435  70.01972 360.82006 162.638320
799  270.70537 193.74153  73.79200   5.530949
800   17.41203  72.79395 362.76203 157.051229
801  266.35063 197.58961  76.85076   5.796383
802   16.63435  70.01972 360.82006 162.638320
803  270.70537 193.74153  73.79200   5.530949
804   17.41203  72.79395 362.76203 157.051229
805  266.35063 197.58961  76.85076   5.796383
806   16.63435  70.01972 360.82006 162.638320
807  270.70537 193.74153  73.79200   5.530949
808   17.41203  72.79395 362.76203 157.051229
809  266.35063 197.58961  76.85076   5.796383
810   16.63435  70.01972 360.82006 162.638320
811  270.70537 193.74153  73.79200   5.530949
812   17.41203  72.79395 362.76203 157.051229
813  266.35063 197.58961  76.85076   5.796383
814   16.63435  70.01972 360.82006 162.638320
815  270.70537 193.74153  73.79200   5.530949
816   17.41203  72.79395 362.76203 157.051229
817  266.35063 197.58961  76.85076   5.796383
818   16.63435  70.01972 360.82006 162.638320
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820   17.41203  72.79395 362.76203 157.051229
821  266.35063 197.58961  76.85076   5.796383
822   16.63435  70.01972 360.82006 162.638320
823  270.70537 193.74153  73.79200   5.530949
824   17.41203  72.79395 362.76203 157.051229
825  266.35063 197.58961  76.85076   5.796383
826   16.63435  70.01972 360.82006 162.638320
827  270.70537 193.74153  73.79200   5.530949
828   17.41203  72.79395 362.76203 157.051229
829  266.35063 197.58961  76.85076   5.796383
830   16.63435  70.01972 360.82006 162.638320
831  270.70537 193.74153  73.79200   5.530949
832   17.41203  72.79395 362.76203 157.051229
833  266.35063 197.58961  76.85076   5.796383
834   16.63435  70.01972 360.82006 162.638320
835  270.70537 193.74153  73.79200   5.530949
836   17.41203  72.79395 362.76203 157.051229
837  266.35063 197.58961  76.85076   5.796383
838   16.63435  70.01972 360.82006 162.638320
839  270.70537 193.74153  73.79200   5.530949
840   17.41203  72.79395 362.76203 157.051229
841  266.35063 197.58961  76.85076   5.796383
842   16.63435  70.01972 360.82006 162.638320
843  270.70537 193.74153  73.79200   5.530949
844   17.41203  72.79395 362.76203 157.051229
845  266.35063 197.58961  76.85076   5.796383
846   16.63435  70.01972 360.82006 162.638320
847  270.70537 193.74153  73.79200   5.530949
848   17.41203  72.79395 362.76203 157.051229
849  266.35063 197.58961  76.85076   5.796383
850   16.63435  70.01972 360.82006 162.638320
851  270.70537 193.74153  73.79200   5.530949
852   17.41203  72.79395 362.76203 157.051229
853  266.35063 197.58961  76.85076   5.796383
854   16.63435  70.01972 360.82006 162.638320
855  270.70537 193.74153  73.79200   5.530949
856   17.41203  72.79395 362.76203 157.051229
857  266.35063 197.58961  76.85076   5.796383
858   16.63435  70.01972 360.82006 162.638320
859  270.70537 193.74153  73.79200   5.530949
860   17.41203  72.79395 362.76203 157.051229
861  266.35063 197.58961  76.85076   5.796383
862   16.63435  70.01972 360.82006 162.638320
863  270.70537 193.74153  73.79200   5.530949
864   17.41203  72.79395 362.76203 157.051229
865  266.35063 197.58961  76.85076   5.796383
866   16.63435  70.01972 360.82006 162.638320
867  270.70537 193.74153  73.79200   5.530949
868   17.41203  72.79395 362.76203 157.051229
869  266.35063 197.58961  76.85076   5.796383
870   16.63435  70.01972 360.82006 162.638320
871  270.70537 193.74153  73.79200   5.530949
872   17.41203  72.79395 362.76203 157.051229
873  266.35063 197.58961  76.85076   5.796383
874   16.63435  70.01972 360.82006 162.638320
875  270.70537 193.74153  73.79200   5.530949
876   17.41203  72.79395 362.76203 157.051229
877  266.35063 197.58961  76.85076   5.796383
878   16.63435  70.01972 360.82006 162.638320
879  270.70537 193.74153  73.79200   5.530949
880   17.41203  72.79395 362.76203 157.051229
881  266.35063 197.58961  76.85076   5.796383
882   16.63435  70.01972 360.82006 162.638320
883  270.70537 193.74153  73.79200   5.530949
884   17.41203  72.79395 362.76203 157.051229
885  266.35063 197.58961  76.85076   5.796383
886   16.63435  70.01972 360.82006 162.638320
887  270.70537 193.74153  73.79200   5.530949
888   17.41203  72.79395 362.76203 157.051229
889  266.35063 197.58961  76.85076   5.796383
890   16.63435  70.01972 360.82006 162.638320
891  270.70537 193.74153  73.79200   5.530949
892   17.41203  72.79395 362.76203 157.051229
893  266.35063 197.58961  76.85076   5.796383
894   16.63435  70.01972 360.82006 162.638320
895  270.70537 193.74153  73.79200   5.530949
896   17.41203  72.79395 362.76203 157.051229
897  266.35063 197.58961  76.85076   5.796383
898   16.63435  70.01972 360.82006 162.638320
899  270.70537 193.74153  73.79200   5.530949
900   17.41203  72.79395 362.76203 157.051229
901  266.35063 197.58961  76.85076   5.796383
902   16.63435  70.01972 360.82006 162.638320
903  270.70537 193.74153  73.79200   5.530949
904   17.41203  72.79395 362.76203 157.051229
905  266.35063 197.58961  76.85076   5.796383
906   16.63435  70.01972 360.82006 162.638320
907  270.70537 193.74153  73.79200   5.530949
908   17.41203  72.79395 362.76203 157.051229
909  266.35063 197.58961  76.85076   5.796383
910   16.63435  70.01972 360.82006 162.638320
911  270.70537 193.74153  73.79200   5.530949
912   17.41203  72.79395 362.76203 157.051229
913  266.35063 197.58961  76.85076   5.796383
914   16.63435  70.01972 360.82006 162.638320
915  270.70537 193.74153  73.79200   5.530949
916   17.41203  72.79395 362.76203 157.051229
917  266.35063 197.58961  76.85076   5.796383
918   16.63435  70.01972 360.82006 162.638320
919  270.70537 193.74153  73.79200   5.530949
920   17.41203  72.79395 362.76203 157.051229
921  266.35063 197.58961  76.85076   5.796383
922   16.63435  70.01972 360.82006 162.638320
923  270.70537 193.74153  73.79200   5.530949
924   17.41203  72.79395 362.76203 157.051229
925  266.35063 197.58961  76.85076   5.796383
926   16.63435  70.01972 360.82006 162.638320
927  270.70537 193.74153  73.79200   5.530949
928   17.41203  72.79395 362.76203 157.051229
929  266.35063 197.58961  76.85076   5.796383
930   16.63435  70.01972 360.82006 162.638320
931  270.70537 193.74153  73.79200   5.530949
932   17.41203  72.79395 362.76203 157.051229
933  266.35063 197.58961  76.85076   5.796383
934   16.63435  70.01972 360.82006 162.638320
935  270.70537 193.74153  73.79200   5.530949
936   17.41203  72.79395 362.76203 157.051229
937  266.35063 197.58961  76.85076   5.796383
938   16.63435  70.01972 360.82006 162.638320
939  270.70537 193.74153  73.79200   5.530949
940   17.41203  72.79395 362.76203 157.051229
941  266.35063 197.58961  76.85076   5.796383
942   16.63435  70.01972 360.82006 162.638320
943  270.70537 193.74153  73.79200   5.530949
944   17.41203  72.79395 362.76203 157.051229
945  266.35063 197.58961  76.85076   5.796383
946   16.63435  70.01972 360.82006 162.638320
947  270.70537 193.74153  73.79200   5.530949
948   17.41203  72.79395 362.76203 157.051229
949  266.35063 197.58961  76.85076   5.796383
950   16.63435  70.01972 360.82006 162.638320
951  270.70537 193.74153  73.79200   5.530949
952   17.41203  72.79395 362.76203 157.051229
953  266.35063 197.58961  76.85076   5.796383
954   16.63435  70.01972 360.82006 162.638320
955  270.70537 193.74153  73.79200   5.530949
956   17.41203  72.79395 362.76203 157.051229
957  266.35063 197.58961  76.85076   5.796383
958   16.63435  70.01972 360.82006 162.638320
959  270.70537 193.74153  73.79200   5.530949
960   17.41203  72.79395 362.76203 157.051229
961  266.35063 197.58961  76.85076   5.796383
962   16.63435  70.01972 360.82006 162.638320
963  270.70537 193.74153  73.79200   5.530949
964   17.41203  72.79395 362.76203 157.051229
965  266.35063 197.58961  76.85076   5.796383
966   16.63435  70.01972 360.82006 162.638320
967  270.70537 193.74153  73.79200   5.530949
968   17.41203  72.79395 362.76203 157.051229
969  266.35063 197.58961  76.85076   5.796383
970   16.63435  70.01972 360.82006 162.638320
971  270.70537 193.74153  73.79200   5.530949
972   17.41203  72.79395 362.76203 157.051229
973  266.35063 197.58961  76.85076   5.796383
974   16.63435  70.01972 360.82006 162.638320
975  270.70537 193.74153  73.79200   5.530949
976   17.41203  72.79395 362.76203 157.051229
977  266.35063 197.58961  76.85076   5.796383
978   16.63435  70.01972 360.82006 162.638320
979  270.70537 193.74153  73.79200   5.530949
980   17.41203  72.79395 362.76203 157.051229
981  266.35063 197.58961  76.85076   5.796383
982   16.63435  70.01972 360.82006 162.638320
983  270.70537 193.74153  73.79200   5.530949
984   17.41203  72.79395 362.76203 157.051229
985  266.35063 197.58961  76.85076   5.796383
986   16.63435  70.01972 360.82006 162.638320
987  270.70537 193.74153  73.79200   5.530949
988   17.41203  72.79395 362.76203 157.051229
989  266.35063 197.58961  76.85076   5.796383
990   16.63435  70.01972 360.82006 162.638320
991  270.70537 193.74153  73.79200   5.530949
992   17.41203  72.79395 362.76203 157.051229
993  266.35063 197.58961  76.85076   5.796383
994   16.63435  70.01972 360.82006 162.638320
995  270.70537 193.74153  73.79200   5.530949
996   17.41203  72.79395 362.76203 157.051229
997  266.35063 197.58961  76.85076   5.796383
998   16.63435  70.01972 360.82006 162.638320
999  270.70537 193.74153  73.79200   5.530949
1000  17.41203  72.79395 362.76203 157.051229

fitG <- vglm(cbind(y1, y2, y3, y4, y5) ~ gender, family = cumulative(parallel = T), 
    data = politics)
summary(fitG)


Call:
vglm(formula = cbind(y1, y2, y3, y4, y5) ~ gender, family = cumulative(parallel = T), 
    data = politics)

Pearson residuals:
  logitlink(P[Y<=1]) logitlink(P[Y<=2]) logitlink(P[Y<=3])
1             0.7597              2.630              7.617
2            -1.9804             -4.638             -9.106
3             2.3841              5.030              5.566
4            -1.6530             -4.616             -7.692
  logitlink(P[Y<=4])
1              3.048
2             -3.548
3              2.525
4             -3.495

Coefficients: 
              Estimate Std. Error z value             Pr(>|z|)    
(Intercept):1 -2.54175    0.16421 -15.478 < 0.0000000000000002 ***
(Intercept):2 -0.55945    0.10036  -5.574         0.0000000248 ***
(Intercept):3  0.42279    0.09929   4.258         0.0000206120 ***
(Intercept):4  2.19962    0.14055  15.651 < 0.0000000000000002 ***
gendermale    -0.18894    0.14138  -1.336                0.181    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: logitlink(P[Y<=1]), logitlink(P[Y<=2]), 
logitlink(P[Y<=3]), logitlink(P[Y<=4])

Residual deviance: 413.054 on 11 degrees of freedom

Log-likelihood: -236.8266 on 11 degrees of freedom

Number of Fisher scoring iterations: 5 

No Hauck-Donner effect found in any of the estimates


Exponentiated coefficients:
gendermale 
 0.8278392

lrtest(fit, fitG)

Likelihood ratio test

Model 1: cbind(y1, y2, y3, y4, y5) ~ party + gender
Model 2: cbind(y1, y2, y3, y4, y5) ~ gender
  #Df   LogLik Df  Chisq            Pr(>Chisq)    
1  10  -35.203                                    
2  11 -236.827  1 403.25 < 0.00000000000000022 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

confint(fit, method = "profile")

                    2.5 %     97.5 %
(Intercept):1 -2.46596771 -1.8014863
(Intercept):2 -0.05464962  0.3937706
(Intercept):3  1.56835767  2.1616181
(Intercept):4  4.20347943  5.1226824
partyrepub    -4.07163810 -3.2178566
gendermale    -0.24638939  0.3414038

fitP <- vglm(cbind(y1, y2, y3, y4, y5) ~ party, family = cumulative(parallel = T), 
    data = politics)
summary(fitP)


Call:
vglm(formula = cbind(y1, y2, y3, y4, y5) ~ party, family = cumulative(parallel = T), 
    data = politics)

Pearson residuals:
  logitlink(P[Y<=1]) logitlink(P[Y<=2]) logitlink(P[Y<=3])
1            -0.3225            -0.7410             0.5691
2            -0.7109             0.6058            -0.1568
3             0.6127             1.1209            -0.4354
4            -0.5147            -1.3155            -0.2420
  logitlink(P[Y<=4])
1           -1.13849
2            0.65161
3           -1.25496
4           -0.06279

Coefficients: 
              Estimate Std. Error z value            Pr(>|z|)    
(Intercept):1 -2.10343    0.15795 -13.317 <0.0000000000000002 ***
(Intercept):2  0.18683    0.09892   1.889              0.0589 .  
(Intercept):3  1.87472    0.13958  13.432 <0.0000000000000002 ***
(Intercept):4  4.66772    0.22813  20.460 <0.0000000000000002 ***
partyrepub    -3.62900    0.21742 -16.691 <0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: logitlink(P[Y<=1]), logitlink(P[Y<=2]), 
logitlink(P[Y<=3]), logitlink(P[Y<=4])

Residual deviance: 9.9069 on 11 degrees of freedom

Log-likelihood: -35.253 on 11 degrees of freedom

Number of Fisher scoring iterations: 4 

No Hauck-Donner effect found in any of the estimates


Exponentiated coefficients:
partyrepub 
0.02654265

lrtest(fit, fitP)

Likelihood ratio test

Model 1: cbind(y1, y2, y3, y4, y5) ~ party + gender
Model 2: cbind(y1, y2, y3, y4, y5) ~ party
  #Df  LogLik Df  Chisq Pr(>Chisq)
1  10 -35.203                     
2  11 -35.253  1 0.0997     0.7522

# PO: Happiness and Family Income pp.172-174
happy <- read.table("/Users/Patrick/Dropbox/data/agresti/Happy.dat", header = T)

fit <- vglm(cbind(y1, y2, y3) ~ income, family = cumulative(parallel = T), data = happy)
summary(fit)


Call:
vglm(formula = cbind(y1, y2, y3) ~ income, family = cumulative(parallel = T), 
    data = happy)

Pearson residuals:
  logitlink(P[Y<=1]) logitlink(P[Y<=2])
1             0.4175           -0.06952
2            -0.6911           -0.04246
3             0.5631            0.21889

Coefficients: 
              Estimate Std. Error z value   Pr(>|z|)    
(Intercept):1  -1.1023     0.2754  -4.003 0.00006251 ***
(Intercept):2   1.3048     0.2775   4.701 0.00000259 ***
income         -0.2668     0.1510  -1.768     0.0771 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: logitlink(P[Y<=1]), logitlink(P[Y<=2])

Residual deviance: 1.0168 on 3 degrees of freedom

Log-likelihood: -14.5664 on 3 degrees of freedom

Number of Fisher scoring iterations: 3 

No Hauck-Donner effect found in any of the estimates


Exponentiated coefficients:
   income 
0.7657883

fit0 <- vglm(cbind(y1, y2, y3) ~ 1, family = cumulative(parallel = T), data = happy)

lrtest(fit, fit0)

Likelihood ratio test

Model 1: cbind(y1, y2, y3) ~ income
Model 2: cbind(y1, y2, y3) ~ 1
  #Df  LogLik Df Chisq Pr(>Chisq)  
1   3 -14.566                      
2   4 -16.121  1 3.109    0.07786 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

fit2 <- vglm(cbind(y1, y2, y3) ~ factor(income), family = multinomial, data = happy)
summary(fit2)


Call:
vglm(formula = cbind(y1, y2, y3) ~ factor(income), family = multinomial, 
    data = happy)

Coefficients: 
                  Estimate Std. Error z value Pr(>|z|)    
(Intercept):1      -0.1957     0.2219  -0.882 0.377753    
(Intercept):2       0.6931     0.1826   3.797 0.000147 ***
factor(income)2:1  -0.6107     0.3273  -1.866 0.062023 .  
factor(income)2:2  -0.1859     0.2489  -0.747 0.455104    
factor(income)3:1  -0.5774     0.5411  -1.067 0.285936    
factor(income)3:2  -0.3677     0.4072  -0.903 0.366496    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])

Residual deviance: -0.000000000000004108 on 0 degrees of freedom

Log-likelihood: -14.058 on 0 degrees of freedom

Number of Fisher scoring iterations: 4 

No Hauck-Donner effect found in any of the estimates


Reference group is level  3  of the response

lrtest(fit2, fit0)

Likelihood ratio test

Model 1: cbind(y1, y2, y3) ~ factor(income)
Model 2: cbind(y1, y2, y3) ~ 1
  #Df  LogLik Df  Chisq Pr(>Chisq)
1   0 -14.058                     
2   4 -16.121  4 4.1258     0.3892

fit3 <- vglm(cbind(y1, y2, y3) ~ income, family = multinomial, data = happy)
summary(fit3)


Call:
vglm(formula = cbind(y1, y2, y3) ~ income, family = multinomial, 
    data = happy)

Pearson residuals:
  log(mu[,1]/mu[,3]) log(mu[,2]/mu[,3])
1             0.2688           -0.06912
2            -0.5878            0.15301
3             0.7148           -0.18204

Coefficients: 
              Estimate Std. Error z value Pr(>|z|)   
(Intercept):1   0.1754     0.4191   0.418  0.67559   
(Intercept):2   0.8704     0.3240   2.686  0.00723 **
income:1       -0.4247     0.2428  -1.749  0.08033 . 
income:2       -0.1805     0.1787  -1.010  0.31242   
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: log(mu[,1]/mu[,3]), log(mu[,2]/mu[,3])

Residual deviance: 0.9568 on 2 degrees of freedom

Log-likelihood: -14.5364 on 2 degrees of freedom

Number of Fisher scoring iterations: 4 

No Hauck-Donner effect found in any of the estimates


Reference group is level  3  of the response

lrtest(fit3, fit0)

Likelihood ratio test

Model 1: cbind(y1, y2, y3) ~ income
Model 2: cbind(y1, y2, y3) ~ 1
  #Df  LogLik Df  Chisq Pr(>Chisq)
1   2 -14.536                     
2   4 -16.121  2 3.1691      0.205

# Chronic Respiratory Disease
airPollHigh <- c(rep(0, 6), rep(1, 6))
jobExposureY <- rep(c(rep(0, 3), rep(1, 3)), 2)
smokestatusEx <- rep(c(0, 1, 0), 4)
smokestatusCurrent <- rep(c(0, 0, 1), 4)
y1 <- c(158, 167, 307, 26, 38, 94, 94, 67, 184, 32, 39, 77)
y2 <- c(9, 19, 102, 5, 12, 48, 7, 8, 65, 3, 11, 48)
y3 <- c(5, 5, 83, 5, 4, 46, 5, 4, 33, 6, 4, 39)
y4 <- c(0, 3, 68, 1, 4, 60, 1, 3, 36, 1, 2, 51)
ChronRespDis <- data.frame(airPollHigh, jobExposureY, smokestatusEx, smokestatusCurrent, 
    y1, y2, y3, y4)

fit <- vglm(cbind(y1, y2, y3, y4) ~ airPollHigh + jobExposureY + smokestatusEx + 
    smokestatusCurrent, family = cumulative(parallel = T), data = ChronRespDis)
summary(fit)


Call:
vglm(formula = cbind(y1, y2, y3, y4) ~ airPollHigh + jobExposureY + 
    smokestatusEx + smokestatusCurrent, family = cumulative(parallel = T), 
    data = ChronRespDis)

Pearson residuals:
                       Min      1Q   Median      3Q   Max
logitlink(P[Y<=1]) -0.9786 -0.6485 -0.06329 0.09382 1.164
logitlink(P[Y<=2]) -2.0789 -0.9668  0.23074 0.77460 1.502
logitlink(P[Y<=3]) -0.4941 -0.3384  0.32979 1.04577 1.773

Coefficients: 
                   Estimate Std. Error z value            Pr(>|z|)    
(Intercept):1       2.08844    0.16329  12.790 <0.0000000000000002 ***
(Intercept):2       2.96964    0.16927  17.544 <0.0000000000000002 ***
(Intercept):3       3.89385    0.17786  21.893 <0.0000000000000002 ***
airPollHigh         0.03929    0.09370   0.419              0.6750    
jobExposureY       -0.86476    0.09546  -9.059 <0.0000000000000002 ***
smokestatusEx      -0.40003    0.20187  -1.982              0.0475 *  
smokestatusCurrent -1.85271    0.16503 -11.227 <0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Names of linear predictors: logitlink(P[Y<=1]), logitlink(P[Y<=2]), 
logitlink(P[Y<=3])

Residual deviance: 29.9969 on 29 degrees of freedom

Log-likelihood: -85.8245 on 29 degrees of freedom

Number of Fisher scoring iterations: 4 

No Hauck-Donner effect found in any of the estimates


Exponentiated coefficients:
       airPollHigh       jobExposureY      smokestatusEx 
         1.0400744          0.4211522          0.6703009 
smokestatusCurrent 
         0.1568115

Chapter 7: Loglinear Models for Contingency Tables & Counts

Ch. 7 Code

# p.195

HappyHeaven <- read.table("http://www.stat.ufl.edu/~aa/cat/data/HappyHeaven.dat", 
    header = T)
HappyHeaven

   happy heaven count
1    not     no    32
2    not    yes   190
3 pretty     no   113
4 pretty    yes   611
5   very     no    51
6   very    yes   326

HH3by2 = xtabs(count ~ happy + heaven, data = HappyHeaven)
HH3by2

        heaven
happy     no yes
  not     32 190
  pretty 113 611
  very    51 326

fitLogLinear <- glm(count ~ happy + heaven, family = poisson, data = HappyHeaven)
summary(fitLogLinear)


Call:
glm(formula = count ~ happy + heaven, family = poisson, data = HappyHeaven)

Deviance Residuals: 
       1         2         3         4         5         6  
-0.15570   0.06459   0.54947  -0.23152  -0.65897   0.27006  

Coefficients:
            Estimate Std. Error z value             Pr(>|z|)    
(Intercept)  3.49313    0.09408   37.13 < 0.0000000000000002 ***
happypretty  1.18211    0.07672   15.41 < 0.0000000000000002 ***
happyvery    0.52957    0.08460    6.26       0.000000000386 ***
heavenyes    1.74920    0.07739   22.60 < 0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 1019.87238  on 5  degrees of freedom
Residual deviance:    0.89111  on 2  degrees of freedom
AIC: 49.504

Number of Fisher Scoring iterations: 3

fitLogLineari <- glm(count ~ happy + heaven + happy * heaven, family = poisson, 
    data = HappyHeaven)
summary(fitLogLineari)


Call:
glm(formula = count ~ happy + heaven + happy * heaven, family = poisson, 
    data = HappyHeaven)

Deviance Residuals: 
[1]  0  0  0  0  0  0

Coefficients:
                      Estimate Std. Error z value             Pr(>|z|)    
(Intercept)            3.46574    0.17678  19.605 < 0.0000000000000002 ***
happypretty            1.26165    0.20025   6.300       0.000000000297 ***
happyvery              0.46609    0.22552   2.067               0.0388 *  
heavenyes              1.78129    0.19108   9.322 < 0.0000000000000002 ***
happypretty:heavenyes -0.09358    0.21679  -0.432               0.6660    
happyvery:heavenyes    0.07378    0.24329   0.303               0.7617    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 1019.872382855081355046  on 5  degrees of freedom
Residual deviance:    0.000000000000029532  on 0  degrees of freedom
AIC: 52.613

Number of Fisher Scoring iterations: 2

# Dayton, Ohio High Schoolers - A, C and M, p.198

druggy <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Substance.dat", header = T)
druggy

  alcohol cigarettes marijuana count
1     yes        yes       yes   911
2     yes        yes        no   538
3     yes         no       yes    44
4     yes         no        no   456
5      no        yes       yes     3
6      no        yes        no    43
7      no         no       yes     2
8      no         no        no   279

druggy$A <- druggy$alcohol
druggy$C <- druggy$cigarettes
druggy$M <- druggy$marijuana
fitHAmodel <- glm(count ~ A + C + M + A * C + A * M + C * M, family = poisson, 
    data = druggy)
druggy$resHAmodel <- rstandard(fitHAmodel, type = "pearson")
summary(fitHAmodel)


Call:
glm(formula = count ~ A + C + M + A * C + A * M + C * M, family = poisson, 
    data = druggy)

Deviance Residuals: 
       1         2         3         4         5         6         7  
 0.02044  -0.02658  -0.09256   0.02890  -0.33428   0.09452   0.49134  
       8  
-0.03690  

Coefficients:
            Estimate Std. Error z value             Pr(>|z|)    
(Intercept)  5.63342    0.05970  94.361 < 0.0000000000000002 ***
Ayes         0.48772    0.07577   6.437       0.000000000122 ***
Cyes        -1.88667    0.16270 -11.596 < 0.0000000000000002 ***
Myes        -5.30904    0.47520 -11.172 < 0.0000000000000002 ***
Ayes:Cyes    2.05453    0.17406  11.803 < 0.0000000000000002 ***
Ayes:Myes    2.98601    0.46468   6.426       0.000000000131 ***
Cyes:Myes    2.84789    0.16384  17.382 < 0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 2851.46098  on 7  degrees of freedom
Residual deviance:    0.37399  on 1  degrees of freedom
AIC: 63.417

Number of Fisher Scoring iterations: 4

fit_AC_CM <- glm(count ~ A + C + M + A * C + C * M, family = poisson, data = druggy)
summary(fit_AC_CM)


Call:
glm(formula = count ~ A + C + M + A * C + C * M, family = poisson, 
    data = druggy)

Deviance Residuals: 
      1        2        3        4        5        6        7        8  
 0.8401  -1.0667   2.4964  -0.6743  -6.0678   5.0235  -4.5440   0.8867  

Coefficients:
            Estimate Std. Error z value             Pr(>|z|)    
(Intercept)  5.57765    0.06032  92.463 < 0.0000000000000002 ***
Ayes         0.57625    0.07456   7.729   0.0000000000000108 ***
Cyes        -2.69414    0.16257 -16.572 < 0.0000000000000002 ***
Myes        -2.77123    0.15199 -18.233 < 0.0000000000000002 ***
Ayes:Cyes    2.87373    0.16730  17.178 < 0.0000000000000002 ***
Cyes:Myes    3.22431    0.16098  20.029 < 0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 2851.461  on 7  degrees of freedom
Residual deviance:   92.018  on 2  degrees of freedom
AIC: 153.06

Number of Fisher Scoring iterations: 6

anova(fit_AC_CM, fitHAmodel, test = "LRT")

Analysis of Deviance Table

Model 1: count ~ A + C + M + A * C + C * M
Model 2: count ~ A + C + M + A * C + A * M + C * M
  Resid. Df Resid. Dev Df Deviance              Pr(>Chi)    
1         2     92.018                                      
2         1      0.374  1   91.644 < 0.00000000000000022 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

library(car)
Anova(fitHAmodel)

Analysis of Deviance Table (Type II tests)

Response: count
    LR Chisq Df            Pr(>Chisq)    
A    1281.71  1 < 0.00000000000000022 ***
C     227.81  1 < 0.00000000000000022 ***
M      55.91  1   0.00000000000007575 ***
A:C   187.38  1 < 0.00000000000000022 ***
A:M    91.64  1 < 0.00000000000000022 ***
C:M   497.00  1 < 0.00000000000000022 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

fit_AM_CM <- glm(count ~ A + C + M + A * M + C * M, family = poisson, data = druggy)
summary(fit_AM_CM)


Call:
glm(formula = count ~ A + C + M + A * M + C * M, family = poisson, 
    data = druggy)

Deviance Residuals: 
      1        2        3        4        5        6        7        8  
 0.0584   4.5702  -0.2619  -4.3441  -0.8663  -9.7716   2.2287   6.8353  

Coefficients:
            Estimate Std. Error z value             Pr(>|z|)    
(Intercept)  5.19207    0.06088  85.285 < 0.0000000000000002 ***
Ayes         1.12719    0.06412  17.579 < 0.0000000000000002 ***
Cyes        -0.23512    0.05551  -4.235            0.0000228 ***
Myes        -6.62092    0.47370 -13.977 < 0.0000000000000002 ***
Ayes:Myes    4.12509    0.45294   9.107 < 0.0000000000000002 ***
Cyes:Myes    3.22431    0.16098  20.029 < 0.0000000000000002 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 2851.46  on 7  degrees of freedom
Residual deviance:  187.75  on 2  degrees of freedom
AIC: 248.8

Number of Fisher Scoring iterations: 5

druggy$res_AM_CM <- rstandard(fit_AM_CM, type = "pearson")
data.frame(druggy$A, druggy$C, druggy$M, druggy$count, fitted(fitHAmodel), druggy$resHAmodel, 
    fitted(fit_AM_CM), druggy$res_AM_CM)

  druggy.A druggy.C druggy.M druggy.count fitted.fitHAmodel.
1      yes      yes      yes          911          910.38317
2      yes      yes       no          538          538.61683
3      yes       no      yes           44           44.61683
4      yes       no       no          456          455.38317
5       no      yes      yes            3            3.61683
6       no      yes       no           43           42.38317
7       no       no      yes            2            1.38317
8       no       no       no          279          279.61683
  druggy.resHAmodel fitted.fit_AM_CM. druggy.res_AM_CM
1         0.6333249       909.2395833         3.695518
2        -0.6333249       438.8404255        12.804588
3        -0.6333249        45.7604167        -3.695518
4         0.6333249       555.1595745       -12.804588
5        -0.6333250         4.7604167        -3.695589
6         0.6333249       142.1595745       -12.804589
7         0.6333250         0.2395833         3.695589
8        -0.6333249       179.8404255        12.804589

options(scipen = 4)
round(exp(confint(fitHAmodel)), 4)

               2.5 %   97.5 %
(Intercept) 248.1623 313.6240
Ayes          1.4050   1.8911
Cyes          0.1088   0.2062
Myes          0.0017   0.0114
Ayes:Cyes     5.6015  11.0971
Ayes:Myes     8.8140  56.6436
Cyes:Myes    12.6458  24.0693

# Maine accidents, p.204

accidents <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Accidents2.dat", 
    header = T)
accidents$G <- accidents$gender
accidents$L <- accidents$location
accidents$S <- accidents$seatbelt
accidents$I <- accidents$injury
accidents

   gender location seatbelt injury count      G     L   S   I
1  female    rural       no     no  3246 female rural  no  no
2  female    rural       no    yes   973 female rural  no yes
3  female    rural      yes     no  6134 female rural yes  no
4  female    rural      yes    yes   757 female rural yes yes
5  female    urban       no     no  7287 female urban  no  no
6  female    urban       no    yes   996 female urban  no yes
7  female    urban      yes     no 11587 female urban yes  no
8  female    urban      yes    yes   759 female urban yes yes
9    male    rural       no     no  6123   male rural  no  no
10   male    rural       no    yes  1084   male rural  no yes
11   male    rural      yes     no  6693   male rural yes  no
12   male    rural      yes    yes   513   male rural yes yes
13   male    urban       no     no 10381   male urban  no  no
14   male    urban       no    yes   812   male urban  no yes
15   male    urban      yes     no 10969   male urban yes  no
16   male    urban      yes    yes   380   male urban yes yes

fitME <- glm(count ~ G + L + S + I, family = poisson, data = accidents)
summary(fitME)


Call:
glm(formula = count ~ G + L + S + I, family = poisson, data = accidents)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-26.8552   -7.3020    0.5095    7.2791   19.4902  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept)  8.480603   0.008683  976.68   <2e-16 ***
Gmale        0.152155   0.007653   19.88   <2e-16 ***
Lurban       0.525589   0.007896   66.57   <2e-16 ***
Syes         0.201277   0.007669   26.24   <2e-16 ***
Iyes        -2.297472   0.013244 -173.47   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 61709.5  on 15  degrees of freedom
Residual deviance:  2792.8  on 11  degrees of freedom
AIC: 2956.2

Number of Fisher Scoring iterations: 4

fitHA <- glm(count ~ G + L + S + I + G * L + G * S + G * I + L * S + L * I + 
    S * I, family = poisson, data = accidents)
summary(fitHA)


Call:
glm(formula = count ~ G + L + S + I + G * L + G * S + G * I + 
    L * S + L * I + S * I, family = poisson, data = accidents)

Deviance Residuals: 
       1         2         3         4         5         6         7  
-1.87206  -0.50333   1.91168  -0.89501   1.42101   0.09462  -1.49166  
       8         9        10        11        12        13        14  
 1.39155   0.99713   1.41493  -1.43845  -0.23069  -0.88563  -1.14680  
      15        16  
 1.25748  -0.38522  

Coefficients:
             Estimate Std. Error z value    Pr(>|z|)    
(Intercept)   8.11786    0.01453 558.535     < 2e-16 ***
Gmale         0.58918    0.01620  36.359     < 2e-16 ***
Lurban        0.75930    0.01602  47.399     < 2e-16 ***
Syes          0.57924    0.01623  35.692     < 2e-16 ***
Iyes         -1.22138    0.02637 -46.320     < 2e-16 ***
Gmale:Lurban -0.20992    0.01612 -13.019     < 2e-16 ***
Gmale:Syes   -0.45992    0.01568 -29.328     < 2e-16 ***
Gmale:Iyes   -0.54053    0.02722 -19.859     < 2e-16 ***
Lurban:Syes  -0.08493    0.01619  -5.244 0.000000157 ***
Lurban:Iyes  -0.75503    0.02695 -28.017     < 2e-16 ***
Syes:Iyes    -0.81400    0.02762 -29.473     < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 61709.521  on 15  degrees of freedom
Residual deviance:    23.351  on  5  degrees of freedom
AIC: 198.81

Number of Fisher Scoring iterations: 3

fit3 <- glm(count ~ G + L + S + I + G * L + G * S + G * I + L * S + L * I + 
    S * I + G * L * S + G * L * I + G * S * I + L * S * I, family = poisson, 
    data = accidents)
summary(fit3)


Call:
glm(formula = count ~ G + L + S + I + G * L + G * S + G * I + 
    L * S + L * I + S * I + G * L * S + G * L * I + G * S * I + 
    L * S * I, family = poisson, data = accidents)

Deviance Residuals: 
       1         2         3         4         5         6         7  
-0.17993   0.33015   0.13110  -0.37131   0.12027  -0.32405  -0.09531  
       8         9        10        11        12        13        14  
 0.37418   0.13122  -0.31071  -0.12537   0.45614  -0.10069   0.36166  
      15        16  
 0.09801  -0.52177  

Coefficients:
                  Estimate Std. Error z value Pr(>|z|)    
(Intercept)        8.08834    0.01731 467.290  < 2e-16 ***
Gmale              0.62979    0.02129  29.588  < 2e-16 ***
Lurban             0.80410    0.02071  38.831  < 2e-16 ***
Syes               0.63159    0.02128  29.681  < 2e-16 ***
Iyes              -1.21855    0.03467 -35.150  < 2e-16 ***
Gmale:Lurban      -0.27351    0.02579 -10.607  < 2e-16 ***
Gmale:Syes        -0.53937    0.02710 -19.903  < 2e-16 ***
Gmale:Iyes        -0.50174    0.04423 -11.344  < 2e-16 ***
Lurban:Syes       -0.16551    0.02561  -6.463 1.03e-10 ***
Lurban:Iyes       -0.75989    0.04460 -17.038  < 2e-16 ***
Syes:Iyes         -0.85855    0.04673 -18.373  < 2e-16 ***
Gmale:Lurban:Syes  0.12646    0.03288   3.846  0.00012 ***
Gmale:Lurban:Iyes -0.08176    0.05469  -1.495  0.13491    
Gmale:Syes:Iyes   -0.01144    0.05603  -0.204  0.83821    
Lurban:Syes:Iyes   0.09685    0.05547   1.746  0.08080 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 61709.5207  on 15  degrees of freedom
Residual deviance:     1.3253  on  1  degrees of freedom
AIC: 184.78

Number of Fisher Scoring iterations: 3

library(car)
Anova(fit3)

Analysis of Deviance Table (Type II tests)

Response: count
      LR Chisq Df   Pr(>Chisq)    
G          396  1    < 2.2e-16 ***
L         4585  1    < 2.2e-16 ***
S          692  1    < 2.2e-16 ***
I        53243  1    < 2.2e-16 ***
G:L        170  1    < 2.2e-16 ***
G:S        868  1    < 2.2e-16 ***
G:I        405  1    < 2.2e-16 ***
L:S         28  1 0.0000001503 ***
L:I        788  1    < 2.2e-16 ***
S:I        902  1    < 2.2e-16 ***
G:L:S       15  1    0.0001187 ***
G:L:I        2  1    0.1347340    
G:S:I        0  1    0.8381830    
L:S:I        3  1    0.0809032 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

fit4 <- glm(count ~ G + L + S + I + G * L + G * S + G * I + L * S + L * I + 
    S * I + G * L * S, family = poisson, data = accidents)
summary(fit4)


Call:
glm(formula = count ~ G + L + S + I + G * L + G * S + G * I + 
    L * S + L * I + S * I + G * L * S, family = poisson, data = accidents)

Deviance Residuals: 
       1         2         3         4         5         6         7  
-0.15190   0.27851   0.51823  -1.44646   0.16160  -0.43483  -0.42327  
       8         9        10        11        12        13        14  
 1.69037  -0.34700   0.83292  -0.05675   0.20564   0.21675  -0.76754  
      15        16  
 0.09329  -0.49684  

Coefficients:
                  Estimate Std. Error z value Pr(>|z|)    
(Intercept)        8.08784    0.01654 488.884  < 2e-16 ***
Gmale              0.63640    0.02015  31.579  < 2e-16 ***
Lurban             0.80411    0.01966  40.891  < 2e-16 ***
Syes               0.62713    0.02027  30.940  < 2e-16 ***
Iyes              -1.21640    0.02649 -45.918  < 2e-16 ***
Gmale:Lurban      -0.28274    0.02441 -11.584  < 2e-16 ***
Gmale:Syes        -0.54186    0.02590 -20.925  < 2e-16 ***
Gmale:Iyes        -0.54483    0.02727 -19.982  < 2e-16 ***
Lurban:Syes       -0.15752    0.02441  -6.453 1.09e-10 ***
Lurban:Iyes       -0.75806    0.02697 -28.105  < 2e-16 ***
Syes:Iyes         -0.81710    0.02765 -29.551  < 2e-16 ***
Gmale:Lurban:Syes  0.12858    0.03228   3.984 6.78e-05 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 61709.5207  on 15  degrees of freedom
Residual deviance:     7.4645  on  4  degrees of freedom
AIC: 184.92

Number of Fisher Scoring iterations: 3

fit5 <- glm(count ~ G + L + S + I + G * L + G * S + G * I + L * S + L * I + 
    S * I + G * L * I, family = poisson, data = accidents)
summary(fit5)


Call:
glm(formula = count ~ G + L + S + I + G * L + G * S + G * I + 
    L * S + L * I + S * I + G * L * I, family = poisson, data = accidents)

Deviance Residuals: 
      1        2        3        4        5        6        7        8  
-2.1237   0.2232   1.5747  -0.2517   1.6044  -0.6465  -1.2595   0.7526  
      9       10       11       12       13       14       15       16  
 1.2431   0.5745  -1.1770  -0.8203  -1.0793  -0.1823   1.0572   0.2683  

Coefficients:
                  Estimate Std. Error z value    Pr(>|z|)    
(Intercept)        8.12222    0.01466 554.207     < 2e-16 ***
Gmale              0.58165    0.01656  35.124     < 2e-16 ***
Lurban             0.75277    0.01629  46.224     < 2e-16 ***
Syes               0.57920    0.01623  35.690     < 2e-16 ***
Iyes              -1.24900    0.02941 -42.472     < 2e-16 ***
Gmale:Lurban      -0.19834    0.01697 -11.690     < 2e-16 ***
Gmale:Syes        -0.45991    0.01568 -29.328     < 2e-16 ***
Gmale:Iyes        -0.48396    0.03754 -12.892     < 2e-16 ***
Lurban:Syes       -0.08488    0.01619  -5.241 0.000000159 ***
Lurban:Iyes       -0.70182    0.03631 -19.327     < 2e-16 ***
Syes:Iyes         -0.81393    0.02762 -29.473     < 2e-16 ***
Gmale:Lurban:Iyes -0.11762    0.05383  -2.185      0.0289 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 61709.521  on 15  degrees of freedom
Residual deviance:    18.569  on  4  degrees of freedom
AIC: 196.03

Number of Fisher Scoring iterations: 3

fit6 <- glm(count ~ G + L + S + I + G * L + G * S + G * I + L * S + L * I + 
    S * I + G * S * I, family = poisson, data = accidents)
summary(fit6)


Call:
glm(formula = count ~ G + L + S + I + G * L + G * S + G * I + 
    L * S + L * I + S * I + G * S * I, family = poisson, data = accidents)

Deviance Residuals: 
       1         2         3         4         5         6         7  
-1.94157  -0.30012   1.96771  -1.13125   1.31737   0.29853  -1.41353  
       8         9        10        11        12        13        14  
 1.16157   1.05786   1.19487  -1.49857   0.09192  -0.80665  -1.34275  
      15        16  
 1.18219  -0.10646  

Coefficients:
                Estimate Std. Error z value    Pr(>|z|)    
(Intercept)      8.11906    0.01463 554.966     < 2e-16 ***
Gmale            0.58719    0.01644  35.712     < 2e-16 ***
Lurban           0.75931    0.01602  47.397     < 2e-16 ***
Syes             0.57731    0.01645  35.090     < 2e-16 ***
Iyes            -1.22907    0.02855 -43.044     < 2e-16 ***
Gmale:Lurban    -0.20993    0.01612 -13.020     < 2e-16 ***
Gmale:Syes      -0.45649    0.01641 -27.816     < 2e-16 ***
Gmale:Iyes      -0.52528    0.03467 -15.150     < 2e-16 ***
Lurban:Syes     -0.08494    0.01619  -5.245 0.000000156 ***
Lurban:Iyes     -0.75503    0.02695 -28.017     < 2e-16 ***
Syes:Iyes       -0.79710    0.03644 -21.872     < 2e-16 ***
Gmale:Syes:Iyes -0.03941    0.05553  -0.710       0.478    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 61709.521  on 15  degrees of freedom
Residual deviance:    22.847  on  4  degrees of freedom
AIC: 200.31

Number of Fisher Scoring iterations: 3

fit7 <- glm(count ~ G + L + S + I + G * L + G * S + G * I + L * S + L * I + 
    S * I + L * S * I, family = poisson, data = accidents)
summary(fit7)


Call:
glm(formula = count ~ G + L + S + I + G * L + G * S + G * I + 
    L * S + L * I + S * I + L * S * I, family = poisson, data = accidents)

Deviance Residuals: 
      1        2        3        4        5        6        7        8  
-1.6873  -0.9683   1.7286  -0.2460   1.2758   0.6134  -1.3477   0.6613  
      9       10       11       12       13       14       15       16  
 1.2473   0.9358  -1.6318   0.3011  -1.0599  -0.6697   1.3971  -0.9129  

Coefficients:
                 Estimate Std. Error z value     Pr(>|z|)    
(Intercept)       8.11465    0.01467 552.959      < 2e-16 ***
Gmale             0.58918    0.01620  36.360      < 2e-16 ***
Lurban            0.76422    0.01630  46.879      < 2e-16 ***
Syes              0.58480    0.01658  35.271      < 2e-16 ***
Iyes             -1.20338    0.02844 -42.309      < 2e-16 ***
Gmale:Lurban     -0.20992    0.01612 -13.019      < 2e-16 ***
Gmale:Syes       -0.45993    0.01568 -29.328      < 2e-16 ***
Gmale:Iyes       -0.54059    0.02722 -19.858      < 2e-16 ***
Lurban:Syes      -0.09353    0.01702  -5.496 0.0000000388 ***
Lurban:Iyes      -0.79124    0.03478 -22.748      < 2e-16 ***
Syes:Iyes        -0.85778    0.03839 -22.345      < 2e-16 ***
Lurban:Syes:Iyes  0.09016    0.05468   1.649       0.0992 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 61709.521  on 15  degrees of freedom
Residual deviance:    20.633  on  4  degrees of freedom
AIC: 198.09

Number of Fisher Scoring iterations: 3

fitLogistic <- glm(I ~ G + L + S, family = binomial, weights = count, data = accidents)
summary(fitLogistic)


Call:
glm(formula = I ~ G + L + S, family = binomial, data = accidents, 
    weights = count)

Deviance Residuals: 
   Min      1Q  Median      3Q     Max  
-44.07  -39.16   11.46   58.98   65.79  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -1.21640    0.02649  -45.92   <2e-16 ***
Gmale       -0.54483    0.02727  -19.98   <2e-16 ***
Lurban      -0.75806    0.02697  -28.11   <2e-16 ***
Syes        -0.81710    0.02765  -29.55   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 41987  on 15  degrees of freedom
Residual deviance: 40082  on 12  degrees of freedom
AIC: 40090

Number of Fisher Scoring iterations: 6

# see pages 208-209 for the following approach
injury <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Injury_binom.dat", 
    header = T)
injury$G <- injury$gender
injury$L <- injury$location
injury$S <- injury$seatbelt
injury$nn <- injury$yes + injury$no
injury$pct <- injury$yes/injury$nn
injury

  gender location seatbelt    no  yes      G     L   S    nn        pct
1 female    urban       no  7287  996 female urban  no  8283 0.12024629
2 female    urban      yes 11587  759 female urban yes 12346 0.06147740
3 female    rural       no  3246  973 female rural  no  4219 0.23062337
4 female    rural      yes  6134  757 female rural yes  6891 0.10985343
5   male    urban       no 10381  812   male urban  no 11193 0.07254534
6   male    urban      yes 10969  380   male urban yes 11349 0.03348313
7   male    rural       no  6123 1084   male rural  no  7207 0.15040932
8   male    rural      yes  6693  513   male rural yes  7206 0.07119067

fitLogistic2 <- glm(pct ~ G + L + S, family = binomial, weights = nn, data = injury)
summary(fitLogistic2)


Call:
glm(formula = pct ~ G + L + S, family = binomial, data = injury, 
    weights = nn)

Deviance Residuals: 
      1        2        3        4        5        6        7        8  
-0.4639   1.7426   0.3172  -1.5365  -0.7976  -0.5055   0.9023   0.2133  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -1.21640    0.02649  -45.92   <2e-16 ***
Gmale       -0.54483    0.02727  -19.98   <2e-16 ***
Lurban      -0.75806    0.02697  -28.11   <2e-16 ***
Syes        -0.81710    0.02765  -29.55   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1912.4532  on 7  degrees of freedom
Residual deviance:    7.4645  on 4  degrees of freedom
AIC: 82.167

Number of Fisher Scoring iterations: 3

# p.216 Linear by Linear Association for Sex Opinions

teenagers <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Teenagers.dat", 
    header = T)
teenagers

   sex birth count
1    1     1    81
2    1     2    68
3    1     3    60
4    1     4    38
5    2     1    24
6    2     2    26
7    2     3    29
8    2     4    14
9    3     1    18
10   3     2    41
11   3     3    74
12   3     4    42
13   4     1    36
14   4     2    57
15   4     3   161
16   4     4   157

fit <- glm(count ~ factor(sex) + factor(birth) + sex:birth, family = poisson, 
    data = teenagers)
summary(fit)


Call:
glm(formula = count ~ factor(sex) + factor(birth) + sex:birth, 
    family = poisson, data = teenagers)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.35834  -0.91606   0.07972   0.61648   1.57618  

Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
(Intercept)     4.10684    0.08951  45.881  < 2e-16 ***
factor(sex)2   -1.64596    0.13473 -12.216  < 2e-16 ***
factor(sex)3   -1.77002    0.16464 -10.751  < 2e-16 ***
factor(sex)4   -1.75369    0.23432  -7.484 7.20e-14 ***
factor(birth)2 -0.46411    0.11952  -3.883 0.000103 ***
factor(birth)3 -0.72452    0.16201  -4.472 7.74e-06 ***
factor(birth)4 -1.87966    0.24910  -7.546 4.50e-14 ***
sex:birth       0.28584    0.02824  10.122  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 431.078  on 15  degrees of freedom
Residual deviance:  11.534  on  8  degrees of freedom
AIC: 118.21

Number of Fisher Scoring iterations: 4

library(car)
Anova(fit)

Analysis of Deviance Table (Type II tests)

Response: count
              LR Chisq Df Pr(>Chisq)    
factor(sex)    201.042  3  < 2.2e-16 ***
factor(birth)   91.243  3  < 2.2e-16 ***
sex:birth      116.119  1  < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# pp.218-219 Death Rates for Lung Cancer Patients

cancer <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Cancer.dat", header = T)
cancer

   time histology stage count risktime
1     1         1     1     9      157
2     1         2     1     5       77
3     1         3     1     1       21
4     2         1     1     2      139
5     2         2     1     2       68
6     2         3     1     1       17
7     3         1     1     9      126
8     3         2     1     3       63
9     3         3     1     1       14
10    4         1     1    10      102
11    4         2     1     2       55
12    4         3     1     1       12
13    5         1     1     1       88
14    5         2     1     2       50
15    5         3     1     0       10
16    6         1     1     3       82
17    6         2     1     2       45
18    6         3     1     1        8
19    7         1     1     1       76
20    7         2     1     2       42
21    7         3     1     0        6
22    1         1     2    12      134
23    1         2     2     4       71
24    1         3     2     1       22
25    2         1     2     7      110
26    2         2     2     3       63
27    2         3     2     1       18
28    3         1     2     5       96
29    3         2     2     5       58
30    3         3     2     3       14
31    4         1     2    10       86
32    4         2     2     4       42
33    4         3     2     1       10
34    5         1     2     4       66
35    5         2     2     2       35
36    5         3     2     0        8
37    6         1     2     3       59
38    6         2     2     1       32
39    6         3     2     0        8
40    7         1     2     4       51
41    7         2     2     4       28
42    7         3     2     2        6
43    1         1     3    42      212
44    1         2     3    28      130
45    1         3     3    19      101
46    2         1     3    26      136
47    2         2     3    19       72
48    2         3     3    11       63
49    3         1     3    12       90
50    3         2     3    10       42
51    3         3     3     7       43
52    4         1     3    10       64
53    4         2     3     5       21
54    4         3     3     6       32
55    5         1     3     5       47
56    5         2     3     0       14
57    5         3     3     3       21
58    6         1     3     4       39
59    6         2     3     3       13
60    6         3     3     3       14
61    7         1     3     1       29
62    7         2     3     2        7
63    7         3     3     3       10

cancer$logrisktime <- log(cancer$risktime)
fit <- glm(count ~ factor(histology) + factor(stage) + factor(time), family = poisson, 
    offset = logrisktime, data = cancer)
summary(fit)


Call:
glm(formula = count ~ factor(histology) + factor(stage) + factor(time), 
    family = poisson, data = cancer, offset = logrisktime)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.00333  -0.74769  -0.03194   0.46468   1.70832  

Coefficients:
                   Estimate Std. Error z value Pr(>|z|)    
(Intercept)        -3.00928    0.16651 -18.073  < 2e-16 ***
factor(histology)2  0.16244    0.12195   1.332  0.18285    
factor(histology)3  0.10754    0.14745   0.729  0.46580    
factor(stage)2      0.47001    0.17444   2.694  0.00705 ** 
factor(stage)3      1.32431    0.15205   8.709  < 2e-16 ***
factor(time)2      -0.12745    0.14908  -0.855  0.39259    
factor(time)3      -0.07973    0.16352  -0.488  0.62585    
factor(time)4       0.11892    0.17107   0.695  0.48694    
factor(time)5      -0.66511    0.26061  -2.552  0.01071 *  
factor(time)6      -0.35015    0.24348  -1.438  0.15040    
factor(time)7      -0.17518    0.24985  -0.701  0.48321    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 175.718  on 62  degrees of freedom
Residual deviance:  43.923  on 52  degrees of freedom
AIC: 251.74

Number of Fisher Scoring iterations: 5

library(car)
Anova(fit)

Analysis of Deviance Table (Type II tests)

Response: count
                  LR Chisq Df Pr(>Chisq)    
factor(histology)    1.876  2    0.39132    
factor(stage)       99.155  2    < 2e-16 ***
factor(time)        11.383  6    0.07724 .  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

# pp. 220-221 Horseshoe Crabs Reprise

crabs <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Crabs.dat", header = T)
head(crabs)

  crab sat y weight width color spine
1    1   8 1   3.05  28.3     2     3
2    2   0 0   1.55  22.5     3     3
3    3   9 1   2.30  26.0     1     1
4    4   0 0   2.10  24.8     3     3
5    5   4 1   2.60  26.0     3     3
6    6   0 0   2.10  23.8     2     3

tail(crabs)

    crab sat y weight width color spine
168  168   2 1  2.175  26.2     3     3
169  169   3 1  2.750  26.1     3     3
170  170   4 1  3.275  29.0     3     3
171  171   0 0  2.625  28.0     1     1
172  172   0 0  2.625  27.0     4     3
173  173   0 0  2.000  24.5     2     2

fit.pois <- glm(sat ~ width, family = poisson, data = crabs)
summary(fit.pois)


Call:
glm(formula = sat ~ width, family = poisson, data = crabs)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.8526  -1.9884  -0.4933   1.0970   4.9221  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -3.30476    0.54224  -6.095  1.1e-09 ***
width        0.16405    0.01997   8.216  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 632.79  on 172  degrees of freedom
Residual deviance: 567.88  on 171  degrees of freedom
AIC: 927.18

Number of Fisher Scoring iterations: 6

library(MASS)
fit.NB <- glm.nb(sat ~ width, data = crabs)
summary(fit.NB)


Call:
glm.nb(formula = sat ~ width, data = crabs, init.theta = 0.90456808, 
    link = log)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.7798  -1.4110  -0.2502   0.4770   2.0177  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -4.05251    1.17143  -3.459 0.000541 ***
width        0.19207    0.04406   4.360 0.000013 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for Negative Binomial(0.9046) family taken to be 1)

    Null deviance: 213.05  on 172  degrees of freedom
Residual deviance: 195.81  on 171  degrees of freedom
AIC: 757.29

Number of Fisher Scoring iterations: 1

              Theta:  0.905 
          Std. Err.:  0.161 

 2 x log-likelihood:  -751.291

Chapter 8: Models for Matched Pairs

Ch. 8 Code

# p.229 Environmental Opinions

Opinions <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Envir_opinions.dat", 
    header = T)
head(Opinions)

  person y1 y2
1      1  1  1
2      2  1  1
3      3  1  1
4      4  1  1
5      5  1  1
6      6  1  1

tail(Opinions)

     person y1 y2
1139   1139  2  2
1140   1140  2  2
1141   1141  2  2
1142   1142  2  2
1143   1143  2  2
1144   1144  2  2

tab <- xtabs(~y1 + y2, data = Opinions)
tab

   y2
y1    1   2
  1 227 132
  2 107 678

mcnemar.test(tab, correct = FALSE)


    McNemar's Chi-squared test

data:  tab
McNemar's chi-squared = 2.6151, df = 1, p-value = 0.1059

library(PropCIs)
# ?diffpropci.Wald.mp
diffpropci.Wald.mp(107, 132, 1144, 0.95)




data:  

95 percent confidence interval:
 -0.004602847  0.048309140
sample estimates:
[1] 0.02185315

diffpropci.Wald.mp(tab[2, 1], tab[1, 2], sum(tab), 0.95)




data:  

95 percent confidence interval:
 -0.004602847  0.048309140
sample estimates:
[1] 0.02185315

scoreci.mp(tab[2, 1], tab[1, 2], sum(tab), 0.95)




data:  

95 percent confidence interval:
 -0.004661338  0.048494581

n12 <- tab[1, 2]
n21 <- tab[2, 1]
n <- sum(tab)
diff <- (n12 - n21)/n
SE <- sqrt(n12 + n21 - (n12 - n21)/n)/n
diff + c(-1, 1) * qnorm(0.975) * SE

[1] -0.004631925  0.048338219

# Fitting GEE model
Opinions2 <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Opinions.dat", 
    header = T)
head(Opinions2)

  person question y
1      1        1 1
2      1        0 1
3      2        1 1
4      2        0 1
5      3        1 1
6      3        0 1

tail(Opinions2)

     person question y
2283   1142        1 0
2284   1142        0 0
2285   1143        1 0
2286   1143        0 0
2287   1144        1 0
2288   1144        0 0

# install.packages('gee')
library(gee)
fit2 <- gee(y ~ question, id = person, family = binomial, data = Opinions2)

(Intercept)    question 
 -0.8858933   0.1035319

summary(fit2)


 GEE:  GENERALIZED LINEAR MODELS FOR DEPENDENT DATA
 gee S-function, version 4.13 modified 98/01/27 (1998) 

Model:
 Link:                      Logit 
 Variance to Mean Relation: Binomial 
 Correlation Structure:     Independent 

Call:
gee(formula = y ~ question, id = person, data = Opinions2, family = binomial)

Summary of Residuals:
       Min         1Q     Median         3Q        Max 
-0.3138112 -0.3138112 -0.2919580  0.6861888  0.7080420 


Coefficients:
              Estimate Naive S.E.    Naive z Robust S.E.   Robust z
(Intercept) -0.8858933 0.06505597 -13.617401  0.06502753 -13.623357
question     0.1035319 0.09107816   1.136737  0.06397794   1.618244

Estimated Scale Parameter:  1.000875
Number of Iterations:  1

Working Correlation
     [,1] [,2]
[1,]    1    0
[2,]    0    1

exp(coefficients(fit2)[2])

question 
1.109081

# Fitting the Logistic-Normal Model of Chapter 10 (pp.274-277)
# install.packages('lme4')
library(lme4)
fit <- glmer(y ~ (1 | person) + question, family = binomial, nAGQ = 50, data = Opinions2)
summary(fit)

Generalized linear mixed model fit by maximum likelihood (Adaptive
  Gauss-Hermite Quadrature, nAGQ = 50) [glmerMod]
 Family: binomial  ( logit )
Formula: y ~ (1 | person) + question
   Data: Opinions2

     AIC      BIC   logLik deviance df.resid 
  2526.8   2544.0  -1260.4   2520.8     2285 

Scaled residuals: 
    Min      1Q  Median      3Q     Max 
-0.8872 -0.2691 -0.2423  0.4646  1.2519 

Random effects:
 Groups Name        Variance Std.Dev.
 person (Intercept) 8.143    2.854   
Number of obs: 2288, groups:  person, 1144

Fixed effects:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  -1.8343     0.1624 -11.295   <2e-16 ***
question      0.2100     0.1301   1.614    0.106    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Correlation of Fixed Effects:
         (Intr)
question -0.452

exp(0.2099747)

[1] 1.233647

n12/n21

[1] 1.233645

# Section 8.3 Comparing Proportions for Nominal Matched-Pairs Responses
# Coffee Brand Market Share example

drinkCoffee <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Coffee.dat", 
    header = T)
head(drinkCoffee)

  person purchase y
1      1        1 1
2      1        0 1
3      2        1 1
4      2        0 1
5      3        1 1
6      3        0 1

tail(drinkCoffee)

     person purchase y
1077    539        1 5
1078    539        0 5
1079    540        1 5
1080    540        0 5
1081    541        1 5
1082    541        0 5

# install.packages('multgee')
library(multgee)
fit <- nomLORgee(y ~ purchase, id = person, LORstr = "independence", data = drinkCoffee)
summary(fit)

GEE FOR NOMINAL MULTINOMIAL RESPONSES 
version 1.6.0 modified 2017-07-10 

Link : Baseline Category Logit 

Local Odds Ratios:
Structure:         independence

call:
nomLORgee(formula = y ~ purchase, data = drinkCoffee, id = person, 
    LORstr = "independence")

Summary of residuals:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
-0.4270 -0.2495 -0.1386  0.0000 -0.0610  0.9390 

Number of Iterations: 1 

Coefficients:
           Estimate   san.se   san.z Pr(>|san.z|)    
beta10      0.81093  0.15516  5.2265      < 2e-16 ***
purchase:1  0.32340  0.16830  1.9215      0.05466 .  
beta20      0.31237  0.16989  1.8387      0.06596 .  
purchase:2 -0.00222  0.18662 -0.0119      0.99051    
beta30      1.34807  0.14490  9.3035      < 2e-16 ***
purchase:3 -0.03729  0.15807 -0.2359      0.81353    
beta40     -0.59784  0.21672 -2.7585      0.00581 ** 
purchase:4  0.17402  0.23531  0.7396      0.45957    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Local Odds Ratios Estimates:
     [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
[1,]    0    0    0    0    1    1    1    1
[2,]    0    0    0    0    1    1    1    1
[3,]    0    0    0    0    1    1    1    1
[4,]    0    0    0    0    1    1    1    1
[5,]    1    1    1    1    0    0    0    0
[6,]    1    1    1    1    0    0    0    0
[7,]    1    1    1    1    0    0    0    0
[8,]    1    1    1    1    0    0    0    0

pvalue of Null model: <0.0001

fit0 <- nomLORgee(y ~ 1, id = person, LORstr = "independence", data = drinkCoffee)
# doesn't work: lrtest(fit0, fit)
waldts(fit0, fit)

Goodness of Fit based on the Wald test 

Model under H_0: y ~ 1
Model under H_1: y ~ purchase

Wald Statistic=12.4869, df=4, p-value=0.0141

with(drinkCoffee, mantelhaen.test(purchase, y, person))


    Cochran-Mantel-Haenszel test

data:  purchase and y and person
Cochran-Mantel-Haenszel M^2 = 12.291, df = 4, p-value = 0.01531

# Fitting Quasi-Symmetry (QS) Model p.239
drinkMoreCoffee <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Coffee2.dat", 
    header = T)
drinkMoreCoffee

   H  T  S  N  B nij nji
1  1 -1  0  0  0  17   9
2  1  0 -1  0  0  44  17
3  1  0  0 -1  0   7   6
4  1  0  0  0 -1  10  10
5  0  1 -1  0  0  11  11
6  0  1  0 -1  0   0   4
7  0  1  0  0 -1   9   4
8  0  0  1 -1  0   9   9
9  0  0  1  0 -1  12  12
10 0  0  0  1 -1   2   2

symmMod <- glm(nij/(nij + nji) ~ -1, family = binomial, weights = nij + nji, 
    data = drinkMoreCoffee)
summary(symmMod)


Call:
glm(formula = nij/(nij + nji) ~ -1, family = binomial, data = drinkMoreCoffee, 
    weights = nij + nji)

Deviance Residuals: 
   Min      1Q  Median      3Q     Max  
-2.355   0.000   0.000   1.123   3.518  

No Coefficients

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 22.473  on 10  degrees of freedom
Residual deviance: 22.473  on 10  degrees of freedom
AIC: 52.433

Number of Fisher Scoring iterations: 0

QSMod <- glm(nij/(nij + nji) ~ -1 + H + T + S + N + B, family = binomial, weights = nij + 
    nji, data = drinkMoreCoffee)
summary(QSMod)


Call:
glm(formula = nij/(nij + nji) ~ -1 + H + T + S + N + B, family = binomial, 
    data = drinkMoreCoffee, weights = nij + nji)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.1010  -0.2902  -0.0804   0.7165   1.4120  

Coefficients: (1 not defined because of singularities)
   Estimate Std. Error z value Pr(>|z|)  
H  0.595440   0.293658   2.028   0.0426 *
T -0.004004   0.329359  -0.012   0.9903  
S -0.113299   0.285082  -0.397   0.6911  
N  0.302115   0.401589   0.752   0.4519  
B        NA         NA      NA       NA  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 22.473  on 10  degrees of freedom
Residual deviance:  9.974  on  6  degrees of freedom
AIC: 47.934

Number of Fisher Scoring iterations: 4

# Recycle or Drive Less to Help the Environment? pp.240-242

motherEarth <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Envir.dat", 
    header = T)
head(motherEarth)

  person question y
1      1        1 1
2      1        0 1
3      2        1 1
4      2        0 1
5      3        1 1
6      3        0 1

tail(motherEarth)

     person question y
2455   1228        1 4
2456   1228        0 4
2457   1229        1 4
2458   1229        0 4
2459   1230        1 4
2460   1230        0 4

library(multgee)
fit <- ordLORgee(y ~ question, id = person, LORstr = "independence", data = motherEarth)
summary(fit)

GEE FOR ORDINAL MULTINOMIAL RESPONSES 
version 1.6.0 modified 2017-07-10 

Link : Cumulative logit 

Local Odds Ratios:
Structure:         independence

call:
ordLORgee(formula = y ~ question, data = motherEarth, id = person, 
    LORstr = "independence")

Summary of residuals:
     Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
-0.354916 -0.264742 -0.059210 -0.002021 -0.033859  0.966141 

Number of Iterations: 1 

Coefficients:
         Estimate   san.se    san.z Pr(>|san.z|)    
beta10   -3.35111  0.08289 -40.4287    < 2.2e-16 ***
beta20   -2.27673  0.07430 -30.6424    < 2.2e-16 ***
beta30   -0.58488  0.05882  -9.9443    < 2.2e-16 ***
question  2.75361  0.08147  33.7985    < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Local Odds Ratios Estimates:
     [,1] [,2] [,3] [,4] [,5] [,6]
[1,]    0    0    0    1    1    1
[2,]    0    0    0    1    1    1
[3,]    0    0    0    1    1    1
[4,]    1    1    1    0    0    0
[5,]    1    1    1    0    0    0
[6,]    1    1    1    0    0    0

pvalue of Null model: <0.0001

motherEarth2 <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Environment.dat", 
    header = T)
motherEarth2

  always often sometimes never x nij nji
1      1    -1         0     0 1  43   4
2      1     0        -1     0 2 163   4
3      1     0         0    -1 3 233   0
4      0     1        -1     0 1  99   8
5      0     1         0    -1 2 185   1
6      0     0         1    -1 1 230  18

fit2 <- glm(nij/(nij + nji) ~ -1 + x, family = binomial, weights = nij + nji, 
    data = motherEarth2)
summary(fit2)


Call:
glm(formula = nij/(nij + nji) ~ -1 + x, family = binomial, data = motherEarth2, 
    weights = nij + nji)

Deviance Residuals: 
       1         2         3         4         5         6  
-0.03567  -1.82023   0.59541   0.33794   0.46512   0.64368  

Coefficients:
  Estimate Std. Error z value Pr(>|z|)    
x   2.3936     0.1508   15.88   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1106.1240  on 6  degrees of freedom
Residual deviance:    4.4139  on 5  degrees of freedom
AIC: 23.35

Number of Fisher Scoring iterations: 4

fit.QS <- glm(nij/(nij + nji) ~ -1 + always + often + sometimes + never, family = binomial, 
    weights = nij + nji, data = motherEarth2)
summary(fit.QS)


Call:
glm(formula = nij/(nij + nji) ~ -1 + always + often + sometimes + 
    never, family = binomial, data = motherEarth2, weights = nij + 
    nji)

Deviance Residuals: 
      1        2        3        4        5        6  
 0.7689  -1.1439   0.6760   0.4572   0.3006  -0.1381  

Coefficients: (1 not defined because of singularities)
          Estimate Std. Error z value Pr(>|z|)    
always      6.9269     0.4708   14.71   <2e-16 ***
often       4.9332     0.3617   13.64   <2e-16 ***
sometimes   2.5817     0.2386   10.82   <2e-16 ***
never           NA         NA      NA       NA    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1106.1240  on 6  degrees of freedom
Residual deviance:    2.6751  on 3  degrees of freedom
AIC: 25.611

Number of Fisher Scoring iterations: 4

# Agreement of Pathologists on p.244

patho <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Pathologists.dat", 
    header = T)
patho

   X Y count diag
1  1 1    22    1
2  1 2     2    0
3  1 3     2    0
4  1 4     0    0
5  2 1     5    0
6  2 2     7    2
7  2 3    14    0
8  2 4     0    0
9  3 1     0    0
10 3 2     2    0
11 3 3    36    3
12 3 4     0    0
13 4 1     0    0
14 4 2     1    0
15 4 3    17    0
16 4 4    10    4

# Fit independence model
fitI <- glm(count ~ factor(X) + factor(Y), family = poisson, data = patho)
summary(fitI)


Call:
glm(formula = count ~ factor(X) + factor(Y), family = poisson, 
    data = patho)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-4.2771  -2.2089  -0.7300   0.6699   5.0439  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept)  1.783e+00  2.589e-01   6.888 5.66e-12 ***
factor(X)2  -3.218e-12  2.773e-01   0.000  1.00000    
factor(X)3   3.795e-01  2.545e-01   1.491  0.13595    
factor(X)4   7.411e-02  2.724e-01   0.272  0.78554    
factor(Y)2  -8.109e-01  3.469e-01  -2.337  0.01942 *  
factor(Y)3   9.383e-01  2.270e-01   4.133 3.58e-05 ***
factor(Y)4  -9.933e-01  3.702e-01  -2.683  0.00729 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 190.40  on 15  degrees of freedom
Residual deviance: 117.96  on  9  degrees of freedom
AIC: 172.78

Number of Fisher Scoring iterations: 6

# Fit Quasi-independence model
fitQI <- glm(count ~ factor(X) + factor(Y) + factor(diag), family = poisson, 
    data = patho)
summary(fitQI)


Call:
glm(formula = count ~ factor(X) + factor(Y) + factor(diag), family = poisson, 
    data = patho)

Deviance Residuals: 
       1         2         3         4         5         6         7  
 0.00000   1.19586  -0.74802  -0.00006   1.48661   0.00000  -0.66364  
       8         9        10        11        12        13        14  
-0.00014  -1.23667   0.63080   0.00000  -0.00008  -1.93254  -1.35095  
      15        16  
 1.02517   0.00000  

Coefficients:
               Estimate Std. Error z value    Pr(>|z|)    
(Intercept)     -0.7700     0.6979  -1.103     0.26986    
factor(X)2       1.6321     0.5604   2.912     0.00359 ** 
factor(X)3       0.5017     0.9261   0.542     0.58801    
factor(X)4       1.3946     0.5551   2.512     0.01199 *  
factor(Y)2       0.4796     0.6552   0.732     0.46414    
factor(Y)3       1.9493     0.4971   3.921 0.000088004 ***
factor(Y)4     -19.3097  4988.8789  -0.004     0.99691    
factor(diag)1    3.8611     0.7297   5.291 0.000000122 ***
factor(diag)2    0.6042     0.6900   0.876     0.38119    
factor(diag)3    1.9025     0.8367   2.274     0.02298 *  
factor(diag)4   20.9877  4988.8789   0.004     0.99664    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 190.398  on 15  degrees of freedom
Residual deviance:  13.178  on  5  degrees of freedom
AIC: 75.997

Number of Fisher Scoring iterations: 17

# Tennis on p.248

tennis <- read.table("http://www.stat.ufl.edu/~aa/cat/data/Tennis.dat", header = T)
tennis

   Djokovic Federer Murray Nadal Wawrinka nij nji
1         1      -1      0     0        0   9   6
2         1       0     -1     0        0  14   3
3         1       0      0    -1        0   9   2
4         1       0      0     0       -1   4   3
5         0       1     -1     0        0   5   0
6         0       1      0    -1        0   5   1
7         0       1      0     0       -1   7   2
8         0       0      1    -1        0   2   4
9         0       0      1     0       -1   2   2
10        0       0      0     1       -1   4   3

fitTennis <- glm(nij/(nij + nji) ~ -1 + Djokovic + Federer + Murray + Nadal + 
    Wawrinka, family = binomial, weights = nij + nji, data = tennis)
summary(fitTennis)


Call:
glm(formula = nij/(nij + nji) ~ -1 + Djokovic + Federer + Murray + 
    Nadal + Wawrinka, family = binomial, data = tennis, weights = nij + 
    nji)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.1225  -0.1262   0.3715   0.5386   1.2928  

Coefficients: (1 not defined because of singularities)
         Estimate Std. Error z value Pr(>|z|)  
Djokovic  1.17612    0.49952   2.354   0.0185 *
Federer   1.13578    0.51095   2.223   0.0262 *
Murray   -0.56852    0.56833  -1.000   0.3172  
Nadal    -0.06185    0.51487  -0.120   0.9044  
Wawrinka       NA         NA      NA       NA  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 26.8960  on 10  degrees of freedom
Residual deviance:  4.3958  on  6  degrees of freedom
AIC: 34.041

Number of Fisher Scoring iterations: 4

Categorical Data Analysis

Graduate Course Notes

Patrick Oster

February 25, 2019

Chapter 1: Introduction

1.1 Categorical Response Data

1.2 Probability Distributions for Categorical Data

1.3 Statistical Inference for a Proportion

Ch. 1 Code

Chapter 2: Analyzing Contingency Tables

Ch. 2 Code

Chapter 3: Generalized Linear Models

Examples in R

Data

Models

Plots

Data 2

Models & Predictions

Other Examples

Chapter 4: Logistic Regression

Basics of LRM (Logistic Regression Modelling)

General

Model

Interpretation

Horshoe Crab Example

Data

Model

Plot

Predictions

Odds/OR’s

LD50/MEL

Retrospective Studies

Section 4.2: Statistical Inference for LRM

General

Testing & CI’s

CI: \(\beta\)

CI: OR

Predictions

Plot

LRM with “real” x, categorical variables, & both

Multiple Logistic Regression

General

Model

Interpretations

MLR Horseshoe Crab Example

Data

Model Fit

OR

Plot

Test: Color

Other Ch 4 Topics

Interaction, ramifications, interpretations w Proportions

AME (Average Marginal Effects)

Classification tables

ROC Curves

Point Predictions

CI Predictions

Miscellaneous points about LRM

Ch. 4 Code

Chapter 5: Building & Applying Logistic Regression Models

Model Selection

Strategies

Stepwise/Canned Procedures

Purposeful Selection

AIC

Bias-Variance Tradeoff & Overfitting

Model Checking

Residuals

GoF

Diagnostics

Other Topics

Sparse Data & Infinite (or near infinite) Estimates

Conditional Likelihood

Probit, other links, & latent variables

Power & sample size

Choosing proper dose scale for Logistic Regression

Ch. 5 Code

Chapter 6: Multicategory Logit Models

Nominal vs Ordinal Response Variable, number of levels

BCL Modeling