This chapter described some of the most common generalized linear models, those used to model counts. It is important to never convert counts to proportions before analysis, because doing so destroys information about sample size. A fundamental difficulty with these models is that parameters are on a different scale, typically log-odds (for binomial) or log-rate (for Poisson), than the outcome variable they describe. Therefore computing implied predictions is even more important than before.
Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Problems are labeled Easy (E), Medium (M), and Hard(H).
Finally, upon completion, name your final output .html file as: YourName_ANLY505-Year-Semester.html and publish the assignment to your R Pubs account and submit the link to Canvas. Each question is worth 5 points.
11E1. If an event has probability 0.35, what are the log-odds of this event?
p = 0.35
log(p/(1-p))## [1] -0.619039211E2. If an event has log-odds 3.2, what is the probability of this event?
logodds = 3.2
exp(logodds)/(1+exp(logodds))## [1] 0.960834311E3. Suppose that a coefficient in a logistic regression has value 1.7. What does this imply about the proportional change in odds of the outcome?
# It means the log odds would increase exp1.7 times.11E4. Why do Poisson regressions sometimes require the use of an offset? Provide an example.
# Offset can bring observations to the same scale. 11M1. As explained in the chapter, binomial data can be organized in aggregated and disaggregated forms, without any impact on inference. But the likelihood of the data does change when the data are converted between the two formats. Can you explain why?
# The likelihood are different since aggregated model has more factors 11M2. If a coefficient in a Poisson regression has value 1.7, what does this imply about the change in the outcome?
# this means the parameter of passion will increase by exp1.7 times when the independent variable increase one unit.11M3. Explain why the logit link is appropriate for a binomial generalized linear model.
# since its values are in (0,1) and a binomial model will generate binary results either 0 or 1.11M4. Explain why the log link is appropriate for a Poisson generalized linear model.
#Since it will always be positive 11M5. What would it imply to use a logit link for the mean of a Poisson generalized linear model? Can you think of a real research problem for which this would make sense?
# It means the likelihood would be>=0.11M6. State the constraints for which the binomial and Poisson distributions have maximum entropy. Are the constraints different at all for binomial and Poisson? Why or why not?
#11M7. Use quap to construct a quadratic approximate posterior distribution for the chimpanzee model that includes a unique intercept for each actor, m11.4 (page 330). Compare the quadratic approximation to the posterior distribution produced instead from MCMC. Can you explain both the differences and the similarities between the approximate and the MCMC distributions? Relax the prior on the actor intercepts to Normal(0,10). Re-estimate the posterior using both ulam and quap. Do the differences increase or decrease? Why?
library(rethinking)
data("chimpanzees")
data <- chimpanzees
data$recipient <- NULL
data$treatment<-1 + data$prosoc_left + 2*data$condition
m1 <- quap(alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a[actor] + b[treatment],
a[actor] ~ dnorm(0,10),
b[treatment] ~ dnorm(0,10)
) ,
data=data)
precis(m1,depth = 2)## mean sd 5.5% 94.5%
## a[1] -1.1004131 3.042441 -5.9628213 3.761995
## a[2] 6.3608727 4.517136 -0.8583825 13.580128
## a[3] -1.4062237 3.043015 -6.2695492 3.457102
## a[4] -1.4062484 3.043015 -6.2695740 3.457077
## a[5] -1.1003945 3.042441 -5.9628027 3.762014
## a[6] -0.1562938 3.042854 -5.0193614 4.706774
## a[7] 1.3947207 3.053729 -3.4857275 6.275169
## b[1] 0.5776665 3.039947 -4.2807554 5.436088
## b[2] 1.1951335 3.040150 -3.6636133 6.053880
## b[3] 0.1598737 3.040204 -4.6989602 5.018708
## b[4] 1.0627504 3.040043 -3.7958255 5.92132611M8. Revisit the data(Kline) islands example. This time drop Hawaii from the sample and refit the models. What changes do you observe?
data(Kline)
data2 <- Kline
data2$P <- scale( log(data2$population) )
data2$contact_id <- ifelse( data2$contact=="high" , 2 , 1 )
summary(data2)## culture population contact total_tools mean_TU
## Chuuk :1 Min. : 1100 high:5 Min. :13.00 Min. :3.20
## Hawaii :1 1st Qu.: 3898 low :5 1st Qu.:22.50 1st Qu.:4.00
## Lau Fiji :1 Median : 7700 Median :30.50 Median :4.85
## Malekula :1 Mean : 34109 Mean :34.80 Mean :4.83
## Manus :1 3rd Qu.: 12050 3rd Qu.:42.25 3rd Qu.:5.30
## Santa Cruz:1 Max. :275000 Max. :71.00 Max. :6.60
## (Other) :4
## P.V1 contact_id
## Min. :-1.2914733 Min. :1.0
## 1st Qu.:-0.4690170 1st Qu.:1.0
## Median :-0.0188353 Median :1.5
## Mean : 0.0000000 Mean :1.5
## 3rd Qu.: 0.2677655 3rd Qu.:2.0
## Max. : 2.3210083 Max. :2.0
## dat <- list(
T = data2$total_tools ,
P = data2$P ,
cid = data2$contact_id )
m2_1 <- ulam(
alist(
T ~ dpois( lambda ),
log(lambda) <- a,
a ~ dnorm(3,0.5)
), data=dat , chains=4 , log_lik=TRUE )##
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## a 3.539815 0.05505236 3.455886 3.62941 583.3882 1.002479data2_1<-data2[data2$culture != c("Hawaii"), ]
data2_1$P <- scale( log(data2_1$population) )
data2_1$contact_id <- ifelse( data2_1$contact=="high" , 2 , 1 )
dat_2 <- list(
T = data2$total_tools ,
P = data2$P ,
cid = data2$contact_id )
m2_2 <- ulam(
alist(
T ~ dpois( lambda ),
log(lambda) <- a,
a ~ dnorm(3,0.5)
), data=dat_2 , chains=4 , log_lik=TRUE )## recompiling to avoid crashing R session##
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## a 3.546568 0.0529969 3.460438 3.629002 579.055 1.001857# compared m2_1 and m2_2, mean and sd are pretty similar.11H1. Use WAIC or PSIS to compare the chimpanzee model that includes a unique intercept for each actor, m11.4 (page 330), to the simpler models fit in the same section. Interpret the results.
data("chimpanzees")
data3 <- chimpanzees
data3$side <- data3$prosoc_left + 1
data3$cond <- data3$condition + 1
data3$recipient<-NULL
dat_3 <- list(
pulled_left = data3$pulled_left,
actor = data3$actor,
side = data3$side,
cond = data3$cond )
m3 <- ulam(
alist(
pulled_left ~ dbinom( 1 , p ) ,
logit(p) <- a[actor] + bs[side] + bc[cond] ,
a[actor] ~ dnorm( 0 , 1.5 ),
bs[side] ~ dnorm( 0 , 0.5 ),
bc[cond] ~ dnorm( 0 , 0.5 )
) , data=dat_3 , chains=4 , log_lik=TRUE )##
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## Chain 4:compare(m1,m3,func=WAIC)## Warning in compare(m1, m3, func = WAIC): Not all model fits of same class.
## This is usually a bad idea, because it implies they were fit by different algorithms.
## Check yourself, before you wreck yourself.## WAIC SE dWAIC dSE pWAIC weight
## m3 530.4649 19.08713 0.00000 NA 7.580332 0.9998529092
## m1 548.1135 19.00082 17.64863 2.304658 15.752937 0.0001470908