# load data
require(faraway)
## Loading required package: faraway
require(car)
## Loading required package: car
##
## Attaching package: 'car'
## The following objects are masked from 'package:faraway':
##
## logit, vif
data(fpe)
species <- gala$Species
endemics <- gala$Endemics
area <- gala$Area
elevation <- gala$Elevation
nearest <- gala$Nearest
scruz <- gala$Scruz
adjacent <- gala$Adjacent
poisson_gala <- glm(
species
~ endemics
+ area
+ elevation
+ nearest
+ scruz
+ adjacent,
family = poisson(link = "log"))
summary(poisson_gala)
##
## Call:
## glm(formula = species ~ endemics + area + elevation + nearest +
## scruz + adjacent, family = poisson(link = "log"))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -4.9919 -2.9305 -0.4296 1.3254 7.4735
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 2.828e+00 5.958e-02 47.471 < 2e-16 ***
## endemics 3.388e-02 1.741e-03 19.459 < 2e-16 ***
## area -1.067e-04 3.741e-05 -2.853 0.00433 **
## elevation 2.638e-04 1.934e-04 1.364 0.17264
## nearest 1.048e-02 1.611e-03 6.502 7.91e-11 ***
## scruz -6.835e-04 5.802e-04 -1.178 0.23877
## adjacent 4.539e-05 4.800e-05 0.946 0.34437
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 3510.73 on 29 degrees of freedom
## Residual deviance: 313.36 on 23 degrees of freedom
## AIC: 488.19
##
## Number of Fisher Scoring iterations: 5
g <- lm(species ~ endemics + area + elevation + nearest + scruz + adjacent, data = gala)
lambda <- powerTransform(g)
lam <- lambda$lambda
boxcox_gala <- lm(species^lam ~ endemics + area + elevation + nearest + scruz + adjacent, data = gala)
summary(boxcox_gala)
##
## Call:
## lm(formula = species^lam ~ endemics + area + elevation + nearest +
## scruz + adjacent, data = gala)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.6863 -1.0684 -0.3981 1.0026 3.8970
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.3931431 0.6368955 3.758 0.00103 **
## endemics 0.3350357 0.0325224 10.302 4.35e-10 ***
## area -0.0002086 0.0007707 -0.271 0.78907
## elevation -0.0008701 0.0032168 -0.270 0.78920
## nearest 0.0248540 0.0338516 0.734 0.47024
## scruz -0.0028778 0.0071562 -0.402 0.69130
## adjacent 0.0001690 0.0008028 0.210 0.83516
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.958 on 23 degrees of freedom
## Multiple R-squared: 0.9623, Adjusted R-squared: 0.9525
## F-statistic: 97.87 on 6 and 23 DF, p-value: 3.33e-15
compareCoefs(poisson_gala, boxcox_gala, se = FALSE)
##
## Call:
## 1: glm(formula = species ~ endemics + area + elevation + nearest +
## scruz + adjacent, family = poisson(link = "log"))
## 2: lm(formula = species^lam ~ endemics + area + elevation + nearest +
## scruz + adjacent, data = gala)
## Est. 1 Est. 2
## (Intercept) 2.83e+00 2.39e+00
## endemics 3.39e-02 3.35e-01
## area -1.07e-04 -2.09e-04
## elevation 2.64e-04 -8.70e-04
## nearest 1.05e-02 2.49e-02
## scruz -6.83e-04 -2.88e-03
## adjacent 4.54e-05 1.69e-04
ei <- fpe$EI
a <- fpe$A
b <- fpe$B
c <- fpe$C
d <- fpe$D
e <- fpe$E
f <- fpe$F
g <- fpe$G
h <- fpe$H
j <- fpe$J
k <- fpe$K
a2 <- fpe$A2
b2 <- fpe$B2
n <- fpe$N
poisson_fpe <- glm(
ei
~ a
+ b
+ c
+ d
+ e
+ f
+ g
+ h
+ j
+ k
+ a2
+ b2
+ n,
family = poisson(link = "log"))
summary(poisson_fpe)
##
## Call:
## glm(formula = ei ~ a + b + c + d + e + f + g + h + j + k + a2 +
## b2 + n, family = poisson(link = "log"))
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -5.6115 -1.0870 -0.0518 1.3408 4.1528
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.5477568 0.0528856 85.992 < 2e-16 ***
## a -0.0418217 0.0125528 -3.332 0.000863 ***
## b -0.0005237 0.0099014 -0.053 0.957818
## c 0.0140079 0.0056616 2.474 0.013354 *
## d -0.0371494 0.0092584 -4.013 6.01e-05 ***
## e 0.0072545 0.0193684 0.375 0.707992
## f -0.0222218 0.0089555 -2.481 0.013088 *
## g -0.0002333 0.0242565 -0.010 0.992327
## h -0.0663190 0.0303968 -2.182 0.029126 *
## j 0.0226055 0.0408362 0.554 0.579877
## k -0.1286581 0.0216346 -5.947 2.73e-09 ***
## a2 0.0371011 0.0105182 3.527 0.000420 ***
## b2 0.0012190 0.0080790 0.151 0.880066
## n NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 3389.10 on 23 degrees of freedom
## Residual deviance: 113.21 on 11 degrees of freedom
## AIC: 320.87
##
## Number of Fisher Scoring iterations: 4
WLS_fpe1 <- lm(ei ~., fpe, weight = 1 / n)
WLS_fpe2 <- lm(ei ~., fpe, weight = 1 / (n^2))
summary(WLS_fpe1)
## Warning in summary.lm(WLS_fpe1): essentially perfect fit: summary may be
## unreliable
##
## Call:
## lm(formula = ei ~ ., data = fpe, weights = 1/n)
##
## Weighted Residuals:
## Min 1Q Median 3Q Max
## -7.425e-14 -8.890e-15 -4.160e-16 1.593e-14 3.890e-14
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.205e-13 7.469e-14 -4.292e+00 0.00158 **
## EI 1.000e+00 7.323e-15 1.366e+14 < 2e-16 ***
## A -5.452e-14 2.232e-14 -2.443e+00 0.03469 *
## B 4.307e-14 2.018e-14 2.134e+00 0.05864 .
## C 1.331e-14 1.466e-14 9.080e-01 0.38528
## D -3.773e-14 1.718e-14 -2.196e+00 0.05280 .
## E -7.335e-14 3.924e-14 -1.869e+00 0.09111 .
## F -3.554e-14 1.791e-14 -1.984e+00 0.07533 .
## G -1.533e-13 5.539e-14 -2.767e+00 0.01988 *
## H 1.032e-13 5.991e-14 1.722e+00 0.11574
## J 1.478e-13 7.577e-14 1.950e+00 0.07969 .
## K 4.055e-14 5.636e-14 7.190e-01 0.48837
## A2 5.422e-14 2.256e-14 2.403e+00 0.03709 *
## B2 -2.484e-14 2.187e-14 -1.136e+00 0.28239
## N NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.772e-14 on 10 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 5.44e+30 on 13 and 10 DF, p-value: < 2.2e-16
summary(WLS_fpe2)
## Warning in summary.lm(WLS_fpe2): essentially perfect fit: summary may be
## unreliable
##
## Call:
## lm(formula = ei ~ ., data = fpe, weights = 1/(n^2))
##
## Weighted Residuals:
## Min 1Q Median 3Q Max
## -4.047e-15 -1.257e-15 9.030e-17 1.344e-15 3.752e-15
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -8.229e-14 1.633e-14 -5.039e+00 0.000507 ***
## EI 1.000e+00 2.139e-15 4.675e+14 < 2e-16 ***
## A -1.656e-14 5.917e-15 -2.798e+00 0.018857 *
## B 1.071e-14 4.545e-15 2.357e+00 0.040189 *
## C -1.621e-16 3.959e-15 -4.100e-02 0.968153
## D -1.096e-14 4.548e-15 -2.409e+00 0.036741 *
## E -2.665e-14 1.029e-14 -2.590e+00 0.026969 *
## F -1.177e-14 4.899e-15 -2.404e+00 0.037089 *
## G -5.091e-14 1.459e-14 -3.489e+00 0.005834 **
## H 3.640e-14 1.539e-14 2.365e+00 0.039584 *
## J 4.467e-14 1.737e-14 2.572e+00 0.027798 *
## K 1.240e-14 1.653e-14 7.500e-01 0.470526
## A2 1.748e-14 6.398e-15 2.732e+00 0.021131 *
## B2 -3.700e-15 5.378e-15 -6.880e-01 0.507107
## N NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.007e-15 on 10 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 7.903e+31 on 13 and 10 DF, p-value: < 2.2e-16
compareCoefs(poisson_fpe, WLS_fpe1, WLS_fpe2)
##
## Call:
## 1: glm(formula = ei ~ a + b + c + d + e + f + g + h + j + k + a2 + b2
## + n, family = poisson(link = "log"))
## 2: lm(formula = ei ~ ., data = fpe, weights = 1/n)
## Warning in summary.lm(object): essentially perfect fit: summary may be
## unreliable
##
## 3: lm(formula = ei ~ ., data = fpe, weights = 1/(n^2))
## Warning in summary.lm(object): essentially perfect fit: summary may be
## unreliable
##
## Est. 1 SE 1 Est. 2 SE 2 Est. 3 SE 3
## (Intercept) 4.55e+00 5.29e-02 -3.21e-13 7.47e-14 -8.23e-14 1.63e-14
## a -4.18e-02 1.26e-02
## b -5.24e-04 9.90e-03
## c 1.40e-02 5.66e-03
## d -3.71e-02 9.26e-03
## e 7.25e-03 1.94e-02
## f -2.22e-02 8.96e-03
## g -2.33e-04 2.43e-02
## h -6.63e-02 3.04e-02
## j 2.26e-02 4.08e-02
## k -1.29e-01 2.16e-02
## a2 3.71e-02 1.05e-02
## b2 1.22e-03 8.08e-03
## n
## EI 1.00e+00 7.32e-15 1.00e+00 2.14e-15
## A -5.45e-14 2.23e-14 -1.66e-14 5.92e-15
## B 4.31e-14 2.02e-14 1.07e-14 4.55e-15
## C 1.33e-14 1.47e-14 -1.62e-16 3.96e-15
## D -3.77e-14 1.72e-14 -1.10e-14 4.55e-15
## E -7.34e-14 3.92e-14 -2.67e-14 1.03e-14
## F -3.55e-14 1.79e-14 -1.18e-14 4.90e-15
## G -1.53e-13 5.54e-14 -5.09e-14 1.46e-14
## H 1.03e-13 5.99e-14 3.64e-14 1.54e-14
## J 1.48e-13 7.58e-14 4.47e-14 1.74e-14
## K 4.05e-14 5.64e-14 1.24e-14 1.65e-14
## A2 5.42e-14 2.26e-14 1.75e-14 6.40e-15
## B2 -2.48e-14 2.19e-14 -3.70e-15 5.38e-15
## N