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# Remove most objects from the working environment
rm(list = ls())
options(stringsAsFactors = F)
# 13.2. Logistic regression
# install.packages("AER")
data(Affairs, package = "AER")
head(Affairs)
summary(Affairs)
# affairs gender age yearsmarried children religiousness
# Min. : 0.000 female:315 Min. :17.50 Min. : 0.125 no :171 Min. :1.000
# 1st Qu.: 0.000 male :286 1st Qu.:27.00 1st Qu.: 4.000 yes:430 1st Qu.:2.000
# Median : 0.000 Median :32.00 Median : 7.000 Median :3.000
# Mean : 1.456 Mean :32.49 Mean : 8.178 Mean :3.116
# 3rd Qu.: 0.000 3rd Qu.:37.00 3rd Qu.:15.000 3rd Qu.:4.000
# Max. :12.000 Max. :57.00 Max. :15.000 Max. :5.000
# education occupation rating
# Min. : 9.00 Min. :1.000 Min. :1.000
# 1st Qu.:14.00 1st Qu.:3.000 1st Qu.:3.000
# Median :16.00 Median :5.000 Median :4.000
# Mean :16.17 Mean :4.195 Mean :3.932
# 3rd Qu.:18.00 3rd Qu.:6.000 3rd Qu.:5.000
# Max. :20.00 Max. :7.000 Max. :5.000
table(Affairs$affairs)
# 0 1 2 3 7 12
# 451 34 17 19 42 38
Affairs$ynaffair[Affairs$affairs > 0] <- 1
Affairs$ynaffair[Affairs$affairs == 0] <- 0
Affairs$ynaffair <- factor(Affairs$ynaffair,
levels = c(0, 1),
labels = c("No", "Yes"))
table(Affairs$ynaffair)
# No Yes
# 451 150
colnames(Affairs)
# [1] "affairs" "gender" "age" "yearsmarried" "children"
# [6] "religiousness" "education" "occupation" "rating" "ynaffair"
fit.full <- glm(ynaffair ~ gender + age + yearsmarried + children +
religiousness + education + occupation + rating,
data = Affairs, family = binomial())
summary(fit.full)
# Call:
# glm(formula = ynaffair ~ gender + age + yearsmarried + children +
# religiousness + education + occupation + rating, family = binomial(),
# data = Affairs)
#
# Deviance Residuals:
# Min 1Q Median 3Q Max
# -1.5713 -0.7499 -0.5690 -0.2539 2.5191
#
# Coefficients:
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) 1.37726 0.88776 1.551 0.120807
# gendermale 0.28029 0.23909 1.172 0.241083
# age -0.04426 0.01825 -2.425 0.015301 *
# yearsmarried 0.09477 0.03221 2.942 0.003262 **
# childrenyes 0.39767 0.29151 1.364 0.172508
# religiousness -0.32472 0.08975 -3.618 0.000297 ***
# education 0.02105 0.05051 0.417 0.676851
# occupation 0.03092 0.07178 0.431 0.666630
# rating -0.46845 0.09091 -5.153 2.56e-07 ***
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# (Dispersion parameter for binomial family taken to be 1)
#
# Null deviance: 675.38 on 600 degrees of freedom
# Residual deviance: 609.51 on 592 degrees of freedom
# AIC: 627.51
#
# Number of Fisher Scoring iterations: 4
fit.reduced <- glm(ynaffair ~ age + yearsmarried + religiousness +
rating, data = Affairs, family = binomial())
summary(fit.reduced)
# Call:
# glm(formula = ynaffair ~ age + yearsmarried + religiousness +
# rating, family = binomial(), data = Affairs)
#
# Deviance Residuals:
# Min 1Q Median 3Q Max
# -1.6278 -0.7550 -0.5701 -0.2624 2.3998
#
# Coefficients:
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) 1.93083 0.61032 3.164 0.001558 **
# age -0.03527 0.01736 -2.032 0.042127 *
# yearsmarried 0.10062 0.02921 3.445 0.000571 ***
# religiousness -0.32902 0.08945 -3.678 0.000235 ***
# rating -0.46136 0.08884 -5.193 2.06e-07 ***
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# (Dispersion parameter for binomial family taken to be 1)
#
# Null deviance: 675.38 on 600 degrees of freedom
# Residual deviance: 615.36 on 596 degrees of freedom
# AIC: 625.36
#
# Number of Fisher Scoring iterations: 4
anova(fit.reduced, fit.full, test = "Chisq")
# Analysis of Deviance Table
#
# Model 1: ynaffair ~ age + yearsmarried + religiousness + rating
# Model 2: ynaffair ~ gender + age + yearsmarried + children + religiousness +
# education + occupation + rating
# Resid. Df Resid. Dev Df Deviance Pr(>Chi)
# 1 596 615.36
# 2 592 609.51 4 5.8474 0.2108
#=============================================================================
# 13.2.1. Interpreting the model parameters
coef(fit.reduced)
# (Intercept) age yearsmarried religiousness rating
# 1.93083017 -0.03527112 0.10062274 -0.32902386 -0.46136144
coef(fit.full)
# (Intercept) gendermale age yearsmarried childrenyes religiousness
# 1.37725816 0.28028665 -0.04425502 0.09477302 0.39767213 -0.32472063
# education occupation rating
# 0.02105086 0.03091971 -0.46845426
exp(coef(fit.reduced))
# (Intercept) age yearsmarried religiousness rating
# 6.8952321 0.9653437 1.1058594 0.7196258 0.6304248
exp(confint(fit.reduced))
# 13.2.2. Assessing the impact of predictors on the probability of an outcome
testdata <- data.frame(rating=c(1, 2, 3, 4, 5), age=mean(Affairs$age),
yearsmarried=mean(Affairs$yearsmarried),
religiousness=mean(Affairs$religiousness))
testdata
# rating age yearsmarried religiousness
# 1 1 32.48752 8.177696 3.116473
# 2 2 32.48752 8.177696 3.116473
# 3 3 32.48752 8.177696 3.116473
# 4 4 32.48752 8.177696 3.116473
# 5 5 32.48752 8.177696 3.116473
testdata$prob <- predict(fit.reduced, newdata = testdata, type = "response")
testdata
# rating age yearsmarried religiousness prob
# 1 1 32.48752 8.177696 3.116473 0.5302296
# 2 2 32.48752 8.177696 3.116473 0.4157377
# 3 3 32.48752 8.177696 3.116473 0.3096712
# 4 4 32.48752 8.177696 3.116473 0.2204547
# 5 5 32.48752 8.177696 3.116473 0.1513079
testdata <- data.frame(rating=mean(Affairs$rating),
age=seq(17, 57, 10),
yearsmarried=mean(Affairs$yearsmarried),
religiousness=mean(Affairs$religiousness))
testdata
# rating age yearsmarried religiousness
# 1 3.93178 17 8.177696 3.116473
# 2 3.93178 27 8.177696 3.116473
# 3 3.93178 37 8.177696 3.116473
# 4 3.93178 47 8.177696 3.116473
# 5 3.93178 57 8.177696 3.116473
testdata$prob <- predict(fit.reduced, newdata=testdata, type="response")
testdata
# rating age yearsmarried religiousness prob
# 1 3.93178 17 8.177696 3.116473 0.3350834
# 2 3.93178 27 8.177696 3.116473 0.2615373
# 3 3.93178 37 8.177696 3.116473 0.1992953
# 4 3.93178 47 8.177696 3.116473 0.1488796
# 5 3.93178 57 8.177696 3.116473 0.1094738
# 13.2.3. Overdispersion
deviance(fit.reduced)/df.residual(fit.reduced)
fit <- glm(ynaffair ~ age + yearsmarried + religiousness +
rating, family = binomial(), data = Affairs)
fit.od <- glm(ynaffair ~ age + yearsmarried + religiousness +
rating, family = quasibinomial(), data = Affairs)
pchisq(summary(fit.od)$dispersion * fit$df.residual,
fit$df.residual, lower = F)
# [1] 0.340122
# 13.3. Poisson regression
# install.packages("robust")
data(breslow.dat, package="robust")
colnames(breslow.dat)
# [1] "ID" "Y1" "Y2" "Y3" "Y4" "Base" "Age" "Trt" "Ysum" "sumY" "Age10"
# [12] "Base4"
summary(breslow.dat[c(6:10)])
# Base Age Trt Ysum sumY
# Min. : 6.00 Min. :18.00 placebo :28 Min. : 0.00 Min. : 0.00
# 1st Qu.: 12.00 1st Qu.:23.00 progabide:31 1st Qu.: 11.50 1st Qu.: 11.50
# Median : 22.00 Median :28.00 Median : 16.00 Median : 16.00
# Mean : 31.22 Mean :28.34 Mean : 33.05 Mean : 33.05
# 3rd Qu.: 41.00 3rd Qu.:32.00 3rd Qu.: 36.00 3rd Qu.: 36.00
# Max. :151.00 Max. :42.00 Max. :302.00 Max. :302.00
opar <- par(no.readonly = T)
par(mfrow=c(1, 2))
attach(breslow.dat)
hist(sumY, breaks = 20,
xlab = "Seizure Count",
main = "Distribution of Seizures")
boxplot(sumY ~ Trt, xlab = "Treatment", main = "Group Comparison")
par(opar)
fit <- glm(sumY ~ Base + Age + Trt, data=breslow.dat, family=poisson())
summary(fit)
# Call:
# glm(formula = sumY ~ Base + Age + Trt, family = poisson(), data = breslow.dat)
#
# Deviance Residuals:
# Min 1Q Median 3Q Max
# -6.0569 -2.0433 -0.9397 0.7929 11.0061
#
# Coefficients:
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) 1.9488259 0.1356191 14.370 < 2e-16 ***
# Base 0.0226517 0.0005093 44.476 < 2e-16 ***
# Age 0.0227401 0.0040240 5.651 1.59e-08 ***
# Trtprogabide -0.1527009 0.0478051 -3.194 0.0014 **
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# (Dispersion parameter for poisson family taken to be 1)
#
# Null deviance: 2122.73 on 58 degrees of freedom
# Residual deviance: 559.44 on 55 degrees of freedom
# AIC: 850.71
#
# Number of Fisher Scoring iterations: 5
#=======================================================
#13.3.1. Interpreting the model parameters
coef(fit)
# (Intercept) Base Age Trtprogabide
# 1.94882593 0.02265174 0.02274013 -0.15270095
summary(fit)
exp(coef(fit))
# (Intercept) Base Age Trtprogabide
# 7.0204403 1.0229102 1.0230007 0.8583864
#===================================================
# 13.3.2. Overdispersion
deviance(fit)/df.residual(fit)
# [1] 10.1717
# install.packages("qcc")
library(qcc)
qcc.overdispersion.test(breslow.dat$sumY, type = "poisson")
# Overdispersion test Obs.Var/Theor.Var Statistic p-value
# poisson data 62.87013 3646.468 0
fit.od <- glm(sumY ~ Base + Age + Trt, data=breslow.dat,
family=quasipoisson())
summary(fit.od)
# Call:
# glm(formula = sumY ~ Base + Age + Trt, family = quasipoisson(),
# data = breslow.dat)
#
# Deviance Residuals:
# Min 1Q Median 3Q Max
# -6.0569 -2.0433 -0.9397 0.7929 11.0061
#
# Coefficients:
# Estimate Std. Error t value Pr(>|t|)
# (Intercept) 1.948826 0.465091 4.190 0.000102 ***
# Base 0.022652 0.001747 12.969 < 2e-16 ***
# Age 0.022740 0.013800 1.648 0.105085
# Trtprogabide -0.152701 0.163943 -0.931 0.355702
# ---
# Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# (Dispersion parameter for quasipoisson family taken to be 11.76075)
#
# Null deviance: 2122.73 on 58 degrees of freedom
# Residual deviance: 559.44 on 55 degrees of freedom
# AIC: NA
#
# Number of Fisher Scoring iterations: 5
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