In this exercise you will:
Agresti (2007) presented an example on the counts of murderers given the death penalty in the state of Florida. The data can be used to judge the level of racial prejudice that might exist in the justice system as the race of the murderer and their victims are noted. Use a log-linear model to determine the appropriate model for this scenario and draw a conclusion on the possible prejudice that may exist. Do your results match those that could be found using a logistic regression approach?
The data can be obtained using:
data(DeathPenalty, package="ELMER")
str(DeathPenalty)'data.frame': 4 obs. of 4 variables:
$ Victim : Factor w/ 2 levels "Black","White": 2 2 1 1
$ Defendant: Factor w/ 2 levels "Black","White": 2 1 2 1
$ Yes : int 53 11 0 4
$ No : int 414 37 16 139We need to reshape the data using:
library(tidyr)
DeathPenalty2 = DeathPenalty |> pivot_longer(Yes:No, names_to="DP", values_to="Count") |> glimpse()Rows: 8
Columns: 4
$ Victim <fct> White, White, White, White, Black, Black, Black, Black
$ Defendant <fct> White, White, Black, Black, White, White, Black, Black
$ DP <chr> "Yes", "No", "Yes", "No", "Yes", "No", "Yes", "No"
$ Count <int> 53, 414, 11, 37, 0, 16, 4, 139before fitting the log-linear model.
DP2.glm <- glm(Count~Victim*Defendant*DP, data=DeathPenalty2, family=poisson())
DP2.glm |> summary()
Call:
glm(formula = Count ~ Victim * Defendant * DP, family = poisson(),
data = DeathPenalty2)
Deviance Residuals:
[1] 0 0 0 0 0 0 0 0
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 4.934e+00 8.482e-02 58.177 < 2e-16 ***
VictimWhite -1.324e+00 1.850e-01 -7.155 8.38e-13 ***
DefendantWhite -2.162e+00 2.640e-01 -8.189 2.63e-16 ***
DPYes -3.548e+00 5.071e-01 -6.996 2.63e-12 ***
VictimWhite:DefendantWhite 4.577e+00 3.149e-01 14.536 < 2e-16 ***
VictimWhite:DPYes 2.335e+00 6.125e-01 3.813 0.000137 ***
DefendantWhite:DPYes -2.153e+01 4.225e+04 -0.001 0.999593
VictimWhite:DefendantWhite:DPYes 2.068e+01 4.225e+04 0.000 0.999609
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 1.2251e+03 on 7 degrees of freedom
Residual deviance: 4.1226e-10 on 0 degrees of freedom
AIC: 54.04
Number of Fisher Scoring iterations: 20DP2.glm|> anova(test="Chisq")Analysis of Deviance Table
Model: poisson, link: log
Response: Count
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev Pr(>Chi)
NULL 7 1225.08
Victim 1 197.93 6 1027.15 < 2.2e-16 ***
Defendant 1 130.79 5 896.36 < 2.2e-16 ***
DP 1 493.52 4 402.84 < 2.2e-16 ***
Victim:Defendant 1 380.57 3 22.27 < 2.2e-16 ***
Victim:DP 1 16.87 2 5.39 3.999e-05 ***
Defendant:DP 1 5.01 1 0.38 0.02514 *
Victim:Defendant:DP 1 0.38 0 0.00 0.53769
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Use the original data for the logistic regression model.
DP.glm <- glm(cbind(Yes,No)~Victim*Defendant, data=DeathPenalty, family=binomial())
DP.glm |> summary()
Call:
glm(formula = cbind(Yes, No) ~ Victim * Defendant, family = binomial(),
data = DeathPenalty)
Deviance Residuals:
[1] 0 0 0 0
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.5482 0.5071 -6.996 2.63e-12 ***
VictimWhite 2.3352 0.6125 3.813 0.000137 ***
DefendantWhite -21.9957 53403.2302 0.000 0.999671
VictimWhite:DefendantWhite 21.1531 53403.2302 0.000 0.999684
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 2.2266e+01 on 3 degrees of freedom
Residual deviance: 2.5803e-10 on 0 degrees of freedom
AIC: 20.92
Number of Fisher Scoring iterations: 22DP.glm|> anova(test="Chisq")Analysis of Deviance Table
Model: binomial, link: logit
Response: cbind(Yes, No)
Terms added sequentially (first to last)
Df Deviance Resid. Df Resid. Dev Pr(>Chi)
NULL 3 22.2659
Victim 1 16.8719 2 5.3940 3.999e-05 ***
Defendant 1 5.0142 1 0.3798 0.02514 *
Victim:Defendant 1 0.3798 0 0.0000 0.53769
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1