case1902: Death Penalty and Race

In Sleuth3: Data Sets from Ramsey and Schafer's "Statistical Sleuth (3rd Ed)"

Description

Lawyers collected data on convicted black murderers in the state of Georgia to see whether convicted black murderers whose victim was white were more likely to receive the death penalty than those whose victim was black, after accounting for aggravation level of the murder. They categorized murders into 6 progressively more serious types. Category 1 comprises barroom brawls, liquor-induced arguments lovers' quarrels, and similar crimes. Category 6 includes the most vicious, cruel, cold=blooded, unprovoked crimes.

Usage

case1902

Format

A data frame with 12 observations on the following 4 variables.

Aggravation

the aggravation level of the crime, a numerical variable ranging from 1 to 6

Victim

a factor indicating race of murder victim, with levels "White" and "Black"

Death

number in the aggravation and victim category who received the death penalty

NoDeath

number in the aggravation and victim category who did not receive the death penalty

Source

Ramsey, F.L. and Schafer, D.W. (2013). The Statistical Sleuth: A Course in Methods of Data Analysis (3rd ed), Cengage Learning.

References

Woodworth, G.C. (1989). Statistics and the Death Penalty, Stats 2: 9–12.

Examples

str(case1902)
attach(case1902)
        
## EXPLORATION
proportionDeath  <- Death/(Death + NoDeath)
myPointCode <- ifelse(Victim=="White",22,24)
myPointColor <- ifelse(Victim=="White","white","black")
plot(proportionDeath ~ Aggravation, pch=myPointCode, bg=myPointColor)
oddsOfDeath <- Death/(NoDeath + .5)  # Add .5 to the demoninator to avoid 0's 
plot(oddsOfDeath ~ Aggravation, pch=myPointCode, bg=myPointColor)
plot(oddsOfDeath ~ Aggravation, log="y", pch=myPointCode, bg=myPointColor)
# Use logistic regression (Ch 21) to see if the 6 odds ratios are constant
myGlm1 <- glm(cbind(Death,NoDeath) ~ Aggravation + Victim + 
  Aggravation:Victim, family=binomial) # Logistic reg with interaction
myGlm2  <- update(myGlm1, ~ . - Aggravation:Victim) # without interaction
anova(myGlm2, myGlm1) # no evidence of interaction.
                                              

## INFERENCE
# Mantel Haenszel
myTable <- array(rbind(Death, NoDeath), dim=c(2,2,6),
  dimnames=list(Penalty=c("Death","No Death"), Victim=c("White","Black"),
  Aggravation=c("1","2","3","4","5","6")))
myTable   # Show the 6 2x2 tables
mantelhaen.test(myTable, alternative="greater", correct=FALSE) # 1-sided p-value
mantelhaen.test(myTable, alternative="greater") # with continuity correction
mantelhaen.test(myTable) # two.sided (default) for confidence interval

# Logistic Regression (Ch 21) (treating aggravation level as numerical)
summary(myGlm2)
beta  <- myGlm2$coef
exp(beta[3])   #   6.1144 
exp(confint(myGlm2,3)) # 2.23040 18.72693
# Interpretation: The odds of death penalty for white victim murderers are 
# estimated to be 6 times the odds of death penalty for black victim murderers 
# with similar aggravation level(95% confidence interval: 2.2 to 18.7 times).

 
## GRAPHICAL DISPLAY FOR PRESENTATION 
myPointColor     <- ifelse(Victim=="White","green", "orange")
plot(jitter(proportionDeath,.1) ~ jitter(Aggravation,.1), 
  xlab="Aggravation Level of the Murder", 
  ylab="Proportion of Murderers Who Received Death Penalty",
  pch=myPointCode, bg=myPointColor, cex=2, lwd=2)
legend(1,1, c("White Victim Murderers","Black Victim Murderers"), pch=c(21,22),
  pt.cex=c(2,2), pt.bg=c("green","orange"), pt.lw=c(2,2))
# Include logistic regression fit on plot
dummyAg <- seq(min(Aggravation),max(Aggravation),length=50)
etaB    <- beta[1] + beta[2]*dummyAg
etaW    <- etaB + beta[3]
pB      <- exp(etaB)/(1 + exp(etaB))  # Estimated prob of DP; Black victim 
pW      <- exp(etaW)/(1 + exp(etaW))  # Estimated prob of DP; White victim
lines(pB ~ dummyAg,lty=1)   
lines(pW ~ dummyAg,lty=2)              

detach(case1902)

Example output

'data.frame':	12 obs. of  4 variables:
 $ Aggravation: int  1 1 2 2 3 3 4 4 5 5 ...
 $ Victim     : Factor w/ 2 levels "Black","White": 2 1 2 1 2 1 2 1 2 1 ...
 $ Death      : int  2 1 2 1 6 2 9 2 9 4 ...
 $ NoDeath    : int  60 181 15 21 7 9 3 4 0 3 ...
Analysis of Deviance Table

Model 1: cbind(Death, NoDeath) ~ Aggravation + Victim
Model 2: cbind(Death, NoDeath) ~ Aggravation + Victim + Aggravation:Victim
  Resid. Df Resid. Dev Df Deviance
1         9     3.8816            
2         8     3.3438  1  0.53778
, , Aggravation = 1

          Victim
Penalty    White Black
  Death        2     1
  No Death    60   181

, , Aggravation = 2

          Victim
Penalty    White Black
  Death        2     1
  No Death    15    21

, , Aggravation = 3

          Victim
Penalty    White Black
  Death        6     2
  No Death     7     9

, , Aggravation = 4

          Victim
Penalty    White Black
  Death        9     2
  No Death     3     4

, , Aggravation = 5

          Victim
Penalty    White Black
  Death        9     4
  No Death     0     3

, , Aggravation = 6

          Victim
Penalty    White Black
  Death       17     4
  No Death     0     0


	Mantel-Haenszel chi-squared test without continuity correction

data:  myTable
Mantel-Haenszel X-squared = 11.26, df = 1, p-value = 0.000396
alternative hypothesis: true common odds ratio is greater than 1
95 percent confidence interval:
 2.264218      Inf
sample estimates:
common odds ratio 
          5.49258 


	Mantel-Haenszel chi-squared test with continuity correction

data:  myTable
Mantel-Haenszel X-squared = 9.6983, df = 1, p-value = 0.0009222
alternative hypothesis: true common odds ratio is greater than 1
95 percent confidence interval:
 2.264218      Inf
sample estimates:
common odds ratio 
          5.49258 


	Mantel-Haenszel chi-squared test with continuity correction

data:  myTable
Mantel-Haenszel X-squared = 9.6983, df = 1, p-value = 0.001844
alternative hypothesis: true common odds ratio is not equal to 1
95 percent confidence interval:
  1.910687 15.789312
sample estimates:
common odds ratio 
          5.49258 


Call:
glm(formula = cbind(Death, NoDeath) ~ Aggravation + Victim, family = binomial)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.93570  -0.22548   0.05142   0.65620   1.01444  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  -6.6760     0.7574  -8.814  < 2e-16 ***
Aggravation   1.5397     0.1867   8.246  < 2e-16 ***
VictimWhite   1.8106     0.5361   3.377 0.000732 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 212.2838  on 11  degrees of freedom
Residual deviance:   3.8816  on  9  degrees of freedom
AIC: 31.747

Number of Fisher Scoring iterations: 4

VictimWhite 
     6.1144 
Waiting for profiling to be done...
   2.5 %   97.5 % 
 2.23040 18.72693 

Sleuth3 documentation built on May 2, 2019, 6:41 a.m.