9.3 Two-way Tables and Simpson’s Paradox

Learning Objectives

  • Review risks, relative risks and conditional risks and relative risks
  • Illustrate Simpson’s Paradox

Useful Functions

  • Use prop.table() to calculate proportions/risks and conditional proportions/risks.
  • Use margin.table() to calculate table margins.

Dataset: Race and Capital Punishment

An influential 1991 study by Radelet and Pierce analyzed data on the association of race and sentence outcomes in capital-punishment (death sentence) cases in Florida, summarized in the table table below (from Alan Agresti). Note: the table may be slightly modified to change the numerical answers for the lab each year, but the central findings are unchanged.

At first glance, their data suggested white defendants are more likely to receive a sentence of capital punishment than black defendants. But they noted that the race of the victim was an important additional variable, and that once this variable was included the evidence actually showed clear and consistent racial disparities in capital punishment. So, is there racial discrimination in sentencing outcomes? And if so, what is the direction of the association?

To better understand how people can use the same data to support arguments on both sides of this controversy, we will analyze the data in this Lab.

First, we’ll need to key in the data for the 2 x 2 x 2 table. The structure of the table will be:

  • rows = race of defendant
  • columns = sentence
  • stratification variable = race of victim
vic.white <- matrix(c(57, 412, 14,  33), nrow=2, byrow=TRUE) # table for victim=white
vic.black <- matrix(c( 0,  18,  6, 144), nrow=2, byrow=TRUE) # table for victim=black
vic.all <- vic.white + vic.black # combined table

rownames(vic.white) <- rownames(vic.black) <- rownames(vic.all) <- c("White Defendant","Black Defendant")
colnames(vic.white) <- colnames(vic.black) <- colnames(vic.all) <- c("Capital Punishment","Other")

Let’s start by looking at the data for the combined table.

print(vic.all)
##                 Capital Punishment Other
## White Defendant                 57   430
## Black Defendant                 20   177

We see that, for example, the study observes 57 white defendants who were sentenced to capital punishment and 177 black defendants who were sentenced to non-capital punishment.

Working with the Combined Data

Round all numeric answers to 2 decimal places unless otherwise noted.

  1. What is the marginal risk of a capital punishment sentence for black defendants?

  2. What is the marginal risk of a capital punishment sentence for white defendants?

  3. What is the relative risk of a capital punishment sentence for black defendants compared to white defendants?

  4. Looking at the evidence above, which of the statements below is supported by the findings?

    • The defendant’s race does not influence the probability of a capital punishment sentence.
    • Black defendants are 13% less likely to receive a capital punishment sentence.
    • Black defendants are 87% less likely to receive a capital punishment sentence.

Conditioning on the Victim’s Race

  1. Conditioning on the victim being white, what is the risk of a capital punishment sentence?
# This question uses the table for white victims
print(vic.white)
##                 Capital Punishment Other
## White Defendant                 57   412
## Black Defendant                 14    33
  1. Conditioning on the victim being black, what is the risk of a capital punishment sentence?
# This question uses the table for black victims
print(vic.black)
##                 Capital Punishment Other
## White Defendant                  0    18
## Black Defendant                  6   144

  1. What is the relative risk of a capital punishment sentence, when the victim is white vs. when the victim is black?

  2. What is the interpretation of this relative risk?

    • When the victim is white, almost half of defendants get the death penalty.
    • Capital punishment is almost 4 times more likely when the victim is white.
    • Capital punishment is almost 4 times more likely when the defendant is white.
    • Capital punishment is almost 4 times more likely when both the victim and the defendant are white.

Comparing Relative Risks by the Defendant’s Race

  1. Conditioning on the victim being white, what is the relative risk of a capital punishment sentence for black defendants vs. white defendants?

  2. What is the interpretation of this relative risk?

    • When the victim is white, half of defendants get the death penalty.
    • A black defendant is over twice as likely to get the death penalty as a white defendant.
    • A black defendant is over twice as likely to get the death penalty as a white defendant, when the victim is black
    • A black defendant is over twice as likely to get the death penalty as a white defendant, when the victim is white.

  3. Conditioning on the victim being black, what is the relative risk of a capital punishment sentence for black defendants vs. white defendants?

  4. Are black defendants more or less likely (than white defendants) to get a sentence of capital punishment when the vicitim is black?

    • More likely
    • Less likely
    • It is impossible to tell with these data

  5. Is the evidence consistent with the claim that there are racial disparities in capital punishment sentencing decisions?

    • Yes
    • No

  6. Does the evidence prove there is racial discrimination in capital punishment sentencing decisions?

    • Yes
    • No