Directions (read before starting)

  1. Please work together with your assigned partner. Make sure you both fully understand something before moving on.
  2. Record your answers to lab questions separately from the lab’s examples. You and your partner should only turn in responses to lab questions, nothing more and nothing less.
  3. Ask for help, clarification, or even just a check-in if anything seems unclear.

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Lab

At this point you should begin working independently with your assigned partner(s).

Death Penalty Sentencing

A widely cited study, published in 1981, analyzed data on all murders which took place during a felony that were committed in the state of Florida between 1972 and 1977. The study recorded numerous attributes pertaining to each of these murders, with the outcome of interest being whether the offender was sentenced to the death penalty. The researchers were interested in investigating potential racial bias in death penalty sentencing.

Data from this study are available at this URL: https://remiller1450.github.io/data/DeathPenaltySentencing.csv

These data contain the following variables:

  • OffenderRace - Whether the offender tried in the case identified as White or Black
  • VictimRace - Whether the murder victim in the case was identified White or Black
  • DeathPenalty - Whether or not the person tried in the case was sentenced to receive the death penalty

Question 1: This question explores whether the explanatory variable OffenderRace is associated with the response variable DeathPenalty.

  • Part A: Construct a two-way frequency table where the rows are values of OffenderRace and the columns are the values of DeathPenalty.
  • Part B: Report the proportion of White offenders who received a death penalty sentence as well as the proportion of Black offenders who received a death penalty sentence.
  • Part C: Report the odds ratio that compares the odds of a White offender receiving a death penalty sentence with the odds of a Black offender receiving a death penalty sentence.
  • Part D: Based upon your answers to Parts A - C, do these data provide compelling evidence of racially biased death penalty sentencing in Florida during the 1970s?

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Stratification

These data contain a third variable, the race of the victim. For reasons that will soon be apparent, it common for statisticians to analyze subgroups of cases, or segments of the data created by conditioning on a variable that is not directly involved in the primary research question.

The technique of splitting up a data set according to a variable that is neither the explanatory nor response variable is called stratification. After stratification, conditional analyses are performed within each stratum.

For the death penalty sentencing data, we can perform a stratified analysis using the third variable VictimRace by using the filter() function (from the dplyr package) before repeating the steps performed in Question 1.

Question 2: In this question you’ll perform a stratified analysis on the subgroups created by the variable VictimRace.

  • Part A: Use the filter() function in the dplyr package to create a subset of data that contains only the cases involving a White victim
  • Part B: Using the results of Part A, construct a two-way frequency table where the rows are values of OffenderRace and the columns are the values of DeathPenalty.
  • Part C: Using the table created in Part B, report the proportion of White offenders who received a death penalty sentence as well as the proportion of Black offenders who received a death penalty sentence.
  • Part D: Using the table created in Part B, report the conditional odds ratio for cases involving a White victim that compares the odds of a Black offender receiving a death penalty sentence to the odds of a White offender receiving a death penalty sentence.
  • Part E: Now, use the filter() function to create a subset of data containing only cases involving a Black victim. Use this data to create a two-way frequency table where the rows are values of OffenderRace and the columns are the values of DeathPenalty.
  • Part F: Based upon your answers to Parts A - E, do these data appear to provide compelling evidence of racially biased death penalty sentencing in Florida during the 1970s?

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Confounding Variables

In this study, “VictimRace” is a confounding variable, or a third variable that is associated with both the explanatory and response variables in the primary analysis. A consequence of confounding is that the real relationship between the explanatory and response variables can be obscured.

Stratification can neutralize the impact of a confounding variable, as every case belonging to a particular stratum has identical values of the confounding variable. Thus, what was a confounding variable no longer meets the definition of confounding within a stratum because it is no longer associated with variable involved in the primary analysis.

Question 3: This question explores why the variable VictimRace confounds the relationship between OffenderRace and DeathPenalty.

  • Part A: Using the full data set, create a bar chart that shows the proportion of cases in each category of DeathPenalty for each value of VictimRace (this is a conditional bar chart). Provide a brief interpretation of the relationship you see in this bar chart.
  • Part B: Using the full data set, create a bar chart that shows the proportion of cases in each category of OffenderRace for each value of VictimRace (this is a conditional bar chart). Provide a brief interpretation of the relationship you see in this bar chart.
  • Part C: The definition of confounding requires a third variable to be associated with both the explanatory and response variables in an analysis. Based upon your answers to Parts A and B, does VictimRace satisfy the definition of confounding?
  • Part D: Briefly explain how it’s possible that when the data are analyzed all together the death penalty sentencing rate was higher for White offenders, but when the data were stratified by the race of the victim it was Black offenders that had a higher death penalty sentencing rate in both strata.