what types of data are collected?

What Types Of Data Are Collected?

What Kinds Of Question Can Be

Asked Of Those Data?

Do people who say they study for more hours also think they’ll finish their doctorate earlier?

Are computer literates less anxious about statistics?

…. ?

Are men more likely to study part-time?

Are women more likely to enroll in CCE?

…. ?

Questions that Require Us To

Examine Relationships

Between Features of the

Participants.

How tall are class members, on average?

How many hours a week do class members report that they study?

…. ?

How many members of the class are women?

What proportion of the class is fulltime?

…. ?

Questions That Require Us To

DescribeSingle Features

of the Participants

“Continuous”

Data

“Categorical”

Data

Research Is A Partnership Of

Questions And Data

Research Is A Partnership Of

Questions And Data

© Willett, Harvard University Graduate School of Education, 04/21/23 S010Y/C06 – Slide 1

S010Y: Answering Questions with Quantitative Data Class 6/II.3: Examining More Complex Relationships Among Categorical Variables

S010Y: Answering Questions with Quantitative Data Class 6/II.3: Examining More Complex Relationships Among Categorical Variables


S010Y: Answering Questions with Quantitative DataClass 6/II.3: Examining More Complex Relationships Among Categorical Variables

S010Y: Answering Questions with Quantitative DataClass 6/II.3: Examining More Complex Relationships Among Categorical Variables

Recall that there is an additional variable in the DEATHPEN.txt dataset – the Race of

the Defendant, RDEFEND, :

Recall that there is an additional variable in the DEATHPEN.txt dataset – the Race of

the Defendant, RDEFEND, :

0 1 10 1 10 1 10 1 10 1 1.

(2475 cases)

.1 2 21 2 21 2 2

0 1 10 1 10 1 10 1 10 1 1.

(2475 cases)

.1 2 21 2 21 2 2

How can we incorporate this additional variable into the contingency table analyses, to support

and extend our original analyses?

How can we incorporate this additional variable into the contingency table analyses, to support

and extend our original analyses?

One straightforward approach is to simply replicate the original contingency table analysisreplicate the original contingency table analysis in interesting “slices of the sampleslices of the sample” defined by

the new “third” variable.

One straightforward approach is to simply replicate the original contingency table analysisreplicate the original contingency table analysis in interesting “slices of the sampleslices of the sample” defined by

the new “third” variable.

Today’s class!Today’s class!


S010Y : Answering Questions with Quantitative DataClass 6/II.3: Examining More Complex Relationships Among Categorical Variables

S010Y : Answering Questions with Quantitative DataClass 6/II.3: Examining More Complex Relationships Among Categorical Variables

Original sample of all defendants, in Georgia:Are DEATH and RVICTIM related?Are DEATH and RVICTIM related?

Original sample of all defendants, in Georgia:Are DEATH and RVICTIM related?Are DEATH and RVICTIM related?

Sub-sample of Black defendants:Are DEATH and RVICTIM related?Are DEATH and RVICTIM related?

Sub-sample of Black defendants:Are DEATH and RVICTIM related?Are DEATH and RVICTIM related?

Sub-sample of White defendants:Are DEATH and RVICTIM related?Are DEATH and RVICTIM related?

Sub-sample of White defendants:Are DEATH and RVICTIM related?Are DEATH and RVICTIM related?

Then, inspect and compare answers across Then, inspect and compare answers across sub-samplessub-samples Would these supplementary analyses be able to further confirm

a theory of racism in death penalty allocation? What would we expect to find in the comparison across the

supplementary analyses, if a racism theory were appropriate?

Then, inspect and compare answers across Then, inspect and compare answers across sub-samplessub-samples Would these supplementary analyses be able to further confirm

a theory of racism in death penalty allocation? What would we expect to find in the comparison across the

supplementary analyses, if a racism theory were appropriate?


S010Y : Answering Questions with Quantitative Data Class 6/II.3: Examining More Complex Relationships Among Categorical Variables


The new “sliced into sub-samples” contingency table analysis is carried out in today’s Data-Analytic Handout … here’s the usual beginning of the PC-SAS program …The new “sliced into sub-samples” contingency table analysis is carried out in today’s Data-Analytic Handout … here’s the usual beginning of the PC-SAS program …

OPTIONS Nodate Pageno=1; TITLE1 ‘S010Y: Answering Questions with Quantitative Data';TITLE2 'Class 6/Handout 1: Examining More Complex Relationships Among Categorical Variables';TITLE3 'Death Penalty and Victim Race, Accounting for Defendant Race';TITLE4 'Data in DEATHPEN.txt'; *--------------------------------------------------------------------------------*Input data, name and label variables in dataset*--------------------------------------------------------------------------------*; DATA DEATHPEN; INFILE 'C:\DATA\A010Y\DEATHPEN.txt'; INPUT DEATH RDEFEND RVICTIM; LABEL DEATH = 'Sentenced to death?' RDEFEND = 'Race of defendant' RVICTIM = 'Race of victim'; *--------------------------------------------------------------------------------*Format labels for values of categorical variables*--------------------------------------------------------------------------------*; PROC FORMAT; VALUE DFMT 0 = 'No' 1 = 'Yes'; VALUE RFMT 1 = 'Black‘ 2 = 'White';

OPTIONS Nodate Pageno=1; TITLE1 ‘S010Y: Answering Questions with Quantitative Data';TITLE2 'Class 6/Handout 1: Examining More Complex Relationships Among Categorical Variables';TITLE3 'Death Penalty and Victim Race, Accounting for Defendant Race';TITLE4 'Data in DEATHPEN.txt'; *--------------------------------------------------------------------------------*Input data, name and label variables in dataset*--------------------------------------------------------------------------------*; DATA DEATHPEN; INFILE 'C:\DATA\A010Y\DEATHPEN.txt'; INPUT DEATH RDEFEND RVICTIM; LABEL DEATH = 'Sentenced to death?' RDEFEND = 'Race of defendant' RVICTIM = 'Race of victim'; *--------------------------------------------------------------------------------*Format labels for values of categorical variables*--------------------------------------------------------------------------------*; PROC FORMAT; VALUE DFMT 0 = 'No' 1 = 'Yes'; VALUE RFMT 1 = 'Black‘ 2 = 'White';

In the options, titling, data-input and formatting parts of the PC-SAS program there is nothing new

– any questions?

In the options, titling, data-input and formatting parts of the PC-SAS program there is nothing new

– any questions?




*---------------------------------------------------------------------------*Carrying out the contingency table analyses separately by defendant race*---------------------------------------------------------------------------*;PROC SORT DATA=DEATHPEN; BY RDEFEND; PROC CHART DATA=DEATHPEN; TITLE5 'Displaying the Relationship Between DEATH & RVICTIM, by Defendant Race'; FORMAT DEATH DFMT. RVICTIM RDEFEND RFMT.; BLOCK RVICTIM / GROUP=DEATH DISCRETE; BY RDEFEND; PROC FREQ DATA=DEATHPEN; TITLE5 'Summarizing the Relationship Between DEATH & RVICTIM, by Defendant Race'; FORMAT DEATH DFMT. RVICTIM RDEFEND RFMT.; TABLES DEATH*RVICTIM / EXPECTED DEVIATION CHISQ CELLCHI2 NOROW; BY RDEFEND;RUN;

*---------------------------------------------------------------------------*Carrying out the contingency table analyses separately by defendant race*---------------------------------------------------------------------------*;PROC SORT DATA=DEATHPEN; BY RDEFEND; PROC CHART DATA=DEATHPEN; TITLE5 'Displaying the Relationship Between DEATH & RVICTIM, by Defendant Race'; FORMAT DEATH DFMT. RVICTIM RDEFEND RFMT.; BLOCK RVICTIM / GROUP=DEATH DISCRETE; BY RDEFEND; PROC FREQ DATA=DEATHPEN; TITLE5 'Summarizing the Relationship Between DEATH & RVICTIM, by Defendant Race'; FORMAT DEATH DFMT. RVICTIM RDEFEND RFMT.; TABLES DEATH*RVICTIM / EXPECTED DEVIATION CHISQ CELLCHI2 NOROW; BY RDEFEND;RUN;

Here’s the part that does the new “sliced into sub-samples” analyses ….Here’s the part that does the new “sliced into sub-samples” analyses ….

Before you can conduct separateseparate analyses by the values of by the values of a third variablea third variable, like RDEFEND, you have to sort the data

by that variable, using the PROC SORT procedure.

Notice how I have used the “race format” definition, RFMT, to provide value labels for both RVICTIM and

RDEFEND.

Notice how I have used the “race format” definition, RFMT, to provide value labels for both RVICTIM and

RDEFEND.

Any procedure in PC-SAS can be carried out in “slices” defined by the values of

another variable, like RDEFEND, or even in slices defined by the values of several

variables at the same time.




Black MurderersBlack Victims

When a Black victim is killed by a Black murderer,

in Georgia

18/(1420+18) or 1.25%

of the murderers are sentenced to death.

Black MurderersWhite Victims

When a White victim is killed by a Black murderer,

in Georgia

50/(178+50) or 21.93%


Descriptive Analysis of Cases with Descriptive Analysis of Cases with Black Black MurderersMurderers, in Georgia, in Georgia

Descriptive Analysis of Cases with Descriptive Analysis of Cases with Black Black MurderersMurderers, in Georgia, in Georgia

“The percentage of Black murderers sentenced to death for killing a White

victim is about seventeen and half times the percentage of Black murderers

sentenced to death for killing a Black victim, in Georgia!!”




Contingency Table Analyses of Cases with Contingency Table Analyses of Cases with Black MurderersBlack Murderers, in Georgia, in Georgia

Contingency Table Analyses of Cases with Contingency Table Analyses of Cases with Black MurderersBlack Murderers, in Georgia, in Georgia

HH00:: DEATH & RVICTIM are not DEATH & RVICTIM are not

related, in the population of related, in the population of Black murderers in Georgia.Black murderers in Georgia.

PearsonPearson χχ22 statistic: statistic: 214.93214.93

p-value:p-value: <.0001 <.0001 (highly unlikely could’ve gotten a χ2 statistic as large as 214.93, or larger, by an accident of sampling from a null population).

DecisionDecision:: Reject H0Reject H0

ConclusionConclusion:: There is a statistically significant There is a statistically significant relationship between the relationship between the allocation of the death penalty and allocation of the death penalty and the race of the victim, in the the race of the victim, in the population of Black murderers in population of Black murderers in Georgia (p Georgia (p < < .0001)..0001).


related, in the population of related, in the population of Black murderers in Georgia.Black murderers in Georgia.


p-value:p-value: <.0001 <.0001 (highly unlikely could’ve gotten a χ2 statistic as large as 214.93, or larger, by an accident of sampling from a null population).

DecisionDecision:: Reject H0Reject H0

ConclusionConclusion:: There is a statistically significant There is a statistically significant relationship between the relationship between the allocation of the death penalty and allocation of the death penalty and the race of the victim, in the the race of the victim, in the population of Black murderers in population of Black murderers in Georgia (p Georgia (p < < .0001)..0001).




When the Murderer is Black…”When the Murderer is Black…”The largest contribution to the χ2 statistic is

made when White victims are killed and a Black murderer has been sentenced to death (cell contribution to χ2 statistic=177.95). In this cell, if there were no relationship

between DEATH & RVICTIM, we would expect just over 9 death penalties. However, 50 death penalties were handed down.

The next largest contribution to the χ2 statistic is made when Black victims are killed and a Black murderer has been sentenced to death (cell contribution to χ2 statistic=28.21). In this cell, if there were no relationship

between DEATH & RVICTIM, we would expect around 59 death penalties. However, only 18 were given.

When the Murderer is Black…”When the Murderer is Black…”The largest contribution to the χ2 statistic is

made when White victims are killed and a Black murderer has been sentenced to death (cell contribution to χ2 statistic=177.95). In this cell, if there were no relationship

between DEATH & RVICTIM, we would expect just over 9 death penalties. However, 50 death penalties were handed down.

The next largest contribution to the χ2 statistic is made when Black victims are killed and a Black murderer has been sentenced to death (cell contribution to χ2 statistic=28.21). In this cell, if there were no relationship

between DEATH & RVICTIM, we would expect around 59 death penalties. However, only 18 were given.

Contingency Table Analyses of the Cases Contingency Table Analyses of the Cases with with Black MurderersBlack Murderers, in Georgia, in Georgia

Contingency Table Analyses of the Cases Contingency Table Analyses of the Cases with with Black MurderersBlack Murderers, in Georgia, in Georgia




Descriptive Analysis of Cases with Descriptive Analysis of Cases with White MurderersWhite Murderers, in Georgia, in Georgia

Descriptive Analysis of Cases with Descriptive Analysis of Cases with White MurderersWhite Murderers, in Georgia, in Georgia

White MurderersBlack Victims

When a Black victim is killed by a White murderer

in Georgia,

2/(62+2) or 3.13%


White MurderersWhite Victims

When a White victim is killed by a White murderer,

in Georgia

58/(687+58) or 7.79%


“The percentage of White murderers sentenced to death for killing a White victim is about two and half times the

percentage of White murderers sentenced to death for killing a Black

victim, in Georgia!!”




Contingency Table Analyses of the Cases Contingency Table Analyses of the Cases with with White MurderersWhite Murderers, in Georgia, in Georgia

Contingency Table Analyses of the Cases Contingency Table Analyses of the Cases with with White MurderersWhite Murderers, in Georgia, in Georgia


related, in the population of White related, in the population of White murderers in Georgia.murderers in Georgia.


p-value:p-value: .1722.1722 (quite like that we could’ve (quite like that we could’ve gotten a gotten a χχ2 statistic as large as 2 statistic as large as 1.86, or larger, by an accident of 1.86, or larger, by an accident of sampling from a null population).sampling from a null population).

DecisionDecision:: Do Not Reject HDo Not Reject H00

ConclusionConclusion:: There is no statistically significant There is no statistically significant relationship between the relationship between the allocation of the death penalty and allocation of the death penalty and the race of the victim, in the the race of the victim, in the population of White murderers in population of White murderers in Georgia (Georgia (pp==.1722).1722)..


related, in the population of White related, in the population of White murderers in Georgia.murderers in Georgia.


p-value:p-value: .1722.1722 (quite like that we could’ve (quite like that we could’ve gotten a gotten a χχ2 statistic as large as 2 statistic as large as 1.86, or larger, by an accident of 1.86, or larger, by an accident of sampling from a null population).sampling from a null population).

DecisionDecision:: Do Not Reject HDo Not Reject H00

ConclusionConclusion:: There is no statistically significant There is no statistically significant relationship between the relationship between the allocation of the death penalty and allocation of the death penalty and the race of the victim, in the the race of the victim, in the population of White murderers in population of White murderers in Georgia (Georgia (pp==.1722).1722)..




Contingency Table Analyses of the Cases Contingency Table Analyses of the Cases with with White DefendantsWhite Defendants

Contingency Table Analyses of the Cases Contingency Table Analyses of the Cases with with White DefendantsWhite Defendants

When the Murderer Is White …When the Murderer Is White …The cell contributions to the χ2 statistic are all

vanishingly small:

The observed frequency in each cell of the contingency table is indistinguishable from the frequency we would expect if there were no relationship between DEATH & RVICTIM.

When we do not reject the null hypothesis, we do not bother interpreting the ancillary statistics because they are probably just be consequence of sampling idiosyncrasy!

When the Murderer Is White …When the Murderer Is White …The cell contributions to the χ2 statistic are all

vanishingly small:

The observed frequency in each cell of the contingency table is indistinguishable from the frequency we would expect if there were no relationship between DEATH & RVICTIM.

When we do not reject the null hypothesis, we do not bother interpreting the ancillary statistics because they are probably just be consequence of sampling idiosyncrasy!




Final Conclusion: Georgia Death Penalty AnalysisWhen the death penalty is handed down in Georgia, race makes a difference! We used contingency table analysis to examine categorical data on a sample of 2475 convicted murderers who had been convicted within the Georgia justice system. We knew the race of each of these murderers, the race of their victims and whether the death penalty had been awarded. Our principal research question asked whether there was a relationship between allocation of the death penalty and the race of the victim, in the population of murderers, and whether that relationship differed by the race of the murderer. While it is true that most murderers and their victims are of the same race and that only around 5% of all murderers are sentenced to death for their crimes, it is also true that the race of the victim plays an important role in the allocation of the most extreme penalty, as does the race of the defendant, as we now explain.

In the sub-population of Black murderers, we detected a statistically significant relationship between the allocation of the death penalty and the race of the victim (χ2 statistic=214.93, p<.0001). We display the sample relationship in Figure 1. Frequencies listed in the figure reveal that, when a Black person was the victim, only 1.3% of convicted murderers were sentenced to death. When a White person was the victim, on the other hand, about 22% of murderers were sentenced to death –almost thirteen times the prior percentage! These differences are dramatic, and particularly so when a Black murderer has killed a White victim. Out of a total of 228 Black murderers in our sample who killed White victims, we would have expected the death penalty to be awarded to only nine of them, if there had been no relationship between the allocation of the death and the race of the victim. However, the most extreme penalty was awarded on in 50 cases!

In the sub-population of White murderers, on the other hand, we found that we could not reject a null hypothesis of no relationship between the allocation of the death penalty and the race of the victim, in the population (χ2 statistic=1.86, p<.1722). In other words, whether their victim was Black or White, the proportion of White murderers who were sentenced to death was independent of the victim’s race.

Finally, while our research design was observational, and not experimental, and cannot therefore support causal inferences, we believe that our findings are strong enough to be taken seriously. They reveal clearly the presence of a gross racial imbalance in the allocation of the death penalty in Georgia.

what types of data are collected?

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