1. CATEGORICAL VARIABLES
Testing hypotheses using
CATEGORICAL VARIABLES

2. Assessing relationships between categorical variables
Hypothesis: formal statement of cause and effect, derived from research question and literature review
Better demeanor  more leniency
To test a relationship between categorical variables, we build TWO tables
Frequencies table
Dependent variable values in COLUMNS
Independent variable values in ROWS
Frequencies (no. of cases with corresponding values) in the cells
Percentages table
Calculated row by row, top to bottom
ROW percentages - each row totals 100%
This is the only acceptable way to build tables in this class. Must do it this way to earn credit on the exams.
Officer’s Disposition
Youth’s
demeanor
Admonish
Reprimand
Citation
Arrest n
Cooperative 24 15 4 2 45
Uncooperative 1 5 14 21
Officer’s Disposition
Youth’s
demeanor
Admonish
Reprimand
Citation
Arrest
Cooperative
53%
33% 9% 5%
100%
Uncooperative
23%
67%

3. Recoding continuous to categorical variables
Hypothesis: higher income persons drive more expensive cars
Independent variable: Income, measured categorically (nominal variable)
Two values: low income and high income
Income is measured by where a car is parked - student lot (low income) and faculty-staff lot (high income)
Dependent variable: Car value, measured categorically (ordinal variable)
1, 2, 3, 4 or 5 (1- cheapest, 5 - most expensive)
Sampling
Stratified, disproportionate, systematic sampling of 10 cars from a student lot, and 10 cars from a faculty lot
Coding
Income is automatically coded by car location (faculty-staff or student lot)
Car values must be recoded from original five-point scale to three categories - Low, Medium, High

4. Step 1: Recoding Step 2: Percentaging
Recode continuous 1-5 car value measure into three categories, L/M/H
1-2: LOW 3: MED : HIGH
DV - Car value
IV - Income
LOW
MED
HIGH n
LOW (student lot) 10
HIGH (F/S lot) 4 6
Step 2:
Percentaging
Convert frequencies into percentages
Convert each row separately so the cells add to 100 percent
DV - Car value
IV - Income
LOW
MED
HIGH n
LOW (student lot) 10
HIGH (F/S lot) 4 6
DV - Car value
IV - Income
LOW
MED
HIGH %
LOW (student lot)
100%
HIGH (F/S lot)
40%
60%

5. Draw the distributions and examine their shapes
DV - Car value
IV - Income
LOW
MED
HIGH %
LOW (student lot)
100%
HIGH (F/S lot)
40%
60%
Step 3:
Distributions
Draw the distributions and examine their shapes
Student
lot L
Faculty
lot H L
Step 4:
Analysis
Switch values of the independent variable. Does the distribution of cases substantially change in the predicted direction?
All the cars in the student lot are low value.
Sixty percent of the cars in the F/S lot are high value.
As we “switch” values of the IV, the distribution changes in the hypothesized direction, with larger proportions of more expensive cars in the F/S lot. Hypothesis confirmed!

6. An example where the hypothesis is NOT confirmed
Student
lot H L
Faculty
lot L M H
DV - Car value
IV - Income
LOW
MED
HIGH %
LOW (student lot)
30%
20%
50%
100%
HIGH (F/S lot)
40%
60%
Differences between rows are relatively slight
For both income levels, distribution is biased to higher car values
Is there a more accurate way to assess possible differences? You bet! That’s where we’re heading…

7. Hypothesis: poverty  crime
Remember: if there appears to be an effect, is it in the hypothesized direction?
Hypothesis: poverty  crime
IV Poverty is measured by income, DV crime by arrests
Income has two values, low and high
Arrests has two values, never arrested and arrest record
To test the hypothesis, switch from one category of the IV to the other.
Does the distribution of cases along the DV change?
Is the change substantial?
Is the change in the hypothesized direction?
Never Arrested
Arrest Record
Low
Income
50%
100%
High
Never Arrested
Arrest Record
Low
Income
20%
80%
100%
High
Never Arrested
Arrest Record
Low
Income
80%
20%
100%
High
Distribution is unaffected. There seems to be no connection between income and arrest record. The hypothesis is rejected.
Distribution flips in the expected direction.
High income persons seem
much less likely to have an
arrest record. The hypothesis is confirmed.
Distribution flips in the opposite direction. High income persons seem much more likely to have an arrest record. The hypothesis is rejected.

8. Elaboration analysis Zero-order table First-order partial tables
Replication
Explanation
Specification

9. Could another IV be at play?
Remember our poverty  crime example?
Intervening variable: Could lack of education or living in a violent area be the more proximate (closer) cause of crime?
Poverty  poor education  crime
Here poverty still affects crime, but mostly through intervening variable education, which is the more proximate cause
Spurious relationship: What seems to be a relationship isn’t - it’s bogus!
Often caused by a strong association between the independent variable of interest (e.g., poverty) and another independent variable (e.g., poor social controls) which turn out to be the real cause
Poor social controls  crime
Poverty X

10. PRACTICAL EXERCISE Hypothesis: Higher rank  Less cynicism
Using “elaboration analysis” to determine if another IV might be at play
PRACTICAL EXERCISE Hypothesis: Higher rank  Less cynicism
Sample of 100 officers and 100 supervisors
Twenty officers scored low on cynicism; 80 were high cynicism
Fifty supervisors scored low on cynicism; 50 were high cynicism
Build a frequency table, then convert it to percentages
Be sure to place the categories of the dependent variable in columns, and the categories of the independent variable in rows

11. PRACTICAL EXERCISE Hypothesis: Higher rank  Less cynicism
Of course, we can’t stop here. There are many variables floating around.
Are there any other variables that are related to our independent variable, rank, and which could possibly affect dependent variable cynicism? A literature review reveals that gender is associated with rank, and that gender may also affect cynicism.
Gender could be an “intervening” variable, mediating the relationship with cynicism…
Rank  Gender  Cynicism
Or, it could be the real cause of changes in cynicism
If it was, the apparent (but non-existing) relationship between rank and cynicism would be called “spurious”
How do we disentangle the possibilities? Coming up!

12. Hypothesis: Higher rank  Less cynicism
Elaboration analysis - using first-order partial tables to analyze the effects of a “control” variable
Let’s “elaborate” (dig deeper). Does the effect of rank on cynicism hold regardless of gender?
Gender is used as a “control” variable. We will test the original, “zero-order” relationship between rank and cynicism, “controlling” for each value of gender.
Gender is categorical, so we keep using tables
Create two tables - one with frequencies, the other with percentages - for each value of control variable gender
Tables are just like the original, but each pair only includes cops of one gender (M or F)
These tables are called “first order partial tables” because…
“First order”: It’s our first “control” variable
“Partial”: Each table only includes cases at one value of the control variable gender (Male or Female)
To distinguish, the original tables are called “zero-order”
Data for gender
MALES: Officers: 10 low cynicism, 50 high Supervisors: 35 low, 35 high
FEMALES: Officers: 10 low, 30 high Supervisors: 15 low, 15 high
Analyze the percentage tables - what do you find?
Hypothesis: Higher rank  Less cynicism
“zero-order” tables
Cynicism
Rank
Low
High n
Officers 20 80
100
Supervisors 50
200
Cynicism
Rank
Low
High n
Officers
20%
80%
100%
Supervisors
50%

13. First-order partial tables
Original
“zero-order”
tables
Cynicism
Rank
Low
High n
Officers 20 80
100
Supervisors 50
200
Cynicism
Rank
Low
High
Officers
20%
80%
100%
Supervisors
50%
First-order partial tables
Both levels of control variable gender - male and female - yield about the same findings as the zero-order table. Knowing an officer’s gender changes nothing. We have replicated the zero-order findings.
Gender is not a factor in rank  cynicism. Our hypothesis is confirmed. That’s called:
REPLICATION

14. Hypothesis: Higher rank  Less cynicism
But we’re not done yet! Our literature review suggested that another variable associated with rank – time on the job – may also affect cynicism.
Let’s “control” for time on the job. Here’s the data:
LESS THAN FIVE YEARS ON THE JOB
Officers: 0 low cynicism, 75 high cynicism
Supervisors: 2 low cynicism, 40 high cynicism
MORE THAN FIVE YEARS ON THE JOB
Officers: 20 low, 5 high
Supervisors: 48 low, 10 high
Create first-order partial tables for time on the job, convert tables to percentages, and analyze the results...

15. Original “zero-order” tables
Cynicism
Rank
Low
High n
Officers 20 80
100
Supervisors 50
200
Cynicism
Rank
Low
High
Officers
20%
80%
100%
Supervisors
50%
Both levels of control variable time on the job are strongly associated with cynicism. This completely “explains” (wipes away) the apparent relationship between rank and cynicism. Our hypothesis (higher rank  less cynicism) is rejected. It’s less time on the job  more cynicism. This outcome is called:
EXPLANATION

16. Hypothesis: Higher rank  Less cynicism
But we’re not done yet! Our literature review suggested that another variable associated with rank – education – may also affect cynicism.
Let’s “control” for education. Here’s the data:
HIGH SCHOOL
Officers: 10 low cynicism, 50 high cynicism
Supervisors: 9 low cynicism, 11 high cynicism
COLLEGE
Officers: 30 low, 10 high
Supervisors: 65 low, 15 high
Create first-order partial tables for educational level, convert tables to percentages, and analyze the results...

17. Original “zero-order” tables
Cynicism
Rank
Low
High n
Officers 20 80
100
Supervisors 50
200
Cynicism
Rank
Low
High
Officers
20%
80%
100%
Supervisors
50%
For education level, our findings are split. The cynicism of high school grads resembles that of the zero-order table. But college-educated cops are much less cynical. One value of the control variable - education - teaches us something new. This outcome is called:
SPECIFICATION

18. Review: first-order partial analysis has three possible outcomes
First-order (and second-order, and so on) partial analyses yield three possible outcomes:
Replication (first example): The original, “zero-order” relationship persists at each value of the control variable. Coding for this variable teaches us nothing. The hypothesis is confirmed.
Explanation (second example): The original, “zero-order” relationship is absent from each value of the control variable. The effect of the IV on the DV in the zero-order table has been “explained away” by the control variable. The hypothesis is rejected.
Specification (final example): The zero-order relationship persists for some but not all values of the control variable. Coding for this variable teaches us something. The hypothesis must be modified.

19. But isn’t our analysis too “loosey-goosey”?
Assume there is a relationship between variables. When we “switch” the value of the IV, will the change in the DV always be obvious?
No. And when the DV has multiple categories, such as in our parking lot exercise, visually discerning an effect can be difficult or impossible. Bottom line - changes in percentage are not enough.
Great. Now what?
We can use the original frequencies table to build a second table, which projects what the frequencies would be if there was absolutely NO relationship between variables. We then compare the two tables with a “test statistic” (statistic used to test hypotheses) known as “Chi-square”, X2.
If this statistic is of sufficient magnitude (large enough), the tables are deemed “significantly” different, meaning that there IS a relationship between variables. More on this during the third part of the semester!
Cynicism - Males
Rank
Low
High
Officers
17%
83%
100%
Supervisors
50%
Cynicism - Males
Rank
Low
High n
Officers 10 50 60
Supervisors 35 70
130

20. Dependent variable: arrest (Y/N)
Where we headed?
Dependent variable: arrest (Y/N)
Hypothesis testing requires that we assess the influence, on the DV, of your IV of interest, and of other variables that may be related to your IV and which could have their own effect on the DV.
Tables in your articles parcel out the effects of all variables on the DV. A “test statistic” informs you exactly how much influence on the DV is exerted by your IV of interest and by its cousins.
We’ll be learning about various test statistics during the semester
Dependent variable: death sentence (Y/N)

21. REMINDER: The ONLY acceptable way to build a table in this class
Dependent variable values in COLUMNS
Independent variable values in ROWS
Frequencies (number of cases with corresponding values) in the cells
Two tables
One table contains the frequencies
A second table, calculated from the first, contains ROW percentages
This is the only acceptable way to build tables in this class. Must do it this way to earn credit on the exams.
Better demeanor  More lenient disposition
Officer’s Disposition
Youth’s
demeanor
Admonish
Reprimand
Citation
Arrest n
Cooperative 24 15 4 2 45
Uncooperative 1 5 14 21
Officer’s Disposition
Youth’s
demeanor
Admonish
Reprimand
Citation
Arrest
Cooperative
53%
33% 9% 5%
100%
Uncooperative
23%
67%