CATEGORICAL VARIABLES Testing hypotheses using. Independent variable: Income, measured categorically...
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Transcript of CATEGORICAL VARIABLES Testing hypotheses using. Independent variable: Income, measured categorically...
CATEGORICAL VARIABLESTesting hypotheses using
• 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 random sampling of 10 cars from a student lot, and 10 cars from a faculty lot
• Coding– Income is automatically coded by a car’s
location (faculty-staff or student lot)– A 5-level categorical measure is used to
code car values
Hypothesis: Higher income persons drivemore expensive cars
DV - Car value
IV - Income 1 2 3 4 5 n
LOW (student lot) 2 2 0 3 3 10
HIGH (F/S lot) 3 1 1 1 4 10
Car value
Student lot
Car value
Faculty/staff lot
Team A
For the purposes of this class, always place the values of the DV along the horizontal axis,and the values of the IV along the vertical axis
Each value of the DV has its own column Each value of the IV has its own row
Step 1:Coding
Hypothesis:Higher income persons drive more expensive cars
DV - Car value
IV - Income 1 2 3 4 5 n
LOW (student lot) 6 4 0 0 0 10
HIGH (F/S lot) 5 1 2 2 0 10
Team B
Student lot Faculty/staff lot
For the purposes of this class, always place the values of the DV along the horizontal axis,and the values of the IV along the vertical axis
Each value of the DV has its own column Each value of the IV has its own row
Step 1:Coding
Hypothesis:Higher income persons drive more expensive cars
DV - Car value
IV - Income 1 2 3 4 5 %
LOW (student lot) 20% 20% 0% 30% 30% 100%
HIGH (F/S lot) 30% 10% 10% 10% 40% 100%
For accurate analysis, frequencies must be converted to percentages
Convert each row separately so the cells add to 100 percent
DV - Car value
IV - Income 1 2 3 4 5 %
LOW (student lot) 60% 40% 0% 0% 0% 100%
HIGH (F/S lot) 50% 10% 20% 20% 0% 100%
Team A
Team B
DV - Car value
IV - Income 1 2 3 4 5 n
LOW (student lot) 2 2 0 3 3 10
HIGH (F/S lot) 3 1 1 1 4 10
DV - Car value
IV - Income 1 2 3 4 5 n
LOW (student lot) 6 4 0 0 0 10
HIGH (F/S lot) 5 1 2 2 0 10
Step 2:Percentaging
DV - Car value
IV - Income 1 2 3 4 5 %
LOW (student lot) 20% 20% 0% 30% 30% 100%
HIGH (F/S lot) 30% 10% 10% 10% 40% 100%
Switch values of the independent variable. Does the distribution of car values
change? If so, is the difference in the predicted direction?
DV - Car value
IV - Income 1 2 3 4 5 %
LOW (student lot) 60% 40% 0% 0% 0% 100%
HIGH (F/S lot) 50% 10% 20% 20% 0% 100%
Team A
Team B
Step 3:Analysis
• Forty percent of the cars in the student lot are value 1 and 2. Same for the F/S lot.
• There are differences between rows in values 3-5, but they seem minimal.
• All the cars in the student lot are value 1 and 2.
• But forty percent of the cars in the F/S lot are value 3 and 4.
• As we “switch” values of the IV from low to high income, the proportion of expensive cars substantially increases. The direction of the effect is consistent with the hypothesis.
• 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 substantially? – If so, is the change in the hypothesized direction?
Another exampleHypothesis: poverty crime
NeverArrested
ArrestRecord
Low
Income80% 20% 100%
High
Income20% 80% 100%
Distribution flip-flops in an unexpected direction. High income persons seem much more likely to have an arrest record. The hypothesis is rejected.
Distribution remains the same. There seems to be no connection between income and arrest record. The hypothesis is rejected.
NeverArrested
ArrestRecord
Low
Income50% 50% 100%
High
Income50% 50% 100%
NeverArrested
ArrestRecord
Low
Income20% 80% 100%
High
Income80% 20% 100%
Distribution flip-flops in the expected direction.High income persons seemmuch less likely to have anarrest record. The hypothesis is confirmed.
Cranking it up a notch with “elaboration analysis”
• Hmmm, interesting! Sergeants are more stressed than patrol officers. • But is it possible that another variable - one closely associated with position - either mediates
the relationship with job stress or is the real driving force? In other words…
Position on police force other variable job stress OR other variable job stress
position on police force
Hypothesis: position on police force determines job stress
Job Stress
Position Low High n
Sergeant 30 60 90
Patrol officer 86 24 110
Source: Fitzgerald
Job Stress
Position Low High
Sergeant 33% 67% 100%
Patrol officer 78% 22% 100%
Elaboration analysis - using first-order partial tablesto analyze the effect of a “control” variable
• So…what variables might be associated with position and with job stress?
– Data indicates that females are less likely to be police supervisors.
– The literature review also suggests that males and females may have different stress responses
• Let’s “elaborate” (dig deeper)
– Does the effect of position on job stress hold regardless of gender?
• Gender is used as a “control” variable. We will test the original, “zero-order” relationship between position and job stress, “controlling” for each value of gender.
– Gender is categorical, so we keep using tables
• Create one table just like the one we originally designed (position job stress) for each value of control variable gender
– One table for males, another for females
– Each table is identical to the zero-order table, except it only includes cops of that gender
• These tables are called “first order partial tables” because they represent our first attempt to introduce a “control” variable.
– Each table is “partial” - only part of the sample - because it only includes cases with a certain value of the control variable
Original “zero-order”tables
First order partial tables - one for each value of the control variable
Job Stress
Position Low High n
Sergeant 30 60 90
Patrol officer 86 24 110
Job Stress - 130 male officers
Position Low High n
Sergeant 14 46 60
Patrol officer 58 12 70
Job Stress - 70 female officers
Position Low High n
Sergeant 16 14 30
Patrol officer 28 12 40
Job Stress - 130 male officers
Position Low High
Sergeant 23% 77% 100%
Patrol officer 83% 17% 100%
Job Stress - 70 female officers
Position Low High
Sergeant 53% 47% 100%
Patrol officer 70% 30% 100%
Job Stress
Position Low High
Sergeant 33% 67% 100%
Patrol officer 78% 22% 100%
Zero-order table,
all cops
First-order partial table,male cops
No, the percentages aren’t exactly the same. But, overall, the relationship in the first-order partial table is in the same direction as in the zero-order table, perhaps stronger. Most male sergeants report being highly stressed, and most male patrol officers report very low stress. Knowing that an officer is male is consistent with the hypothesis that higher position leads to more job stress.
Does the zero-order relationship between position and job stress persist for males?
Job Stress - Male officers
Position Low High
Sergeant 23% 77% 100%
Patrol officer 83% 17% 100%
Job Stress
Position Low High
Sergeant 33% 67% 100%
Patrol officer 78% 22% 100%
OUTCOME: SPECIFICATION
Knowing that officers were male didn’t change our opinion about the effects of position on job stress. So for male officers, the “zero-order” relationship between position and job stress holds. But knowing that officers were female gave us a new insight. Only 47% of female sergeants report being highly stressed, a far smaller proportion than 77% of male sergeants.
So our opinion of the effects of position on job stress is moderated by one value of the control variable, female. Knowing that a supervisor is female tells us something we didn’t know.
Does the zero-order relationship between position and job stress persist for females?
Zero-order table,
all cops
Job Stress
Position Low High
Sergeant 33% 67% 100%
Patrol officer 78% 22% 100%
Job Stress - Female officers
Position Low High
Sergeant 53% 47% 100%
Patrol officer 70% 30% 100%
Job Stress - Male officers
Position Low High
Sergeant 23% 77% 100%
Patrol officer 83% 17% 100%
First-order partial analysis:three outcomes
• Doing a first-order partial analysis yields three possible interpretive outcomes:– Specification (prior example): The zero-order relationship persists for
some but not all values of the new variable. Coding this variable teaches us something.
– Replication (next example): The original relationship from the zero-order table persists at both values of the new variable. Coding for the new variable teaches us nothing.
– Explanation (final example): The zero-order relationship is not present at any value of the new variable. The apparent effect of the original independent variable - the one in the hypothesis - has been completely “explained away.”
We just covered specification. Let’s turn to the other twopossible outcomes of elaboration analysis.
PRACTICAL EXERCISEHypothesis: 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 (zero-order) 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
• According to our literature review, a variable associated with rank – gender – may affect cynicism.
• Let’s “control” for gender. We get data on cynicism for officers and supervisors, broken down by gender:
MALES
Officers: 10 low cynicism, 50 high cynicism
Supervisors: 35 low, 35 high
FEMALES
Officers: 10 low, 30 high
Supervisors: 15 low, 15 high
• Create first-order partial tables for gender, convert tables to percentages, and analyze the results...
PRACTICAL EXERCISEHypothesis: Higher rank Less cynicism
• But the literature suggests that still another variable associated with rank – time on the job – may 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
FIVE YEARS OR MORE 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...
PRACTICAL EXERCISEHypothesis: Higher rank Less cynicism
Cynicism - Males
Rank Low High n
Officers 10 50 60
Supervisors 35 35 70
130
But isn’t this too “loosey-goosey”?• Assume there is a relationship between variables.
When we “switch” the value of the IV, will thechange in the DV always be this obvious?
• No. And when the DV has multiple categories,such as in our parking lot exercise, visuallydiscerning an effect can be impossible. Bottomline - changes in percentage are not enough.
• Great. Now what?• Fortunately, we can use the cell frequencies to
calculate a statistic known as “Chi-square”, X2.This statistic assigns a numerical measure tothe relationship between variables. We thenlook up that number in a table to determine ifit is large enough to be statistically “significant.”
• All we need is the original frequency table?• We use the table to build a second table, which projects what the frequencies
would be if there was NO relationship between variables. We then compare the two frequency tables. More on that during the third part of the semester!
Cynicism - Males
Rank Low High
Officers 17% 83% 100%
Supervisors 50% 50% 100%