Measures of Association Quiz1. What do phi and b (the slope) have in common?

2. Which measures of association are chi square based?

3. What do gamma, lambda & r2 have in common?

4. When is it better to use Cramer’s V instead of lambda?

Statistical Control

Conceptual FrameworkElaboration for Crosstabs (Nom/Ord)Partial Correlations (IR)

3 CRITERIA OF CAUSALITY When the goal is to explain whether X

causes Y the following 3 conditions must be met:Association

X & Y vary together Direction of influence

X caused Y and not vice versaElimination of plausible rival explanations

Evidence that variables other than X did not cause the observed change in Y

Synonymous with “CONTROL”

CONTROLExperiments are the best research method in

terms of eliminating rival explanations Experiments have 2 key features:

Manipulation. . . Of the independent variable being studied

Control. . . Over conditions in which the study takes place

CONTROL VIA EXPERIMENT

Example:Experiment to examine the effect of type of

film viewed (X) on mood (Y) Individuals are randomly selected & randomly

assigned to 1 of 2 groups: Group A views The Departed (drama) Group B views Harold and Kumar (comedy)

Immediately after each film, you administer an instrument that assesses mood. Score on this assessment is D.V. (Y)

CONTROL VIA EXPERIMENT BASIC FEATURES OF THE EXPERIMENTAL DESIGN:

1. Subjects are assigned to one or the other group randomly

2. A manipulated independent variable (film viewed)

3. A measured dependent variable (score on mood assessment)

4. Except for the experimental manipulation, the groups are treated exactly alike, to avoid introducing extraneous variables and their effects.

CONSIDER ANALTERNATIVE APPROACH…

Instead of conducting an experiment, you interviewed moviegoers as they exited a theater to see if what they saw influenced their mood.

Many RIVAL CAUSAL FACTORS are not accounted for here

STATISTICAL CONTROL

Multivariate analysis simultaneously considering the relationship among

3+ variables

The Elaboration Method Process of introducing control variables into a bivariate

relationship in order to better understand (elaborate) the relationship

Control variable – a variable that is held constant in an attempt to understand better

the relationship between 2 other variables

Zero order relationship in the elaboration model, the original relationship between 2 nominal

or ordinal variables, before the introduction of a third (control) variable

Partial relationships the relationships found in the partial tables

3 Potential Relationships between x, y & z1. Spuriousness

a relationship between X & Y is SPURIOUS when it is due to the influence of an extraneous variable (Z)

(X & Y are mistaken as causally linked, when they are actually only correlated)

SURVEY OF DULUTH RESIDENTS BICYCLING PREDICTS VANDALISM

Does bicycling cause you to be a vandal?

extraneous variable a variable that influences both the independent and

dependent variables, creating an association that disappears when the extraneous variable is controlled

AGE relates to both bicycling and vandalism Controlling for age should make the bicycling/vandalism relationship go away.

Examples of spurious relationship

XZ

Ya. X (# of fire trucks) Y ($ of fire damage)

Spurious variable (Z) – size of the fire

b. X (hair length) Y (performance on exam)Spurious variable (Z) – sex (women, who tend to have

longer hair) did better than men

“Real World” ExampleResearch Question: What is the difference in

rates of recidivism between ISP and regular probationers?

Ideal way to study: Randomly assign 600 probationers to either ISP or regular probation.

300 probationers experience ISP 300 experience regular Follow up after 1 year to see who recidivates

Problem: CJ folks do not like this idea—reluctant to randomly assign.

“Real World” Example If all we have is preexisting groups (random assignment is not

possible) we can use STATISTICAL control

Bivariate (zero-order) relationship between probation type & recidivism:

Re-Arrest Regular Probation

ISP Totals

Yes 100 (33%) 135 (45%) 235

No 200 165 365

Totals 300 300 600

2 = 8.58 (> critical value: 3.841)

CONCLUSION FROM THIS TABLE?

“Real World” Example• 2 partial tables that control for risk: LOW RISK (2 = 0.03)

Re-arrest Regular ISP Totals

Yes 30 (17%) 15 (17%) 45

No 150 71 220

180 86 266

Re-arrest Regular ISP Totals

Yes 70 (58%) 120 (56%) 190

No 50 94 144

120 214 334

HIGH RISK (2 = 0.09)

“Real World” Example Conclusion: after controlling for risk, there is no

causal relationship between probation type and recidivism. This relationship is spurious.

Instead, probationers who were “high risk” tended to end up in ISP

High risk probationers fail more (get arrested more) than low risk probationers

After “controlling for risk,” there is no relationship between type of probation and arrest.

IN OTHER WORDS….

XZ

Y

X = ISP/Regular Y = RecidivismZ = Risk for Recidivism

3 Potential Relationships between x, y & z#2

Identifying an intervening variable (interpretation)

Clarifying the process through which the original bivariate relationship functions

The variable that does this is called the INTERVENING VARIABLE

a variable that is influenced by an independent variable, and that in turn influences a dependent variable

REFINES the original causal relationship; DOESN’T INVALIDATE it

Intervening (mediating) relationships X Z Y

Examples of intervening relationships:a. Children from broken homes (X) are more likely to

become delinquent (Y)Intervening variable (Z): Parental supervision

b. Low education (X) crime (Y)Intervening variable (Z): lack of $ opportunity

Spuriousness vs. Mediating

Mathematically, these effects will look the sameControlling for a “third” variable will

dramatically reduce or eliminate the original “zero order” relationship

Intervening vs. Mediating effects are determined through theory (prior expectations) and sometimes logic (common sense)

3 Potential Relationships between x, y & z #3

Specifying the conditions for a relationship – determining WHEN the bivariate relationship occurs

aka “specification” or “interaction”

Occurs when the association between the IV and DV varies across categories of the control variable

One partial relationship can be stronger, the other weaker. AND/OR,

One partial relationship can be positive, the other negative

Example: The effect of delinquent peers on a person’s crime depends upon the individuals’ IQ

“Real World” Example II Bivariate (zero-order) relationship between treatment type &

recidivism Cognitive behavioral treatment is out “best technology” for

rehabilitating offenders

New Arrest? Cognitive Behavioral

Control Group Totals

Yes 35 (25%) 45 (31%) 80

No 110 100 210

Totals 145 145 290

CONCLUSION FROM THIS TABLE?

“ 2 partial tables that control for risk: LOW RISK

New Arrest? Cog-Behavioral Control Totals

Yes 25 (24%) 15 (16%) 40

No 80 80 160

105 95 200

New Arrest? Cog-Behavioral Control Totals

Yes 10 (25%) 30 (60%) 40

No 30 20 50

40 50 90

HIGH RISK

An Interaction Effect This would be an example of an interaction

between treatment and risk for recidivism Treatment had a small positive impact on recidivism

overall Treatment had a strong positive impact for high risk

offenders, but not low risk offenders In other words, the effect of treatment

depended upon the risk level of the offenders

Limitations of Table Elaboration:

1. Can quickly become awkward to use if controlling for 2+ variables or if 1 control variable has many categories

2. Greater # of partial tables can result in empty cells, making it hard to draw conclusions from elaboration

Partial Correlation

“Zero-Order” Correlation Correlation coefficients for bivariate relationships

Pearson’s r

 

Statistical Control with Interval-Ratio Variables

Partial Correlation Partial correlation coefficients are symbolized as

ryx.z This is interpreted as partial correlation coefficient that

measures the relationship between X and Y, while controlling for Z

Like elaboration of tables, but with I-R variables

Partial Correlation Interpreting partial correlation coefficients:

Can help you determine whether a relationship is direct (Z has little to no effect on X-Y relationship) or (spurious/ intervening)

The more the bivariate relationship retains its strength after controlling for a 3rd variable (Z), the stronger the direct relationship between X & Y

If the partial correlation coefficient (ryx.z) is much lower than the zero-order coefficient (ryx) then the relationship is EITHER spurious OR intervening

Partial Correlation Example: What is the partial correlation coefficient

for education (X) & crime (Y), after controlling for lack of opportunity (Z)?

ryx (r for education & crime) = -.30 ryz (r for opportunity & crime) = -.40 rxz (r for education and opportunity) = .50

ryx.z = -.125 The correlation between education and crime, after

controlling for opportunity Interpretation?

Comparing the zero order to the new “partial” correlation

Partial Correlation Based on temporal ordering & theory, we would

decide that in this example Z is intervening (X Z Y) instead of extraneous

If we had found the same partial correlation for firetrucks (X) and fire damage (Y), after controlling for size of fire (Z), we should conclude that this relationship is spurious.

Partial CorrelationAnother example:

What is the relationship between hours studying (X) and GPA (Y) after controlling for # of memberships in campus organizations(Z)?

ryx (r for hours studying & GPA) = .80 ryz (r for # of organizations & GPA) = .20 rxz (r for hrs studying & # organizations) = .30

ryx.z = .795 Interpretation?