Measures of Association Quiz 1. What do phi and b (the slope) have in common? 2. Which measures of association are chi square based? 3. What do gamma, lambda & r 2 have in common? 4. When is it better to use Cramer’s V instead of lambda?

Statistical Control Conceptual Framework Elaboration 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 versa  Elimination of plausible rival explanations Evidence that variables other than X did not cause the observed change in Y Synonymous with “CONTROL”

CONTROL  Experiments 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 AN ALTERNATIVE 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 & z 1. 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 X Z Y a. 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” Example  Research 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: RecidivismRegularISPTotals Yes100 (33%)135 (45%)235 No Totals  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) Recid.RegularISPTotals Yes30 (17%)15 (17%)45 No Recid.RegularISPTotals Yes70 (58%)120 (56%)190 No 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

IN OTHER WORDS…. X Z Y X = ISP/Regular Y = Recidivism Z = 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 same  Controlling 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

“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 GroupTotals Yes35 (25%)45 (31%)235 No Totals CONCLUSION FROM THIS TABLE?

“ 2 partial tables that control for risk: LOW RISK New Arrest?Cog-BehavioralControlTotals Yes25 (24%)15 (16%)40 No New Arrest?Cog-BehavioralControlTotals Yes10 (25%)30 (60%)40 No HIGH RISK

An Interaction Effect This would be an example of an interaction between treatment and risk for recidivism  Treatment had little impact on recidivism overal  Treatment did have 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 r yx.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 (r yx.z ) is much lower than the zero-order coefficient (r yx ) 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)?  r yx (r for education & crime) = -.30  r yz (r for opportunity & crime) = -.40  r xz (r for education and opp) =.50 r yx.z =  Interpretation?

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 Correlation  Another example: What is the relationship between hours studying (X) and GPA (Y) after controlling for # of memberships in campus organizations(Z)?  r yx (r for hours studying & GPA) =.80  r yz (r for # of organizations & GPA) =.20  r xz (r for hrs studying & # organizations) =.30 r yx.z =.795  Interpretation?