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Measures of Association Quiz. What do phi and b (the slope) have in common? Which measures of association are chi square based? What do gamma, lambda & r 2 have in common? When is it better to use Cramer’s V instead of lambda ?. Statistical Control. Conceptual Framework
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Measures of Association Quiz • What do phi and b (the slope) have in common? • Which measures of association are chi square based? • What do gamma, lambda & r2 have in common? • 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 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 & 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: 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) 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 CONCLUSION FROM THIS TABLE?
“ • 2 partial tables that control for risk: LOW RISK 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 impactfor 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: • Can quickly become awkward to use if controlling for 2+ variables or if 1 control variable has many categories • 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 opp) = .50 • ryx.z = -.125 • 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)? • 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?