Moderation, Mediation, and Other Issues in Regression

Regression Assumptions Dependent variable a linear function of the IVs Each observation drawn independently (No relationships between cases such that errors for each case are independent from others)(HLM) Variance of errors around line should be fairly constant: Homoscedascticity Errors are normally distributed Critical Robust

To Interpret Regression as Effects The DV does not influence any of the IVs (cause and effect) The IVs are measured without error Must include all common causes of the presumed cause and presumed effect

Common Cause A common cause is a variable that affects both a presumed cause and a presumed effect. Here, variable a and variable b are correlated because they are both affected by the common cause. See chapter 11 for more detail

No Multicollinearity Tolerance = ranges from 0 (no independence from other variables) to 1 (complete independence) Multicollinearity - when Tolerance is low (closer to 0) and VIF is high (around 6, 7)

Multicollinearity Present ?

Moderation = Interaction Interaction effects = Moderation, or when the magnitude of the effect of one variable depends on another There is also a handout on this topic in the files containing answers to exercises

TV viewing by Ability Level The effect of TV viewing on achievement is moderated by ability level

Pretest - Posttest The effect of pre-test scores is moderated by condition

Intervening Variables Think of a relationship between two variables (one IV1 and one DV) Can you think of an intervening variable (IV2) that might help to explain the relationship between IV1 and your DV? Write down example and discuss with your neighbor

Mediation Mediation = indirect effect. Mediators are intervening variables that partially account for the relationship between an influence and outcome

Mediation can be total… …or partial. Here, variable 1 continues to have a direct effect on variable 2, in addition to its indirect effect through the mediator

Conditions for Mediation Step 1: Is influence a significant predictor of outcome? Regress DV on IV1 Influence Outcome

Conditions for Mediation Step 2: Is the mediator a significant predictor of outcome? Regress DV on Mediator Outcome Mediator

Conditions for Mediation Step 3: Is there a relationship between the influence variable and the mediator? Regress Mediator on Influence Influence Mediator

Conditions for Mediation Step 4: Does the magnitude of the direct path from influence to outcome decrease in the presence of the mediator? Regress Outcome on both Mediator and IV1. Influence Outcome Mediator

Conditions for Mediation Step 5: Is the mediation effect meaningful and significant? (more later) Influence Outcome Mediator Websites: http://www.psych.ku.edu/preacher/sobel/sobel.htm http://quantpsy.org/sobel/sobel.htm

Example of Mediation

In SPSS Regress Achievement on Ability (Path B) Regress Achievement on Motivation (Path C) Regress Motivation on Ability (Path A) Regress Achievement on both Ability (Path B’) AND Motivation (Path C’) Compare Path B to Path B’

SATMath TotalCoursePts

TotalCoursePts Self-Efficacy

SATMath Self-efficacy

SAT Math TotalCoursePts Self-efficacy

Sobel Statistic a = .004 (Sa) = .001 b = 33.143 (Sb) = 4.551 http://quantpsy.org/sobel/sobel.htm a = .004 (Sa) = .001 b = 33.143 (Sb) = 4.551 http://quantpsy.org/sobel/sobel.htm

Jargon Structural Equation Modeling (SEM) AKA: Analysis of Covariance Structures; Causal Analysis; LISREL Includes: Latent variable SEM; CFA; Path Analysis Path Analysis Simplest form of SEM Uses measured variables Can be solved via MR

Correlations Among 3 Variables

A Causal Model

The Tracing Rule The correlation between two variables X and Z is equal to the sum of the product of all paths from each possible tracing between X and Z [in Figure 2]. EXCEPT: the same variable is not entered twice per tracing and a variable is not both entered and exited through an arrowhead.

The Solved Model

Path Analysis with MR Regress Achievement on Ability and Motivation Regress Motivation on Ability See SPSS output

Regression Results Paths to Achievement: Path to Ability:

A More Complete Model Disturbances = error = residuals = SQRT(1-R2)

HW 9 The purpose of this homework is to test a mediational model. Based on theory, test a mediational model using your HSLS dataset. Write up your results. In doing so, make sure you: Briefly justify your model using theory and past research Provide a brief data analytic plan (in other words, describe what analyses you will be running to test your mediational model) Report and interpret statistics related to your mediational style using APA style Interpret your findings