Regression Assumptions of OLS

Assumptions of multiple regression Equal probability of selection (SRS) Linearity (visible and invisible variables) Independence of observations: Errors are uncorrelated The mean of error term is ALWAYS zero: Mean does not depend on x. Normality (of the error term) Homoskedasticity Variance does not depend on x. No multicollinearity

Homoskedasticity The variance of the error term is fixed (equal across all cases). Compliance with this assumption can be empirically checked. Consequences if violated: SE will be upward biased.

Multicollinearity Has to do with the quality of the information matrix. No linear combination of independent variables should be able to predict any other independent variable.

Multicollinearity - Dx Tolerance: VIF: Inverse of tolerance Indicates inflated standard errors >2 >2.5 Multiple correlation among IVs

Example: Regression with SPSS

Regression exercise Maternal aggression Child aggression Paternal Harsh parenting

Correlations

SPSS output

SPSS output

Regression exercise Maternal aggression Harsh parenting Child Paternal aggression

SPSS step 1: Harsh parenting

Step 2: Direct effects of mom

Step 3: Mediated effects of mom

Multicollinearity check

Including nominal Or ordinal Variables Regression Including nominal Or ordinal Variables

Categorical variables in regression

Association with DV

Dummy variables

Regression with dummy variables

ANOVA UNIANOVA kidagr BY harsh_o WITH momagr dadagr /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /CRITERIA = ALPHA(.05) /DESIGN = momagr dadagr harsh_o .

Regression Interaction effects

Moderated regression Maternal aggression Child aggression Paternal Harsh parenting

Moderated regression momdadagr = momagr*dadagr

Issues Related to Regression Homework

Issues Interpreting regression coefficients when measurement units are not meaningful Interval level, different units of measurement Legend of conceptual framework Test of mediated effects XYZ Atheoretical regression models Write-up Hypotheses Less…than According to conceptual framework Regression equations in text Decimal points

Regression & ANOVA: Wrap up

Common elements All of these models are linear: DV(s)=b1*IV1 + b2*IV2 + b3*IV3 All of these models assume a interval/ratio level DV. All of these models can handle categorical or interval/ratio IVs. All of these models use some form of least squares method (squared deviations from the mean).

Common elements All of these methods assume SRS (independence of observations). All of these methods assume homoskedasticity. All of these methods can only model “flat” and unidirectional effects.

ANOVA / Regression Differences arise from “traditions.” ANOVA  Experimental design Regression  Non-experimental/survey design. Differences in the yield of information: Regression is superior.

Within Subjects Designs Regression with fixed or random effects:

Factor Analyses Regression with an “unknown” IV:

HLMs Regression coefficients themselves are DVs.