Multivariate Statistics Week 5 - Multiple Regression pt 2

Adjusted R²

calculates R² while taking into account the number of variables in the multiple regression model - more variables penalize the adjusted R²

why? because every variable added to the MR will increase R² but that’s not always meaningful.

Adjusted R² only increases when new variables improve the model more than would be expected by chance

what multiple regression to use when asking if the model could be improved by adding new variables or changing old ones

Hierarchical multiple regression or

Stepwise multiple regression

what multiple regression to use when asking if there is a chain of prediction

mediated multiple regression

what model to use when asking does one predictor change the effect of another predictor

moderated multiple regression

Standard Multiple Regression

“simultaneous” or “forced entry”

all predictors are entered simultaneously

calculates how much variance in the DV is independently accounted by each predictor

tells us what each predictor adds to the prediction beyond that afforded by all others

Hierarchical Multiple Regression

“sequential”

predictors are entered sequentially in pre-specified order based on study-relevant theory or logic

predictors are evaluated at their point of entry so the first entered predictor gets “first bite”

every predictor after can only get credit for the variance that was NOT accounted for by the previous predictor(s)

no overlapping/sharing credit for variance

logic behind sequential order of predictors in hierarchical mutliple regression

controlling for nuisance variables

• ex. get any variance resulting from age ‘out of the way’ if age is irrelevant to the research question

causal priority

• shared variance between predictors may reflect causality between predictors (one predictor may cause the other)

• inputting the predictor that has causal priority makes more logical sense

multiple models

run MR in steps and examine at each step:

• size/significance of full model

• size/significant of predictors

• change in size/significant

suppressor variables

variables that when added to a MR increase significance of another variable by reducing noise/error

confounding variables

variables that shrink the relationship between a predictor and the DV by absorbing a shared signal

importance of the order in which predictors are entered

critically determines results

total variance accounted for is not affected by order but the contributions of each predictor entered at each step is affected

Stepwise MR

“statistical” multiple regression

predictors are entered in steps like in hierarchical BUT order of entry is selected by computer in order to maximize R² with minimum predictors

not the ideal way to select order of entry because by ignoring the theoretical reasons for a pre-specified order, this model capitalizes on chance and may give inflated results

Mediated Multiple Regression

useful for testing plausbility of a chain of prediction

possibility that predictor 1 causes predictor 2 which causes the DV thus predictor 1 indirectly causes the DV

evidence/requirements for mediation

predictor x must predict mediator z

mediator z must predict DV (y)

predictor x must predict DV (y)

the relationship between predictor x and DV (Y) must shrink when taking mediator Z into account

Moderation

when a relationship between the DV and a predictor varies as a function of a second predictor

often answers the question “when” and/or “for whom” the predictor predicts the DV

Moderated Multiple Regression

tests for interactions between predictors (presence of moderators)

requires follow up analysis to clarify the relationship between X,Z (predictor, mediator)

Moderated MR follow up analyses

• multiple z-scores of X and Z (the main effect predictors) to compute interaction term (XZ) and test for significance

  • use z-scores to standardize data and prevent covariation with other predictors

• examine x → y (DV) relationship at different values of Z and analyze simple slopes

testing significance of interaction terms

does interaction term make significant contribution to prediction beyond the main effects of individual predictors

test using hierarchical MR - enter XZ after X and Z to see if R² significantly improves

Simple Slopes analysis

similar to examining main effects of ANOVA at different levels of IV

simple slopes examines the X → Y relationship at critical values of Z (often +1 and -1 SD)

limitations of moderated MR

power - difficult to detect interactions

reliability of predictors

alternatives to moderated MR

• Structural Equation Modeling

• Median Split procedure

Structural equation modeling

Median split Procedure

splitting continuous variables artifiically at the median into ‘high’ or ‘low’ to allow for data to be analyzed with an ANOVA (requires categorical variable)

less powerful than moderated MR and less theoretically compelling