multiple regression. the problem using several predictors to predict the dependent variableusing...
TRANSCRIPT
Multiple Regression
The Problem
• Using several predictors to predict the dependent variable
• Finding a measure of overall fit
• Weighting each predictor
An Example
• Study by Kliewer et al. (1998) on effect of violence on internalizing behavior Define internalizing behavior
• Predictors Degree of witnessing violence Measure of life stress Measure of social support
Violence and Internalizing
• Subjects are children 8-12 years Lived in high-violence areas Hypothesis: violence and stress lead to
internalizing behavior.
• Data available at www.uvm.edu/~dhowell/StatPages/
More_Stuff/Kliewer.dat
Intercorrelation Matrix
Preliminary Stuff
• Note that both Stress and Witnessing Violence are significantly correlated with Internalizing.
• Note that predictors are largely independent of each other.
Multiple Correlation
• Directly analogous to simple r
• Always capitalized (e.g. R)
• Always positive Correlation of with observed Y
• where is computed from regression equation
Often reported as R 2 instead of R
YY
R 2
Regression Coefficients
• Slopes and an intercept.
• Each variable adjusted for all others in the model.
• Just an extension of slope and intercept in simple regression
• SPSS output on next slide
Slopes and Intercept
Regression Equation
• A separate coefficient for each variable These are slopes
• An intercept (here called b0 instead of a)
517.0076.0272.0038.0
ˆ0332211
SocSuppStressWit
bXbXbXbY
Interpretation
• Note slope for Witness and Stress are positive, but slope for Social Support is negative. Does this make sense?
• If you had two subjects with identical Stress and SocSupp, a one unit increase in Witness would produce 0.038 unit increase in Internal.
Cont.
Interpretation--cont.
• The same holds true for other predictors.
• t test on two slopes are significant SocSupp not significant. Elaborate
• R 2 has same interpretation as r 2. 13.6% of variability in Internal
accounted for by variability in Witness, Stress, and SocSupp.
Interpretation--cont.
• Intercept usually not meaningful. Prediction when all predictors are 0.0
Predictions
• Assume Witness = 20, Stress = 5, and SocSupp = 35.
023.
517.0)35(076.)5(272.)20(038.
517.0*074.*273.*038.ˆ
SocSuppStressWitY
Hypothesis Testing
• Test on R 2 given in Analysis of Variance table
Cont.
Testing--cont.
• Tests on regression coefficients given along with the coefficients.
• See next slide
• Note tests on each coefficient.
Testing Slopes and Intercept
Raw vs. Standardized Betas
• Raw beta weights are in their original units (not standardized) Use for prediction
• Standardized beta weights are not in their original units Use for comparing predictors
Procedures for Entering Predictor Variables
• Simultaneous Enter all predictors simultaneously
• Forward Enter largest predictor first and test. If
sig. enter next largest partial r. Continue until you no more sig. predictors (sig. level can change after entered but predictor remains)
Procedures for Entering Predictor Variables
Continue• Backward
Enter all predictors first. Remove smallest nonsig. predictor and retest. Continue until model contains no nonsig predictors (sig. level can change after entered but predictor remains)
• Stepwise Combing forward and backward
Procedures for Entering Predictor Variables
Continued• Hierarchical
Enter predictors in pre-determined order as a means of hypothesis testing• E.g., does social support influence
internalizing above and beyond stress and witnessing violence?
Stepwise Results for Kliewer data
Potential Problems
• Multicollinearity – large correlations between predictors VIF (Variance Inflation Factor) < 10 Doesn’t qualify overall model but
estimates of individual predictors can be unstable
• Uncorrelated residuals Durban-Watson test (0-4): 1.5 – 2.5
generally acceptable