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Multiple Regression

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The Problem Using several predictors to predict the dependent variableUsing several predictors to predict the dependent variable Finding a measure of overall fitFinding a measure of overall fit Weighting each predictorWeighting each predictor

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An Example Study by Kliewer et al. (1998) on effect of violence on internalizing behaviorStudy by Kliewer et al. (1998) on effect of violence on internalizing behavior XDefine internalizing behavior PredictorsPredictors XDegree of witnessing violence XMeasure of life stress XMeasure of social support

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Violence and Internalizing Subjects are children 8-12 yearsSubjects are children 8-12 years XLived in high-violence areas XHypothesis: violence and stress lead to internalizing behavior. Data available atData available at Xwww.uvm.edu/~dhowell/StatPages/ More_Stuff/Kliewer.dat More_Stuff/Kliewer.datwww.uvm.edu/~dhowell/StatPages/ More_Stuff/Kliewer.dat

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Intercorrelation Matrix

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Preliminary Stuff Note that both Stress and Witnessing Violence are significantly correlated with Internalizing.Note that both Stress and Witnessing Violence are significantly correlated with Internalizing. Note that predictors are largely independent of each other.Note that predictors are largely independent of each other.

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Multiple Correlation Directly analogous to simple rDirectly analogous to simple r Always capitalized (e.g. R)Always capitalized (e.g. R) Always positiveAlways positive XCorrelation of with observed Y where is computed from regression equationwhere is computed from regression equation XOften reported as R 2 instead of R

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R 2R 2R 2R 2

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Regression Coefficients Slopes and an intercept.Slopes and an intercept. Each variable adjusted for all others in the model.Each variable adjusted for all others in the model. Just an extension of slope and intercept in simple regressionJust an extension of slope and intercept in simple regression SPSS output on next slideSPSS output on next slide

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Slopes and Intercept

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Regression Equation A separate coefficient for each variableA separate coefficient for each variable XThese are slopes An intercept (here called b 0 instead of a)An intercept (here called b 0 instead of a)

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Interpretation Note slope for Witness and Stress are positive, but slope for Social Support is negative.Note slope for Witness and Stress are positive, but slope for Social Support is negative. XDoes this make sense? If you had two subjects with identical Stress and SocSupp, a one unit increase in Witness would produce unit increase in Internal.If you had two subjects with identical Stress and SocSupp, a one unit increase in Witness would produce unit increase in Internal. Cont.

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Interpretation--cont. The same holds true for other predictors.The same holds true for other predictors. t test on two slopes are significantt test on two slopes are significant XSocSupp not significant. XElaborate R 2 has same interpretation as r 2.R 2 has same interpretation as r 2. X13.6% of variability in Internal accounted for by variability in Witness, Stress, and SocSupp.

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Interpretation--cont. Intercept usually not meaningful.Intercept usually not meaningful. XPrediction when all predictors are 0.0

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Predictions Assume Witness = 20, Stress = 5, and SocSupp = 35.Assume Witness = 20, Stress = 5, and SocSupp = 35.

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Hypothesis Testing Test on R 2 given in Analysis of Variance tableTest on R 2 given in Analysis of Variance table Cont.

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Testing--cont. Tests on regression coefficients given along with the coefficients.Tests on regression coefficients given along with the coefficients. See next slideSee next slide Note tests on each coefficient.Note tests on each coefficient.

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Testing Slopes and Intercept

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Raw vs. Standardized Betas Raw beta weights are in their original units (not standardized)Raw beta weights are in their original units (not standardized) XUse for prediction Standardized beta weights are not in their original unitsStandardized beta weights are not in their original units XUse for comparing predictors

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Procedures for Entering Predictor Variables SimultaneousSimultaneous XEnter all predictors simultaneously ForwardForward XEnter 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)

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Procedures for Entering Predictor Variables Continue BackwardBackward XEnter 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) StepwiseStepwise XCombing forward and backward

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Procedures for Entering Predictor Variables Continued HierarchicalHierarchical XEnter 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?E.g., does social support influence internalizing above and beyond stress and witnessing violence?

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Stepwise Results for Kliewer data

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Potential Problems Multicollinearity – large correlations between predictorsMulticollinearity – large correlations between predictors XVIF (Variance Inflation Factor) < 10 XDoesn’t qualify overall model but estimates of individual predictors can be unstable Uncorrelated residualsUncorrelated residuals XDurban-Watson test (0-4): 1.5 – 2.5 generally acceptable

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