Multiple Regression Models• Advantages of multiple regression• Important preliminary analyses• Parts of a multiple regression model & interpretation• Raw score vs. Standardized models• Differences between r, bbiv, bmult& βmult• Steps in examining & interpreting a full regression model• Underspecification & Proxy Variables• Searching for “the model”Advantages of Multiple RegressionPractical issues …• better prediction from multiple predictors• can “avoid” picking/depending on a single predictor• can “avoid” non-optimal combinations of predictors (e.g., total scores)Theoretical issues …• even when we know in our hearts that the design will not support causal interpretation of the results, we have thoughts and theories of the causal relationships between the predictors and the criterion -- and these thoughts are about multi-causal relationships• multiple regression models allow the examination of more sophisticated research hypotheses than is possible using simple correlations• gives a “link” among the various correlation and ANOVA modelsBefore launching into the various hypotheses tests and other types of analyses, be sure to “get familiar” with your data and determine if it has any “problems” …1. Perform appropriate data checking & cleaning • non-normality, outliers & nonlinearities?2. Get means and standard deviations for each variable • do they “make sense” for these measures & this population?3. Consider the correlations of each variable with the criterion• do they “make sense” for these measures & this population?4. Consider the correlations among the predictors (collinearities)• do they make sense for these measures & this population? • will there be a “collinearity problem” ?raw score regression y’ = b1x1+ b2x2 + b3x3+ aeach b• represents the unique and independent contribution of that predictor to the model• for a quantitative predictor tells the expected direction and amount of change in the criterion for a 1-unit change in that predictor, while holding the value of all the other predictors constant • for a binary predictor (with unit coding -- 0,1 or 1,2, etc.), tells direction and amount of group mean difference on the criterion variable, while holding the value of all the other predictors constant a• the expected value of the criterion if all predictors have a valueof 0 Let’s practice -- Tx (0 = control, 1 = treatment)depression’ = (2.0 * stress) - (1.5 * support) - (3.0 * Tx) + 35• apply the formula patient has stress score of 10, support score of 4 and was in the treatment group dep’ = • interpret “b” for stress -- for each 1-unit increase in stress, depression is expected to by , when holding all other variables constant• interpret “b” for support -- for each 1-unit increase in support, depression is expected to by , when holding all other variables constant• interpret “b” for tx – those in the Tx group are expected to have a mean depression score that is than the control group, when holding all other variables constant• interpret “a” -- if a person has a score of “0” on all predictors, their depression is expected to be standard score regression Zy’= βZx1+ βZx2+ βZx3each β• for a quantitative predictor the expected Z-score change in the criterion for a 1-Z-unit change in that predictor, holding the values of all the other predictors constant • for a binary predictor, tells size/direction of group mean difference on criterion variable in Z-units, holding all other variable values constantAs for the standardized bivariate regression model there is no “a”or “constant” because the mean of Zy’ always = Zy = 0The most common reason to refer to standardized weights is when you (or the reader) is unfamiliar with the scale of the criterion. A second reason is to promote comparability of the relative contribution of the various predictors (but see the important caveat to this discussed below!!!).It is important to discriminate among the information obtained from ...bivariate r & bivariate regression model weightsr -- simple correlationtells the direction and strength of the linear relationship between two variables (r = β for bivariate models)r2-- squared correlation tells how much of the Y variability is “accounted for,”. “predicted from” or “caused by” X (r = β for bivariate models)b -- raw regression weight from a bivariate modeltells the expected change (direction and amount) in the criterion for a 1-unit change in the predictorβ -- standardized regression wt. from a bivariate modeltells the expected change (direction and amount) in the criterion in Z-score units for a 1-Z-score unit change in that predictor, holding the value of all the other predictorsconstantIt is important to discriminate among the information obtained from ...multivariate R & multivariate regression model weightsR2-- squared multiple correlation tells how much of the Y variability is “accounted for,”. “predicted from” or “caused by” the multiple regression modelR -- multiple correlation (not used that often)tells the strength of the relationship between Y and the . multiple regression modelbi-- raw regression weight from a multivariate modeltells the expected change (direction and amount) in the criterion for a 1-unit change in that predictor, holding the value of all the other predictors constantβi-- standardized regression wt. from a multivariate modeltells the expected change (direction and amount) in the criterion in Z-score units for a 1-Z-score unit change in that predictor, holding the value of all the other predictors constantyx1x2x3Venn diagrams representing r, b and R2ry,x1ry,x2ry,x3yx1x2x3Remember that the b of each predictor represents the part of that predictor shared with the criterion that is not shared with any other predictor -- the unique contribution of that predictor to the modelbx1 & βx1bx2& βx2bx3& βx2yx1x2x3Remember R2is the total variance shared between the model (all of the predictors) and the criterion (not just the accumulation of the parts uniquely attributable to each predictor).R2= + + +Inspecting and describing the results of a multiple regression formula …0. Carefully check the bivariate correlations/regressions1. Does the