Regression Models w/ k-group & Quant Variables• Sources of data for this model • Variations of this model• Main effects version of the model– Interpreting the regression weights– Plotting and interpreting the model• Interaction version of the model– Composing the interaction terms– Testing the interaction term = testing homogeneity of regression slope assumption– Interpreting the regression weights– Plotting and interpreting the modelAs always, “the model doesn’t care where the data come from”. Those data might be …• a measured k-group variable (e.g., single, married, divorced) and a measured quant variable (e.g., age)• a manipulated k-group variable (Tx1 vs. Tx2 vs. Cx) and a measured quant variable (e.g., age)• a measured k-group variable (e.g., single, married, divorced) and a manipulated quant variable (e.g., 0, 1, 2, 5,10 practices)• a manipulated binary k-group variable (Tx1 vs. Tx2 vs. Cx) and a manipulated quant variable (e.g., 0, 1, 2, 5, 10 practices)Like nearly every model in the ANOVA/regression/GLM family –this model was developed for and originally applied to experimental designs with the intent of causal interpretability !!!As always, causal interpretability is a function of design (i.e., assignment, manipulation & control procedures) – not statistical model or the constructs involved !!!There are two important variations of this model1. Main effects model• Terms for the k-group variable & quant variable• No interaction – assumes regression slope homgeneity• b-weights for k-group & quant variables each represent main effect of that variable2. Interaction model• Terms for k-group variable & quant variable• Term for interaction - does not assume reg slp homogen !!• b-weights for k-group & quant variables each represent the simple effect of that variable when the other variable = 0• b-weight for the interaction term represented how the simple effect of one variable changes with changes in the value of the other variable (e.g., the extent and direction of the interaction)a Æ regression constant• expected value of y if all predictors = 0• mean of the control group (G3)• height of control group Y-X regression lineb2Æ regression weight for dummy coded comparison of G1 vs G3• expected direction and extent of change in Y for a 1-unit increase in Z1, after controlling for the other variable(s) in the model • main effect of Z1• Y-X reg line height difference for G1 & G3Models with a centered quantitative predictor & a dummy coded k-category predictor y’ = b1X + b2Z1+ b3Z2+ ab1Æ regression weight for centered quant predictor• expected direction and extent of change in Y for a 1-unit increase in X aftercontrolling for the other variable(s) in the model• main effect of X• slope of Y-X regression line for all groupsb3Æ regression weight for dummy coded comparison of G2 vs. G3• expected direction and extent of change in Y for a 1-unit increase in Z2, after controlling for the other variable(s) in the model• main effect of Z2• Y-X regline height difference for G2 & G3Group Z1Z21 1 02 0 13* 0 0This is called a main effects model Æ there are no interaction terms.To plot the model we need to get separate regression formulas for each Z group. We start with the multiple regression model…For the Comparison Group coded Z1= 0 & Z2= 0Model Æ y’ = b1X + b2Z1+ b3Z2+ aSubstitute the Z code values y’ = b1X + b2*0 + b3*0 + aSimplify the formula y’ = b1X + aFor the Target Group coded Z1= 1 & Z2= 0Substitute the Z code values y’ = b1X + b2*1 + b3*0 + aSimplify the formula y’ = b1X + (b2+ a)For the Target Group coded Z1= 0 & Z2= 1Substitute the Z code values y’ = b1X + b2*0 + b3*1 + aSimplify the formula y’ = b1X + (b3+ a)Group Z1Z21 1 02 0 13* 0 0slopeheightslopeheightslopeheight0 10 20 30 40 50 60ab1-b2CxTx1Tx2b3-20 -10 0 10 20 Å XcenZ1= Tx1 vs. Cx(0) Z2= Tx2 vs. Cx (0)Xcen= X – XmeanPlotting & Interpreting Models with a centered quantitative predictor & a dummy coded k-category predictor y’ = b1X + -b2Z1+ b3Z2+ aa = ht of Cx lineÆ mean of Cxb1= slp of Cx lineCx slp = Tx1slp = Tx2slpNo interactionb2= htdif Cx & Tx1Æ Cx & Tx1mean difb3= htdif Cx & Tx2Æ Cx & Tx2mean difThis is called a main effects model Æ no interaction Æ the regression lines are parallel.0 10 20 30 40 50 60ab1 = 0CxTx1Tx2b3b2 = 0-20 -10 0 10 20 Å XcenZ1= Tx1 vs. Cx(0) Z2= Tx2 vs. Cx (0)Xcen= X – XmeanPlotting & Interpreting Models with a centered quantitative predictor & a dummy coded k-category predictor y’ = b1X + b2Z1+ b3Z2+ aa = ht of Cx lineÆ mean of Cxb1= slp of Cx lineCx slp = Tx1slp = Tx2slpNo interactionb2= htdif Cx & Tx1Æ Cx & Tx1mean difb3= htdif Cx & Tx2Æ Cx & Tx2mean difThis is called a main effects model Æ no interaction Æ the regression lines are parallel.0 10 20 30 40 50 60aa-b1CxTx2Tx1b3b2 = 0-20 -10 0 10 20 Å XcenZ1= Tx1 vs. Cx(0) Z2= Tx2 vs. Cx (0)Xcen= X – XmeanPlotting & Interpreting Models with a centered quantitative predictor & a dummy coded k-category predictor y’ = -b1X + b2Z1+ b3Z2+ aa = ht of Cx lineÆ mean of Cxb1= slp of Cx lineCx slp = Tx1slp = Tx2slpNo interactionb2= htdif Cx & Tx1Æ Cx & Tx1mean difb3= htdif Cx & Tx2Æ Cx & Tx2mean difThis is called a main effects model Æ no interaction Æ the regression lines are parallel.Models with InteractionsAs in Factorial ANOVA, an interaction term in multiple regression is a “non-additive combination”• there are two kinds of combinations – additive & multiplicative• main effects are “additive combinations”• an interaction is a “multiplicative combination”In SPSS you have to compute the interaction term – as the product of each dummy code for the k-group variable & the centered quantitative variableThe 3-group variable coded as on the rightand a centered quant variable age_cen, thenyou would compute 2 interaction terms as …compute age_mar1_int = mar1 * age_cen.compute age_mar2_int = mar2 * age_cenGroup