Regression Models w/ 2-group & Quant Variables• Sources of data for this model • Variations of this model• Main effects version of the model– Interpreting the regression weight– Plotting and interpreting the model• Interaction version of the model– Composing the interaction term– Testing the interaction term = testing homogeneity of regression slope assumption– Interpreting the regression weight– Plotting and interpreting the model• Plotting more complex modelsAs always, “the model doesn’t care where the data come from”. Those data might be …• a measured binary variable (e.g., ever- vs. never-married) and a measured quant variable (e.g., age)• a manipulated binary variable (Tx vs. Cx) and a measured quant variable (e.g., age)• a measured binary variable (e.g., ever- vs. never-married) and a manipulated quant variable (e.g., 0, 1, 2, 5, or 10 practices)• a manipulated binary variable (Tx vs. Cx) and a manipulatedquant variable (e.g., 0, 1, 2, 5, or 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 binary variable & quant variable• No interaction – assumes regression slope homogeneity• b-weights for binary & quant variables each represent main effect of that variable2. Interaction model• Terms for binary variable & quant variable• Term for interaction - does not assume reg slp homogen !!• b-weights for binary & 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 binary predictor• expected direction and extent of change in Y for a 1-unit increase in Z, after controlling for the other variable(s) in the model • main effect of Z• group Y-X regression line height differenceModels with a centered quantitative predictor & a dummy coded binary predictor y’ = b1X + b2Z + aThis is called a main effects model Æ there are no interaction terms.b1Æ regression weight for centered quant predictor• expected direction and extent of change in Y for a 1-unit increase in X after controlling for the other variable(s) in the model• main effect of X • Slope of Y-X regression line for both groupsModel Æ y’ = b1X + b2Z + ay’ = b1X + b2*0 + ay’ = b1X + ay’ = b1X + b2*1 + ay’ = b1X + ( b2+ a)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 Z = 0Substitute the 0 in for Z Simplify the formulaFor the Target Group coded Z = 1Substitute the 1 in for Z Simplify the formulaslopeheightslopeheight0 10 20 30 40 50 60ab1b2CxTx-20 -10 0 10 20 Å Xcena = ht of Cx lineÆ mean of Cxb1= slp of Cx lineb2= htdif Cx & TxÆ Cx & Tx mean difCx slp = Tx slpNo interactionXcen= X – XmeanZ = Tx1 vs. Cx(0)y’ = b1X + b2 Z + aPlotting & Interpreting Models with a centered quantitative predictor & a dummy coded binary predictorThis is called a main effects model Æ no interaction Æ the regression lines are parallel.0 10 20 30 40 50 60a-b1b2 = 0CxTx-20 -10 0 10 20 Å XcenXcen= X – XmeanZ = Tx1 vs. Cx(0)a = ht of Cx lineÆ mean of Cxb1= slp of Cx lineb2= htdif Cx & TxÆ Cx & Tx mean difCx slp = Tx slpNo interactiony’ = -b1X + -b2 Z + aPlotting & Interpreting Models with a centered quantitative predictor & a dummy coded binary predictorThis is called a main effects model Æ no interaction Æ the regression lines are parallel.0 10 20 30 40 50 60ab1= 0ab2CxTx-20 -10 0 10 20 Å Xcena = ht of Cx lineÆ mean of Cxb1= slp of Cx lineb2= htdif Cx & TxÆ Cx & Tx mean difCx slp = Tx slpNo interactionXcen= X – XmeanZ = Tx1 vs. Cx(0)Plotting & Interpreting Models with a centered quantitative predictor & a dummy coded binary predictory’ = b1X + b2 Z + aThis is called a main effects model Æ no interaction Æ the regression lines are parallel.In SPSS you have to compute the interaction term – as the product of the binary variable dummy code & the centered quantitative variableSo, if you have sex_dc (0=male & 1=female) and age_cencentered at its mean, you would compute the interaction as…compute age_sex_int = sex_dc * age_cen.• males will have age_sex_int values of 0• females will have age_sex_int values = their age_cen valuesModels 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”Testing the interaction/regression homogeneity assumption…There are two equivalent ways of testing the significance of theinteraction term:1. The t-test of the interaction term will tell whether or not b=02. A nested model comparison, using the R2Δ F-test to compare the main effect model (dummy-coded binary variable & centered quant variable) with the full model (also including the interaction product term)These are equivalent because t2= F, both with the same df & p.Retaining H0: means that • the interaction term does not contribute to the model, after controlling for the main effects• which can also be called regression homogeneity.Interpreting the interaction regression weightIf the interaction contributes to the model, we need to know how to interpret the regression weight for the interaction term.We are used to regression weight interpretations that read like,“The direction and extent of the expected change in Y for a 1-unit increase in X, holding all the