INTERPRETING INTERACTIONS AMONG QUANTITATIVE VARIABLES Interactions 2 way X x Z Association between DV and X changes in magnitude and or direction across values of Z Association between DV and X is conditional on values of Z 3 way X x W x Z Association between DV and X x W changes in magnitude and or direction across values of Z Association between DV and X is conditional on values of W and Z Forming an Interaction 2 Step Process 1 Form Product Term XZ X Z This is not the interaction It is confounded with X and Z 2 Partial all lower order constituent effects Y B0 B1 X B2 Z B3 XZ B3 reflects the interaction Process creates multicollinearity that negatively affects test of lower order terms when product term is included in model Today s procedure assumes centered lower order terms Decomposing a 2 way Interaction Y B0 B1 X B2 Z B3 XZ A significant B3 indicates that association b w Y and X changes across values of Z vice versa for Y Z To explain the interaction w will test the simple slope of X at various values of Z Choosing Conditional Values of Z At least 2 simple slopes of X should be tested Conditional values of Z can be derived from 1 Clinical practice If Z is assessed on a normed scale with cut off values distinguishing between at risk vs normal population can test simple slope above and below cutoff scores on Z 2 Theory might dictate values of Z at which slope of X should be examined 3 Can assess simple slope of X at high and low values of Z such as 1 SD above and below the mean of Z Testing Simple Slope of X at Values of Z Each simple slope will create a separate regression of Y on X at values of Z E g Y B0 B1 X Y B0 B1 X at a high value of Z and at a low value of Z We will statistically test B1 at each value of Z to determine if each simple slope is significantly different than 0 Centering Provides Intuitive Meaning of 0 Y B0 B1 X B2 Z B3 XZ B1 holds only when Z 0 When Z is mean centered a value of zero corresponds to the mean of Z We will use this same logic of centering to obtain the simple slope of Y on X at low and high values of Z i e 1 SD below and above mean of Z Re center so 0 1SD below Above Z mean of Z 2 and SD 1 Subtracting 1 SD from each mean centered Z re centers Z at 1SD above Z mean i e the simple slope of Y on X is estimated at 1SD above Z mean Adding 1 SD to each mean centered Z re centers Z at 1SD below Z mean i e the simple slope of Y on X is estimated at 1SD below Z mean An Example Pretend we are interested in why persons like their own social groups Further pretend that self extension theory suggests that persons internalize their group as an aspect of the self i e a social identity and evaluate their group in regard to how they evaluate their self Such a theory predicts that self esteem and group identification interact to predict liking for the ingroup such that liking and self esteem are positively related for persons who identify strongly with the group and unrelated for persons who identify weakly with the group Bogus Data To test the theory we assessed liking for a group identification with the group and self esteem for 28 persons Each construct was measured on a valid scale that ranges from 1 7 with higher values indicating greater degrees of the construct Center Predictors and Form Product Term data temp set liking proc standard out center1 m 0 var ident se run data center1 set center1 id se ident se st Check Corr b w 1 Order Product Term proc corr var ident se id se run Centering reduced collinearity Simult Regression to Test Interaction proc reg title like ident se Ident x SE model like ident se id se run Like 4 14 0 49SE 0 13Ident 0 21Se Ident The interaction is significant B 21 t 24 5 68 p 0001 Decompose Interaction Test simple slope of Esteem at high low values of Ident Mean of Ident 0 centered SD can be obtained via proc means var ident run SD of Ident 2 0522281 Regress Liking on Self esteem at Ident 2 0522281 high identification Ident 2 052281 low identification Test Simple Slope 1 SD Above Mean Ident Re center Ident so it is centered 1 SD above the mean idhigh ident 2 0522281 Form new product term se idhigh se idhigh Liking B0 B1SE B2Idhigh B3SE Idhigh B1 simple slope of SE at high identification B0 Y intercept for simple slope Together B0 and B1 form the model at Ident High Test Simple Slope 1 SD above Mean Ident data simple set center1 idhigh ident 2 0522281 se idhigh se idhigh proc reg title simple slope of like se at high IDENT model like se idhigh se idhigh run Like 4 41 0 92SE At 1 SD above the mean of ident liking is positively and significantly associated with self esteem B 92 t 24 8 48 p 0001 Test Simple Slope 1 SD Below Mean Ident Re center Ident so it is centered 1 SD below the mean idlow ident 2 0522281 Form new product term se idlo se idlow Liking B0 B1SE B2Idlow B3SE Idlow B1 simple slope of SE at low identification B0 Y intercept for simple slope Together B0 and B1 form the model at Ident Low Test Simple Slope 1 SD Below Mean Ident data simple set center1 idlow ident 2 0522281 se idlow se idlow proc reg title simple slope of like se at low IDENT model like se idlow se idlow run Like 3 87 0 07SE At 1 SD below the mean of ident liking is not significantly associated with self esteem B 07 t 24 0 66 p 5178 Plotting the Interaction Liking and SE are significantly positively related for persons higher in identification and unrelated for persons lower in identification High Identification Like 4 41 0 92SE Low Identification Like 3 87 0 07SE Can use models to graphically display the interaction Plug centered SE scores into above models to generate predicted values To obtain predicted score for high identifiers who report SE score of 7 plug 3 i e 7 4 0 3 0 4 0 is mean of uncentered SE into equation Like 4 41 0 92 3 00 7 17 Repeat with various values of SE for both models and plot predicted scores Plotting the Interaction Summarizing the Results for a Publication We simultaneously regressed liking for the group on self esteem identification and the Esteem x Identification interaction Self esteem and identification were centered before the forming the product term to reduce collinearity among the main effects and the product …
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