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UT Knoxville STAT 201 - 7) sld_interaction_quantitative

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INTERPRETING INTERACTIONSAMONG QUANTITATIVE VARIABLESInteractions-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 ZForming an Interaction-2-Step Process(1) Form Product TermXZ = X*Z(This is not the interaction. It is confounded with X and Z)(2) Partial all lower order constituent effectsXZBZBXBBY3210ˆ(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 InteractionXZBZBXBBY3210ˆ-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 ZTesting Simple Slope of X at Values of Z-Each simple slope will create a separate regression of Y on X at values of ZE.g, XBBY10ˆ at a high value of ZandXBBY10ˆ at a low value of ZWe 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 0XZBZBXBBY3210ˆ-B1 holds only when Z = 0-When Z is mean centered, a value of zero corresponds to themean of Z.-We will use this same logic of centering to obtain the simpleslope 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 theyevaluate 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 Termdata temp;set liking;proc standard out=center1 m=0; var ident se;run;data center1;set center1;id_se=ident*se;Check Corr. b/w 1st Order & Product Termproc corr; var ident se id_se; run;Centering reduced collinearitySimult. Regression to Test Interactionproc reg;title "like = ident se Ident x SE";model like = ident se id_se;run;Like = 4.14 + 0.49SE + 0.13Ident + 0.21Se_IdentThe interaction is significant (B=.21), t(24) = 5.68, p =.0001Decompose 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 meanidhigh = ident – 2.0522281-Form new product termse_idhigh = se*idhighLiking = B0 + B1SE +B2Idhigh + B3SE_IdhighB1 = simple slope of SE at high identificationB0 = Y-intercept for simple slopeTogether B0 and B1 form the model at Ident HighTest Simple Slope 1 SD above Mean Identdata 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 meanidlow = ident – (-2.0522281)-Form new product termse_idlo = se*idlowLiking = B0 + B1SE +B2Idlow + B3SE_IdlowB1 = simple slope of SE at low identificationB0 = Y-intercept for simple slopeTogether B0 and B1 form the model at Ident LowTest Simple Slope 1 SD Below Mean Identdata 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 identificationHigh 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) = 7.17Repeat with various values of SE for both models and plot predicted scoresPlotting the InteractionSummarizing the Results for a PublicationWe simultaneously regressed liking for the group on self-esteem, identification, and the Esteem x Identification interaction. Self-esteem andidentification were centered before the forming the product term to reducecollinearity among the main effects and the product


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UT Knoxville STAT 201 - 7) sld_interaction_quantitative

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