Testing SE x Ident 1 SD ABOVE Mean of Threat Using SASTesting SE x Ident 1 SD ABOVE Mean of Threat Using SPSSTest se*id Under High Threat Using SPSSTest se*id Under High Threat Using SASTesting SE x Ident 1 SD BELOW Mean of Threat Using SASTesting SE x Ident 1 SD BELOW Mean of Threat Using SPSSTesting Simple Effect Of SE At 1 SD ABOVE Ident And 1 SD ABOVE Threat Using SASTesting Simple Effect Of SE At 1 SD ABOVE Ident And 1 SD ABOVE Threat Using SPSSSimple Effect Of SE At High Ident And High Threat Using SASSimple Effect Of SE At High Ident And High Threat Using SPSSTesting Simple Effect Of SE At 1 SD BELOW Ident And 1 SD ABOVE Threat Using SASTesting Simple Effect Of SE At 1 SD BELOW Ident And 1 SD ABOVE Threat Using SPSSSimple Effect Of SE At Low Ident And High Threat Using SASSimple Effect Of SE At Low Ident And High Threat Using SPSSTesting Simple Effect Of SE At 1 SD ABOVE Ident And 1 SD BELOW Threat Using SASTesting Simple Effect Of SE At 1 SD ABOVE Ident And 1 SD BELOW Threat Using SPSSSimple Effect Of SE At High Ident And Low Threat Using SASSimple Effect Of SE At High Ident And Low Threat Using SPSSTesting Simple Effect Of SE At 1 SD BELOW Ident And 1 SD BELOW Threat Using SASTesting Simple Effect Of SE At 1 SD BELOW Ident And 1 SD BELOW Threat Using SPSSSimple Effect Of SE At Low Ident And Low Threat Using SASSimple Effect Of SE At Low Ident And Low Threat Using SPSSHOW INTERACTION TERMS ARE FORMEDChoosing Conditional Values of ZA Bogus Data SetBivariate CorrelationsPlotting The InteractionSummarizing the Results for a Journal ArticleA Bogus Data SetBivariate Correlations Among PredictorsPlotting The 3-Way InteractionSummarizing the Results for a Journal ArticleCourse: Multiple Regression Topic: Interaction Among Quantitative Predictors 1Interpreting Interactions Among Quantitative PredictorsAs we’ve discussed previously, a two-way interaction among predictors X and Z indicates that the association between the DV and X changes in magnitude and / or direction across values of Z. In other words, the association between X and the DV is conditional on values of Z. Likewise, a three-way interaction among predictors W, X, and Z indicates that the association between the DV and the 2-way interaction between W and X changes in magnitude and/or direction across values of Z. In other words, with a 3-way interaction the association between the DV and W is conditional on values of X and Z. In today’s lecture, we will examine how to statistically probe (i.e., decompose) a significant interaction among continuous predictor variables. We will begin by decomposing a significant 2-way interaction and then demonstrate how the process generalizes to higher order interactions such as a 3-way interaction. Finally, we will conclude by demonstrating how to obtain the correct standardized solution from computer software for models that include interaction terms. Before beginning our discussion, however, let’s quickly review how interaction terms are formed.HOW INTERACTION TERMS ARE FORMEDInteraction terms are formed with a two-step process. First, a product term (e.g. XZ) is formed by multiplying the variables that are involved in the interaction (e.g., XZ=X*Z). Keep in mind that this product term is highly correlated with its lower order constituent effects and is not the interaction. The second step partials from the product term all lower order constituent effects.For example, the test of B3 in the following model reflects a test of the interaction because the lower order constituent effects have been partialled:XZBZBXBBY3210ˆKeep in mind, including the product term XZ in the model increases multicollinearity and poses a problem for testing the lower order effects (e.g., B1 and B2). One potential solution to this problem, as recommended by Aiken and West (1992), is to first center all first order predictors before forming product terms. The technique that we discuss today assumes that all first-order quantitative predictors have been centered.DECOMPOSING A 2-WAY INTERACTION BETWEEN CONTINUOUS PREDICTORSA significant B3, in the above model, indicates that the association between Y and X changes across values of Z (and the association between Y and Z changes across values of X). After determining that the interaction term is significant, the next step is to decompose it to determine how the association between Y and X changes across levels of Z (and/or vice versa forthe association between Y and Z). If X and Z were both categorical variables, as we thoroughly discussed in the ANOVA class, we could interpret the XxZ interaction by testing the simple effects of X in levels of Z. In the current context X and Z are both continuous variables. Nonetheless, we will follow a similar procedure for interpreting the XxZ interaction and test the slope of Y on X at different values of Z. The slope of Y on X at a specific value of Z is referred to as a simple slope.Course: Multiple Regression Topic: Interaction Among Quantitative Predictors 2Choosing Conditional Values of ZThe first step in testing simple slopes of Y on X is determining at what values of Z the slopes should be tested. Keep in mind that at least two simple slopes should be tested. The values of Z at which the slopes of Y on X are tested can be derived from clinical practice. If for example, Z is assessed with a normed scale with cutoff values that distinguish, say, at risk persons from not-at-risk persons or highly depressed from moderately depressed persons, the slopes can be examined above and below those cutoff values. Alternatively, if theory dictates specific values of Z at which the slopes should be examined, then values derived from theory should be used. Finally, in the absence of clinical practice and theory, the simple slopes can be assessed at high and low values of Z, such as at 1 standard deviation above and below the mean of Z.Keeping in mind that an X x Z interaction indicates that the slope of Y on X is different across values of Z will facilitate an understanding of how we will calculate the simple slopes of Yon X at different values of Z (i.e., low and high values of Z). Take for example the following regression equation:XZBZBXBBY3210ˆThe XZ interaction indicates that B1 (i.e., the slope of Y on X) holds only when Z = 0. That is, if we plug zero into the above equation for Z, the B2 and B3 terms drop out (i.e., multiplied by a Z value of 0) and
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