Course Multiple Regression Topic Interaction Among Quantitative Predictors 1 Interpreting Interactions Among Quantitative Predictors As 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 FORMED Interaction 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 Y B0 B1 X B2 Z B3 XZ Keep 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 PREDICTORS A 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 for the 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 2 Choosing Conditional Values of Z The 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 Y B0 B1 X B2 Z B3 XZ The 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 the equation simplifies to Y B0 B1 X As we discussed previously mean centering the X and Z values rescales X and Z such that a score of zero on each variable corresponds to their respective means That is mean centering provides an intuitive interpretation of B1 in the interaction model such that the slope of Y on X i e B1 occurs at the mean of Z i e 0 corresponds to the mean of Z when Z is mean centered We will use the logic and intuitive interpretations of centering when we estimate the simple slope of Y on X at low and high values of Z To estimate the simple slope of Y on X at a low and high value of Z respectively we will re center Z such that 0 corresponds to the desired low and high value In other words rather than centering Z at its mean we will center Z at the desired low and high values In the examples that we cover in the lecture we will use 1SD below and above the mean of Z to represent low and high values respectively That is we will explore how to estimate and test the slope of Y on X at 1SD below the mean of Z and at 1 SD above the mean of Z The following table provides an intuitive example of the various centering techniques we will use when testing and decomposing interactions I ve included this table because students are often confused as to why we add 1 SD to the Z variable to center 1 SD below the mean of Z and why we subtract 1 SD from the Z variable to center 1 SD above the mean of Z Course Multiple Regression Topic Interaction Among Quantitative Predictors 3 RAW Z 1 2 3 MEAN CENTERED Z 1 0 1 2 1 0 0 1 2 Z 2 2 is mean of Z CENTERED 1SD ABOVE MEAN Z centered Z 1SD CENTERED 1SD BELOW MEAN Z centered Z 1 SD The top row contains the distribution of a variable e g Z which has raw scores of 1 2 and 3 Obviously the mean of this …
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