Suppose we obtain a MLR equation as follows:

Transcription

1 Psychology 8 Lecture #9 Outline Probing Interactions among Continuous Variables Suppose we carry out a MLR analysis using a model that includes an interaction term and we find the interaction effect to be statistically significant. It then becomes important to be able to interpret, understand, and communicate the meaning of the interaction effect. Brief review: Given IVs and, define 3 = MLR model: Y ˆ = B0 + B + B 3 Y ˆ = B0 + B + B Representing moderator effect: Y ˆ = B + B + ( B + B Review of example: 0 3 ) Suppose we obtain a MLR equation as follows: Y ˆ = Y ˆ =

2 Let us consider as the moderator and rewrite this equation as Y ˆ = B0 + B + ( B ) Y ˆ = (3+ 8 ) To assess the nature of the interaction, let us select two distinctly different values of the IV ; say and 0. We substitute each of these values of into the above regression equation. For =: Y ˆ = 5+ (6 ) + (3+ (8 )) Which reduces to: Y ˆ = This equation represents the regression of Y on when =. For =0: Y ˆ = 5+ (6 0) + (3+ (8 0)) Which reduces to: Y ˆ = This equation represents the regression of Y on when =0. The two simple regression equations just shown represent the regression of Y on for selected values of, the moderator variable. The difference in the coefficient for reflects the nature and strength of the interaction.

3 We now consider how to apply this approach in general. Given a regression model of the form Y ˆ = B0 + B + B 3 Y ˆ = B0 + B + B Y ˆ = B0 + B + ( B ) Select several values of the moderator variable. Typically we choose values representing high, medium, and low values from the sample range. If IVs are standardized, it is common to select values representing the mean (0) and one standard deviation above and below the mean (+.0 and -.0). Once values of are chosen, for each selected value in turn we substitute it into the last equation shown above and simplify the equation. The result will be a simple linear regression equation representing Y as a function of. We will obtain one such equation for each selected value of. Depending on the strength of the interaction effect, these equations will exhibit distinctly different regression coefficients, which are called simple slopes. It is then especially useful to plot these equations as lines on a single plot of and Y. The difference in the lines shows the nature of the interaction effect. 3

4 4 An excellent online calculator and plotter are available to carry out this process. ( Comment on centering: The simple slopes obtained by this process will not be affected by centering of the IVs. The intercepts in the separate simple regression equations will be affected. Finally, note that the procedure described above can be reconfigured reversing the roles of and so that is the moderator and the simple slopes are associated with the effect of on Y. Testing significance of simple slopes For any given equation containing a simple slope it may be of interest to determine whether the simple slope is significantly different from zero. Conducting this test requires that we obtain the standard error of the simple slope. Designating as the moderator, we need the standard error of the regression coefficient for at a given value of. The procedure for obtaining this standard error is described in the text. Call this value SE B at.

5 5 Once this value is obtained, we can test the significance of a simple slope in the usual fashion using a t-statistic: t ( B SE B at ) = with (n-k-) degrees of freedom This test can be applied to a simple slope obtained for any selected value of. Obviously there would be an infinite number of possible values of that could be selected, with an outcome of this test available for each such value. This concept suggests that there may be a range of across which the simple slope for would be significant, and a range across which it would not be significant. That is, we can consider the question: Across what range of would the simple slope for be significant? More substantively, across what range of would the effect of on Y be significant? This region of significance can be computed and plotted using an online calculator to be demonstrated. (

6 6 Confidence intervals for simple slopes For any selected value of we can obtain a confidence interval for the simple slope associated with by usual procedures: * * ( B ) tα / SEB ( B + B ) ( B + B ) + t SEB at at 3 3 α / This CI provides an interval estimate of the effect of on Y at a selected value of. Obviously we could obtain such a CI for an infinite number of values of. At each value of the CI would exhibit a different width. Such an infinite series of CIs could be represented as a confidence band, showing the CI for the simple slope as it changes across values of. These confidence bands can also be computed and plotted using an online calculator to be demonstrated in class. ( In general, we can use the tools described here to probe interactions by computing simple slopes, plotting and comparing the resulting regression lines, and investigating regions of significance and confidence bands for simple slopes.