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Currently I am analyzing a multiple regression model to test the effect of 2 continuous predictors (P1 and P2) and 1 discrete predictor (P3, two levels) on a continuous response variable (RV). The model has the following form:

lm(RV~ P1*P2*P3, data)

The assumptions of homoscedasticity and normality do not seem to be seriously violated, as shown by the diagnostic plots (residual plot and QQ plot). Furthermore, there is no serious case of multicollinearity (variance inflation factor VIF <10). I decided to construct all possible interaction terms since it makes sense conceptually.

Now for my questions: How should I visualize and interpret the interation terms when (1) all predictors as well as all 2-way interactions are significant and (2) all predictors, all 2-way interactions as well as the 3-way interaction are significant. For multiple regression models with only 2 predictors, visualization and interpretation of a significant interaction is relatively straightforward and well documented. However, for a case with 3 predictors (especially with multiple interactions), I have not been able to find a satisfactory solution. In case of a 3-way-ANOVA i would decompose it into simple two-way interactions, simple simple main effect or simple simple pairwise comparisons (depending on the significance of the interactions) with appropriate posthoc tests. Does anyone know how to proceed in these two multiple regression cases?

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Choose sets of predictor values based on your understanding of the subject matter, and display estimated results for them. Make sure that the combinations of predictor values make sense and aren't extrapolations beyond the data you have.

Consider a set of plots.

For example, you could show side-by-side contour plots of the estimated outcome as a function of the two continuous predictors, one plot for each level of your discrete predictor.

You could plot estimated outcome versus one continuous predictor for each level of the binary predictor, at a set of representative discrete values for the second continuous predictor. Use color coding or line types to distinguish the levels of the binary predictor and the chosen discrete values of the second continuous predictor. Then produce another plot that switches the continuous/discrete representation of the two continuous predictors. That has the advantage that confidence intervals could also be displayed.

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