Jarrett Byrnes writes,

A group of us are working through your Multilevel book, and a question has come up regarding models incorporating multiple predictors. We were working some of the chapters on using simulation to draw inference, but have been been puzzling over how one can represent their data with fitted and simulated lines, one factor at a time. True, one can show the fitted and simulated model for a variety of other factors that are not of interest, but this seems unsatisfactory, particularly if you are incorporating three or more predictors on your model. Do you have any suggestions as to how one can best present data and models for these more complex models, such that a reader can assess the relationship between the model and the data for each single predictor?

My reply: I think you have two questions:

1. How to display a fitted regression that has many input variables?

2. How to display a sequence of fitted models?

I’ll discuss each issue in turn. (Warning: for neither problem do I have a great answer yet.)

1. How to display a fitted regression that has many input variables? I’d start with curves of the expected value of y as a function of each input, using a separate plot for each input variable and multiple curves as necessary to show interactions. See, for example, the graphs on page 91 of our new book. With more than two inputs, I’d probably stack the graphs vertically. We’re still working on our general R function for this. And I’d also display the estimated coefficients (as in the lower graph on page 337 of our book), probably after standardizing the inputs by subtracting means and dividing by two standard deviations.

When a model has three-way interactions or many two-way interactions, the displays start to get tricky, and I have no great answer yet. I do think, though, that if we try harder we’ll gradually make progress here. Traditionally, graphical methods have focused on displaying raw data; as that same ingenuity is used to display inferences and fitted models, I think good new general plots will arise.

2. How to display a sequence of fitted models? This is a really important question, and again I don’t see anything great right now. The series of graphs for the arsenic example in chapter 5 of our book give some sense of what can be done, but we’re pretty disorganized there. It would be good to have a coherent display to visualize what happens to a model when a predictor is added, something like the graphs on page 12 of this paper. I will add, though, that I am not particularly interested in model selection or model averaging, at least as these concepts are typically formulated statistically. I’m more interested in putting together a good model and using simpler models as steps in understanding what the ultimate model is doing.