# Smooth bivariate interaction decomposition in GAM models

by Isabella Ghement   Last Updated October 19, 2019 20:19 PM - source

Background

Consider the following additive model containing a smooth bivariate interaction term:

$$y = f(x,z) + \epsilon$$

where y is a continuous outcome variable, x and z are continuous predictor variables and $$\epsilon$$ is a random error term following a Normal distribution with mean 0 and unknown variance $$\sigma^2$$. Furthermore, $$f$$ is an unknown smooth bivariate function. In R, such a model would be fitted with the commands:

library(mgcv)

model.1 <- gam(y ~ te(x,z), family=gaussian, data=data)


The mgcv help file https://www.rdocumentation.org/packages/mgcv/versions/1.8-24/topics/gam.models mentions that sometimes it is interesting to allow $$f$$ to have a main effects + interaction structure of the form $$f(x,z) = f_1(x) + f_2(z) + f_3(x,z)$$, meaning that the model would be expressed as:

$$y = f_1(x) + f_2(z) + f_3(x,z) + \epsilon$$

and fitted with the R command:

model.1 <- gam(y ~ ti(x) + ti(z) + ti(x,z), family=gaussian, data=data)


Questions

Q1: Generally speaking, under what circumstances would it be interesting to consider the main effects + interaction formulation for a bivariate smooth interaction? How would we know for a specific data set whether those circumstances were applicable? In particular, are those circumstances supposed to be driven by the data or by the research questions?

Q2: Assuming the circumstances in question are applicable for a data set, what is the interpretation of the main effects $$f_1(x)$$ and $$f_2(z)$$? For example, can we say that $$f_1(x)$$ represents the overall nonlinear effect of $$x$$ on the the outcome variable $$y$$, averaged across the values of $$z$$, while $$f_2(x)$$ represents the overall nonlinear effect of $$z$$ on the outcome variable $$y$$, averaged across the values of $$x$$? Also, would we need to worry in this context about the main smooth effects not necessarily being interpretable in the presence of a smooth interaction?

Q3: What is the interpretation of the smooth interaction term $$f_3(x, z)$$? Intuitively, how is this interpretation different from the interpretation of the smooth interaction term $$f(x, z)$$?

Many thanks for any insights you can share.

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