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I am modeling time-series data (30 measurements) at the individual-level by a grouping factor (5 levels) and I have the following model specification from a generalized additive model (GAM):

bam(response ~ s(Time) + s(Time, fac, bs = "fs", m = 1))

My specific question concerns what the intercept term in such a model indicates. Here (GAM in R; an intercept term), Simpson indicated that with a smooth term, the intercept is the mean of the response. My concrete question is: With the inclusion of the factor smooth, what does the intercept represent, exactly? It doesn't appear to be the mean of the response, but close to (though not exactly) the mean of the means of the factor levels of the response variable. How can we interpret the intercept in such a model?

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  • $\begingroup$ So sorry, I did not phrase the first sentence well; I did not mean by = fac, but rather that the s(Time, fac, bs = "fs", m =1) is the non-linear random effect. $\endgroup$ Mar 27, 2023 at 16:02

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