GAM(M) for longitudinal measurements from paired control treatment samples in R

Question

I have a test dataset looking like;

ID	Sample	Group	Week
X1	Sample1	Healthy	0
X1	Sample2	Disease	0
X1	Sample3	Healthy	2
X1	Sample4	Disease	2
X1	Sample5	Healthy	5
X1	Sample6	Healthy	16
X1	Sample7	Disease	16
X2	Sample8	Healthy	0
X2	Sample9	Disease	0
X2	Sample10	Healthy	2
X2	Sample11	Disease	2
X2	Sample12	Healthy	5
X2	Sample13	Disease	5
X2	Sample14	Healthy	16
X2	Sample15	Disease	16
X3	Sample16	Healthy	0
X3	Sample17	Disease	0
X3	Sample18	Healthy	2
X3	Sample19	Disease	2
X3	Sample20	Healthy	5
X3	Sample21	Disease	5
X3	Sample22	Healthy	16
X3	Sample23	Disease	16
X4	Sample24	Disease	0
X4	Sample25	Healthy	2
X4	Sample26	Disease	2
X4	Sample27	Healthy	5
X4	Sample28	Disease	5
X4	Sample29	Healthy	16
X4	Sample30	Disease	16

The ID columns define each subject that paired samples collected from. There are total of four time points; 0, 2, 5, and 16.

As some part of the data were exampled above, I have 200 different log-transformed measurements taken from each paired sites at each time point, but not all subjects have complete sample sets. Assuming the measurements are not following linear trend, can you please suggest (to someone who is pretty new to the GAMs/GAMMs) how can I perform testing whether measurements have significantly different longitudinal trend between healthy and disease groups by taking care of the paired sampling strategy using GAM or GAMMs in R?

PS: I tried some models such as

gam(Measurement1 ~ Group + Week +
            s(ID, bs = 're'),
          data = data, method = 'REML')

but I need some other families to test rather than Gaussian.

Answer 1

You probably don't need a generalized additive model (GAM) for this study.

A smooth in a GAM can nicely handle a nonlinear association between a predictor and an outcome when your knowledge of the subject matter doesn't suggest a particular functional form. For example, if you had a large number of time points varying among subjects, a GAM for time (perhaps in the simple form of a restricted cubic spline) could be a good choice.

The model you propose, however, doesn't even include Week in a GAM smooth term. The way the model is written, you are assuming a strictly linear association between Week and your outcome. Without an interaction between Week and Group, you are further assuming that the only difference between the Group values is in the intercepts, not in the slopes.

As there are only 4 distinct Week values and two Group values in your data, there wouldn't be much to gain here by using a GAM smooth for Week and its interaction with Group. You can include Group and Week as fixed effects, along with a term for their interaction, to capture the fundamental structure of your data.

With such a model and default coding of predictors:

• the intercept will be the estimated outcome at Week0 for the reference level of Group;
• the Group coefficient will represent the difference between groups at Week0;
• the Week coefficients will be the differences from Week0 associated with the other Week values, for the reference level of Group
• the interaction coefficients will document whether there are differences over time between the healthy and disease groups, evaluating whether the Group difference is significant at each Week.

You do need to handle the within-individual correlations. Instead of the s(ID, bs = 're') term used for that in the gam() function of the R mgcv package, you could proceed instead with a mixed model with random effects for ID, or with a generalized least squares model if you have a reasonably well behaved continuous outcome measure. If you do have a reasonably well behaved continuous outcome measure, you won't need to use a generalized linear model with a family different from Gaussian.

Chapter 7 of Frank Harrell's Regression Modeling Strategies provides a useful overview of longitudinal data analysis. Although the emphasis is on generalized least squares, it nicely outlines the principles and the strengths and weaknesses of different approaches.