How to interpret the grouping variable when having an lme with multiple random intercepts?

Question

I'm doing a linear mixed effects regression model. There are 80 subjects that are being observed over three days (some subjects are just observed on day 1, others on day 1 and 2, and others on day 1, 2 and 3). The total amount of observations is 202

I am rather new to mixed effect modelling but I understand that since observations among the same subject are somehow correlated, it is appropriate to add the Subject as a random intercept in the model. This is called the grouping variable if I am correct?

However, I also see that there is quite a difference on the response variable when making a boxplot for each of the days (subjects that are only examined for 1 day have a lower response variable than those that are examined on 2 or 3 days). It seems to me that this variable which indicates the day should also be included as a random intercept effect. Is it useful to do this, or rather pointless? Moreover, if you'd add the two as random intercepts, which one is seen as "the grouping variable"?

Answer 1

In principle you can add day as a random effect, but I would suggest adding it as a fixed effect for several reasons:

• only 3 levels (i.e. only three separate days on which individuals are measured) might not be enough to reliably estimate an among-day variance;
• similarly, with only 3 levels you don't gain much in the parsimony of the model (intercept + variance = 2 parameters for the random model vs intercept + (day2-day1) + (day3-day1) = 3 parameters for the fixed model)
• the 3 days might not be exchangeable, e.g. there might be a trend over time
• with crossed random effects of Subject and day you'd probably have to switch from nlme::lme to lme4::lmer. (With crossed random effects there are multiple grouping variables, which makes it hard for lme.)

So I'd suggest something like response~...+day, random = ~1|Subject ; if each individual is measured at most once per day, then it's not worth nesting day within Subject - that term will be confounded with the residual variance.

I'm not quite sure I understand your last paragraph. Do you have separate observations for each Subject-day combination, or just the total for each individual and the days on which it was observed? In other words, do your data look like this:

Subject day response
a       1   17
b       1   19
b       2   10
c       2   12
c       3   11

or this?

Subject days total
a       1    17
b       1,2  29
c       2,3  23

... or something else ?