Modelling treatment strength in a mixed-effects model?

Question

I'm analyzing an experiment where the strength of the treatment varied by treatment day in an uncontrolled way. Specifically, we tested the response time of 11 non-overlapping groups honey bees on 4+ days, each. On any given day, only one group was tested under two experimental conditions: first unfed, then fed. How much a group of bees fed was out of our hands and varied considerably, hence the variable treatment strength.

My lmer() model currently implicitly includes treatment strength via the random effect (1 | group_id/day):

response_time ~ experimental_condition + 
(1 | group_id/subject_id) + (1 | group_id/day)

I'm interested in estimating the effect of experimental_condition on response_time and am wondering whether this model could be improved by incorporating how much a group fed per day? If so, how would I do that, and how can I handle the small number of days on which we were unable to measure how much the bees had consumed?

Answer 1

From the description, I don't see any problem with adding the variable treatment strength as a fixed effect. However, I think there are a few other issues here.

- Duplication of random intercepts:

Your random structure is:

(1 | group_id/subject_id) + (1 | group_id/day)

which expands to

(1 | group_id) + (1 | group_id:subject_id) + (1 | group_id) + (1 | group_id:day)

so the random intercepts for group_id are duplicated. This can cause problems with convergence. A more appropriate structure is:

(1 | group_id) + (1 | group_id:subject_id) + (1 | group_id:day)

- How to handle missing data:

For the days with missing treatment strength data, you could consider multiple imputation. It seems plausible that the missingness is NMAR (not missing at random, which is a problem for multiple imputation), particularly if the conditions that led to the missing data (like rain) are believed to directly affect the feeding behaviour or the amount of food consumed by the bees, which is the unobserved data. However, if you have recorded data on conditions like weather, and these conditions fully explain the missingness, then it can be considered MAR, in which case multiple imputation should be

- Consider interaction between treatment strength and experimental_condition:

You might want to consider an interaction term between experimental_condition and treatment_strength. This would allow the model to differentiate the effect of feeding amount based on the experimental condition (fed vs unfed).

- Treatment of the day variable`:

I'm a little bit puzzled by the day variable. You seem to be sure that it's nested within group, but that means that each day "belongs" to one and only one group, there are multiple days "belonging" to each group, and no other groups were tested on the same day. If that's not the case then they should be crossed random effects, not nested.