Can a linear mixed model handle missing data that is not “missing-at-random”?

Question

I'm using a linear mixed model (R,lme4) to evaluate the effect of drug dose on auditory thresholds in birds across a range of 14 frequencies: lmer(Threshold ~ Frequency + Condition + (1|Subject)

We are missing observations at the higher frequencies for a variety of reasons. A few cases are because we tested the first few subjects with a narrower range of frequencies. The majority of missing observations are due to the drug treatment increasing thresholds outside the range at which we can test them (our speaker can only get so load). Missing observations increase with frequency. At the highest 3 frequencies, we have 0 observations for the drug conditions. The attached plot compares our controls to one dose of the drug. The numbers in parentheses indicate proportion of subjects for which we have a response at the given frequency (we have 100% of responses below 500 Hz for all subjects).

Is there a way for the LMM to handle this data and allow us to make an inference about the difference in thresholds between conditions at the higher frequencies or would it best to use a different kind of model that deals with binary outcomes?

Here is the data: CSV file

Answer 1

As you say, linear mixed-effects (LME) models are assuming that any missing data are missing at random (MAR). A way to account for suspected violations of the MAR assumption of LME models is to use joint models. Under this framework, the longitudinal response and the survival probability are associated by jointly estimating a single likelihood function. Using a joint model, you can get an estimate of how much a possible difference in survival is affecting the longitudinal response and how much the fluctuations in the longitudinal response are associated with survival.