# Longitudinal data with nonrandom missing values

I'm working on experiment where two types of terminally ill mice are treated with two different drugs. Then physiological measurements are taken in four time points. Due to the disease some mice dropout during the experiment. My question is if I should treat this absence of data as missing at random or not missing at random? Eventually the drop out ratio is equal between the groups but obviously not equal between time points (i.e. if mouse is removed in time point 2 then it misses time point 3 and 4.

If I can treat this as missing at random I'd use mixed effect model. Otherwise I cannot see the way how to treat this lack of data.

For some clarity below is the data snippet.

ID Drug Time Value
1 D1 T1 20.1
1 D1 T2 17.5
1 D1 T3 17.4
1 D1 T4 16.2
2 D1 T1 24.5
2 D1 T2 14.3
2 D1 T3 NA
2 D1 T4 NA

• What do you know about would happen after the drop-out? E.g. would one think the same trends as before would continue? Or are animals doing so badly that results would be especially bad thereafter? I guess the animals would not have continued to be given the drug or were they, just the measurement is not there? Commented Jan 27, 2023 at 12:31
• I should have been more precise. The mice just die so further measurements are not possible to obtain. Commented Jan 27, 2023 at 13:02
• Ah, in that case, are you interested in the number you would measure, if you measured it in a dead mouse? Are you interested in the number, if the mouse had been doing as badly as it really did, but it somehow just about survived to be measured? Or if hypothetically it was doing about as well as when it was last measured and had hypothetically not declined, even though we know that is not what really happened and the mouse did do so badly that it died? Commented Jan 27, 2023 at 16:41
• Thanks @Björn for your help. The death of mouse is the worst outcome. I would definitely prefer situation were I have complete measurements in all time points. However due to the disease it is rather expected that some mice would die. Hypothetically I would be interested in measuring dead mouse because the main question is whether the measured value is different between the two drugs. Commented Jan 29, 2023 at 20:39

This is a situation where the estimand (that which is to be estimated by as suitable estimator to get an estimate for the specific dataset) concept is useful. What would we even want, if we had the complete data?

Since it's the actual measurements including the ones after the "intercurrent event" of death (I assume the measurement is still possible after death, otherwise the situation is even more complicated and e.g. composite or other approaches may be needed), the question would be whether the trends on the data would be extrapolated sensibly into that situation by a mixed effects model. If you don't have any data in that situation, you'd have to base such a decision on judgment.

You could instead of a mixed effects model for repeated measures (MMRM) with an unstructured covariance matrix (which would implicitly impute missing data under an assumption that not much changes after data become missing and that extrapolates the trends in the data) use multiple imputation (MI) to see what a MMRM implicitly imputes (with a suitable setup including the right interactions in the MMRM, it behaves almost exactly like MI with a very large number of imputations). If those imputations don't look sensible/don't capture what should change after the intercurrent event, then you know that a mixed effects model will not behave sensibly.

Instead, a common strategy is then to set-up an imputation model that imputes in some sensible way. One typical idea in clinical trials in humans (if one is interested in a treatment policy estimand of "what really happened irrespective of whether patients stayed on their assigned treatment) would be to impute data after patients discontinue treatment and leave the trial either based on those that discontinued treatment and stayed in the trial, or based on what happens without treatment such as in a placebo group. Instead, a MMRM applied to on-treatment data would in this situation estimate what would happen, if hypothetically everyone had stayed on the treatment (while using MMRM with a mix of on- and off-treatment data results in an uninterpretable mess).

It's a little less clear how to impute what happens after death (which one would assume to be a major event that could really change what happens to the mouse's body thereafter), if you don't have data on that from any of the mice (but perhaps there's literature on that?).

• Thank you. I guess I will check if MMRM and MI look sensible with the data (What should be done if neither MI nor MMRM would give reasonable outcome? Should I just simply discard mice with any missing time point?). Of course since mice don't have option to opt out from trail I don't have any data on discontinued treatment. Commented Jan 30, 2023 at 14:54
• Discarding mice with missing data (aka "complete case analysis") is almost certainly a terrible option. It would only be valid, if the reason for missingness had not relationship whatsoever with how the values that are missing look like ("missing completely at random"), this would mean that the presumably disease-related measurements must not be higher or lower (or more variable or anything else) in individuals that die of the disease (before or after their death). I'd assume that's extremely unlikely to be satisfied. Commented Jan 30, 2023 at 15:49
• A keyword for the scenario where MMRM/MI out of the box don't do something sensible is "missing not at random" (MNAR). There's many MNAR related approaches to what one can do, e.g. using MI makes it easier to adjust the imputed values in a suitable way after their imputation (assuming you know how to/can justify the adjustment). E.g. if you think that after some intercurrent event values should be lower by X than those imputed by MI, you can subtract X from the imputed values. The difficulty is more in justifying what X should be, which why it's more popular to find relevant data for Commented Jan 30, 2023 at 15:51
• the missing data would have been like (e.g. any historically available, or made-in-your-study measurements on dead mice). That corresponds to (to return to the clinical trial in humans) imputing after treatment discontinuation based on either the control group of the experiment or based on those that discontinued treatment ("jump-to-reference imputation") & still had measurements ("retrieved dropout imputation"), but it's hard to know whether there is an obvious corresponding option in your situation. Commented Jan 30, 2023 at 15:54
• Other approaches include switching to a composite outcome, where outcomes are some number of ordered categories, where death is the worst category (e.g. like in this example that also covers longitudinal aspects). Multiple categories is usually better than just "good" vs. "bad" outcome. There have also been attempts to make this work with continuous data, but I don't know how exactly that would work with longitudinal data. Commented Jan 30, 2023 at 15:58