# Linear Mixed Models for Missing Data in Pre-Post Studies

In the past few months, I’ve gotten the same question from a few clients about using linear mixed models for repeated measures data.  They want to take advantage of its ability to give unbiased results in the presence of missing data.  In each case the study has two groups complete a pre-test and a post-test measure.  Both of these have a lot of missing data.

The research question is whether the groups have different improvements in the dependent variable from pre to post test.

As a typical example, say you have a study with 160 participants.

90 of them completed both the pre and the post test.

Another 48 completed only the pretest and 22 completed only the post-test.

Repeated Measures ANOVA will deal with the missing data through listwise deletion. That means keeping only the 90 people with complete data.  This causes problems with both power and bias, but bias is the bigger issue.

Another alternative is to use a Linear Mixed Model, which will use the full data set.  This is an advantage, but it’s not as big of an advantage in this design as in other studies.

The mixed model will retain the 70 people who have data for only one time point.  It will use the 48 people with pretest-only data along with the 90 people with full data to estimate the pretest mean.

Likewise, it will use the 22 people with posttest-only data along with the 90 people with full data to estimate the post-test mean.

If the data are missing at random, this will give you unbiased estimates of each of these means.

But most of the time in Pre-Post studies, the interest is in the change from pre to post across groups.

The difference in means from pre to post will be calculated based on the estimates at each time point.  But the degrees of freedom for the difference will be based only on the number of subjects who have data at both time points.

So with only two time points, if the people with one time point are no different from those with full data (creating no bias), you’re not gaining anything by keeping those 72 people in the analysis.

Compare this to a study I also saw in consulting with 5 time points.  Nearly all the participants had 4 out of the 5 observations.  The missing data was pretty random–some participants missed time 1, others, time 4, etc.  Only 6 people out of 150 had full data.  Listwise deletion created a nightmare, leaving only 6 people in the data set.

Each person contributed data to 4 means, so each mean had a pretty reasonable sample size.  Since the missingness was random, each mean was unbiased.  Each subject fully contributed data and df to many of the mean comparisons.

With more than 2 time points and data that are missing at random, each subject can contribute to some change measurements.  Keep that in mind the next time you design a study.

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1. Mariam says

Hi Karen,

I would like to know if you know in R using lme or lmer how to specify to the software how to deal with missingness and predict the missingness.

• Mariam,
You don’t have to do anything for this to work, at least not for the outcome variable. It’s inherent in how the model is estimated.

2. Daisy says

This is very helpful, thanks! You mentioned, “So with only two time points, if the people with one time point are no different from those with full data(creating no bias), you’re not gaining anything by keeping those 72 people in the analysis.” May I please ask what analysis should I run to test if there is any difference between people with one-time point and those with full data?

3. Sandra says

Hi,

Thank you for this clear explanation. I am still slightly unclear on one point – when you say “The difference in means from pre to post will be calculated based on the estimates at each time point. But the degrees of freedom for the difference will be based only on the number of subjects who have data at both time points.”

Do you mean that the results of the model do take all the data into account (including maximum likelihood for missing data) – but when you look at the degrees of freedom this won’t be reflected, since this will only be based on the cases that have data at both timepoints?

I’m trying to work out whether it makes more sense to impute missing values across a dataset before feeding this into a mixed model – or whether to just do the analysis using a mixed model with the missing data included. Do you have any guidance on this?

• Yes, the results of the model take into account all the data.

If I am missing data only on level one variables (including the outcome) I would not impute and instead would rely on the mixed model’s maximum likelihood.

The missingness mechanism assumption is MAR for both, so there is no advantage there. The one exception might be if there are auxiliary variables that could help predict the missing values in the multiple imputation.

4. Emma says

Hi there,

Do you have any references that support linear mixed models can handle missing outcome data?

5. Nathan says

Hi Karen,

This explanation really helped me, so thanks!

If I was interested in better understanding the justification for a mixed linear model in the case of missing data, would you recommend any sources?

Thanks a lot,

Nathan

6. mary says

I will compare a continuous variable between two different treatment in 5 time points. But I have the data of 47 patients in the first time point, 15 patients in the third time point, only 8 patients in the 5th time point. Missingness is not random. Can I use a linear mixed model? And what is the non-parametric alternative test to linear mixed model?
Thanks

7. Igbiks Tamuno says

I have a data of over 200 patients followed up for 12 months with creatinine measurements at six time points. missing data is over 60%. multiple imputation was used to deal with missing data. can i have guidance in analyzing this data using linear mixed model?

• It’s definitely something we could help you with in our membership or in consulting. We’d have to dig into the details with you to give solid guidance.

8. Kim says

Hi Karen,

This article is very helpful. Given the information you’ve provided above, do you recommend a different statistical approach for handling missing data in a study using a pre-post design where data are missing at random?

Thanks,
Kim

• Hi Kim,
Not necessarily. This is still going to give you the most unbiased results. The only other option is multiple imputation, and you only get limited information from that when you have the impute the outcome variable.

9. John says

In the above you state that:

“The difference in means from pre to post will be calculated based on the estimates at each time point. But the degrees of freedom for the difference will be based only on the number of subjects who have data at both time points.”

Are these the estimates of those people with posttest-only data along with the people with full data?

Similarly is the “mixed model” you described above the same as a random effects logit regression?

Very best,

John

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