# Linear mixed model for repeated measures analysis with missing values

we have ran an experiment where we compared three interaction techniques for a 3d docking task. So we had two factors: the aforementioned technique type and a factor representing the direction translation (i.e.: if participants had to move an object that appeared close to their viewpoint and move it in depth or vice versa). Each trial was repeated 5 times.

Some of those trials were skipped because of the difficulty. If I run a regular repeated measures anova then each participant that even a single missing value will be dropped from the analysis. This means that I'd have to remove more than half of the participants. By reading around it seems I can use a linear mixed model instead.

My doubt is, can I use a mixed model for this type of situation? I am confused as to whether mixed models are only relevant when you have a between-subjects factor such as the classic treatment/control groups. In my case every participant was subjected to the same conditions. There were no between-subjects factors.

I ran the mixed model analysis by using Technique, Direction and Repetition as Repeated, the ID of each participant as the subject and technique and direction as fixed factors.

Are my assumptions correct or did I do a terrible mistake?

If so, what alternatives do I have when dealing with missing values?

Thanks!

• Yes this sounds like an ideal situation to use mixed-effects models. You'll have fixed effects as usual, and then at least random intercepts that vary across participants. You won't have to drop people with missing trials (but your data have to be arranged in "long" format, with one row per trial). – Patrick Coulombe Oct 21 '17 at 12:40

Here is a background paper, "Normal distribution based pseudo ML for missing data: With applications to mean and covariance structure analysis" with full text available at http://www.sciencedirect.com/science/article/pii/S0047259X09001079 .

Per this reference to quote some pertinent comments:

"When the population follows a confirmatory factor model, and data are missing due to the magnitude of the factors, the MLE may not be consistent even when data are normally distributed. When data are missing due to the magnitude of measurement errors/uniqueness, MLEs for many of the covariance parameters related to the missing variables are still consistent."

The aforementioned paper identifies and discusses factors that impact the asymptotic biases of the MLE for data that are not missing at random.

A technique that I believe has value in providing a solution is to replace the missing value, see "What to Do about Missing Values in Time-Series Cross-Section Data", full text available at https://www.google.com/url?sa=t&source=web&rct=j&ei=Ecb4U5rgEfLfsAS_BQ&url=http://gking.harvard.edu/files/pr.pdf&cd=10&ved=0CDcQFjAJ&usg=AFQjCNGpnhU_8okmEeNqnvRLCppKDEromw&sig2=Q5pAcvzVVH4IWmzFQFPe8A

To quote the author, the paper suggests the "concept of “multiple imputation,” a well-accepted and increasingly common approach to missing data problems in many fields. The idea is to extract relevant information from the observed portions of a data set via a statistical model,to impute multiple (around five) values for each missing cell,and to use these to construct multiple “completed” datasets."

• Validity of multiple imputation requires that data be "missing at random"; that is, the probability of a data point's being missing does not depend on the value of the missing data point. In this case, with the missing data points being those that are most difficult, this necessary assumption is almost certainly not met. – EdM Jul 12 '15 at 1:20
• if you were able to come up with a censoring threshold (e.g., you were willing to assert that you thought the score on a skipped trial would be >= the observed worst-case (maximum?) value for that participant), you could do a relatively crude analysis in that direction using a survival analysis package that allowed for Gaussian responses with right-censored data (e.g. survreg in the R survival package ...) – Ben Bolker Dec 24 '15 at 15:29