# Is GEE the simplest choice for a 2x2 factorial design with repeated measures and binary dependent variable?

I have a 2x2 factorial design. Both independent variables are binary (age: young/old and condition: experimental/control). Age is between subject and condition is within subject. The dependent variable is binary (pass/fail). The hypothesis is that there will be interaction between the two factors.
Is Generalized Estimated Equations (GEE) the simplest correct way to test the hypothesis? Or are there simpler methods that will give fully equivalent results of the statistical test?

Edit: Each subject receives only one experimental and one control task. All included subjects contribute score from both task.

jan

• Is the each patient randomized to receive experimental treatment and control treatment only one time? Has each patient completed an equal number of trials? Mar 5, 2021 at 17:30
• Thanks @AdamO Yes and yes. I added this info to the main question above.
– jan
Mar 5, 2021 at 17:38

This is a case for conditional logistic regression, not GEE.

GEE is a widely touted general method for handling repeated measures dependent data. But one struggles to justify its low power and handling of the response under control condition in an experimental design. The power to detect an effect of treatment will suffer because GEE doesn't fully condition on the outcome during the "control" portion of the study. Rather, the outcome while receiving control is considered a random response that has residual variation - some of which attributed to the patient - but some of which is random binomial variation. In all, there's a loss of power.

With pre-post analyses, like with paired t-test, ANCOVA, or McNemar's test, the powerful analysis results from conditioning on the "pre" response. Had you no strata/blocking, the McNemar test would readily present as the obvious choice. But recently, methods have been established for a stratified version.

The conditional logistic regression is a powerful and computationally complex algorithm. The internal working of conditional logistic regression is irrelevant. But essentially the conditional likelihood of the binomial response is identical to the partial likelihood of a Cox proportional hazards model for time-to-event outcomes where each matched set comprises a stratum in an analysis of a single event time with patients achieving response marked as "events" and those not marked as "censored". Since the number of strata are relatively high compared to standard Cox models.

This is why in R, the conditional logistic regression lives inside the survival package as clogit with detailed documentation.

• Thank you @AdamO. I am looking into the conditional logistic regression. I did not know of it, so I am very happy to learn. I see two problems: 1. The study will have a moderate-size sample to start with: 50 participants total in two equal groups of 25. 2. I expect high performance (mostly pass score) in the control task overall and high performance in one group but not other. So after dropping participants with same performance in control and experimental task (mostly "pass" on both task), the sample left will be small and likely imbalanced across conditions. Will this affect inference?
– jan
Mar 5, 2021 at 18:37
• Are you saying 25 are randomized to receive control then treatment, and 25 are randomized to receive treatment then control? Mar 5, 2021 at 18:42
• No. I will have two groups of 25. I called them above "young" and "old". Maybe these labels are confusing, because this is a cross-sectional design, not longitudinal. We can call them group 1 and group 2.Both groups are tested at the same time. The within subject factor is type of task: experimental vs. control. And in fact they are also administered at the same time. And their order cannot be counterbalanced. So time and order play no role in this design.
– jan
Mar 5, 2021 at 19:07
• @jan you are not giving a clear enough example to make this design obvious to someone who is not actually involved in the research. You can't administer two conditions at the "same time" unless they're, for example, in two parts of the body, like the right and left eye. Mar 5, 2021 at 20:14
• Ok sorry. I realize I should have given less abstract description. This is an imitation paradigm for toddlers. They see a two element sequence: Action-outcome. When they imitate they can produce one or the other or both. By theory producing the outcome is the control task. Producing the action is the experimental task.The sequence has to be presented to participants in a fixed order. But children can produce any or both elements in either order.
– jan
Mar 5, 2021 at 20:33