# Appropriate model choice for analyzing a cluster based longitudinal randomized controlled trial

I am performing a randomized controlled trial (RCT) of an educational intervention to improve knowledge, belief and practice among healthcare workers in hospitals. One hospital is assigned to the intervention and the other hospital is selected as a control group. I will measure all outcomes at baseline data from both groups. I wish to compare the effects of my intervention at 1 month post-intervention, then 6 months post-intervention.

I looked at distributions of baseline values. They appear approximately normally distributed and no difference between control and intervention group. Loss to follow up was minimal (110 respondents per group and only 6 respondents loss to follow up in each group).

I know of three modeling approaches: generalized linear models, generalized linear mixed models, and generalized estimating equations. Which is better for this analysis? How do I see the effects of intervention within-and-between group and also if taking into account covariates (age, gender, position etc etc).

• Ordinarily, the selection of a statistical procedure is made before conducting the RCT. This suggests you might be engaged in exploration of secondary, post hoc hypotheses. Would that be the case or not? – whuber May 21 at 12:36
• GEEs are well celebrated for their robustness and applicability to cluster randomized trials. – AdamO May 21 at 12:36
• Do you have an RCT with just two units of assignment? – Heteroskedastic Jim May 21 at 14:12

This cluster-based design is severely limited by the intervention being aliased entirely by the hospital in which it's performed. Ideally you would have included at least two more hospitals, possibly matched on some factors, and randomly permuted treatment assignment between pairs. This design demands adjustment for within-level covariates, adjustment for between level covariates risks reducing power and yet valid inference may not be possible without some control of them such as facility type (teaching hospital, etc.), regional SES, etc.

The collection of 1 and 6 months' post-baseline outcome is redundant except as it may contribute to a secondary hypothesis. The hypothesis to be tested is whether intervention increases education. So we don't expect anything but random variability between the 1 and 6 months' assessment. One could secondarily test a hypothesis of growth by adjusting for time and it's interaction with the outcome.

If we selected only 1 month outcome data, we could test the hypothesis by using a between-within F-test for cluster randomized trials as described in Baldwin 2011 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3987820/ The best design for analyzing pre-post outcome data with one follow-up data point is an ANCOVA model where one adjusts for baseline values as a covariate in the model. The residuals of such a model are conditionally independent of one another. The test-statistic for the models which do and do not adjust for the indicator of receipt of intervention are best compared to an F distribution with 1 numerator degree of freedom and 2 denominator degrees of freedom which is very low power but appropriate to the design.

• But if we have missing data that are missing at random I think the ANCOVA approach could provide biased results. – Dimitris Rizopoulos May 21 at 14:44
• @DimitrisRizopoulos if it's a consideration, multiple imputation is feasible, easy, and more generalizable than any other method. As mentioned in the question, less than 10% means the extent of possible "bias" due to non-response is quite low. More often than not, the missingness is non-ignorable, and one must simply note that non-response could mean any number of things. I generally interpret results strictly among responders (consider for instance those who did not agree to participate in the baseline sample). – AdamO May 21 at 14:50

A couple of points:

• Repeated measurements over time on the same subjects/patients are expected to be correlated. Hence, a GLM that assumes independent observations will not be appropriate.
• GEEs and GLMMs are typically used when you have non-normal outcome data, e.g., when your outcome is dichotomous or a count. In the case of normal data, you can use a linear mixed model or a marginal model. Since you only have three time points, and assuming that all subjects came at these specific time points and there were not many fluctuations, the more typical model to use is a marginal model (i.e., multivariate regression model) with unstructured covariance matrix for the error terms. If you work in R, this can be done with function gls() from package nlme.
• You should better check the appropriateness of the normal distribution for your data using the residuals of your model.
• Some observations on this answer: it's important to note that pre/post measures correlate differently than repeated post/post measures, GEEs and GLMMs do include normal outcomes as a special case ("linear model with identity link"). – AdamO May 21 at 14:23
• @AdamO indeed the linear mixed model I mentioned in my answer is a special type of GLMM and the marginal model is (almost) equivalent to a GEE for continuous data and identity link. As the linear regression is a special case of a GLM. The unstructured covariance matrix can capture the correlations between the 3 measurements. – Dimitris Rizopoulos May 21 at 14:39