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I have a question about the differences between two forms of logistic regression.

Let's say that I have data that is collected with some nesting. For concreteness, we'll say that I've got data across a few thousand elementary schools. For each school, I have data on each of the classrooms. My outcome is the number of students in each classroom who received some type of disciplinary action in an academic year. So the outcome is a number of binomial counts, where I have the number of students who were disciplined and the total number of students in each classroom. My 'explanatory variable' is measured at the level of school. We'll say that the explanatory variable is some kind of school-level socioeconomic status (SES).

My first instinct would be to estimate a multilevel model. In lmer syntax, it would look something like this:

m <- glmer(cbind(n_disciplined, n_students) ~ SES + (1|school), family='binomial')

But perhaps I can save myself some trouble by just aggregating the counts of the number of students disciplined and the total number of students up to the level of school. So instead of a multilevel model, I can just estimate a standard logistic regression:

m <- glm(cbind(tot_disciplined, tot_students) ~ SES, family='binomial')

If I'm only interested in how SES is associated with disciplinary differences across schools, is there a reason why I should use the former, more complicated model instead of the latter? Typically, I would avoid aggregating the data, but I'm interested in if doing so would change either the estimated values, or the conclusions I would/could draw from the model fits.

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    $\begingroup$ Well, with lower level aggregated you cannot use individual-level covariates. If you do not have such covariates, there should not be a difference. $\endgroup$ – kjetil b halvorsen Aug 31 '18 at 20:19
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The two models do not give you coefficients with the same interpretation. The mixed effects model will give log odds ratios conditional on the random effects / school, whereas the GLM or a GEE approach would give you coefficients that are the log odds ratios across schools. Even though there are approaches to obtain coefficients with a marginal interpretation from the GLMM.

As in the comment you received, if you have invividual level covariates these could not be included in the GLM approach, but they could in the GEE approach.

Another difference is about missing data, the GLMMs will work if you have missing data in your outcome that are of the missing at random type, whereas working with GEE (standard version not weighted) or the aggregates approach would only be valid under the missing completely at random assumption.

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  • $\begingroup$ Thanks, this is great. I'm wondering if you can expand a bit on the first paragraph here. In particular, what does it mean for the estimates to be conditional on the RE's, and how does it contrast with the marginal interpretation under the situation where there are no individual-level covariates? These are all characteristics of glmms that I'm familiar with, but I'm not totally sure how these characteristics translate into plain english interpretations. $\endgroup$ – triddle Sep 3 '18 at 14:04
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    $\begingroup$ Because of the nonlinear link function (i.e., the logit function) used in the definition of a mixed effects logistic regression, the coefficients we obtain have an interpretation conditionally on the random effects. In your example, say you're interested in the effect of the sex. From the mixed effect logistic regression, you get the odds ratio for sex for students from the same school. If you would fit a GEE you get the odds ratio for sex across schools. For more details on this topic, you can have a look at my course notes (Section 5.2): drizopoulos.com/courses/EMC/CE08.pdf $\endgroup$ – Dimitris Rizopoulos Sep 3 '18 at 17:58

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