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Would anyone be able to explain conceptually why GLMMs with a random intercept at the study level are occasionally used for meta-analysis of proportions / counts, when the observations are already at the level of the study? I only have one observation per study in my dataset, so I'm not sure where the within study variation / hierarchical structure is coming from for the model.

To provide a little more detail, I'm conducting a meta-regression of factors associated with loss to follow-up in a sample of RCTs. Each RCT has an associated number of patients lost to follow-up over a follow-up period. I am likely going to model this using a poisson-normal model in metafor (which has been a really amazing package), using code similar to the below:

mf.poisson <- rma.glmm(measure="IRLN", xi=lost.to.fu, ti=patient.years, mods = moderator, data=test)

For a small fabricated dataset, this produces an estimate of 1.255 (95%CI 0.64, 1.87), or when exponentiated, a relative rate of 3.51 (95%CI 1.90, 6.49). Tau^2 = 0.51, I^2 = 86%.

To try to understand the modelling, I also ran the following using lme4 with an offset for log patient years, and random intercept at the study level:

lme.poisson <- glmer(lost.to.fu ~ moderator + offset(log.patient.years) + (1 | study.number), data=test, family = poisson(link = "log”))

confint(lme.poisson, method="Wald")

Which, for the same fabricated data, produced an estimate of 1.254 (95%CI 0.64, 1.87). Essentially the same. Random effects variance of 0.51 (tau^2). (However not sure how to calculate I^2 using lme4, so thank goodness for metafor!)

Lastly, I ran the following is using a naive poisson model:

naive.poisson <- glm(lost.to.fu ~ moderator + offset(log.patient.years), data=test, family = poisson(link = "log"))

And the estimate was quite similar, although not exactly the same: 1.22 (95% CI 0.98, 1.47).

So I'm not sure why mixed models are used, when I only have 1 observation per study.

Thank you very much for your help! And I'm sorry for my ignorance - I'm a medical resident doing a graduate program in clinical epidemiology, and am very new to these concepts and R!

Sincerely, Richard

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    $\begingroup$ Study variation means between-study variation, not within-study. So it doesn’t matter if you have one observation per study as long as you’ve aggregated more than one study for your meta-analysis. The variance term of the random intercept encodes how heterogeneous your collated studies are $\endgroup$
    – Daeyoung
    Commented Mar 28, 2022 at 13:33
  • $\begingroup$ Thanks Daeyoung - that makes sense for the random intercept. However where does the within study variation come from? If 100% of the variation is between studies (ICC=1) wouldn't a GLM also work? As in this comment: stats.stackexchange.com/questions/65371/… $\endgroup$
    – Richard Walker
    Commented Mar 28, 2022 at 13:47
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    $\begingroup$ The within-study variation is the standard error (or confidence interval) that accompanies the point estimate. It's usually calculated by $\frac{\text{upper CI - lower CI}}{2 * 1.96}$ $\endgroup$
    – Eli
    Commented Mar 28, 2022 at 13:54
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    $\begingroup$ @RichardWalker If the true ICC=1, I believe you’re right. Meta-analytic data AFAIK include study sample sizes, and certain quantities’ variances can be derived from available information. For example, log relative risk has a variance $\frac{IN}{IE(IE+IN)}+\frac{CN}{CE(CE+CN)}$. Even if there’s only one observation, the information is sufficient to calculate the within-study variance of the statistic. $\endgroup$
    – Daeyoung
    Commented Mar 28, 2022 at 13:59
  • $\begingroup$ Thanks Eli - so if I understand correctly, the mixed model is estimating a standard error for each study (somehow based on event count and person-years of exposure, assuming a Poisson distribution), and allowing each study to have it's own effect size along that sampling distribution, as opposed to assuming a fixed effect size for all. $\endgroup$
    – Richard Walker
    Commented Mar 28, 2022 at 14:09

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