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Decision record · DR-003

GEE (population-averaged) rather than a GLMM for repeated wheeze measurements

Accepted (decision made in 2023; recorded in 2026, with sensitivity analyses added in 2026)

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Context

The Ohio data follow 537 children in Steubenville, recording whether each wheezed at ages 7, 8, 9 and 10 (2,148 binary observations), with maternal smoking recorded once per child. Answers from the same child are correlated, so ordinary logistic regression understates the uncertainty of anything that varies between children. Assignment 3 asked for a GEE fit with an exchangeable working correlation. Two families of models handle the clustering: marginal models fitted by generalised estimating equations (GEE) and conditional models with random effects (GLMMs).

Decision

I used a logistic GEE with an exchangeable working correlation and robust (sandwich) standard errors as the main analysis, and in 2026 added a random-intercept GLMM as a sensitivity analysis rather than a replacement.

Options considered

  1. Naive logistic regression treating all 2,148 observations as independent.
  2. GEE with independence, exchangeable (chosen), AR(1) or unstructured working correlation.
  3. Random-intercept logistic GLMM, which models each child's latent propensity to wheeze.
  4. Transition (Markov) models that condition on the previous year's wheeze.

Why

  • The question is population-level: how much more common is wheeze among children of smoking mothers? GEE coefficients answer that directly, as population-averaged odds ratios.
  • Maternal smoking varies between children, not within them. A GLMM coefficient compares two children with the same latent propensity, which is a harder quantity to explain and to act on for a between-child exposure.
  • GEE's robust standard errors stay valid even if the working correlation is wrong; the GLMM relies on the random intercept being normally distributed.
  • The naive model gets the point estimates roughly right but the uncertainty wrong, which is exactly the error this assignment was about.

What happened

  • GEE (exchangeable): smoking log odds ratio 0.265, robust SE 0.178, odds ratio 1.30 (95% CI 0.92 to 1.85), p = 0.136. The naive model's SE of 0.123 would have given p = 0.028, a false sense of precision. The estimated working correlation is α = 0.354.
  • The working correlation barely matters here. Smoking estimates are 0.272 (independence), 0.265 (exchangeable) and 0.234 (AR(1)), with robust SEs of 0.178 to 0.181. QIC is 1829.49, 1829.48 and 1830.26: differences under one unit, too small to prefer one structure.
  • A cluster bootstrap that resamples whole children (1,000 replicates for each of five seeds) gives 95% percentile intervals for the smoking odds ratio from 0.90 to 0.92 at the lower end and 1.82 to 1.88 at the upper end across seeds, close to the sandwich interval.
  • The GLMM (maximum likelihood with Gauss–Hermite quadrature, checked against lme4::glmer) estimates a large between-child spread (σ = 2.16; latent-scale intraclass correlation 0.59) and a conditional smoking log odds ratio of 0.399 (SE 0.273; odds ratio 1.49). Shrinking it by the usual marginal approximation, 1/√(1 + 0.346σ²), gives 0.246, close to the GEE estimate of 0.265. The two models agree once their different scales are taken into account.
  • Neither model finds clear evidence of a smoking effect: every interval includes an odds ratio of 1. The 2023 write-up stated this correctly.
  • The residual correlation falls with the gap between visits (about 0.40 one year apart, 0.31 two years and 0.30 three years), so the exchangeable structure is a simplification, harmless here because the robust standard errors do not depend on it.
  • The 2023 analysis did not compare working correlations or check the robust SEs against a bootstrap; those checks are new.

What I'd change

  • State the estimand first (population-averaged or child-specific), then pick the model, and say so in the write-up.
  • Report the working-correlation comparison and a cluster bootstrap as routine checks, as /diagnostics now does.
  • Present the GLMM next to the GEE when readers might ask about individual children, with the scale difference explained.
  • Treat maternal smoking as a baseline exposure measured once, and be explicit that this is an observational association, not a causal effect.