Marginal model

Definition taught in Marginal models, and two responses at once, part of Correlated data

Marginal model

Schematic, not a recorded data set. From Marginal models, and two responses at once.

Type	Definition
Taught in	Marginal models, and two responses at once
Part	Correlated data
Worked examples	2
Questions using it	5

A marginal model for clustered responses has three parts.

Definition

1. A mean, g(\mathrm{E}(Y_{ji}\mid {\boldsymbol{X}}_{ji}))={\boldsymbol{X}}_{ji}^\top\boldsymbol{\beta}, the same link and the same linear predictor as for independent data.
2. A variance, \mathrm{Var}(Y_{ji}\mid {\boldsymbol{X}}_{ji})=\phi V(\mu_{ji}), depending on the covariates only through the mean.
3. A working association within a cluster, allowed to depend on the two means and on further association parameters.

Examples

C38. Generalised estimating equations

Question page: C38. Generalised estimating equations

What are generalised estimating equations, and what mean-model parameters do they estimate?

Worked answer (click to show)

Definitions/notation. GEE are estimating equations for the coefficients of a marginal (population-average) mean model for correlated responses, constructed using derivatives of the mean and a working within-cluster covariance. They require no full joint response distribution, but assume independent clusters and, for consistency under standard regularity conditions, a correctly specified marginal mean.

C40. Sandwich robustness

Question page: C40. Sandwich robustness

What misspecification can the GEE sandwich covariance estimator tolerate? Does it repair an incorrect mean model?

Worked answer (click to show)

Definitions/notation. The sandwich estimator estimates the sampling covariance of the GEE coefficient estimator using empirical cluster residual variation, allowing the working covariance to be misspecified. It does not repair an incorrect mean model: its usual robustness is asymptotic with many independent clusters, a correct marginal mean, valid estimating equations and regularity conditions, not a guarantee for small samples or informative missingness.

Figures

The two burn slopes. The dashed rectangle is the pair of marginal 95% intervals from the standard errors reported above. The ellipse is the joint 95% region for those same standard errors if the slopes were uncorrelated. The sandwich covariance is not reported, so the ellipse drawn here is not rotated. A pair can sit inside the rectangle and outside the ellipse, and the corners of the rectangle sit outside the ellipse. From Marginal models, and two responses at once.

Used in

• Taught in Marginal models, and two responses at once, where the derivation and examples are.
• Also used in Random slopes, and a binary mixed model · Three species on one hillside
• Questions that use it (5) — Question 20 (answer) · C38 · C39 · C40 · C41
• All 6 questions on Marginal models, and two responses at once

See also

Glossary and index