Introduction

This document clarifies what a “marginal model” means in the context of hierarchical (mixed) models, with special attention to spatio-temporal random effects. We distinguish between:

  1. Marginalizing (integrating out) random effects from a hierarchical specification.

  2. Marginal models as population-averaged models (Generalized Estimating Equations, GEE).

1. Spatio-Temporal Model

We start with a hierarchical specification:

Data Level :

\[ Y_t \mid w_t \sim N(\mu_t, \sigma^2 I) \tag{1} \]

Process Level:

\[ w_t \sim N(0, \sigma_w^2 S_w) \tag{2} \]

where \(S_w\) is a spatio-temporal correlation matrix.

Marginalizing (Integrating Out) \(w_t\)

To obtain the marginal model for \(Y_t\), we integrate out the random effects \(w_t\):

\[ \begin{aligned} \text{Var}(Y_t) &= \text{Var}(E[Y_t \mid w_t]) + E[\text{Var}(Y_t \mid w_t)] \\ &= \text{Var}(\mu_t + w_t) + E[\sigma^2 I] \\ &= \sigma_w^2 S_w + \sigma^2 I \end{aligned} \]

Resulting Marginal Model:

\[ Y_t \sim N(\mu_t, \sigma^2 I + \sigma_w^2 S_w) \]

So, the spatio-temporal random effect \(w_t\) has been integrated out.

The marginal covariance now has two parts:

  • Nugget (\(\sigma^2 I\)): measurement error or micro-scale variation
  • Spatio-temporal dependence (\(\sigma_w^2 S_w\)): from the latent process

2. Other Types of Marginal Models

The term “marginal model” can mean different things depending on context.

Type A: Marginal over Random Effects

This is common in hierarchical or linear mixed models (LMMs).

  • Approach: Start with conditional model \(E[Y \mid \text{random effects}]\), then integrate out the random effects
  • Result: Marginal likelihood \(L(\beta, \theta; Y)\) where \(\theta\) are variance parameters
  • Software: lme4, brms, INLA, spTimer

Example (Non-spatial, longitudinal):

Conditional: \[ Y_{ij} \mid b_i \sim N(\mu + b_i, \sigma^2), \quad b_i \sim N(0, \tau^2) \]

Marginal: \[ Y_{ij} \sim N(\mu, \sigma^2 + \tau^2) \] with compound symmetry covariance.

Type B: Population-Averaged Models (GEE)

From Generalized Estimating Equations (Liang & Zeger, 1986).

  • Goal: Directly model marginal mean \(E[Y_{ij}]\) and a “working” correlation matrix
  • No integration — no random effects are ever introduced
  • Inference: Robust sandwich estimator protects against misspecified correlation
Feature Your LMM Marginal GEE (Population-Averaged)
Random effects integrated out? Yes N/A (none exist)
Interpretation of \(\beta\) Same as conditional for linear models Strictly population-average
Likelihood-based? Yes (REML/ML) No (quasi-likelihood)
Handles missing data? Requires MAR + correct likelihood Requires MAR + robust

3. Crucial Caveat: Linear vs. Non-Linear Models

For Linear Models (Gaussian)

The conditional and marginal means are identical:

\[ E[Y] = X\beta \quad \text{(both)} \]

Only the covariance structure differs.

For Non-Linear Models (Logistic, Poisson)

This is NOT true. Example with logistic GLMM:

Conditional: \[ \text{logit}(E[Y_{ij} \mid b_i]) = X\beta + b_i \]

Marginal (integrated): \[ E[Y_{ij}] = \int \text{logit}^{-1}(X\beta + b) \, p(b) \, db \neq \text{logit}^{-1}(X\beta) \]

🔴 Marginal effect sizes are attenuated toward zero compared to conditional effects. This is a well-known phenomenon in non-linear mixed models.

Summary

Question Answer
In spatio-temporal LMM, does “marginal model” mean random effects are integrated out? Yes — \(Y_t \sim N(\mu_t, \sigma^2 I + \sigma_w^2 S_w)\) results from integrating out \(w_t\)
Does “marginal model” always mean that? No — GEEs have no random effects to integrate out
For non-Gaussian data, does integration change only covariance? No — the mean structure also changes (attenuation)

When you read “marginal model,” ask:

  • Is it marginal over random effects from a hierarchical specification? (Likelihood-based)
  • Or is it a population-averaged model using GEE? (No random effects, robust inference)

The equations (1)-(2) represent a classic LMM marginal model (Type A).