Time-Varying Predictors In GLMMs And GEE

Why Separate Within-Person Change From Between-Person Differences?

A time-varying predictor is measured repeatedly on the same person: pain at each visit, weekly activity, or a patient’s current medication dose. Its values can differ both within a person over time and between people. Those are distinct comparisons. A person whose activity is usually high may differ from a person whose activity is usually low; separately, a person may have a different outcome at occasions when their activity is above their own usual level.

A model with only one coefficient for the raw predictor can blur these comparisons. A useful approach is to express each observation as a deviation from that person’s average and the person’s average itself. The resulting terms let the model estimate within-person and between-person associations separately. This distinction applies to both conditional GLMM mean models and population-averaged GEE mean models, though their coefficients refer to different estimands.

Two Terms, Two Comparisons

A common way to include a time-varying predictor is to put both components into the linear predictor. For example, a random-intercept GLMM can have conditional mean model

Here \(Y_{it}\) is the outcome, \(g\) is its link function, \(b_i\) is the person-specific random intercept, and \(\mathbf{x}_i\) denotes that person’s predictor history. The model also requires a specified response distribution and random-effects distribution. Its conditional mean interpretation assumes the chosen mean structure and link are appropriate; standard random-effects modeling also requires assumptions about the distribution of \(b_i\) and how it relates to the covariates. In this parameterization, \(\beta_W\) describes the association with being one unit above a person’s observed mean, conditional on the random effect and other included terms. \(\beta_B\) describes the association with a one-unit difference in the person-specific mean, comparing people at the same within-person deviation and other covariates.

In a GEE, the same decomposition can be used in a marginal mean model:

The predictor decomposition is the same, but the mean is population-averaged rather than conditional on a person-specific random effect. Thus, in a logistic model, \(\beta_W\) in a GLMM gives a conditional log-odds association, whereas \(\beta_W\) in a logistic GEE gives a population-averaged log-odds association. These coefficients need not be equal. The distinction between conditional and marginal estimands is essential when comparing the models.

The “within-person” label refers to the contrast represented by the deviation term, not a guarantee that the observed data isolate a causal effect. A causal interpretation would require additional design and identification assumptions. Time trends, treatment changes, measurement error, and time-varying confounders can all affect the association.

How the Decomposition Changes Interpretation

To see why the two terms matter, consider a model with an identity link and a correctly specified additive mean:

At a fixed person-specific mean, a one-unit increase in the deviation changes the linear predictor by \(\beta_W\). At a fixed deviation, a one-unit increase in the person mean changes it by \(\beta_B\). For an identity link these changes are in outcome units. For a logit link, the changes are in log-odds units and exponentiating a coefficient gives the corresponding odds ratio. For a log link, exponentiating gives a rate ratio or mean ratio, as appropriate to the response model.

A model containing only \(x_{it}\) imposes a single slope on both dimensions of variation. In general, that restriction may be scientifically implausible. In a balanced dataset, the issue can be shown directly for an identity-link mean model. Write \(w_{it}=x_{it}-\bar{x}_i\), so \(x_{it}=w_{it}+\bar{x}_i\), and suppose the true mean is \(\beta_0+\beta_Ww_{it}+\beta_B\bar{x}_i\). If we regress the outcome on raw \(x_{it}\) alone, its population slope is

For this expression, assume equal numbers of observations per person, a mean-zero outcome error uncorrelated with the predictors, and use the pooled observation distribution. The within-person deviations sum to zero for every person, so they have zero pooled covariance with the person means: \(\operatorname{Cov}(w,\bar{x})=0\). Substituting \(x=w+\bar{x}\) and the stated mean into \(\operatorname{Cov}(x,Y)\) gives \(\beta_W\operatorname{Var}(w)+\beta_B\operatorname{Var}(\bar{x})\); the same decomposition gives \(\operatorname{Var}(x)=\operatorname{Var}(w)+\operatorname{Var}(\bar{x})\). The raw-predictor slope is therefore a variance-weighted combination of the two slopes, not necessarily either one. With unequal observation counts or a nonlinear link, that simple formula does not generally apply; the central point remains that a single raw-predictor coefficient does not separately identify the two associations.

Choosing an Approach and Checking the Analysis

The decomposition is a choice of predictor representation; it does not by itself choose between a GLMM and a GEE. A GLMM specifies a conditional response model with random effects and is useful when subject-specific conditional means are of interest and its random-effects assumptions are defensible. A GEE specifies a marginal mean model and is useful when the population-averaged mean is the target. Within either framework, including both the deviation and person mean answers a different question from using only the raw time-varying predictor.

Several checks help assess whether the representation is useful and credible:

• Confirm that the predictor varies within people. If it is constant for every person, all deviations are zero and a within-person association cannot be estimated from that predictor.
• Inspect the number and timing of measurements per person. A mean calculated from very few or unevenly timed observations may be a poor summary of a person’s usual level.
• Check whether occasion or calendar-time effects are needed. If both the predictor and outcome trend over time, omitting time can make their association misleading.
• Examine missingness and measurement quality. A mean based only on observed occasions may differ systematically from the intended long-term mean, especially if measurement or follow-up depends on a person’s condition.
• Consider whether the predictor may respond to earlier outcomes or treatment. In that case, the fitted association does not automatically have a causal interpretation.

Centering around each person’s observed mean also has limitations. It can be sensitive to how many occasions are observed and to the observation schedule; it does not remove time-varying confounding or measurement error. A model with both components can estimate two associations, but it cannot guarantee that either is unbiased for a causal effect. The appropriate alternative depends on the scientific target: a raw predictor may be adequate if a common association is a justified modeling restriction, while a within-between decomposition is preferable when the two comparisons are substantively distinct.

A Common Interpretation Pitfall

A frequent mistake is to say, “The coefficient for activity shows what happens when a person becomes more active,” when the fitted model contains only raw activity. That coefficient can reflect both changes in activity within a person and persistent differences between people with different usual activity levels. The correction is to identify which terms the model includes and state the comparison each term represents. With both terms included, describe the deviation coefficient as the within-person association and the person-mean coefficient as the between-person association, while specifying whether the model is conditional or marginal.