Unit-Specific vs. Population-Average Models


Mark Lai


December 28, 2020

One thing that I always felt uncomfortable in multilevel modeling (MLM) is the concept of a unit-specific (US)/subject-specific model vs. a population-average (PA) model. I’ve come across it several times, but for some reason I haven’t really made an effort to fully understand it. I happened to come across this paper by Harring and Blozis, which I read before, and think that why not try to really understand the relationship between the coefficient estimates in a US model and in a PA model in the context of generalized linear mixed-effect model (GLMM). So I have this learning note.

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While MLM/GLMM is a US model, which models the associations between predictors and the outcome for each cluster, PA models are popular in some areas of research, with the popular method of the generalized estimating equation (GEE). Whereas the fixed effect coefficients in US are the same as the coefficients in PA in linear models, when it comes to generalized linear models with nonlinear link functions, the coefficients are not the same. This is because some of the generalized linear models typically assume constant variance on the latent continuous response variable. For example, in a single-level logistic model and a GEE model, the latent response \(Y^*\) has a variance of \(\pi^2 / 3\), but in a two-level model, the variance is \(\pi^2 / 3 + \tau^2_0\).1 Because the coefficients are in the unit of the latent response, it means that the coefficients are on different units for US vs. PA. But how are they different? I will explore four link functions: identity, log, probit, and logit. But first, some notations.

Model Notations

While in actual modeling, the distributional assumptions of the response variables are important (e.g., normal, Poisson), the comparison of US vs. PA mainly concerns the mean of the outcome and the link function. For all models, the random effects are normally distributed.

Conditional (US) Model

\[ \begin{aligned} \mathop{\mathrm{E}}(y_{ij} | u_j) & = \mu_{ij} \\ h(\mu_{ij}) & = \boldsymbol{\mathbf{x}}^\top_{ij} \boldsymbol{\mathbf{\gamma }}+ \boldsymbol{\mathbf{z}}^\top_{ij} \boldsymbol{\mathbf{u}}_j \end{aligned} \] where \(h(\cdot)\) is the link function, \(\boldsymbol{\mathbf{x}}_{ij}\) and \(\boldsymbol{\mathbf{z}}_{ij}\) are the fixed and random covariates for the \(i\)th person in the \(j\)th cluster. The distributional assumption is \(\boldsymbol{\mathbf{u}}_j \sim N_q(\boldsymbol{\mathbf{0}}, \boldsymbol{\mathbf{G}})\)

Marginal (PA) Model

Now one is modeling the marginal mean:

\[ \begin{aligned} \mathop{\mathrm{E}}(y_{ij}) & = \mathop{\mathrm{E}}[\mathop{\mathrm{E}}(y_{ij} | \mu_{ij})] = \mu^\text{PA}_{ij} \\ h(\mu^\text{PA}_{ij}) & = \boldsymbol{\mathbf{x}}^\top_{ij} \boldsymbol{\mathbf{\gamma^\text{PA}}} \end{aligned} \] The above two formulas can be used to find the transformation from the unit-specific coefficients, \(\boldsymbol{\mathbf{\gamma}}\), to the population-average coefficients, \(\boldsymbol{\mathbf{\gamma^\text{PA}}}\).


  1. Snijders & Bosker (2012), chapter 17.↩︎

  2. https://www.johndcook.com/blog/2010/05/18/normal-approximation-to-logistic/↩︎