Random Intercepts in Zero-Inflated GLMMs

Why Add a Group-Level Intercept?

Repeated counts from the same group may be more alike than counts from different groups. For example, patients in the same clinic may share unmeasured features that affect their event rates. A group-level random intercept represents this shared heterogeneity in the count component of a zero-inflated generalized linear mixed model (GLMM). It allows counts in a group to be conditionally independent given the intercept while still being marginally dependent after that intercept is averaged out.

The key modeling choice is which part of the mixture receives the random intercept. Here it enters the count-rate component only. The probability of a structural zero may depend on covariates, but it has no group-level random effect in the model considered here. The earlier material on Random Effects and Marginal Zero Probabilities provides the general distinction between conditional and marginal probabilities; the focus here is the dependence induced by a shared intercept in the count component.

Model Specification and Assumptions

The defining assumptions are that the random intercept is shared by observations in its group, the \(b_g\) values are independent draws from the stated normal distribution, and the random intercept is independent of the covariates and zero-state indicators. Conditional on \(b_g\), observations’ zero-state indicators and count outcomes are independent across observations, with the count outcome governed by its own \(\lambda_{ig}(b_g)\). The fixed-effect predictors and offsets must be specified appropriately for the data; in particular, exposure belongs as a log offset when the rate is per unit exposure.

For a fixed group effect \(b_g\), the expected observed count is \(q_{ig}\lambda_{ig}(b_g)\): the count is zero with probability \(\pi_{ig}\), and has Poisson mean \(\lambda_{ig}(b_g)\) with probability \(q_{ig}\). The count intercept therefore shifts the log rate by the same amount for every observation in a group. Increasing \(b_g\) by one multiplies the conditional count rate by \(e\), holding covariates and exposure fixed. It does not directly change \(\pi_{ig}\).

Marginal Means and Within-Group Covariance

A prediction conditional on a particular \(b_g\) describes a group with that random-intercept value. A marginal prediction averages over the distribution of group effects. Put \(a_{ig}=e_{ig}\exp(\mathbf{x}_{ig}^{\mathsf T}\boldsymbol{\beta})\), so that \(\lambda_{ig}(b_g)=a_{ig}e^{b_g}\). Since \(b_g\sim N(0,\sigma_b^2)\), the lognormal moment formula gives \(E(e^{b_g})=e^{\sigma_b^2/2}\). Therefore,

This is generally not the same as the conditional mean at \(b_g=0\), which is \(q_{ig}a_{ig}\). Setting the random effect to zero describes a group at the center of the random-effects distribution; averaging across groups also accounts for the nonlinear exponential link.

For two distinct observations \(i\) and \(j\) in the same group, the shared \(b_g\) induces marginal covariance. Conditional on \(b_g\), the observations are independent, so their conditional covariance is zero. The law of total covariance then gives

The first term vanishes by conditional independence. For the second, the two rates share the same \(e^{b_g}\). Using \(E(e^{2b_g})=e^{2\sigma_b^2}\),

Thus, for distinct observations in the same group, \(\operatorname{Cov}(Y_{ig},Y_{jg})=q_{ig}q_{jg}a_{ig}a_{jg}e^{\sigma_b^2}(e^{\sigma_b^2}-1)\). This covariance is nonnegative and is zero if \(\sigma_b^2=0\), provided the model’s conditional independence assumptions hold. Its size depends on the rates and zero probabilities, so a single common correlation is not generally implied when observations have different predictors.

For one observation, the marginal variance also includes both zero inflation and random-intercept variation. Conditional on \(b_g\), a zero-inflated Poisson count has variance \(q_{ig}\lambda_{ig}(b_g)+q_{ig}\pi_{ig}\lambda_{ig}(b_g)^2\). Applying the law of total variance yields

The final term is the additional variation in the conditional mean caused by differences among groups. The second term reflects the zero-inflated conditional variance contribution involving \(\lambda_{ig}^2\). In particular, these sources should not be conflated: zero inflation and a random intercept can both increase marginal variance, but the random intercept also creates covariance between repeated counts in the same group.

Likelihood, Diagnostics, and Alternatives

For a group with observations \(y_{ig}\), the likelihood contribution integrates the conditional likelihood over that group’s random-intercept distribution. In the count component, each observation contributes a zero-inflated probability: for \(y_{ig}=0\), the probability is \(\pi_{ig}+q_{ig}\exp\{-\lambda_{ig}(b_g)\}\); for \(y_{ig}>0\), it is \(q_{ig}\exp\{-\lambda_{ig}(b_g)\}\lambda_{ig}(b_g)^{y_{ig}}/y_{ig}!\). Thus the group likelihood is

where \(\mathbb{1}(y_{ig}=0)\) equals one for a zero and zero otherwise, and \(\phi\) is the normal density. The product is inside the integral because observations share the same \(b_g\); replacing it with a product of separately integrated probabilities would discard the induced dependence. Estimation typically evaluates this integral numerically.

Useful checks include inspecting observed counts and zero frequencies by group, comparing the fitted model’s group-level patterns with those observed, and checking whether residual patterns remain associated within groups. Very few groups or very little replication within groups may provide weak information about \(\sigma_b^2\). A variance estimate near zero suggests that the data provide little evidence for this source of group heterogeneity, while numerical convergence problems or a highly uncertain variance estimate call for caution. These checks do not establish that the random-effects distribution is correct; they help reveal conspicuous mismatches and weak support.

A zero-inflated model without a random intercept allows structural and sampling zeros but does not represent this shared group heterogeneity. A model with a group-level fixed intercept instead estimates a separate intercept for each observed group and does not describe groups as draws from a population distribution in the same way. The random-intercept model is useful when groups are viewed as sampled from a broader population, but it relies on a distributional assumption and can be difficult to estimate when there are few groups. Other count-distribution choices may be appropriate for different forms of overdispersion; they do not automatically reproduce the within-group dependence implied by a shared intercept.

Common Pitfalls and Practical Limits

• Calling every zero structural. An observed zero may be a structural zero or a sampling zero from the count distribution. The model assigns probabilities to these routes; the observed value alone does not reveal which route occurred.
• Assuming the random intercept changes both components. In this specification it changes only the count rate. The zero-inflation probability remains determined by its stated predictor.
• Interpreting conditional independence as marginal independence. Conditional independence given \(b_g\) does not imply marginal independence. Integrating over the shared group effect gives the positive covariance derived above.
• Using the zero-random-effect prediction as the population mean. Because of the exponential link, \(E\{\exp(b_g)\}=\exp(\sigma_b^2/2)\), which is greater than one when \(\sigma_b^2>0\). The marginal mean is therefore not generally the prediction at \(b_g=0\).

The model’s assumptions matter as much as its algebra. A normal random-intercept distribution, a Poisson count distribution conditional on the effect, and the stated conditional independence structure may be imperfect approximations. Also, a random intercept captures a shared shift in log rate, not every possible pattern of group heterogeneity. Its estimated association should not be interpreted as evidence that the group effect causes higher counts.