Class Membership Versus Shared Random Effects

Two Representations of Subject Heterogeneity

People can differ in their longitudinal trajectories and their event risks. A model must specify how it represents that variation. A latent-class model represents it using a finite set of qualitatively distinct groups. A shared-random-effects model represents it using continuous subject-specific deviations that enter both the longitudinal and event components. These are different scientific descriptions, not interchangeable labels for “unobserved differences.”

Let \(Y_{ij}\) denote the longitudinal measurement for subject \(i\) at time \(t_{ij}\), and let \(T_i\) and \(\delta_i\) denote the event time and event indicator. Let \(X_i\) denote baseline covariates. A latent class \(C_i\) takes one of \(K\) values, while \(b_i\) is a vector of continuous random effects. The class-specific trajectory and event-risk specifications established in earlier material provide the building blocks for this comparison.

A class is usually treated as stable over the observation period: a subject belongs to one class, even though the class is not directly observed. In contrast, a random effect represents a subject’s position on a continuous scale, such as a higher-than-average intercept or slope. A model with random effects does not, by itself, divide subjects into a finite number of types.

What the Models Assume

A simple class-only model specifies a class probability \(\pi_{ic}=P(C_i=c\mid X_i)\), class-specific longitudinal means, and class-specific event risks. For example, a longitudinal distribution might be written

The event model may likewise depend on \(c\), for example through a class-specific baseline hazard \(h_{0c}(t)\). Conditional on class and modeled covariates, a class-only formulation generally assumes that the remaining longitudinal and event outcomes have no additional subject-level dependence. That is a substantive assumption, not an automatic consequence of using classes.

A shared-random-effects model instead allows continuous deviations. A representative longitudinal component and event hazard are

Here \(Z_{ij}\) specifies how the random effects affect the measurement, and \(\alpha\) specifies how they affect event risk. For example, a random intercept can shift a subject’s measurements upward and, through a nonzero corresponding element of \(\alpha\), increase or decrease that subject’s hazard. The association is modeled through \(b_i\); it does not assert that a measured biomarker causes the event.

These formulations require assumptions to be made explicit. Subjects are typically assumed independent of one another, conditional on the model and covariates. A random-effects model specifies a distribution for \(b_i\), commonly a mean-zero multivariate normal distribution with a covariance matrix. A class model specifies class probabilities and a finite set of class-specific distributions or parameters. For either approach, censoring must be noninformative conditional on the variables included in the model if the event-time analysis is to use the censoring mechanism in the usual way. A shared-effects formulation further assumes that the modeled shared effects adequately capture the relevant residual dependence between longitudinal and event outcomes.

How the Representations Differ

A class-only model can induce an association between measurements and event risk if both vary across classes. But it does so by asserting a finite grouping structure. A shared-random-effects model can induce association without asserting such groups. Its interpretation depends on the random-effects distribution and on which effects are shared. In particular, if \(\alpha=0\), the random effects in the displayed hazard do not link \(b_i\) to event risk, even if they explain longitudinal variation.

Why Mixtures Can Look Like Continuous Variation

The law of total variance makes one distinction precise. Let \(Y\) be a measurement with class-specific mean \(\mu_c\) and within-class variance \(v_c\), and let \(P(C=c)=\pi_c\). Then

The first term is average variation within classes; the second is variation between class means. A continuous random-intercept model can also produce variation in the marginal measurements. Therefore, matching a model to the overall variance alone cannot determine whether the underlying heterogeneity is discrete or continuous. The distribution’s shape, the time pattern, the event association, and the scientific question all matter.

The decomposition is a justification for examining both sources of variation rather than treating an overall variance estimate as evidence for a particular representation. It follows directly from \(\operatorname{Var}(Y)=E(Y^2)-E(Y)^2\), by conditioning the first moment and variance on \(C\).

When Both Forms of Heterogeneity Are Plausible

A scientific setting may suggest both broad subpopulations and meaningful variation within each subpopulation. Conceptually, a combined model assigns a class \(C_i\) and also allows a continuous effect \(b_i\) within that class. The class can distinguish broad trajectory or risk patterns, while \(b_i\) can represent subject-specific departures around the pattern. One possible conceptual specification is

This equation illustrates a combined representation; it is not a claim that every application needs both components. It implies finite class structure and continuous within-class heterogeneity. The distribution of \(b_i\) may itself depend on class, so the analyst must state whether its mean or covariance differs across classes. The substantive interpretation also depends on the conditional-independence assumptions: after conditioning on \(C_i\), \(b_i\), and covariates, any remaining dependence between repeated measurements and the event must be addressed or judged negligible.

Diagnostics, Alternatives, and Limitations

The choice should be guided by the scientific structure and checked against observed patterns. A class model may be inadequate if substantial continuous variation remains within its modeled groups, or if a few classes fail to capture smooth variation. A random-effects model may be inadequate if the assumed continuous distribution is a poor description of sharply separated patterns or if residual dependence with event risk remains. In either case, useful warning signs include systematic trajectory residual patterns, poor fit across event-risk strata, implausible variance estimates, and estimates that change greatly under reasonable modeling choices. These signs motivate investigation; they do not, by themselves, identify the correct representation.

The main alternative to a class representation is a continuous random-effects representation; the main alternative to a shared-effects representation is to model the longitudinal and event components without a shared latent source, if conditional independence is scientifically defensible. A combined model is an additional option when both kinds of heterogeneity are credible, but it requires more assumptions and parameters. It may be difficult to distinguish class separation from broad continuous variation using limited or noisy data. No representation should be selected solely because it yields a preferred scientific story or a better in-sample fit.

A common mistake is to interpret estimated classes as objectively existing biological types. The correction is that classes are features of a specified statistical model: evidence for them depends on the measurement schedule, the assumed distributions, covariates, and other modeling choices. Likewise, a random effect is not automatically a measured trait or a causal mechanism. It is a model-based latent quantity. Interpret either representation as a useful description of heterogeneity only to the extent that its assumptions and diagnostics support that use.