Adding Random Effects Within Trajectory Classes

Why Add Random Effects Within Trajectory Classes?

A latent class describes a subgroup with its own average longitudinal pattern. But subjects assigned to the same class need not follow identical trajectories. Some may begin above the class average; others may change faster or more slowly. A model with class-specific means alone leaves these individual differences to the measurement-error term, which can be an inadequate representation when deviations persist across visits or follow a systematic pattern over time.

Within-class random effects add subject-level departures from the class mean. The two parts answer different questions: class-specific means describe how the average trajectory differs between classes, while random effects describe how subjects vary around their own class's trajectory. In a joint model, these components can also provide a way to relate a subject's longitudinal deviation to their event risk. The central task here is to specify the longitudinal part and understand the heterogeneity it represents.

Model Components and Assumptions

For a standard Gaussian specification, conditional on class \(C_i=g\) and observed covariates, assume \(b_i\mid C_i=g\sim N(0,D_g)\), where \(D_g\) is a positive semidefinite covariance matrix. Assume also that \(\epsilon_{ij}\mid C_i=g\sim N(0,\sigma_g^2)\), with errors independent across visits and independent of \(b_i\). These assumptions imply that the class-specific trajectory is the conditional mean: \(E(Y_{ij}\mid C_i=g,x_i)=m_g(t_{ij},x_i;\beta_g)\). The zero mean for \(b_i\) is important: the class-specific mean is the average trajectory in that class, rather than a mean shifted by an unidentified average random effect.

For example, a linear class-specific mean and a random intercept and slope can be written

In this model, \(\beta_{0g}\) and \(\beta_{1g}\) describe the class's average starting level and rate of change, while \(b_{0i}\) and \(b_{1i}\) describe subject \(i\)'s deviations from those class averages. The covariate effect \(\gamma_g\) may be shared across classes or allowed to differ; that is a modeling choice, not a consequence of including random effects. The event-time submodel can be specified separately by class and may include a function of \(b_i\) when subject-specific longitudinal deviations are intended to relate to event risk. The precise event-time specification is a separate modeling decision.

Random-effects distributions and covariance structures need not differ by class. A common \(D\) can be used when similar within-class variability is plausible and the data cannot support separate covariance matrices. Class-specific \(D_g\) allows different patterns of within-class variability, but requires more information. In either case, a covariance matrix must be valid: its diagonal elements are variances, and its off-diagonal elements must be consistent with a positive semidefinite matrix.

What Variation Do the Components Capture?

The random effects induce dependence between repeated measurements from the same subject. Conditional on class \(g\), the model gives a useful decomposition. Let \(Z_i\) be the matrix whose rows are \(z(t_{ij},x_i)^\top\), and let \(R_i\) be the covariance matrix of the occasion-specific residuals. Under the assumptions above, \(R_i=\sigma_g^2 I\) for independent, equal-variance residuals. The law of total covariance yields the marginal covariance of the repeated measurements within class \(g\).

The first term is persistent, structured between-subject variation: the same random intercept or slope affects multiple visits. The second term is residual variation that remains after class and random effects are accounted for. With independent residuals, it contributes only to the diagonal. This distinction is a feature of the specified model; it does not establish that the assumptions are correct for a particular data set.

Between-class and within-class heterogeneity can coexist. To see the distinction, imagine two classes with different average levels, and subjects who also vary around their respective class averages. The class contrast is a difference in \(m_g\); the within-class spread is governed by \(D_g\) and \(R_i\). If one reports only the overall spread, these sources can be conflated. In particular, pooling classes without accounting for their different means can produce apparent variation that is not within-class subject variability.

Specification, Estimation, and Failure Modes

The model's flexibility depends on the random-effects design. A random intercept allows subjects to differ in level while preserving a common trajectory shape. Adding a random slope allows their rates of change to vary. Higher-order random effects or class-specific covariance matrices can represent richer variation, but also add parameters and estimation demands. The design should reflect scientifically plausible individual departures and the measurement schedule, not simply maximize complexity.

In a joint model, class membership remains latent, so the longitudinal observations and event-time information contribute to uncertainty about class and random effects through the model. Estimation must account for that uncertainty rather than treating a subject's most likely class assignment as known. The random effects are also integrated over their distribution, rather than treated as known subject-level constants. The resulting computation can be more demanding than a class-specific mean model with no random effects.

Several warning signs merit attention. A random-slope variance estimated near zero may indicate that the data do not support slope heterogeneity, although numerical and design limitations should also be considered. A correlation estimate near \(1\) or \(-1\) between random effects can signal a weakly identified or overly elaborate covariance structure. Estimates can be especially unstable when visits are sparse, follow-up times are similar for everyone, or a class contains little information. Patterns in within-class residuals that change over time can indicate a misspecified mean, variance, or residual structure. These checks assess model adequacy; they do not by themselves identify a unique replacement model.

A key comparison is between three representations. A class-specific mean model without random effects permits distinct subgroup averages but no additional persistent subject-level trajectory departures. A model with random effects but no classes permits a continuous distribution of subject deviations around one population mean trajectory. The combined model allows both discrete differences in average trajectory and continuous subject variation within each class. They answer different scientific questions; one is not automatically superior to the others.

There are limitations. A Gaussian random-effects distribution may be a poor description of within-class variation, and normality can affect estimation and inference. A class-specific mean can be too rigid if its shape is wrong; random effects should not be used to disguise a systematic mean misspecification. Conversely, adding many random effects may leave too little information to estimate their covariance. If the class structure and random-effects distribution both have substantial flexibility, their roles can become difficult to distinguish. Results should therefore be interpreted as features of a specified model, not as proof that the population consists of exactly those classes or that all individual differences follow the chosen distribution.

Common Pitfall

A frequent mistake is to say, “The random intercept accounts for the class difference because people in a class start at different levels.” That reasoning confuses variation around a class mean with differences between class means. In the model, the random intercept has mean zero within class \(g\); it describes subject deviations around \(m_g\). If two classes have different average starting levels, those differences belong in their class-specific mean parameters. Allowing arbitrary nonzero class-specific random-effect means without constraints would blur the distinction and make the mean and random-effect components difficult to identify.