Sensitivity to the Number and Shape of Classes

Why Sensitivity Analysis Matters

A latent-class joint model describes a population as a mixture of unobserved groups, with a longitudinal trajectory and a survival process associated with each group. The resulting classes are features of the fitted model; changing the number of classes or the form of their trajectories can change the estimated groups, their proportions, and their event associations. A conclusion that appears compelling under one specification may therefore be fragile.

Sensitivity analysis asks whether the substantive conclusions are similar across a small set of plausible, adequately fitted candidate models. It is not a search for a specification that produces the preferred result. Nor does agreement across models prove that the latent groups are real. The goal is to distinguish stable findings from those that depend on modeling choices.

For a subject in class \(c\), write the longitudinal mean as \(m_c(t)\), where \(t\) is time, and let \(\pi_c\) denote the class probability, with \(\sum_c \pi_c=1\). A common survival specification is \(h_i(t\mid C_i=c,b_i,w_i)=h_{0c}(t)\exp\{\gamma_c^\mathsf{T}w_i+\alpha_c m_i(t)\}\), where \(C_i\) is the latent class, \(b_i\) denotes any within-class random effects, \(w_i\) denotes survival covariates, and \(m_i(t)\) is the subject-specific latent marker trajectory. The exact specification can vary, but comparisons must refer to the same quantities and conditioning assumptions.

Candidate fits should be obtained under the model’s stated assumptions: the class probabilities and component distributions define a valid mixture; the longitudinal and survival contributions are combined according to the chosen conditional-independence or shared-random-effects structure; and the event-time and censoring model is appropriate for the observed data. Comparisons also require credible numerical fits. A failed convergence check or a solution that changes substantially across starting values is not a sound candidate for scientific interpretation.

What to Compare

Compare more than a single fit statistic. First, examine overall fit and complexity using criteria such as AIC or BIC, computed on the same data and likelihood scale. Second, compare class-level results that answer the scientific question: prevalence, predicted trajectories at prespecified times, and survival associations on a common scale. Third, inspect whether classes remain meaningfully distinct and sufficiently represented. The material on class separation, posterior classification uncertainty, local maxima, and within-class residual checks is useful for interpreting these diagnostics; none replaces the sensitivity comparison itself.

Align classes by their estimated pattern, not by their numeric labels. A “class 1” in a two-class fit need not correspond to “class 1” in a three-class fit. Compare trajectories over the observed time range and at scientifically meaningful times, rather than comparing polynomial coefficients in isolation: coefficients can change scale or interpretation when the trajectory form changes.

Comparing Trajectory Shapes

A class count can be held fixed while the trajectory specification changes. For example, a linear mean \(m_c(t)=\beta_{0c}+\beta_{1c}t\) imposes a constant rate of change. A quadratic mean \(m_c(t)=\beta_{0c}+\beta_{1c}t+\beta_{2c}t^2\) allows curvature. These are different assumptions about how the mean evolves, not merely different ways to label the same fitted curve.

Compare fitted means in the outcome’s units at times that have adequate data support. Also compare clinically or scientifically meaningful features, such as the direction of change, whether curves cross, and the size of the contrast between classes. A dramatic difference at the boundary of follow-up may reflect extrapolation rather than reliable evidence. The trajectory model should be assessed alongside residual patterns and the distribution of actual observation times.

Why Survival Results Can Shift

A change in trajectory shape can affect survival results because the longitudinal trajectory may enter the hazard model. A change in class count can also reassign subjects or split a class, changing the fitted class-specific association. Consequently, report comparable survival quantities, not just the raw association coefficients. For a model with a current-marker association \(\alpha_c\), a five-unit marker contrast at a fixed time, holding the other modeled quantities constant, corresponds to a hazard ratio \(\exp(5\alpha_c)\). This is a conditional association, not a causal effect.

Mixture survival also explains why a class-specific result need not describe the population-level pattern. For simplicity, suppose class \(c\) has baseline probability \(\pi_c\) and survival function \(S_c(t)\). The marginal survival function is \(S(t)=\sum_c\pi_cS_c(t)\). Differentiating and using \(h_c(t)=-S'_c(t)/S_c(t)\) gives

The final weights are the class proportions among subjects still event-free at time \(t\), not generally the baseline proportions \(\pi_c\). They change over time when class-specific survival differs. Thus a revised class structure can affect both the component hazards and the mixture’s evolving composition. This derivation assumes the stated mixture survival representation; covariate-specific survival requires conditioning on covariates and using the corresponding conditional component functions.

A Practical Comparison and Its Limits

A disciplined comparison separates numerical fit, predictive adequacy, and scientific interpretation. Information criteria summarize fit with a complexity penalty; they do not show that class trajectories are credible. A likelihood-ratio test can be appropriate for certain nested models under regularity conditions, but adding mixture components can put parameters on boundaries or make them unidentified under the null. The usual reference chi-square distribution may then fail. AIC and BIC, predictive assessment, and substantive examination can complement one another, but none turns an ill-fitting candidate into an acceptable model.

Use the same outcome and time scales, covariate definitions, and analysis sample wherever possible. Check convergence from multiple starting values, class prevalence, uncertainty, residual patterns, and whether each fitted trajectory is supported by observations. A tiny estimated class, highly uncertain trajectory, or markedly different solutions across starts weakens the interpretation of that candidate. If models have different analytic samples or likelihood definitions, raw likelihood criteria may not be directly comparable.

A useful report states which model features were varied, which were held fixed, the fit summaries used, and what changed in the quantities central to the scientific question. Present trajectory predictions at selected times with uncertainty, class proportions with appropriate caution, and hazard associations on a common scale. State plainly when an interpretation depends on the chosen class count or shape. Sensitivity analysis does not eliminate uncertainty about the number or meaning of latent classes; it makes that dependence visible.

Common Mistake