Conditional Independence Within Classes

Why Check Conditional Independence?

A latent class joint model represents individuals as belonging to unobserved trajectory groups and models longitudinal outcomes and event risk within those groups. Its conditional-independence assumption says that, once class membership and the model covariates are accounted for, the longitudinal process supplies no further information about the event time. This assumption is important: it lets the joint contribution for an individual be written as a product of a longitudinal contribution and an event contribution within each class.

The assumption is not that longitudinal measurements and event times are unrelated in the whole population. They may be strongly associated because class membership affects both. The question is whether that association is adequately explained by the specified classes and covariates, or whether a systematic association remains within classes.

Definition: Let $C_i$ denote the latent class for individual $i$, $X_i$ the modeled covariates, $Y_i$ the longitudinal outcome process, and $T_i$ the event time. Conditional independence within classes means $$ Y_i \perp T_i \mid C_i,X_i, $$ or, equivalently, that the conditional joint distribution factors as $$ f(Y_i,T_i\mid C_i=k,X_i) = f_Y(Y_i\mid C_i=k,X_i)\,f_T(T_i\mid C_i=k,X_i) $$ for every class $k$ represented by the model. With right censoring, the event contribution is expressed using the observed event data $(U_i,\Delta_i)$, where $U_i=\min(T_i,D_i)$, $D_i$ is the censoring time, and $\Delta_i=I(T_i\leq D_i)$. The censoring mechanism must be handled under its own assumptions; conditional independence of $Y_i$ and $T_i$ does not itself imply independent censoring.

In an observed-data mixture likelihood, the class is unobserved, so the individual contribution is a sum of class-specific products, weighted by class probabilities. The factorization is within each class, not generally after averaging over classes. The model also needs the stated class structure, covariate effects, longitudinal distribution, and event-time distribution to be adequate for the intended interpretation. A failure of any of these components can look like residual dependence.

What the Assumption Does—and Does Not—Say

Conditional independence is a modeling restriction, not a biological claim that the measurements and event process have no connection. It says that the model’s class membership and covariates capture their dependence. If, for example, individuals with a persistently high biomarker tend to experience earlier events even among individuals with the same class and covariates, the restriction may be inadequate.

Random effects require particular care. Suppose $b_i$ is an individual-specific random effect used to describe variation in trajectories within a class. If the event model also depends on $b_i$, then the two processes may be independent only after conditioning on $C_i$, $X_i$, and $b_i$. Integrating over the unobserved $b_i$ can leave dependence given class and covariates alone. Thus, conditional independence given class and covariates is stronger than conditional independence given class, covariates, and a shared random effect.

Worked Example: Shared Random Effects Can Leave Dependence

Consider one class and no covariates. In a hypothetical model, $b_i$ is either $-1$ or $1$, each with probability $0.5$. A binary longitudinal summary $R_i$ equals $b_i$. Conditional on $b_i$, the indicator $E_i$ of an event by a fixed time is independent of $R_i$, with event probability $0.1$ when $b_i=-1$ and $0.5$ when $b_i=1$.

The marginal event probability is $$ P(E_i=1)=0.5(0.1)+0.5(0.5)=0.30. $$ The mean of $R_i$ is zero. Since $R_i=1$ in the $b_i=1$ group and $R_i=-1$ in the $b_i=-1$ group, $$ E(R_iE_i)=0.5(1)(0.5)+0.5(-1)(0.1)=0.20. $$ Therefore $$ \operatorname{Cov}(R_i,E_i) =E(R_iE_i)-E(R_i)E(E_i) =0.20-(0)(0.30)=0.20. $$ The two outcomes are independent conditional on $b_i$, but associated conditional on class alone (there is only one class). A shared random effect in the event model can therefore explain dependence that a class-only factorization leaves behind.

Why Marginal Dependence Can Arise Even When the Assumption Holds

A mixture of conditionally independent distributions need not itself factor into independent marginal distributions. To see why, suppose there are two classes with equal probability. Within each class, let $A$ indicate a particular longitudinal outcome and $E$ indicate an event by a specified time. Suppose $P(A=1\mid C=1)=0.8$, $P(E=1\mid C=1)=0.7$, $P(A=1\mid C=2)=0.2$, and $P(E=1\mid C=2)=0.1$, with $A$ and $E$ independent within each class.

Worked Example: Marginal Association From Class Mixing

The joint probability of $A=1$ and $E=1$ is the class-weighted sum of the within-class joint probabilities: $$ P(A=1,E=1) =0.5(0.8)(0.7)+0.5(0.2)(0.1) =0.28+0.01=0.29. $$ The marginal probabilities are $$ P(A=1)=0.5(0.8)+0.5(0.2)=0.50, \qquad P(E=1)=0.5(0.7)+0.5(0.1)=0.40. $$ If $A$ and $E$ were marginally independent, their joint probability would be $0.50(0.40)=0.20$, not $0.29$. The marginal covariance is $0.29-0.20=0.09$. This association is entirely compatible with conditional independence within both classes: the classes differ in both outcomes.

This calculation also gives a useful diagnostic principle: observing an overall longitudinal–event association is not evidence by itself against the assumption. The relevant question is whether a pattern remains after accounting for class and covariates. Conversely, a model that forces independence within classes can still show substantial population-level association.

Assessing Residual Dependence

No single diagnostic proves conditional independence, especially because class membership is unobserved. The most useful assessment combines subject-matter reasoning, conditional summaries, and checks of how well the fitted model reproduces relevant patterns. A residual association may indicate a missing shared source of variation, an inadequate trajectory or event model, insufficient covariate adjustment, or a class structure that does not capture the important subgroups.

One focused check uses a longitudinal summary available before a chosen landmark time. For example, let $R_i$ be a trajectory residual or a pre-specified deviation from the fitted longitudinal mean, calculated using measurements before the landmark. Among individuals still at risk at that time, examine whether subsequent event risk varies with $R_i$, within class and covariate strata. A systematic trend suggests that the class and covariates may not have captured all relevant association. The landmark and summary must be chosen so that later measurements are not used to predict earlier events.

Another approach is to fit an event model with a pre-specified longitudinal summary as an additional predictor. For example, within class $k$, one could examine the coefficient $\gamma_k$ in

h_i(t\mid C_i=k,X_i,R_i) = h_{0k}(t)\exp\{X_i^\top\beta_k+\gamma_k R_i\},

where $R_i$ is defined using only longitudinal information available at the relevant prediction time. Under the conditional-independence restriction, this added summary should not provide systematic event information once class and covariates are accounted for; in this particular model that corresponds to $\gamma_k=0$. A nonzero estimate is a diagnostic signal, not an automatic proof of a biological link. It may reflect model misspecification, noisy estimated residuals, class uncertainty, or selection into the landmark risk set.

Worked Example: A Within-Class Landmark Pattern

In a hypothetical diagnostic data set, 40 individuals in the same well-represented class have comparable modeled covariates and are all event-free at a landmark. Their pre-landmark biomarker residuals are divided into low and high groups, with 20 individuals in each group. By the end of the next year, 2 of the 20 low-residual individuals and 8 of the 20 high-residual individuals experience the event.

The observed one-year event proportions are $2/20=0.10$ for the low-residual group and $8/20=0.40$ for the high-residual group. The estimated risk difference is $$ \widehat{P}(E=1\mid R\text{ high})- \widehat{P}(E=1\mid R\text{ low}) = \frac{8}{20}-\frac{2}{20} =0.30. $$ Thus, in these hypothetical data, the high-residual group has a 30-percentage-point higher observed one-year event proportion. This is a potentially important residual pattern, but it is imprecise with only 20 individuals per group. It should prompt further checks, not a causal conclusion or a claim that conditional independence has been disproved. The comparison also assumes that follow-up and event ascertainment are comparable across the two groups.

Posterior class probabilities should be respected when using these diagnostics. Assigning every individual to the class with the largest posterior probability and then treating that assignment as known can distort within-class patterns, particularly when classes overlap. Prefer summaries that account for classification uncertainty, such as posterior-weighted checks or checks based on data simulated from the fitted model. Interpret any diagnostic in light of how well the classes are separated and how many individuals contribute information to each class.

Derivation: Why the Observed Mixture Does Not Usually Factor

Write $\pi_{ik}=P(C_i=k\mid X_i)$. Under conditional independence, the joint contribution for longitudinal and event data is

f(Y_i,T_i\mid X_i) = \sum_k \pi_{ik} f_Y(Y_i\mid C_i=k,X_i)f_T(T_i\mid C_i=k,X_i).

The product of the two marginal contributions instead is

f_Y(Y_i\mid X_i)f_T(T_i\mid X_i) = \left\{\sum_k\pi_{ik}f_Y(Y_i\mid C_i=k,X_i)\right\} \left\{\sum_\ell\pi_{i\ell}f_T(T_i\mid C_i=\ell,X_i)\right\}.

The first expression pairs the longitudinal and event contributions within the same class before summing. The second independently sums each contribution and then multiplies them, including cross-class terms with $k\ne\ell$. They are not generally equal. This is why overall association can arise under the within-class assumption, and why the appropriate diagnostic must target patterns conditional on class and modeled covariates rather than the unadjusted association.

Diagnostics, Alternatives, and Limitations

A useful assessment asks whether the fitted model reproduces the joint features relevant to the analysis, rather than relying on a single test. Possible checks include comparing event experience across pre-specified longitudinal residual groups within classes, examining whether event timing varies systematically with pre-event trajectory deviations, and using posterior predictive simulations to compare observed summaries with summaries generated under the fitted conditional-independence model.

A formal comparison can contrast the conditional-independence model with a model that permits an additional association parameter, such as a shared random effect or a longitudinal summary in the event model. A likelihood-ratio statistic is meaningful only when the models are properly nested and the regularity conditions for its reference distribution hold. In mixture models, weakly identified classes or parameters on boundaries can violate ordinary large-sample conditions; a parametric bootstrap may be more appropriate, with the full estimation procedure repeated for each simulated data set. A test’s failure to reject is not proof of independence, particularly when sample size or class-specific event counts are small.

The main alternatives differ in what they allow: a richer class structure can represent more subpopulations; a shared random effect can represent individual-level variation linked across processes; and a direct residual association term can capture a specified remaining relationship. These alternatives are not interchangeable. They encode different assumptions, can be difficult to identify from sparse data, and should be motivated by the scientific question. Adding complexity solely to remove one diagnostic pattern can make the model unstable or obscure what its parameters represent.

Finally, longitudinal measurements may stop at the event. A post-event biomarker value may be undefined, not merely unobserved. Treating event-truncated measurements as ordinary missing values can create misleading diagnostics. Checks should use a clearly defined observation window and account for the fact that individuals must remain event-free to contribute measurements through a landmark. The introductory material on latent class joint models establishes the mixture framework; here the key additional issue is whether its within-class factorization is credible for the data and purpose at hand.

Common pitfall: “The overall biomarker and event time are associated, so conditional independence is false.” This reasoning ignores association induced by mixing classes. The correction is to assess whether the association remains within classes after covariate adjustment, while accounting for uncertainty in class membership and any shared random effects.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

State the conditional-independence factorization for longitudinal outcomes and event time within class $k$.
In the two-class example above, why is marginal association compatible with conditional independence?
Suppose an event model depends on the same random effect used in the longitudinal trajectory model. What conditioning set may be needed for independence, and what can happen when that random effect is integrated out?
In a landmark check, 5 of 25 individuals in a low-residual group and 10 of 25 in a high-residual group have an event in the following year. Compute the observed risk difference and state one reason not to interpret it as proof of residual dependence.

Solutions. (1) The factorization is $f(Y_i,T_i\mid C_i=k,X_i)=f_Y(Y_i\mid C_i=k,X_i)f_T(T_i\mid C_i=k,X_i)$. (2) The classes have different probabilities for both outcomes; averaging over the same latent class can associate the outcomes even when they are independent within each class. (3) Independence may hold only conditional on class, covariates, and the shared random effect. Integrating out that effect can induce dependence given class and covariates alone. (4) The risk difference is $10/25-5/25=0.20$, or 20 percentage points. The estimate may be imprecise, affected by class uncertainty or model misspecification, and is an observational diagnostic rather than proof of a causal relationship.