Why a Population Survival Curve Is a Mixture
A latent-class joint model describes people as belonging to one of several unobserved trajectory classes. Each class can have its own event-risk model, while the class itself is not directly observed. To describe survival in the population, the class-specific survival curves must therefore be combined according to the class probabilities in the population of interest. This marginal curve answers a different question from any one class-specific curve: it describes survival before the person’s latent class is known.
The key is to use the baseline class probabilities when combining survival probabilities at a fixed covariate value. But those baseline probabilities are not generally the class proportions among people who remain event-free at time \(t\). Classes with higher event risk tend to become less common among survivors, and this changing composition affects the marginal hazard.
This identity follows by partitioning the event \(T>t\) according to class. It does not require the classes to have equal hazards or equal sizes. It does require the class-specific survival functions and class probabilities to refer to the same target population and covariate condition. When the target is the population without conditioning on \(X\), the covariate distribution must also be averaged over:
Thus, a marginal curve for a particular covariate group and a population-standardized curve need not be the same. If class membership depends on baseline covariates, the class weights also vary with those covariates.
Worked Example: Combining Two Exponential Survival Curves
Suppose a hypothetical population has two classes with baseline probabilities \(\pi_1=0.70\) and \(\pi_2=0.30\). Within class 1 the constant hazard is \(0.10\) events per person-year; within class 2 it is \(0.50\) events per person-year. The exponential survival functions are \(S_1(t)=e^{-0.10t}\) and \(S_2(t)=e^{-0.50t}\), where \(t\) is in years.
At year 2, the marginal survival probability is
The model therefore gives a \(68.35\%\) probability of surviving beyond 2 years in this population, or a \(31.65\%\) probability of an event by that time. These are population probabilities, not the survival probability of either class alone.
The marginal median is the time \(t\) at which \(0.70e^{-0.10t}+0.30e^{-0.50t}=0.5\). Substitution at \(t=4.15\) gives \(0.70e^{-0.415}+0.30e^{-2.075}\approx0.4999\), so the marginal median is approximately 4.15 years. It is not the weighted average of the class medians: those are \(\log(2)/0.10=6.931\) years and \(\log(2)/0.50=1.386\) years, whose probability-weighted average is \(5.268\) years. A mixture median must be obtained from the mixture survival function itself.
How the Marginal Hazard Changes Over Time
For a differentiable survival function with positive survival probability, the hazard is the instantaneous event rate among those who have survived to time \(t\). Write \(\lambda_g(t\mid x)\) for the hazard in class \(g\). Differentiating the mixture survival function gives the marginal hazard:
The weights in this expression have a useful interpretation. Among people with covariates \(x\) who have survived beyond \(t\), the probability of class \(g\) is
Consequently, the marginal hazard is a weighted average of the class-specific hazards, but the weights are the class proportions among survivors, not the original baseline proportions. The riskier class generally loses weight as time passes. This survivor-composition change is the central reason a marginal hazard can evolve differently from every class-specific hazard.
Worked Example: A Correctly Weighted Marginal Hazard
Continue with two classes, now comparing a binary covariate \(x\). Assume baseline class probabilities remain \(0.40\) and \(0.60\) at each value of \(x\). In a hypothetical exponential model, the class-specific hazards in events per person-year are \(0.04452\) and \(0.20\) for \(x=0\), and \(0.12\) and \(0.60\) for \(x=1\). Calculate the marginal hazards at year 3.
For \(x=0\), the class-specific survivals at year 3 are \(e^{-0.04452(3)}\approx0.8750\) and \(e^{-0.20(3)}\approx0.5488\). Thus the marginal survival is
The marginal hazard is the survivor-weighted average:
events per person-year. For \(x=1\), the class-specific survivals are \(e^{-0.12(3)}\approx0.6977\) and \(e^{-0.60(3)}\approx0.1653\). The marginal survival is approximately \(0.40(0.6977)+0.60(0.1653)=0.3783\), and the hazard is
events per person-year. The instantaneous marginal hazard ratio at year 3 is therefore approximately \(0.2459/0.1199=2.051\). It describes the ratio of the two marginal hazards at that time, not a constant effect over follow-up. Rounding the class survivals before calculating has little effect here, but the calculations should use adequate precision until the final reported values.
Why a Mixture Is Generally Not Proportional Hazards
A proportional-hazards model requires the hazard ratio between covariate groups to remain constant over time. Class-specific proportional hazards do not, by themselves, guarantee this property after classes are mixed. In fact, even a shared covariate effect across classes is not generally enough: the mixture of class-specific survival functions need not have the form of a single baseline survival function raised to a common covariate-dependent power.
For a simple justification, suppose the two classes have constant baseline hazards \(0.10\) and \(0.50\) per year, equal baseline probabilities, and the same covariate hazard multiplier of 2. At time zero, the marginal hazards are \(0.5(0.10)+0.5(0.50)=0.30\) for \(x=0\) and \(0.5(0.20)+0.5(1.00)=0.60\) for \(x=1\), giving a hazard ratio of 2. At year 2, the survivor weights differ between covariate groups:
The corresponding marginal hazards are approximately \(0.690(0.10)+0.310(0.50)=0.224\) and \(0.832(0.20)+0.168(1.00)=0.334\) events per person-year. Their ratio is about \(1.49\), not 2. Since the ratio changed from 2 at time zero to about 1.49 at year 2, the marginal hazards are not proportional in this example. The shift occurs because the higher-risk class is depleted more rapidly among survivors, and the depletion differs by covariate group.
A marginal proportional-hazards form can hold in special cases. For instance, if the classes have identical event distributions at a given covariate value, mixing does not create distinct survivor compositions. More generally, proportionality needs to be established for the marginal hazards themselves; it cannot be inferred solely from class-specific proportional-hazards assumptions.
Practical Interpretation, Comparisons, and Diagnostics
The marginal survival curve is often the appropriate summary when the scientific question concerns the population rather than a hypothetical member of a known class. It accounts for the fact that class membership is latent and provides an interpretable survival probability for the target covariate distribution. Class-specific curves remain useful for describing heterogeneity, but they answer conditional questions.
| Quantity | Weights used | Interpretation |
|---|---|---|
| Marginal survival at baseline covariates \(x\) | Baseline class probabilities \(\pi_g(x)\) | Survival for the population with covariates \(x\), before class is known |
| Marginal hazard at time \(t\) | Class probabilities among survivors at \(t\) | Instantaneous event rate among the population still event-free |
| Class-specific survival or hazard | One class at a time | Event experience conditional on latent class membership |
A tempting shortcut is to average the class-specific hazards using the baseline probabilities at every time. That is generally wrong: the hazard among survivors uses the time-dependent weights \(\pi_g(x)S_g(t\mid x)/S_{\mathrm{marg}}(t\mid x)\). Another shortcut is to average class medians or other quantiles; mixture quantiles instead come from solving the mixture survival equation. A single Cox model fitted to marginal data can still be used as a descriptive approximation, but if the true marginal hazard ratio varies with time, its coefficient does not represent an exact constant marginal effect.
Useful checks include plotting the model-implied marginal survival curve against an empirical survival estimate for the same target group, and comparing predictions at clinically meaningful times. If a hazard-ratio summary is desired, examine model-implied marginal hazards or survival contrasts over follow-up rather than assuming a constant ratio. Large discrepancies in survival calibration, implausible class-specific predictions, or strong sensitivity of the marginal curve to small changes in class probabilities can signal that the fitted mixture is not adequately describing the target population. These checks do not establish that the latent classes are uniquely identified or that the class structure is correct; those questions require separate assessment.
The marginal calculation also has limits. It inherits the assumptions and uncertainty of the fitted class probabilities and class-specific survival models. If the covariate distribution used for standardization differs from the intended target population, the resulting curve answers the wrong population question. And because classes are latent, a marginal survival curve should not be described as if each person’s class were observed with certainty. For estimation from censored data, the censoring assumptions needed by the fitted model and any empirical comparison must also be appropriate.
Check Your Understanding
Work through each question, then compare with the solutions.
- A two-class population has probabilities \(0.25\) and \(0.75\), with exponential hazards \(0.20\) and \(0.80\) events per year. Calculate marginal survival at year 1.
- At a particular time, baseline class probabilities are \(0.40\) and \(0.60\), and the class-specific survival probabilities are \(0.80\) and \(0.20\). Find the probability of class 1 among survivors.
- Does a common covariate hazard multiplier in every class guarantee marginal proportional hazards? Give the reason.
- Why is it incorrect to calculate a marginal hazard using only the baseline class probabilities at every time?
- If class probabilities depend on baseline covariates, what additional averaging is needed for a population survival curve?
Solutions. (1) \(S_{\mathrm{marg}}(1)=0.25e^{-0.20}+0.75e^{-0.80}=0.25(0.8187)+0.75(0.4493)\approx0.5417\), so the probability of surviving beyond one year is about \(54.17\%\). (2) The survivor probability for class 1 is \(0.40(0.80)/[0.40(0.80)+0.60(0.20)]=0.32/0.44\approx0.7273\). (3) No. Different classes generally contribute different proportions of survivors over time, and those proportions can differ by covariate group; the marginal hazard ratio can therefore vary even when each class has the same multiplier. (4) The baseline weights describe the full population, whereas a hazard is conditional on having survived to that time. The appropriate weights are proportional to baseline probability times class-specific survival. (5) First combine class-specific survival using \(\pi_g(x)\) at each covariate value, then average that conditional mixture over the target distribution \(F_X(x)\).