Flexible Latency Models Beyond Proportional Hazards

Why Extend the Latency Model?

A mixture cure model separates two processes: whether an individual is susceptible to the event during the modeled lifetime, and the event time among susceptible individuals. A latency model describes that second process. Its specification matters: a susceptible proportional hazards model assumes that covariate hazard ratios are constant over time, while a chosen parametric distribution also imposes a particular shape on the event-time distribution.

These restrictions can be inadequate even when the cure component is well specified. For example, a treatment might have little association with the susceptible hazard early in follow-up and a stronger association later. Alternatively, the susceptible event-time distribution may be skewed or have a hazard pattern that a selected parametric family cannot represent. Flexible latency models address these possibilities without changing the basic distinction between cure probability and latency.

The notation allows incidence covariates \(z\) and latency covariates \(x\) to differ, though they may overlap. Unless a model explicitly links them, a latency effect changes the event-time distribution among susceptible individuals, not their probability of being susceptible. Standard cure-model assumptions include a well-defined susceptible event-time distribution, a cure fraction that remains event-free over the modeled horizon, and censoring independent of event time and susceptibility conditional on the modeled covariates. Inference also needs adequate follow-up to distinguish a long latency tail from a cured fraction, as emphasized in the earlier material on identifiability and follow-up.

Time-Varying Covariate Effects

A direct extension of the susceptible proportional hazards model lets a covariate coefficient vary with time:

Here \(h_0(t)\) is a positive baseline hazard and \(\beta(t)\) is a specified or estimated function. For a binary covariate, the susceptible hazard ratio comparing \(x=1\) with \(x=0\) at time \(t\) is \(\exp\{\beta(t)\}\). If \(\beta(t)\) is constant, this reduces to proportional hazards. A smooth function, such as a linear function of \(\log(t/t_0)\), or a spline basis can represent a changing effect. The reference time \(t_0\) makes the logarithm dimensionless.

The model requires more than a convenient formula. The baseline hazard must stay positive, the time-varying effect must be defined over the analysis range, and the amount of flexibility must be supportable by the number and timing of events. A linear log-time effect imposes a particular smooth trend; a spline can be more flexible but can also overfit, especially in the tail where few subjects remain at risk. A changing coefficient is not a substitute for checking coding, functional form, or the incidence model.

Flexible Parametric Latency Distributions

Another strategy specifies a parametric distribution for \(T_u\) that does not impose proportional hazards. A lognormal accelerated failure time model is one example:

For a binary covariate, one may set \(\mu(x)=\gamma_0+\gamma_1x\). The conditional susceptible survival is \(S_u(t\mid x)=1-\Phi\{[\log(t)-\mu(x)]/\sigma\}\), where \(\Phi\) is the standard normal cumulative distribution function. The density is \(f_u(t\mid x)=\phi\{[\log(t)-\mu(x)]/\sigma\}/(t\sigma)\), where \(\phi\) is the standard normal density. A positive \(\gamma_1\) shifts the susceptible event-time distribution toward longer times by a factor \(\exp(\gamma_1)\) at corresponding distributional quantiles. It does not imply a constant hazard ratio.

Likelihood and a Short Justification

For an observed time \(y\) and event indicator \(\delta\), where \(\delta=1\) denotes an observed event and \(\delta=0\) denotes right censoring, the conditional likelihood contribution is

This expression follows by conditioning on the unobserved susceptibility status. If an event is observed, the individual must be susceptible, contributing probability \(\pi(z)\) of susceptibility times the susceptible event-time density. If the observation is censored at \(y\), either the person is cured, with probability \(1-\pi(z)\), or susceptible and event-free beyond \(y\), with probability \(\pi(z)S_u(y\mid x)\). These mutually exclusive possibilities add.

The same mixture logic yields the marginal hazard. Differentiating the marginal survival gives \(-dS_m(t)/dt=\pi(z)f_u(t\mid x)\). Dividing this marginal density by the marginal survival gives

Consequently, even if the susceptible hazard is proportional, the marginal hazard generally is not: the proportion of susceptible individuals among those still event-free changes over time. This distinction is important when interpreting a fitted time-varying coefficient. The coefficient above describes latency conditional on susceptibility; it is not automatically a time-varying effect on the population hazard.

Choosing, Checking, and Limiting Flexibility

The main alternatives address different shortcomings. A susceptible proportional hazards model leaves the baseline hazard flexible while constraining covariate hazard ratios to be constant. A time-varying coefficient model relaxes that covariate-effect restriction, but still depends on its chosen time-function form. A parametric accelerated failure time model such as the lognormal specifies the latency distribution directly and gives interpretable time-scale effects, but its distributional assumptions may be restrictive. More flexible parametric forms can represent a wider range of hazard shapes, at the cost of additional parameters and greater risk of unstable tail behavior.

Model choice should be driven by the suspected inadequacy and supported by the data. Useful warning signs include systematic departures between fitted and observed event-time patterns among susceptible individuals, an estimated time-varying effect that changes sharply where few events occur, convergence problems, implausible fitted hazards, or major changes in the cure estimate under modest latency changes. Diagnostics must respect the mixture: censoring and the latent susceptibility status prevent direct inspection of the susceptible event-time distribution for every subject. Model-based comparisons should therefore examine fit and sensitivity rather than treating one diagnostic plot as decisive.

A useful practical distinction is whether the problem concerns the covariate effect or the latency shape. If the effect appears to change over time, consider a time-varying coefficient. If the chosen susceptible distribution has an implausible or inadequate shape even after accounting for covariates, consider a different or more flexible latency distribution. Using both forms of flexibility can be reasonable, but it increases estimation demands and can make cure and long-latency behavior harder to separate.

These models have finite limits. A flexible curve cannot recover information absent from the follow-up, and long-term event-free observations alone may not distinguish cure from very long latency. A complicated model can also fit random variation and yield unstable cure estimates. Finally, a time-varying association is not evidence that a covariate causally changes event times; causal interpretation requires appropriate design and assumptions beyond the latency model.