Conditional Hazard Interpretation Under Frailty

Why Condition on Frailty?

Individuals in the same clinic, household, or other cluster may have different underlying propensities for an event. A frailty model represents unmeasured multiplicative heterogeneity with a positive latent variable. The resulting hazard describes an individual or cluster at a specified frailty value. It is not automatically the hazard or risk observed after averaging across a population with varying frailty.

The interpretation depends on keeping three quantities distinct: the baseline hazard, the covariate effect, and the frailty multiplier. A hazard is an instantaneous rate among individuals who have not yet had the event; a risk is the probability of having the event over a specified interval. The distinction matters because a rate, even when multiplied by a frailty value, is not itself a probability.

Definition: For a single terminal event in cluster $i$, let $T_{ij}$ be the event time for individual $j$, $x_{ij}$ a covariate, and $Z_i>0$ a shared frailty for the cluster. A proportional shared-frailty model specifies the conditional hazard as $$ \lambda_{ij}(t\mid Z_i,x_{ij})=Z_i\lambda_0(t)\exp(\beta x_{ij}). $$ Here $\lambda_0(t)$ is the baseline hazard, and $\beta$ is a conditional log hazard ratio for a one-unit increase in $x$, holding frailty fixed. The frailty is conventionally scaled so that $E(Z_i)=1$; this sets a reference scale for the baseline hazard.

The conditional interpretation requires $Z_i$ to be held fixed. For two covariate values $x_a$ and $x_b$ in the same frailty stratum, the ratio of hazards at the same time is $\exp\{\beta(x_a-x_b)\}$. The shared multiplier and baseline hazard cancel. In particular, a one-unit increase in $x$ has conditional hazard ratio $\exp(\beta)$. This is a comparison of instantaneous hazards among individuals still event-free at time $t$, conditional on the specified frailty; it is not a risk ratio.

What the Baseline Hazard Means

The baseline hazard $\lambda_0(t)$ is the hazard at time $t$ for the reference covariate value $x=0$ and frailty value $Z_i=1$. Its units are inverse time, such as events per person-month. If $Z_i=2$, the hazard is twice the reference hazard at that time when the covariate is held at its reference value. If $x=1$, the hazard is multiplied by $\exp(\beta)$ as well.

The baseline hazard need not be constant over time. A value of $0.04$ per month at month 6 describes an instantaneous rate at that time, not a 4% probability of an event during a month. To obtain a probability over an interval, the hazard must be accumulated over time and converted to survival or risk under the model.

Worked Example: Interpreting a Conditional Hazard

Suppose a hypothetical study uses $\lambda_{ij}(t\mid Z_i,x_{ij})=Z_i\lambda_0(t)\exp(\beta x_{ij})$. At month 6, let $\lambda_0(6)=0.04$ per month and $\beta=\log(1.5)$. Compare the conditional hazards for $x=0$ and $x=1$ in a cluster with $Z_i=2$.

For $x=0$, the hazard is $$ 2(0.04)\exp\{\log(1.5)(0)\}=0.08\text{ per month}. $$ For $x=1$, it is $$ 2(0.04)\exp\{\log(1.5)(1)\}=0.08(1.5)=0.12\text{ per month}. $$

Thus, at month 6, the $x=1$ individual has 1.5 times the instantaneous hazard of the $x=0$ individual, conditional on the same frailty $Z_i=2$. The baseline hazard is $0.04$ per month for $x=0$ at $Z_i=1$; it is not the hazard for this cluster. Neither $0.08$ nor $0.12$ per month is a probability.

From Conditional Hazard to Conditional Risk

For the terminal-event model above, assume the covariate and frailty are fixed over the interval and the conditional hazard is integrable. Define the cumulative baseline hazard by $\Lambda_0(t)=\int_0^t\lambda_0(u)\,du$. Conditional survival is the probability of remaining event-free through time $t$, given $Z_i$ and $x_{ij}$. It follows from the hazard definition:

\frac{d}{dt}S_{ij}(t\mid Z_i,x_{ij}) =-\lambda_{ij}(t\mid Z_i,x_{ij})S_{ij}(t\mid Z_i,x_{ij}), \qquad S_{ij}(0\mid Z_i,x_{ij})=1.

Dividing by survival and integrating from 0 to $t$ gives the conditional survival function. The conditional risk by time $t$ is one minus that survival probability.

\begin{aligned} \log S_{ij}(t\mid Z_i,x_{ij}) &=-\int_0^t \lambda_{ij}(u\mid Z_i,x_{ij})\,du\\ &=-Z_i\exp(\beta x_{ij})\Lambda_0(t),\\ S_{ij}(t\mid Z_i,x_{ij}) &=\exp\{-Z_i\exp(\beta x_{ij})\Lambda_0(t)\},\\ P(T_{ij}\le t\mid Z_i,x_{ij}) &=1-\exp\{-Z_i\exp(\beta x_{ij})\Lambda_0(t)\}. \end{aligned}

This derivation shows why the cumulative hazard, rather than a hazard at one instant, determines interval risk. The formula also depends on the model being used for a single terminal event from time 0. For recurrent-event intensities, a mean event count is not generally obtained by treating the intensity as a terminal-event hazard and taking one minus survival.

Worked Example: Calculating a Conditional Six-Month Risk

In a hypothetical terminal-event study, suppose $\Lambda_0(6)=0.24$ over six months, $\beta=\log(1.5)$, $x=1$, and the individual's cluster has frailty $Z_i=2$. Calculate the conditional survival and event risk through six months.

The conditional cumulative hazard is $$ Z_i\exp(\beta x)\Lambda_0(6) =2(1.5)(0.24)=0.72. $$ Therefore, $$ S_{ij}(6\mid Z_i=2,x=1)=\exp(-0.72)=0.4868, $$ and $$ P(T_{ij}\le 6\mid Z_i=2,x=1)=1-0.4868=0.5132. $$

Under this model, the individual has a 0.5132 conditional probability of experiencing the event by six months, given $Z_i=2$ and $x=1$. The interval and conditioning are part of the interpretation. This is not the population risk unless the target population is restricted to individuals with those covariate and frailty values.

Conditional Quantities Versus an Observed Risk

A population risk combines the risks of individuals with different covariates and frailties in the target population. Even in a simple setting with a fixed covariate, averaging conditional risks is not the same operation as reporting a conditional risk at one particular frailty value. To make the distinction concrete, consider clusters drawn at baseline from a population with two frailty groups. The calculation below averages their six-month risks using their population proportions; it does not condition on remaining event-free partway through follow-up.

Worked Example: Averaging Risks Across Frailty Groups

Suppose, hypothetically, 75% of clusters have $Z_i=0.4$ and 25% have $Z_i=2.8$. The mean frailty is $0.75(0.4)+0.25(2.8)=1$. Let $x=0$ for everyone and let the six-month cumulative baseline hazard be $\Lambda_0(6)=0.30$. Find the six-month risk in each group and the population-average risk at baseline.

For $Z_i=0.4$, the conditional cumulative hazard is $0.4(1)(0.30)=0.12$, so the risk is $$ 1-\exp(-0.12)=0.1131. $$ For $Z_i=2.8$, it is $2.8(1)(0.30)=0.84$, giving risk $$ 1-\exp(-0.84)=0.5683. $$ Weighting these risks by the group proportions gives $$ 0.75(0.1131)+0.25(0.5683)=0.2269. $$

The average six-month risk across the baseline population is approximately $0.2269$, or 22.69%. It differs from either group-specific conditional risk. Evaluating the conditional formula at $Z_i=1$ would instead give $1-\exp(-0.30)=0.2592$, which is the risk at the reference frailty, not the weighted average calculated here. This finite-group calculation illustrates the distinction without requiring a particular continuous frailty distribution.

Assumptions, Diagnostics, and Limitations

A conditional interpretation is meaningful only relative to the model and its assumptions. The basic model assumes a positive multiplicative frailty that is constant over the modeled period, a specified covariate effect, and a baseline hazard that is nonnegative. In the shared-frailty version, individuals in the same cluster have the same $Z_i$. If their event outcomes are modeled as conditionally independent given the shared frailty and covariates, that additional assumption concerns their joint distribution; it is not needed merely to interpret one individual's conditional hazard.

The mean-one convention $E(Z_i)=1$ fixes the scale of frailty relative to $\lambda_0(t)$. Without a scale convention, multiplying all frailties by a constant and dividing the baseline hazard by that constant would leave the product unchanged. Frailty values are generally latent rather than observed measurements, so a conditional statement at a given $Z_i$ describes the model, not necessarily a directly identifiable subgroup in the data.

Useful checks include asking whether the modeled covariate effect is plausibly proportional over time, whether the shared structure matches the clustering, and whether a multiplicative, time-constant latent effect is reasonable. A covariate effect that changes over time, or substantial event-order variation, can undermine a single proportional interpretation. The earlier material on checking proportional effects and diagnosing event-order heterogeneity provides relevant diagnostics for those features. A frailty model also does not correct an incorrectly specified baseline hazard or remove bias from unmeasured confounding.

A conventional hazard model without frailty gives a different model-based interpretation when important latent heterogeneity is present. Conversely, a cluster-robust standard error changes uncertainty calculations but does not itself introduce a latent multiplicative risk term. These approaches answer different modeling questions; neither turns a conditional hazard ratio into a population risk ratio. A central limitation is that conditional effects require a clear frailty scale and model, while an observed risk depends on the covariate and frailty composition of the target population.

Common Interpretation Errors

Calling a hazard a probability. A value such as $0.12$ per month is an instantaneous rate. Use a cumulative hazard and a specified interval to calculate a model-based event probability.
Calling the baseline hazard the average hazard. It is the hazard at the reference covariate and $Z_i=1$, not generally the population average.
Interpreting $\exp(\beta)$ without conditioning. In this model it is the hazard ratio for a one-unit covariate contrast at fixed frailty, not automatically a marginal population contrast.
Substituting mean frailty for averaging risks. A risk evaluated at $Z_i=E(Z_i)$ need not equal the average of risks across the frailty distribution. Calculate the latter by averaging the conditional risks over the target population, as in the two-group example.

Key takeaway: In a multiplicative frailty model, the baseline hazard and covariate hazard ratio have conditional meanings at specified frailty values. An observed risk over a time interval is a probability; it requires integration over time and, for a population target, averaging over the relevant covariate and frailty distribution.

Check Your Understanding

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

In the shared-frailty model, what does $\exp(\beta)$ compare, and what must be held fixed for that interpretation?
If $\lambda_0(t)=0.03$ per week at a particular time, $Z_i=3$, $x=0$, and $\beta=\log(2)$, what is the conditional hazard at that time?
If the cumulative baseline hazard over an interval is $0.20$, with $Z_i=1.5$, $x=0$, and $\beta=\log(2)$, what is the conditional event risk over the interval?
Why does a risk calculated at $Z_i=1$ not necessarily equal the population-average risk when $E(Z_i)=1$?

Solutions. (1) It compares instantaneous hazards for a one-unit covariate increase at the same time, holding frailty and other modeled covariates fixed. (2) The hazard is $3(0.03)\exp\{\log(2)(0)\}=0.09$ per week. (3) The cumulative conditional hazard is $1.5(1)(0.20)=0.30$, so the risk is $1-\exp(-0.30)=0.2592$, or 25.92% over the stated interval. (4) The risk is a nonlinear function of frailty, so averaging that function over the frailty distribution need not equal evaluating it at the mean frailty.