Proportional Hazards Regression With Interval-Censored Times

Why Interval-Censored Regression Needs Its Own Likelihood

With intermittent inspections, an event may be known to have occurred after one visit and by the next, but its exact time is not observed. A proportional hazards regression model can relate covariates to that event-time distribution, but the likelihood must use the information actually recorded: an interval containing the event, rather than an invented exact event time.

The model describes the event time $T_i$ conditional on covariates $x_i$. Assume independent subjects, a continuous event-time distribution, and a proportional hazards model with a common baseline hazard. The inspection or censoring process is assumed noninformative for the event time conditional on the modeled covariates and observed history. In particular, an observation interval should not be selected in a way that conveys extra information about the unobserved event time once those variables are conditioned on.

Definition: For subject $i$, let $x_i$ be a vector of covariates, $\beta$ the vector of regression coefficients, $h_0(t)$ the baseline hazard, and $H_0(t)=\int_0^t h_0(u)\,du$ the baseline cumulative hazard. The proportional hazards model is $$ h(t\mid x_i)=h_0(t)\exp(\beta^\mathsf{T}x_i), \qquad S(t\mid x_i)=\exp\{-H_0(t)\exp(\beta^\mathsf{T}x_i)\}. $$ The hazard at time $t$ is multiplied by the same covariate-dependent factor at every time. The baseline hazard may be specified parametrically or estimated using a suitable semiparametric approach.

The survival expression follows by integrating the hazard: the cumulative hazard for covariates $x_i$ is $H_0(t)\exp(\beta^\mathsf{T}x_i)$, and survival is the exponential of the negative cumulative hazard. This model permits a positive limiting survival probability if $H_0(\infty)$ is finite; it does not require that every subject eventually experience the event.

Likelihood Contributions From Observed Intervals

Suppose inspections establish that subject $i$ was event-free at $L_i$ and had experienced the event by $R_i$. The observed information is $L_i<T_i\le R_i$, so the probability of the observation, conditional on covariates, is the probability that the event time falls in that interval.

Likelihood contribution: For an interval-censored event observed in $(L_i,R_i]$, $$ P(L_i<T_i\le R_i\mid x_i) =S(L_i\mid x_i)-S(R_i\mid x_i). $$ For a right-censored subject known to be event-free at $C_i$, the contribution is $S(C_i\mid x_i)$. For a left-censored event known to have occurred by $R_i$, the contribution is $1-S(R_i\mid x_i)$, assuming the event time is nonnegative and $S(0\mid x_i)=1$.

The interval formula can also be derived by conditioning on having survived to $L_i$. Among subjects event-free at $L_i$, the probability of experiencing the event by $R_i$ is $1-S(R_i\mid x_i)/S(L_i\mid x_i)$. The observed interval itself includes the probability of surviving to $L_i$, so its unconditional probability is $$ S(L_i\mid x_i)\left\{1-\frac{S(R_i\mid x_i)}{S(L_i\mid x_i)}\right\} =S(L_i\mid x_i)-S(R_i\mid x_i). $$ Thus the conditional probability alone is not the likelihood contribution for an unselected subject; it omits the probability of surviving to the interval's left endpoint.

A right-censored observation at $C_i$ says only that $T_i>C_i$, so its contribution is $P(T_i>C_i\mid x_i)=S(C_i\mid x_i)$, including any probability of never experiencing the event. In contrast, if an observation establishes that the event occurs at a finite time after $C_i$, its contribution is $S(C_i\mid x_i)-S(\infty\mid x_i)$. These statements describe different observed information.

For independent subjects, multiply their contributions. If every subject has an interval-censored event, the likelihood is $$ L(\beta,H_0)=\prod_{i=1}^n \left[ \exp\{-H_0(L_i)\exp(\beta^\mathsf{T}x_i)\} -\exp\{-H_0(R_i)\exp(\beta^\mathsf{T}x_i)\} \right]. $$ With right-censored or left-censored observations, replace the corresponding factors by the contributions above. In practice, maximizing the log-likelihood is more stable than multiplying many small probabilities. The form of the baseline model or estimation method must also be specified to obtain estimates of $\beta$ and the baseline hazard.

Worked Example: One Interval Under Two Covariate Values

Suppose a hypothetical model has a binary exposure $x$, with $x=0$ unexposed and $x=1$ exposed. Let $\exp(\beta)=2$, and suppose $H_0(2)=0.20$ and $H_0(5)=0.60$. Compare the likelihood contributions for subjects whose event is known to lie in $(2,5]$.

For an unexposed subject, $\exp(\beta x)=1$. Therefore, $$ P(2<T\le5\mid x=0)=e^{-0.20}-e^{-0.60} =0.818731-0.548812=0.269919. $$

For an exposed subject, $\exp(\beta x)=2$, so the covariate doubles the cumulative hazard at each time. Thus, $$ P(2<T\le5\mid x=1)=e^{-0.40}-e^{-1.20} =0.670320-0.301194=0.369126. $$

For this interval, the exposed-to-unexposed ratio of interval probabilities is $0.369126/0.269919=1.368$, to three decimal places. It is not 2. The model's hazard ratio is exactly 2 at every time, but an interval probability depends on both endpoints and on the baseline cumulative hazard. It is not itself a hazard ratio.

Worked Example: Different Observation Patterns, Different Contributions

For an unexposed subject, suppose $H_0(2)=0.20$ and $H_0(5)=0.60$. Calculate the contributions for three hypothetical observation patterns: an event in $(0,2]$, an event in $(2,5]$, and right censoring at time 5. Since $x=0$, the covariate multiplier is 1.

For an event in $(0,2]$, use $S(0)-S(2)$, with $S(0)=1$: $$ 1-e^{-0.20}=1-0.818731=0.181269. $$ For an event in $(2,5]$, the contribution is $$ e^{-0.20}-e^{-0.60}=0.818731-0.548812=0.269919. $$

For right censoring at 5, the contribution is the probability of being event-free at 5: $$ S(5)=e^{-0.60}=0.548812. $$ This last observation does not say that an event occurred in a later interval. Its contribution is survival beyond 5, whether that survival is temporary or persists indefinitely.

The three values differ because they represent distinct observed information. In particular, replacing a right-censoring factor by the probability of an event in a later finite interval would use a stronger observation than the data provide.

Interpreting Covariate Effects

For a one-unit increase in covariate $x_j$, holding the other covariates fixed, the model's hazard ratio is $\exp(\beta_j)$. More generally, changing a covariate vector from $x$ to $x'$ gives hazard ratio $\exp\{\beta^\mathsf{T}(x'-x)\}$. These are comparisons of instantaneous hazards at the same time among subjects who remain event-free up to that time. They are not direct comparisons of probabilities of an event within a particular observation interval, nor are they risk ratios over a fixed follow-up period.

A positive coefficient means a higher hazard under the model, and a negative coefficient means a lower hazard. The size and units of the covariate change matter: if a continuous predictor is measured in years, $\exp(\beta_j)$ is the hazard ratio per additional year; a ten-year contrast has hazard ratio $\exp(10\beta_j)$. This interpretation depends on the proportional hazards assumption: the hazard ratio does not vary with time unless the model is extended to allow time-varying effects.

Worked Example: Assemble a Likelihood for Three Subjects

Consider three hypothetical, independent subjects under the model in the previous examples, with $\exp(\beta)=2$, $H_0(2)=0.20$, and $H_0(5)=0.60$. Subject A is unexposed and has an event in $(0,2]$; subject B is exposed and has an event in $(2,5]$; subject C is unexposed and is right-censored at 5. Evaluate the likelihood at these stated model values.

Subject A contributes $1-e^{-0.20}=0.181269$. Subject B contributes $$ e^{-2(0.20)}-e^{-2(0.60)} =e^{-0.40}-e^{-1.20} =0.369126. $$ Subject C contributes $e^{-0.60}=0.548812$. By independence, the joint likelihood is $$ L=(0.181269)(0.369126)(0.548812)\approx0.03672. $$ The log-likelihood is the sum of the log contributions: $$ \ell=\log(0.181269)+\log(0.369126)+\log(0.548812) \approx-1.7078-0.9966-0.6000=-3.3044. $$

This is the likelihood at one proposed parameter value, not a probability that the parameter values are correct. Estimation would compare such likelihood values across candidate regression and baseline parameters and choose values that maximize the likelihood.

Assumptions, Diagnostics, and Alternatives

The likelihood is appropriate when the event-time model is correctly specified and the observed intervals accurately represent what was learned at inspection. Independent subjects justify multiplying individual contributions. Noninformative inspection is also important: if inspection timing depends on the unobserved event time even after conditioning on the modeled information, the observation process may need to be modeled jointly. A proportional hazards model additionally assumes that covariate effects on the hazard are constant over time.

Practical warning signs include systematic differences in event patterns across inspection schedules, estimates that change markedly when the baseline model changes, and implausible fitted survival curves. Broad or sparse inspection intervals can make the event-time distribution and covariate effects weakly identified: many parameter combinations may assign similar probabilities to the observed intervals. Diagnostics should therefore consider the visit process, the fitted survival behavior, and whether proportional hazards is plausible. A successful numerical optimization does not establish these assumptions.

For exact event times with appropriate right censoring, standard Cox regression uses a partial likelihood that conditions on risk sets at observed event times. With interval-censored event times, those exact event-time risk sets are unavailable, so that ordinary partial-likelihood construction does not directly apply. The interval-censored likelihood instead uses probabilities of the observed intervals and requires an approach to the baseline hazard. The Turnbull estimator, from the earlier material on interval-censored survival, provides a nonparametric distributional estimate but does not by itself supply this covariate-adjusted proportional hazards regression. Assigning interval midpoints and treating them as exact times is generally not a sound substitute: it adds artificial precision and can distort both the likelihood and coefficient estimates.

The model is also limited by its structure. A single proportional hazards regression is not automatically suitable for time-varying covariate effects, dependent event times, or informative observation schedules. When covariates or disease states change between visits, a model specifically built for those histories may be needed. The appropriate alternative depends on the scientific question and observation process; an accelerated failure time model is a distinct regression approach, not a reinterpretation of the hazard ratio.

Common Mistakes

Using the conditional interval probability as the full contribution. $1-S(R)/S(L)$ is conditional on surviving to $L$. For an unselected subject observed in $(L,R]$, the contribution is $S(L)-S(R)$.
Calling an interval-probability ratio a hazard ratio. The hazard ratio is $\exp(\beta_j)$ for a one-unit covariate contrast; the ratio of interval probabilities depends on the interval and baseline hazard.
Subtracting the limiting survival from a right-censoring contribution. Right censoring at $C$ contributes $S(C)$. Subtracting $S(\infty)$ would describe a finite event after $C$, not merely being event-free at $C$.
Replacing an interval with its midpoint. The midpoint was not observed. Use the probability over the whole recorded interval unless a justified measurement or event-time model says otherwise.

Key takeaway: An interval-censored proportional hazards model links covariates to the hazard through $\exp(\beta^\mathsf{T}x)$, while each observed interval contributes $S(L\mid x)-S(R\mid x)$. Keep the hazard-ratio interpretation separate from the probability of an event in any particular interval.

Check Your Understanding

Answer each question using the stated model and observation information.

For an interval observation $(L,R]$, explain why $S(L)-S(R)$, rather than $1-S(R)/S(L)$, is the unconditional likelihood contribution.
A binary covariate has coefficient $\beta=\log(1.5)$. State the hazard ratio for exposed versus unexposed subjects, and explain whether this determines the ratio of their probabilities of an event in $(L,R]$.
Suppose $S(C)=0.40$ and $S(\infty)=0.10$. What is the contribution for right censoring at $C$? What probability does $S(C)-S(\infty)$ represent?
Why can ordinary Cox partial likelihood not simply treat an interval-censored event as if it happened at the midpoint of its observation interval?

Solutions. 1. The conditional probability $1-S(R)/S(L)$ is conditional on surviving to $L$. Multiplying it by $S(L)$, the probability of reaching $L$ event-free, gives $S(L)-S(R)$. 2. The hazard ratio is 1.5 at every time under proportional hazards. The interval-probability ratio is not determined by that number alone; it also depends on $L$, $R$, and the baseline cumulative hazard. 3. Right censoring contributes $S(C)=0.40$. The difference $S(C)-S(\infty)=0.30$ is the probability of an event at a finite time after $C$. 4. The midpoint is not an observed event time; assigning it creates artificial exact timing and risk sets. The interval likelihood uses the probability that the event falls anywhere in the observed interval.