Cumulative-Hazard Criteria for Lifetime Comparisons

Why Compare Cumulative Hazards?

A hazard rate describes the instantaneous failure rate among individuals who have survived up to a given time. Comparing two hazard rates directly can be useful, but the functions may be difficult to inspect or may cross. Cumulative hazards provide another route: they aggregate hazard over time, and their difference reveals whether the relative survival of two lifetimes moves consistently in one direction.

The key distinction is between comparing cumulative hazards at a single time and comparing how their difference changes over time. Hazard-rate order is determined by the latter. The resulting criteria are especially convenient for parametric lifetime models, including exponential and Weibull distributions.

Definitions and the Comparison Direction

Let \(X\) and \(Y\) be nonnegative, absolutely continuous lifetimes with densities \(f_X\) and \(f_Y\), survival functions \(S_X(t)=P(X>t)\) and \(S_Y(t)=P(Y>t)\), and \(S_X(t),S_Y(t)>0\) for every finite \(t\geq 0\). Their hazard rates and cumulative hazards are

with analogous definitions for \(Y\). The equality \(H_X(t)=-\log S_X(t)\) follows from absolute continuity and \(S_X(0)=1\). Hazard rates are defined almost everywhere; the integral defining \(H_X\) does not depend on their values on a set of measure zero.

The comparison direction matters: a larger hazard for \(X\) corresponds to \(X\leq_{\mathrm{hr}}Y\). If hazards are continuous, the almost-everywhere inequality is equivalent to the inequality holding at every time. When the lifetime distributions have finite endpoints or other support complications, criteria must be applied on the relevant common survival interval and the supports must be considered separately. The results below use the stated positive-survival assumption.

The Cumulative-Hazard and Survival-Ratio Criteria

Define the cumulative-hazard difference \(D(t)=H_X(t)-H_Y(t)\). Since \(H_X(0)=H_Y(0)=0\), we have \(D(0)=0\). The survival ratio in the direction that matches \(X\leq_{\mathrm{hr}}Y\) is

Thus a nondecreasing difference \(D\) gives a nondecreasing survival ratio \(R\). The next result shows that these conditions are not just convenient consequences; they are equivalent to the hazard-rate comparison.

Proof. The cumulative hazards are locally absolutely continuous and satisfy \(H_X'(t)=h_X(t)\) and \(H_Y'(t)=h_Y(t)\) almost everywhere. Therefore \(D\) is locally absolutely continuous, with \(D'(t)=h_X(t)-h_Y(t)\) almost everywhere. A locally absolutely continuous function is nondecreasing exactly when its derivative is nonnegative almost everywhere. Hence \(h_X\geq h_Y\) almost everywhere if and only if \(D\) is nondecreasing.

For every finite \(t\), \(R(t)=\exp\{D(t)\}\). The exponential function is strictly increasing, so \(D\) is nondecreasing if and only if \(R\) is nondecreasing. This proves all three equivalences. \(\square\)

This result uses the difference of cumulative hazards, not their ratio. Equivalently, on any interval where the hazards are continuous, one can differentiate the log survival ratio:

The derivative form is often the quickest calculation for smooth models. The cumulative-hazard form remains useful when comparing integrated hazards or when derivatives need only be interpreted almost everywhere.

An Interval-by-Interval Criterion

The same comparison can be expressed without checking whether a function is globally increasing. For any \(0\leq s<t\), the survival ratio changes between \(s\) and \(t\) according to the integrated difference in hazards over that interval.

In particular, \(X\leq_{\mathrm{hr}}Y\) if and only if this ratio of survival ratios is at least \(1\) for every \(0\leq s<t\).

Proof. By the definition of cumulative hazard, \(H_X(t)-H_X(s)=\int_s^t h_X(u)\,du\), and the same identity holds for \(Y\). Taking the difference gives \(D(t)-D(s)=\int_s^t(h_X(u)-h_Y(u))\,du\). Since \(R(v)=\exp\{D(v)\}\), dividing \(R(t)\) by \(R(s)\) yields the displayed identity. The ratio is at least \(1\) for all \(s<t\) exactly when \(D(t)\geq D(s)\) for all such pairs, which is precisely that \(D\) is nondecreasing. By the theorem, this is equivalent to hazard-rate order. \(\square\)

This form emphasizes that the comparison is cumulative over each time interval. A brief interval on which \(h_X<h_Y\) rules out the order if it makes the cumulative-hazard difference decrease there; positive differences elsewhere cannot repair a decrease that has already occurred.

Worked Comparisons for Lifetime Models

Common Pitfall: Pointwise Cumulative-Hazard Inequalities

A frequent mistake is to infer \(X\leq_{\mathrm{hr}}Y\) from \(H_X(t)\geq H_Y(t)\) for every \(t\). That inequality says only that \(D(t)\geq0\); it does not say that \(D\) is nondecreasing. Hazard-rate order requires \(D\) to be nondecreasing, or equivalently requires its almost-everywhere derivative \(h_X-h_Y\) to be nonnegative.

For a concrete check, define two valid cumulative hazards by

Both start at zero and are increasing: \(H_Y'(t)=1\), while \(H_X'(t)=1+(1-t)e^{-t}>0\) for all \(t\geq0\). They define proper continuous lifetime distributions through \(S_X=e^{-H_X}\) and \(S_Y=e^{-H_Y}\). Although \(H_X(t)-H_Y(t)=t e^{-t}\geq0\) at every time, its derivative is \((1-t)e^{-t}\), which becomes negative for \(t>1\). Hence the survival ratio \(S_Y/S_X=\exp(t e^{-t})\) eventually decreases, and \(X\leq_{\mathrm{hr}}Y\) does not hold. The cumulative-hazard difference, not its sign at each time, is the decisive quantity.