Diagnosing Increasing, Decreasing, and Bathtub Hazards

Reading Lifetime Patterns from the Hazard

A hazard describes the instantaneous rate of failure among units that have survived up to a given time. Its shape therefore conveys how that rate changes with time: it may increase, decrease, remain constant, or fall and then rise. These patterns are useful summaries of lifetime distributions, but they are not interchangeable with the shape of a density. A density describes where failures accumulate across the whole population; a hazard describes failure at a particular time among those still at risk.

Let \(T\) be a nonnegative, absolutely continuous lifetime, with survival function \(S(t)=P(T>t)\) and density \(f\). At times where \(S(t)>0\), define the hazard by \(h(t)=f(t)/S(t)\), using a density version for which the formula is understood almost everywhere. Equivalently, when the derivatives exist, \(h(t)=-S'(t)/S(t)\). For smooth models below, these identities hold pointwise on the stated time range. The almost-everywhere qualification matters: a density may be changed at an individual point without changing the distribution.

Because \(S(t)\) is the fraction surviving beyond \(t\), the denominator in \(f(t)/S(t)\) adjusts the failure density for the number still exposed to failure. In differentiable models the same definition gives \(h(t)=-\frac{d}{dt}\log S(t)\): the hazard is the negative slope of log survival. Thus a steepening decline in log survival corresponds to an increasing hazard, while a straight line in log survival corresponds to a constant hazard.

Exponential and Weibull Shapes

For an exponential lifetime with rate \(\lambda>0\), \(S(t)=e^{-\lambda t}\) and \(f(t)=\lambda e^{-\lambda t}\) for \(t\geq0\). Their ratio is \(h(t)=\lambda\), so the hazard is constant. The interpretation is that, in this model, the instantaneous failure rate does not change with elapsed survival time.

The Weibull family gives a compact way to represent constant, increasing, or decreasing hazards. Let \(T\) have survival function \(S(t)=\exp[-(t/\eta)^k]\) for \(t\geq0\), with scale \(\eta>0\) and shape \(k>0\). Differentiating the survival function gives its density and hazard on \(t>0\).

To verify the result, differentiate \(S(t)\): $$ f(t)=-S'(t)=\frac{k}{\eta}\left(\frac{t}{\eta}\right)^{k-1} \exp[-(t/\eta)^k]. $$ Dividing by \(S(t)=\exp[-(t/\eta)^k]\) yields the displayed hazard. Its derivative has the sign of \(k-1\), since $$ h'(t)=\frac{k(k-1)}{\eta^2}\left(\frac{t}{\eta}\right)^{k-2},\qquad t>0. $$ All other factors are positive, so the hazard decreases, is constant, or increases according as \(k-1\) is negative, zero, or positive. When \(k<1\), the hazard tends to infinity as \(t\downarrow0\); this does not prevent the Weibull survival from being proper, because its cumulative hazard \((t/\eta)^k\) is finite for each finite \(t\). When \(k=1\), the Weibull reduces to the exponential model with rate \(1/\eta\). \(\square\)

A Bathtub-Shaped Hazard

A bathtub pattern captures an initial decline, a relatively low middle-life hazard, and a later increase. One explicit model can be built by specifying a nonnegative hazard and integrating it to obtain survival. Measure time in years and, for \(t\geq0\), set $$ h(t)=0.2+0.3(t-1)^2. $$ The coefficients are understood in the corresponding time units, so the hazard is measured per year. Integrating from zero to \(t\) gives the cumulative hazard $$ H(t)=\int_0^t h(u)\,du=0.2t+0.1\bigl((t-1)^3+1\bigr). $$ Consequently, define \(S(t)=e^{-H(t)}\).

Here the initial hazard is \(h(0)=0.5\) per year, the minimum is \(h(1)=0.2\), and \(h(2)=0.5\). The construction is valid because \(S'(t)=-h(t)S(t)\), so \(f(t)=-S'(t)=h(t)S(t)\geq0\), and $$ \int_0^\infty f(t)\,dt=S(0)-\lim_{t\to\infty}S(t)=1-0=1. $$ This is an idealized bathtub curve: the hazard falls and then rises, with the minimum representing a low-risk middle period. Real data may show a less symmetric shape or a broad nearly flat middle phase.

Why Mixtures Can Change Hazard Shape

A population can combine subgroups with different lifetime distributions. Even if each subgroup has a constant hazard, the overall population hazard need not be constant. Faster-failing subgroups become less represented among those still surviving, changing the composition of the survivors over time.

For a finite mixture, let subgroup \(i\) have initial probability \(p_i>0\), where \(\sum_{i=1}^m p_i=1\), survival \(S_i(t)\), and density \(f_i(t)\). Suppose each component lifetime is absolutely continuous. The mixture survival and density are \(S(t)=\sum_i p_iS_i(t)\) and \(f(t)=\sum_i p_if_i(t)\). At times where the density identities hold and \(S(t)>0\), the mixture hazard is $$ h(t)=\frac{\sum_{i=1}^m p_i f_i(t)}{\sum_{i=1}^m p_iS_i(t)} =\sum_{i=1}^m w_i(t)h_i(t),\qquad w_i(t)=\frac{p_iS_i(t)}{\sum_{j=1}^m p_jS_j(t)}. $$ The weights are nonnegative and sum to one. They are the subgroup proportions among survivors at time \(t\), not necessarily the initial mixture proportions. The identity holds almost everywhere for absolutely continuous component distributions; if the component survival and density identities hold pointwise, it holds pointwise as well.

To prove the result, write \(w_i(t)=p_i e^{-\lambda_i t}/\sum_j p_j e^{-\lambda_jt}\). The mixture hazard is \(h(t)=\sum_i w_i(t)\lambda_i\). Differentiating the weights gives $$ w_i'(t)=w_i(t)\bigl(h(t)-\lambda_i\bigr). $$ It follows that $$ h'(t)=\sum_i w_i'(t)\lambda_i =h(t)\sum_i w_i(t)\lambda_i-\sum_i w_i(t)\lambda_i^2 =h(t)^2-\sum_i w_i(t)\lambda_i^2 =-\sum_i w_i(t)\bigl(\lambda_i-h(t)\bigr)^2\leq0. $$ The final sum is the negative of the weighted variance of the component rates. It is zero exactly when all rates with positive survivor weight are identical; otherwise it is negative. This proves the claim. \(\square\)

Diagnosis and Common Pitfalls

When a model is presented through survival, first compute \(h=-S'/S\) where differentiability and positivity permit, or use \(f/S\) with the appropriate density version. Then simplify the expression and inspect its derivative or monotonicity. If the model is a mixture, use the survivor-weighted hazards rather than averaging component hazards with their initial mixture proportions. For instance, a mixture hazard may decline although neither subgroup has a declining hazard.

Hazard shape supports a lifetime interpretation, but does not by itself identify a physical cause. An increasing hazard is consistent with risk that rises with age or accumulated use; a decreasing hazard is consistent with early failures removing more vulnerable units. A bathtub shape is consistent with both an early high-risk period and a later rise. These are interpretations of the model’s conditional rate, not proof that a particular mechanism caused the pattern.

• Confusing hazard and density. A density can rise or fall for reasons that do not match the hazard’s trend. Divide by survival before diagnosing the hazard.
• Ignoring survivor composition. In a mixture, the overall hazard uses the proportions among survivors at time \(t\), which change over time.
• Treating a density value as uniquely defined at every point. Densities are defined up to changes on sets of measure zero. State derivative identities almost everywhere unless a suitable pointwise regularity condition is in force.
• Calling a conditional rate a failure probability. The hazard is a rate per unit time; a failure probability over an interval must account for survival across that interval.