Time-to-Event Data, Censoring, and Hazard Functions › Visualizing Survival, Censoring, and Hazards

Before fitting anything, it is worth looking at the shapes involved. The three quantities $S(t)$, $h(t)$, and $H(t)$ move together, and the way they move is easiest to grasp from a figure. We will use a Weibull family because it conveniently covers the three qualitative hazard shapes: constant (exponential as a special case), increasing, and decreasing.

The Weibull survival function with shape $k > 0$ and scale $\lambda > 0$ is $S(t) = \exp\!\left(-(t/\lambda)^k\right)$, and the corresponding hazard is $h(t) = \frac{k}{\lambda}\Big(\frac{t}{\lambda}\Big)^{k-1}$. Here $k$ controls the shape of the hazard and $\lambda$ is the scale parameter measured in the same time units as $t$. Note that this scale $\lambda$ plays a different role from the exponential rate $\lambda$ used in the worked example below, where $\lambda$ is the constant hazard. When $k = 1$ the hazard is constant at $1/\lambda$. When $k > 1$ the hazard increases with time (wear-out). When $k < 1$ the hazard decreases with time (infant mortality). Increasing, decreasing, and approximately constant hazards over particular time ranges have been observed in component reliability data. Understanding these survival curves is a first diagnostic tool.

The next two figure cells are split purely for presentation. Both read the precomputed grids and the scenario dictionaries, and neither recomputes the data. The first shows the survival functions. The second shows the hazards.

The survival plot makes two things visible. First, all three start at $S(0) = 1$ and eventually decay to zero, as required. Second, the shape parameter changes the shape of the decay. The decreasing-hazard case (infant mortality) drops steeply early and then flattens, so its survival curve has a long tail. The increasing-hazard case (wear-out) stays near one for a while and then falls rapidly. These are two ends of the stated monotone-hazard spectrum. Real data require their own hazard diagnostics.

A constant hazard means the same instantaneous failure rate at each age; for an exponential lifetime it also gives a memoryless residual lifetime. A rising hazard can describe wear-out, while a falling hazard can describe early failures followed by lower risk among survivors. A decreasing population hazard alone does not prove that individual devices become stronger. The hazard grid starts at $0.01$ because the shape-$0.5$ hazard diverges as $t\downarrow0$; the survival grid includes zero.

Cumulative hazard $H(t)$ is dimensionless and increases at rate $h(t)$. At a given time, a larger $H(t)$ corresponds to a smaller survival probability through $S(t)=\exp(-H(t))$. The absolute rate of decline of survival is $h(t)S(t)$, so a larger hazard alone does not imply a steeper survival curve when the survival levels differ.

For the censoring plot, imagine devices installed at independent, uniformly distributed calendar dates during a fifteen-unit recruitment period, with follow-up ending at a common administrative cutoff. Each device is followed from installation, so its potential follow-up duration $C$ is uniform from zero to fifteen and independent of its simulated lifetime $T$. This is staggered recruitment without left truncation. Horizontal segments end at $Y=\min(T,C)$; diamonds mark failures and squares mark censorings. A censoring marker locates the end of observation, not an unobserved failure.

This is the visual form of the $(Y_i, \Delta_i)$ representation. Read each line as "device $i$ was observed from 0 to $Y_i$; the marker tells us whether the device failed or was still running." The censored lines are not wrong or missing. They carry important information, namely that the true failure time is greater than the endpoint we see. A correct survival likelihood gives those records weight $S(Y_i)$ for the region to the right of the marker, which is exactly the probability that the event had not yet occurred at the moment we stopped observing.