The Hazard Function

Instantaneous event rate given survival so far; central concept of survival analysis

Introduction

The hazard function \(h(t)\) gives the instantaneous rate at which events occur at time \(t\), given that they have not occurred before \(t\). It is the primary object in survival analysis and reliability engineering: proportional-hazards models, accelerated-failure-time models, and competing-risks methods all reason about hazards rather than distributions directly.

Prerequisites

Continuous distributions; survival function \(S(t) = P(T > t)\).

Theory

For a non-negative continuous random variable \(T\) with density \(f\) and survival function \(S(t) = 1 - F(t)\),

\[h(t) = \frac{f(t)}{S(t)}.\]

Intuitively, \(h(t) \, dt\) is approximately \(P(t \leq T < t + dt \mid T \geq t)\) for small \(dt\).

The cumulative hazard is

\[H(t) = \int_0^t h(u) \, du = -\log S(t).\]

Conversely, \(S(t) = e^{-H(t)}\). Once either of \(f\), \(F\), \(S\), \(h\), \(H\) is known, the others follow.

Hazard shapes for common distributions:

• Exponential: \(h(t) = \lambda\) (constant).
• Weibull with shape \(k\): \(h(t) = (k/\lambda)(t/\lambda)^{k-1}\) – increasing if \(k > 1\), decreasing if \(k < 1\).
• Gompertz: \(h(t) = \alpha e^{\beta t}\) – exponentially increasing; common in human mortality.
• Log-normal: \(h(t)\) rises then falls.

Assumptions

\(T\) is a non-negative continuous random variable with a density. Proportional-hazards models further assume \(h(t \mid x) = h_0(t) \exp(\beta^\top x)\).

R Implementation

library(survival); library(ggplot2)

# Constant hazard (exponential)
x <- seq(0.01, 5, length.out = 400)
h_exp <- rep(0.5, length(x))
h_weibull_inc <- (2.0 / 1.5) * (x / 1.5)^(2.0 - 1)
h_weibull_dec <- (0.5 / 1.5) * (x / 1.5)^(0.5 - 1)

df <- data.frame(
  t  = rep(x, 3),
  h  = c(h_exp, h_weibull_inc, h_weibull_dec),
  kind = factor(rep(c("exp lambda=0.5",
                      "weibull k=2 (incr)",
                      "weibull k=0.5 (decr)"), each = length(x)))
)

ggplot(df, aes(t, h, colour = kind)) +
  geom_line(linewidth = 1) +
  scale_colour_manual(values = c("#6A4C93", "#2A9D8F", "#F4A261")) +
  labs(x = "t", y = "h(t)", colour = "",
       title = "Hazard functions for three canonical models") +
  theme_minimal()

# Empirical cumulative hazard from censored data
set.seed(2026)
T <- rexp(100, rate = 0.3)
status <- rep(1, 100)
fit <- survfit(Surv(T, status) ~ 1)
plot(fit, fun = "cumhaz", col = "#2A9D8F", lwd = 2,
     xlab = "t", ylab = "H(t)",
     main = "Empirical cumulative hazard (Nelson-Aalen)")
abline(0, 0.3, col = "#F4A261", lty = 2)

Output & Results

The plot shows three hazards: exponential (flat), increasing Weibull, decreasing Weibull. The empirical cumulative hazard from 100 exponential event times is approximately linear with slope \(\lambda = 0.3\), matching the theoretical \(H(t) = 0.3 t\).

Interpretation

In survival analysis, hazards are the natural target of regression: “the hazard of death was 40 % higher in the treated group than in controls (HR = 1.40, 95 % CI 1.10-1.78)”. The proportional-hazards model is ubiquitous because it avoids specifying the baseline hazard shape.

Practical Tips

• A constant hazard implies exponentiality; a log-log plot of \(-\log S(t)\) against \(\log t\) that is linear indicates Weibull.
• Empirical hazards are noisy at the tail where few subjects remain; report cumulative hazard (smoother) or kernel-smoothed hazard estimates.
• The Nelson-Aalen estimator is the standard non-parametric estimator of the cumulative hazard under right-censoring.
• In competing risks, cause-specific hazards differ from the sub-distribution hazard of Fine-Gray; the two answer different questions.
• Time-varying hazard ratios invalidate the proportional-hazards assumption; use stratification or time-interactions.