Intuition for cumulative hazard function (survival analysis)

Question

I'm trying to get intuition for each of the main functions in actuarial science (specifically for the Cox Proportional Hazards Model). Here's what I have so far:

• $f(x)$: starting at the start time, the probability distribution of when you will die.
• $F(x)$: just the cumulative distribution. At time $T$, what % of the population will be dead?
• $S(x)$: $1-F(x)$. At time $T$, what % of the population will be alive?
• $h(x)$: hazard function. At a given time $T$, of the people still alive, this can be used to estimate how many people will die in the next time interval, or if interval->0, 'instantaneous' death probability.
• $H(x)$: cumulative hazard. No idea.

What's the idea behind combining hazard values, especially when they are continuous? If we use a discrete example with death rates across four seasons, and the hazard function is as follows:

• Starting at Spring, everyone is alive, and 20% will die
• Now in Summer, of those remaining, 50% will die
• Now in Fall, of those remaining, 75% will die
• Final season is Winter. Of those remaining, 100% will die

Then the cumulative hazard is 20%, 70%, 145%, 245%?? What does that mean, and why is this useful?

Answer 1

Combining proportions dying as you do is not giving you cumulative hazard. Hazard rate in continuous time is a conditional probability that during a very short interval an event will happen:

$$h(t) = \lim_{\Delta t \rightarrow 0} \frac {P(t<T \le t + \Delta t | T >t)} {\Delta t}$$

Cumulative hazard is integrating (instantaneous) hazard rate over ages/time. It's like summing up probabilities, but since $\Delta t$ is very small, these probabilities are also small numbers (e.g. hazard rate of dying may be around 0.004 at ages around 30). Hazard rate is conditional on not having experienced the event before $t$, so for a population it may sum over 1.

You may look up some human mortality life table, although this is a discrete time formulation, and try to accumulate $m_x$.

If you use R, here's a little example of approximating these functions from number of deaths at each 1-year age interval:

dx <-  c(3184L, 268L, 145L, 81L, 64L, 81L, 101L, 50L, 72L, 76L, 50L, 
         62L, 65L, 95L, 86L, 120L, 86L, 110L, 144L, 147L, 206L, 244L, 
         175L, 227L, 182L, 227L, 205L, 196L, 202L, 154L, 218L, 279L, 193L, 
         223L, 227L, 300L, 226L, 256L, 259L, 282L, 303L, 373L, 412L, 297L, 
         436L, 402L, 356L, 485L, 495L, 597L, 645L, 535L, 646L, 851L, 689L, 
         823L, 927L, 878L, 1036L, 1070L, 971L, 1225L, 1298L, 1539L, 1544L, 
         1673L, 1700L, 1909L, 2253L, 2388L, 2578L, 2353L, 2824L, 2909L, 
         2994L, 2970L, 2929L, 3401L, 3267L, 3411L, 3532L, 3090L, 3163L, 
         3060L, 2870L, 2650L, 2405L, 2143L, 1872L, 1601L, 1340L, 1095L, 
         872L, 677L, 512L, 376L, 268L, 186L, 125L, 81L, 51L, 31L, 18L, 
         11L, 6L, 3L, 2L)

x <- 0:(length(dx)-1) # age vector

plot((dx/sum(dx))/(1-cumsum(dx/sum(dx))), t="l", xlab="age", ylab="h(t)", 
     main="h(t)", log="y")
plot(cumsum((dx/sum(dx))/(1-cumsum(dx/sum(dx)))), t="l", xlab="age", ylab="H(t)", 
     main="H(t)")

Hope this helps.

Answer 2

The Book "An Introduction to Survival Analysis Using Stata" (2nd Edition) by Mario Cleves has a good chapter on that topic.

You can find the chapter on google books, p. 13-15. But I would advise on reading the whole chapter 2.

Here is the short form:

• "it measures the total amount of risk that has been accumulated up to time t" (p. 8)
• count data interpretation: "it gives the number of times we would expect (mathematically) to observe failures [or other events] over a given period, if only the failure event were repeatable" (p. 13)

Answer 3

I'd _^HAZARD a guess that it's noteworthy owing to its use in diagnostic plots:

(1) In the Cox proportional hazards model $h(x)=\mathrm{e}^{\beta^\mathrm{T} z}h_0(x)$, where $\beta$ and $z$ are the coefficient and covariate vectors respectively, & $h_0(x)$ is the baseline hazard function; & so (by integrating both sides with respect to $x$ & taking logarithms) $\log H(x)=\beta^\mathrm{T} z + H_0(x)$. If you plot the estimate $\log \hat{H}(x)$ against $x$, different covariate patterns follow parallel curves, provided the proportional hazards assumption is correct.

(2) In the Weibull model $h(x)=\frac{\alpha}{\theta}\left(\frac{x}{\theta}\right)^{\alpha-1}$, where $\theta$ & $\alpha$ are the scale & shape parameters respectively; & so $\log H(x) = \alpha \log x - \alpha \log \theta$. If you plot the estimate $\log \hat{H}(x)$ against $\log x$, you get a straight line with slope $\hat{\alpha}$ & intercept $-\hat{\alpha}\log\hat{\theta}$, provided the Weibull assumption is correct. And of course a slope near to 1 suggests an exponential model might fit.

An intuitive interpretation of $H(x)$ is the expected number of deaths of an individual up to time $x$ if the individual were to be resurrected after each death (without resetting time to zero).

Answer 4

In paraphrasing what @Scortchi is saying, I would emphasize that the cumulative hazard function does not have a nice interpretation, and as such I would not try to use it as a way to interpret results; telling a non-statistical researcher that the cumulative hazards are different will most likely result in an "mm-hm" answer and then they'll never ask about the subject again, and not in a good way.

However, the cumulative hazard function turns out to be very useful mathematically, such as a general way to link the hazard function and the survival function. So it's important to know what the cumulative hazard is and how it can be used in various statistical methods. But in general, I don't think it's particularly useful to think about real data in terms cumulative hazards.