cumulative hazard graph

[tim bradshaw]

Hi there
My understanding is that cumulative hazard is basically the inverse of cumulative survival.

However this does not appear to be the case when I generate a survival curve and a cumulative hazard curve using exactly the same data and very similar commands.

Survival analysis:
command: sts graph, xlab(0(1)15) ylab(0(0.1)1.0, angle(horizontal))
graph: "survival curve" (attached)

Cumulative hazard analysis:
command: sts graph, cumhaz xlab(0(1)15) ylab(0(0.1)1.0, angle(horizontal))
graph: "cumulative hazard curve" (attached)


The total cumulative hazard is about 47%, whereas the total cumulative survival is about 63%. These don't add up to 100%. Am I missing something?

Any thoughts greatly appreciated.

With thanks,
Tim

[Steve Samuels]

Your understanding is incorrect. If \(\lambda(t)\) is the hazard function; \(\Lambda(t)\) is the cumulative, or integrated, hazard; and \(S(t)\) is the survival function, here are the relationships between them:
\[
\Lambda(t) = \int_0^t \negthinspace \lambda(x)dx
\]
\[
\Lambda(t) = -\text{log}(S(t))
\]
\[
S(t) = \text{exp}\left(- \int_0^t \negthinspace \Lambda(x)dx\right)
\]

[Steve Samuels]

It is the (cumulative) failure and survival functions which add to 1.

\[
S(t) = 1-F(t)
\]

[tim bradshaw]

Thanks for the reply Steve Samuels.

And yet the cumulative hazard lifetable is entirely consistent with the survival lifetable (hazard+survival=1), as follows:

ltable intervalyrs recur, hazard

Cum
Interval Cum. Failure
0 1 0.2804
1 2 0.3086
2 3 0.3258
3 4 0.3384
4 5 0.347
5 6 0.3539
6 7 0.359
7 8 0.3632
8 9 0.3669
9 10 0.3699
10 11 0.372
11 12 0.3735
12 13 0.3758

ltable intervalyrs recur

Cum
Interval Cum. Survival
0 1 0.7196
1 2 0.6914
2 3 0.6742
3 4 0.6616
4 5 0.653
5 6 0.6461
6 7 0.641
7 8 0.6368
8 9 0.6331
9 10 0.6301
10 11 0.628
11 12 0.6265
12 13 0.6242


It seems very odd to me that the lifetables are consistent with each other, and that both lifetables are also consistent with the survival graph, but that the hazard graph is inconsistent with all of the others.

Does anyone else get the same problem when using the "sts graph, cumhaz" command....?

[Steve Samuels]

ltable is showing the estimated cumulative failure function F(t). You must have run

ltable intervalyrs recur, failure

not

ltable intervalyrs recur, hazard

But even this version would not list the cumulative hazard, but rather the interval-specific hazard rates.

Please read FAQ 12 and present all commands, listings, and results between CODE delimiters, so that columns align correctly.

[Steve Samuels]

Except for the fact that both functions increase, the cumulative hazard is nothing like the failure curve. In fact the cumulative hazard can exceed 1.0, so is not a probability. Below are the estimated failure function and two different estimates of the cumulative hazard function. The first is the one Stata generates. It is the Nelson-Aalen estimate, shown on page 300 of the Manual Entry for sts. The N-A estimate is the finite sample version of the definition:
\[
\Lambda_1(t) = \int_0^t \lambda(t)dt
\]
A second estimate can be based on the mathematical relationship of the cumulative hazard function to the Survival curve
\[
\Lambda_2(t)= -\textrm{log}(1-S(t))
\]
where the Kaplan-Meier estimate \(\widehat{S}\) is substituted for \(S\). The two estimates are very close

webuse catheter, clear
stset time infect
sts gen cumhaz1 = na km = s
label var cumhaz1 "Cum Haz:Nelson-Aalen"
gen cumhaz2 = -log(km)
label var cumhaz2 "Cum Haz:-log(s)"
gen cumfail = 1 - km
sort _t
label var cumfail "Cumulative Failure Probability"
#delim;
twoway connect cumhaz1 cumhaz2 cumfail _t,
 c(stairstep stairstep)
 title("KM Failure & Two Cum Hazard Estimates")
 saving(g01, replace);
#delim cr
graph use g01
graph export graph.png

[Steve Samuels]

Tim,

You observed that \(\widehat{\Lambda}(t) \approx 1 - \widehat{S}(t) = \widehat{F}(t)\). The example above shows that this isn't true in general. However the approximation does hold for small values of \(F(t)\) and \(\Lambda(t)\) and that's all you have in your data set (\(F(t)<0.30\)). Here's a proof. Start with the mathematical formula:
\[
S(t) = e^{-\Lambda(t)}
\]
Now consider the exponential function \(e^{-x}\) for small values of \(x\). The first term of a Taylor's Series for \(e^{-x}\) expanded about zero is:
\[
e^{-x} \approx 1- x
\]
so that \(x \approx 1- e^{-x}\). Apply this to small values of \( x =\Lambda(t)\) and you get
\[
S(t ) \approx 1- \Lambda(t)
\]
or
\[
\Lambda(t) \approx 1 - S(t) = F(t)
\]
as you observed. Here's the graph of the Nelson-Aalen estimate and the estimated failure curve for the data above with \(F(t) < 0.3\). The curves do look similar and values are close early on.

[tim bradshaw]

Dear Steve - thank you very much for your thoughts and knowledge on this.

I will have to dig deeper to understand this more fully and will post back if I have anything further to contribute. Your input is hugely appreciated and helpful.

Thanks again,
Tim

[tim bradshaw]

Incidentally, I did run:

ltable intervalyrs recur, hazard

and not

ltable intervalyrs recur, failure

...and, interestingly, the hazard lifetable doesn't show cumulative hazard (it only gives interval-specific hazards), but it does give cumulative failure, which was why I quoted the cumulative failure results from the ltable hazard command (which added to my confusion!).

It would surely be useful to see cumulative hazard in tabular (lifetable) format in a way that corresponds roughly to a cumulative hazard graph. At the moment I cannot find a way to generate this.

[Maarten Buis]

You can use sts list with the cumhaz option to see the cumulative hazard values.

[Steve Samuels]

sts list will display the cumulative hazard.

sts list,  survival failure cumhaz

Why was the cumulative hazard omitted from ltable? The cumulative hazard has never been a standard life tble measure; the values by themselves are not meaningful. When needed, they can be estimated by \(\textrm{log}(S(t))\)
I do find the cumulative hazard function useful:
1. For estimating the average hazard rate in an interval \([t_j, t_k]\).
\[
\textrm{average hazard} = \frac{\Lambda(t_k) - \Lambda(t_j)}{t_k - t_j}
\]
2. For checking distributional and model assumptions. For example one of the plots for checking a proportional hazards assumption (stcox postestimation) iis called the "log- log plot of survival". This is a plot of the log cumulative hazard.