Germán Rodríguez

Survival Analysis
Princeton University
The WWS 509 datasets page has a copy of the Gehan survival data, which were analyzed by Cox in his original proportional hazards paper. The data shows the length of remission in weeks for two groups of leukemia patients, treated and controls, click here for more information. We analyze the data using R and Stata.

As usual, the first task is to read the data. We also `stset`

them and label the groups.

. clear . infile group weeks relapse using /// > https://data.princeton.edu/wws509/datasets/gehan.raw (42 observations read) . label define group 1 "control" 2 "treated" . label values group group . stset weeks, failure(relapse) failure event: relapse != 0 & relapse < . obs. time interval: (0, weeks] exit on or before: failure ------------------------------------------------------------------------------ 42 total observations 0 exclusions ------------------------------------------------------------------------------ 42 observations remaining, representing 30 failures in single-record/single-failure data 541 total analysis time at risk and under observation at risk from t = 0 earliest observed entry t = 0 last observed exit t = 35

> gehan <- read.table("https://data.princeton.edu/wws509/datasets/gehan.dat") > library(dplyr) > summarize(gehan, events = sum(relapse), exposure = sum(weeks)) events exposure 1 30 541

As you can see, 30 of the 42 patients had a relapse, with a total observation time of 541 weeks. The weekly relapse rate is 7.8%.

The Kaplan-Meier estimators for the two groups are easily plotted using `sts graph`

with the `by(group)`

option. computed using `survfit`

in the survival package and plotted using the generic function `plot`

, or my own `ggfy`

function, which makes base plots look a bit more like `ggplot`

.

. set scheme plottig . sts graph, by(group) failure _d: relapse analysis time _t: weeks . graph export kmg.png, width(500) replace (file kmg.png written in PNG format)

> library(survival) > kmg <- survfit(Surv(weeks, relapse) ~ group, data=gehan) > source("https://data.princeton.edu/pop509/ggfy.R.txt") > png("kmgr.png", width=500, height=400) > ggfy(kmg) > dev.off() null device 1

This shows that after 23 weeks all patients in the control group had relapsed, but about half those in the treated group remained in remission. This reproduces Figure 1 in the notes.

We can list the estimates using `sts list`

. the `summary()`

method.

. sts list, by(group) failure _d: relapse analysis time _t: weeks Beg. Net Survivor Std. Time Total Fail Lost Function Error [95% Conf. Int.] ------------------------------------------------------------------------------- control 1 21 2 0 0.9048 0.0641 0.6700 0.9753 2 19 2 0 0.8095 0.0857 0.5689 0.9239 3 17 1 0 0.7619 0.0929 0.5194 0.8933 4 16 2 0 0.6667 0.1029 0.4254 0.8250 5 14 2 0 0.5714 0.1080 0.3380 0.7492 8 12 4 0 0.3810 0.1060 0.1831 0.5778 11 8 2 0 0.2857 0.0986 0.1166 0.4818 12 6 2 0 0.1905 0.0857 0.0595 0.3774 15 4 1 0 0.1429 0.0764 0.0357 0.3212 17 3 1 0 0.0952 0.0641 0.0163 0.2612 22 2 1 0 0.0476 0.0465 0.0033 0.1970 23 1 1 0 0.0000 . . . treated 6 21 3 1 0.8571 0.0764 0.6197 0.9516 7 17 1 0 0.8067 0.0869 0.5631 0.9228 9 16 0 1 0.8067 0.0869 0.5631 0.9228 10 15 1 1 0.7529 0.0963 0.5032 0.8894 11 13 0 1 0.7529 0.0963 0.5032 0.8894 13 12 1 0 0.6902 0.1068 0.4316 0.8491 16 11 1 0 0.6275 0.1141 0.3675 0.8049 17 10 0 1 0.6275 0.1141 0.3675 0.8049 19 9 0 1 0.6275 0.1141 0.3675 0.8049 20 8 0 1 0.6275 0.1141 0.3675 0.8049 22 7 1 0 0.5378 0.1282 0.2678 0.7468 23 6 1 0 0.4482 0.1346 0.1881 0.6801 25 5 0 1 0.4482 0.1346 0.1881 0.6801 32 4 0 2 0.4482 0.1346 0.1881 0.6801 34 2 0 1 0.4482 0.1346 0.1881 0.6801 35 1 0 1 0.4482 0.1346 0.1881 0.6801 -------------------------------------------------------------------------------

> summary(kmg) Call: survfit(formula = Surv(weeks, relapse) ~ group, data = gehan) group=control time n.risk n.event survival std.err lower 95% CI upper 95% CI 1 21 2 0.9048 0.0641 0.78754 1.000 2 19 2 0.8095 0.0857 0.65785 0.996 3 17 1 0.7619 0.0929 0.59988 0.968 4 16 2 0.6667 0.1029 0.49268 0.902 5 14 2 0.5714 0.1080 0.39455 0.828 8 12 4 0.3810 0.1060 0.22085 0.657 11 8 2 0.2857 0.0986 0.14529 0.562 12 6 2 0.1905 0.0857 0.07887 0.460 15 4 1 0.1429 0.0764 0.05011 0.407 17 3 1 0.0952 0.0641 0.02549 0.356 22 2 1 0.0476 0.0465 0.00703 0.322 23 1 1 0.0000 NaN NA NA group=treated time n.risk n.event survival std.err lower 95% CI upper 95% CI 6 21 3 0.857 0.0764 0.720 1.000 7 17 1 0.807 0.0869 0.653 0.996 10 15 1 0.753 0.0963 0.586 0.968 13 12 1 0.690 0.1068 0.510 0.935 16 11 1 0.627 0.1141 0.439 0.896 22 7 1 0.538 0.1282 0.337 0.858 23 6 1 0.448 0.1346 0.249 0.807

The convention is to list the survival function immediately *after* each time. In the control group there are no censored observations, and the Kaplan-Meier estimate is simply the proportion alive after each distinct failure time. You can also check that the standard error is the usual binomial estimate. For example just after 8 weeks there are 8 out of 21 alive, the proportion is 8/21 = 0.381 and the standard error is v0.381(1 - 0.381)/21) = 0.106.

In the treated group 12 cases are censored and 9 relapse. Can you compute the estimate by hand? Only the distinct times of death –6, 7, 10, 13, 16, 22 and 23– are relevant. The counts of relapses are 3, 1, 1, 1, 1, 1, 1. When there are ties between event and censoring times, it is customary to assume that the event occurred first; that is, observations censored at *t* are assumed to be exposed at that time, effectively censored just after *t*. The counts of censored observations after each death time (but before the next) are 1, 1, 2, 0, 3, 0 and 5. When the last observation is censored, the K-M estimate is greater than zero, and is usually considered undefined from that point on.

We can also compute the Nelson-Aalen estimate of the cumulative hazard, and then exponentiate minus the cumulative hazard to obtain an alternative estimate of the survival function. Stata’s `sts list`

can compute the Nelson-Aalen estimate using the `na`

option, and `sts generate`

can save either estimate in a new variable as shown below. In R we can estimate the cumulative hazard “by hand” using counts of events and exposure.

. sts generate kmS = s . sts generate naH = na . gen naS = exp(-naH) . twoway (line kmS _t, c(J) sort) (line naS _t, c(J) sort) . graph export kmna.png, width(500) replace (file kmna.png written in PNG format)

> sf <- survfit(Surv(weeks, relapse) ~ 1, data = gehan) > d <- data.frame(time = sf$ time, km = sf $surv, + na = exp(-cumsum(sf$n.event / sf$n.risk))) > dl <- reshape(d, direction="long", idvar="time", timevar="method", + v.names="survival", varying=c("km","na")) %>% + mutate(method = factor(method, labels=c("km","na"))) > library(ggplot2) > ggplot(dl, aes(time, survival, color=method)) + geom_step() > ggsave("kmnar.png", width=500/72, height=400/72, dpi=72)

The two estimates are very similar.

To test equality of survival curves in the two groups we can use the Mantel-Haenszel or log-rank test, available in the `sts test`

command with the default `logrank`

option. via the `survdiff`

function.

. sts test group failure _d: relapse analysis time _t: weeks Log-rank test for equality of survivor functions ¦ Events Events group ¦ observed expected --------+------------------------- control ¦ 21 10.75 treated ¦ 9 19.25 --------+------------------------- Total ¦ 30 30.00 chi2(1) = 16.79 Pr>chi2 = 0.0000

> survdiff(Surv(weeks, relapse) ~ group, data = gehan) Call: survdiff(formula = Surv(weeks, relapse) ~ group, data = gehan) N Observed Expected (O-E)^2/E (O-E)^2/V group=control 21 21 10.7 9.77 16.8 group=treated 21 9 19.3 5.46 16.8 Chisq= 16.8 on 1 degrees of freedom, p= 4.17e-05

We see that the treated have fewer deaths than would be expected (and thus the controls have more) if the two groups had the same survival distribution: 9 instead of 19.25. The Mantel-Haenszel chi-squared statistic of 16.8 on one d.f. is highly significant.

It is instructive to compute this statistic “by hand”.

The Mantel-Haenszel test is just one of several test statistics that we could use. There is a family of test based on a comparison of observed and expected counts at each distinct failure time, which differ in terms of the weights assigned to each time as shown in the table below

Name | Weight |
---|---|

Mantel-Haenszel | 1 |

Wilcoxon | \(n_i\) |

Tarone-Ware | \(\sqrt{n_i}\) |

Peto-Peto-Prentice | \(\tilde{S}(t_i)\) |

Fleming-Harrington | \(\hat{S}(t_i)^p (1-\hat{S}(t_i))^q\) |

The most popular of these is the Wilcoxon test, actually an extension of Wilcoxon’s well-known non-parametric test proposed by Gehan and Breslow, which gives more weight to early failures. Stata can compute all four using the options `wilcoxon`

, `tware`

, `peto`

and `fh(p, q)`

. R implements three using a parameter `rho`

(*p* in our notation), with `rho=0`

for log-rank, `rho=1`

for Peto-Peto, and `rho=p`

for Fleming-Harrington with `q=1`

. The tests using the survival as weight treat it as left-continuous.