Exercise 1: Life tables and Kaplan-Meier estimates of survival
(a) Hand calculation: Life table and Kaplan-Meier estimates of survival
The results are contained in an Excel file and are also shown in the R output below.
(b) Using R to validate the hand calculations done in part 1 (a)
First, load the biostat3 library:
Following are the life table estimates. Note that in the lectures, when we estimated all-cause survival, there were 8 deaths in the first interval. One of these died of a cause other than cancer so in the cause-specific survival analysis we see that there are 7 ‘deaths’ and 1 censoring (Stata uses the term ‘lost’ for lost to follow-up) in the first interval.
## tstart tstop nsubs nlost nrisk nevent surv pdf hazard se.surv se.pdf se.hazard
## 0-1 0 1 35 1 34.5 7 1.00 0.203 0.23 0.000 0.068 0.085
## 1-2 1 2 27 3 25.5 1 0.80 0.031 0.04 0.068 0.031 0.040
## 2-3 2 3 23 4 21.0 5 0.77 0.182 0.27 0.073 0.073 0.120
## 3-4 3 4 14 1 13.5 2 0.58 0.086 0.16 0.090 0.058 0.113
## 4-5 4 5 11 1 10.5 0 0.50 0.000 0.00 0.095 NaN NaN
## 5-6 5 6 10 0 10.0 0 0.50 0.000 0.00 0.095 NaN NaN
## 6-7 6 7 10 3 8.5 0 0.50 0.000 0.00 0.095 NaN NaN
## 7-8 7 8 7 1 6.5 0 0.50 0.000 0.00 0.095 NaN NaN
## 8-9 8 9 6 4 4.0 1 0.50 0.124 0.29 0.095 0.110 0.283
## 9-10 9 10 1 1 0.5 0 0.37 0.000 0.00 0.129 NaN NaN
## 10-Inf 10 Inf 0 0 0.0 0 0.37 NA NA 0.129 NA NAFollowing is a table of Kaplan-Meier estimates. Although it’s not clear from the table, the person censored (lost) at time 2 was at risk when the other person dies at time 2. On the following is a graph of the survival function.
mfit <- survfit(Surv(surv_mm, status == "Dead: cancer") ~ 1, data = colon_sample) # make Kaplan-Meier estimates
summary(mfit) # print Kaplan-Meier table## Call: survfit(formula = Surv(surv_mm, status == "Dead: cancer") ~ 1,
## data = colon_sample)
##
## time n.risk n.event survival std.err lower 95% CI upper 95% CI
## 2 35 1 0.971 0.0282 0.918 1.000
## 3 33 1 0.942 0.0398 0.867 1.000
## 5 32 1 0.913 0.0482 0.823 1.000
## 7 31 1 0.883 0.0549 0.782 0.998
## 8 30 1 0.854 0.0605 0.743 0.981
## 9 29 1 0.824 0.0652 0.706 0.962
## 11 28 1 0.795 0.0692 0.670 0.943
## 22 24 1 0.762 0.0738 0.630 0.921
## 27 22 1 0.727 0.0781 0.589 0.898
## 28 20 1 0.691 0.0823 0.547 0.872
## 32 19 2 0.618 0.0882 0.467 0.818
## 33 16 1 0.579 0.0908 0.426 0.788
## 43 13 1 0.535 0.0941 0.379 0.755
## 46 12 1 0.490 0.0962 0.334 0.720
## 102 4 1 0.368 0.1284 0.185 0.729plot(mfit, # plot Kaplan-Meier curve
ylab="S(t)",
xlab="Time since diagnosis in months",
main = "Kaplan−Meier estimates of cause−specific survival")