In this and the next few entries, we expand upon support in R and SAS for survival (time-to-event) models. We'll start with a small, artificial dataset of 19 subjects. Each subject contributes a pair of variables: the time and an indicator of whether the time is when the event occurred (event=TRUE) or when the subject was censored (event=FALSE).

time event

0.5 FALSE

1 TRUE

1 TRUE

2 TRUE

2 FALSE

3 TRUE

4 TRUE

5 FALSE

6 TRUE

7 FALSE

8 TRUE

9 TRUE

10 FALSE

12 TRUE

14 FALSE

14 TRUE

17 FALSE

20 TRUE

21 FALSE

Until an instant before time=1, no events were observed (only the censored observation), so the survival estimate is 1. At time=1, 2 subjects out of the 18 still at risk observed the event, so the survival function S(.) at time 1 is S(1) = 16/18 = 0.8889. The next failure occurs at time=2, with 16 still at risk, so S(2)=15/16 * 16/18 = 0.8333. Note that in addition to the event at time=2, there is a subject censored then, so the number at risk at time=3 is just 13 (so S(3) = 13/14 * 15*16 * 16/18 = 0.7738). The calculations continue until the final event is observed.

**R**

In R, we use the

`survfit()`function (section 5.1.19) within the

`survival`library to calculate the survival function across time.

library(survival)

time = c(0.5, 1,1,2,2,3,4,5,6,7,8,9,10,12,14,14,17,20, 21)

event = c(FALSE, TRUE, TRUE, TRUE, FALSE, TRUE, TRUE, FALSE,

TRUE, FALSE, TRUE, TRUE, FALSE, TRUE, FALSE, TRUE, FALSE,

TRUE, FALSE)

fit = survfit(Surv(time, event) ~ 1, data=ds)

The returned survival object includes a number of attributes, such as the survival estimates at each timepoint, the standard error of those estimates, and the number of subjects at risk.

> names(fit)

[1] "n" "time" "n.risk" "n.event" "n.censor"

"surv" "type" "std.err"

[9] "upper" "lower" "conf.type" "conf.int" "call"

> summary(fit)

Call: survfit(formula = Surv(time, event) ~ 1, data = ds)

time n.risk n.event survival std.err lower 95% CI upper 95% CI

1 18 2 0.889 0.0741 0.7549 1.000

2 16 1 0.833 0.0878 0.6778 1.000

3 14 1 0.774 0.0997 0.6011 0.996

4 13 1 0.714 0.1084 0.5306 0.962

6 11 1 0.649 0.1164 0.4570 0.923

8 9 1 0.577 0.1238 0.3791 0.879

9 8 1 0.505 0.1276 0.3078 0.829

12 6 1 0.421 0.1312 0.2285 0.775

14 5 1 0.337 0.1292 0.1587 0.714

20 2 1 0.168 0.1354 0.0348 0.815

**SAS**

We can read in the artificial data using an

`input`statement (section 1.1.8).

data ds;

input time event;

cards;

0.5 0

1 1

1 1

2 1

2 0

3 1

4 1

5 0

6 1

7 0

8 1

9 1

10 0

12 1

14 0

14 1

17 0

20 1

21 0

run;

proc lifetest data=ds;

time time*event(0);

run;

Here we denote censoring as being values where event is equal to 0. If we had a censoring indicator coded in reverse (1 = censoring), the second line might read

`time time*censored(1);`.

The survival function can be estimated in

`proc lifetest`(as shown in section 5.1.19). In a break from our usual practice, we'll include all of the output generated by

`proc lifetest`.

The LIFETEST Procedure

Product-Limit Survival Estimates

Survival

Standard Number Number

time Survival Failure Error Failed Left

0.0000 1.0000 0 0 0 19

0.5000* . . . 0 18

1.0000 . . . 1 17

1.0000 0.8889 0.1111 0.0741 2 16

2.0000 0.8333 0.1667 0.0878 3 15

2.0000* . . . 3 14

3.0000 0.7738 0.2262 0.0997 4 13

4.0000 0.7143 0.2857 0.1084 5 12

5.0000* . . . 5 11

6.0000 0.6494 0.3506 0.1164 6 10

7.0000* . . . 6 9

8.0000 0.5772 0.4228 0.1238 7 8

9.0000 0.5051 0.4949 0.1276 8 7

10.0000* . . . 8 6

12.0000 0.4209 0.5791 0.1312 9 5

14.0000 0.3367 0.6633 0.1292 10 4

14.0000* . . . 10 3

17.0000* . . . 10 2

20.0000 0.1684 0.8316 0.1354 11 1

21.0000* . . . 11 0

NOTE: The marked survival times are censored observations.

Summary Statistics for Time Variable time

Quartile Estimates

Point 95% Confidence Interval

Percent Estimate [Lower Upper)

75 20.0000 12.0000 .

50 12.0000 6.0000 20.0000

25 4.0000 1.0000 12.0000

Mean Standard Error

11.1776 1.9241

NOTE: The mean survival time and its standard error were underestimated because

the largest observation was censored and the estimation was restricted to

the largest event time.

Summary of the Number of Censored and Uncensored Values

Percent

Total Failed Censored Censored

19 11 8 42.11

## 4 comments:

awesome thanks!!

Thank you! You made this so easy to get started.

Kudos to Giles Crane for pointing out that R can also compute quantiles, and confidence limits, of survival curves.

The quantile function has a method for survfit objects of the survival package:

quantile( survfit( fit), c(0.25, 0.50, 0.75) )

Nick

Great, much easier to understand. thanks!

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