Manual calculation of the Kaplan-Meier estimator

Question

I thought this would be a trivial thing to do but I still have some trouble adjusting to writing code instead of pointing and clicking on a spreadsheet.

month = as.integer(c(1,2,3,4,5,6,7,8,9,10,11,12))
remaining = c(1000,925,852,790,711,658,601,567,530,501,485,466)
left = c(75, 73, 62, 79, 53, 57, 34, 37, 29, 16, 19, 0)
KPdata = data.frame(month, remaining, left)

> KPdata
   month remaining left    
1      1      1000   75 
2      2       925   73 
3      3       852   62 
4      4       790   79 
5      5       711   53 
6      6       658   57 
7      7       601   34
8      8       567   37 
9      9       530   29 
10    10       501   16 
11    11       485   19 
12    12       466   12 

How do I calculate the Kaplan-Meier survival function at each month? Note that I want to do this manually, I am aware that there are packages which will do it for me.

Answer 1

I was so taken with the question and @bouncyball's inspiring answer, I thought I'd add my ha'penny worth with an attempt at handling censoring. This is intended to be in the spirit of the original question - doing things 'handraulically' to develop key insights.

## rename remaining -> survived; left -> died 
month = as.integer(c(1,2,3,4,5,6,7,8,9,10,11,12))
survived = c(1000,925,852,790,711,658,601,567,530,501,485,466)
died = c(75, 73, 62, 79, 53, 57, 34, 37, 29, 16, 19, 0)

## arbitrary censoring @ 10 per time period
censored <- c(0, rep(10,11))
KPdata3 = data.frame(month, at.risk, censored, died, survived) 

## define those at risk <= those who survived
## awful bit of R fiddling for (something simple like) offsetting the index in base R
len <- length(month)
at.risk <- c(survived[1], 
             survived[-len] - died[-len] - cumsum(censored[-len]) )
## note use of cumsum()

## censoring uses at risk, rather than survived/remained
KPdata3$KM_increment <- (KPdata3$at.risk - KPdata3$died)/ KPdata3$at.risk
## code credit to @bouncyball
KPdata3$KM_cumulative <- cumprod(KPdata3$KM_increment)
KPdata3

Gives this.....

   month at.risk censored died survived KM_increment KM_cumulative
1      1    1000        0   75     1000    0.9250000     0.9250000
2      2     925       10   73      925    0.9210811     0.8520000
3      3     842       10   62      852    0.9263658     0.7892637
4      4     770       10   79      790    0.8974026     0.7082873
5      5     681       10   53      711    0.9221733     0.6531636
6      6     618       10   57      658    0.9077670     0.5929203
7      7     551       10   34      601    0.9382940     0.5563336
8      8     507       10   37      567    0.9270217     0.5157333
9      9     460       10   29      530    0.9369565     0.4832197
10    10     421       10   16      501    0.9619952     0.4648551
11    11     395       10   19      485    0.9518987     0.4424949
12    12     366       10    0      466    1.0000000     0.4424949

Setting rep(0,11) gives the same answer as @bouncyball's.

Answer 2

I think this is what you're trying to do. We use lag and cumprod to get a manual KM estimator:

KPdata$KM_init <- lag((KPdata$remaining - KPdata$left) / KPdata$remaining)
KPdata[1,ncol(KPdata)] <- 1
KPdata$KM_final <- cumprod(KPdata$KM_init)

KPdata
   month remaining left   KM_init KM_final
1      1      1000   75 1.0000000    1.000
2      2       925   73 0.9250000    0.925
3      3       852   62 0.9210811    0.852
4      4       790   79 0.9272300    0.790
5      5       711   53 0.9000000    0.711
6      6       658   57 0.9254571    0.658
7      7       601   34 0.9133739    0.601
8      8       567   37 0.9434276    0.567
9      9       530   29 0.9347443    0.530
10    10       501   16 0.9452830    0.501
11    11       485   19 0.9680639    0.485
12    12       466    0 0.9608247    0.466

Alternatively, I think there's a different form of a KM estimator that looks like this (note that I've added a row corresponding to month = 0):

month = as.integer(c(0,1,2,3,4,5,6,7,8,9,10,11,12))
remaining = c(1000,1000,925,852,790,711,658,601,567,530,501,485,466)
left = c(0,75, 73, 62, 79, 53, 57, 34, 37, 29, 16, 19, 0)
KPdata2 = data.frame(month, remaining, left)

KPdata2$KM_init <- (KPdata2$remaining - KPdata2$left) / KPdata2$remaining
KPdata2$KM_final <- cumprod(KPdata2$KM_init)

KPdata2
   month remaining left   KM_init KM_final
1      0      1000    0 1.0000000    1.000
2      1      1000   75 0.9250000    0.925
3      2       925   73 0.9210811    0.852
4      3       852   62 0.9272300    0.790
5      4       790   79 0.9000000    0.711
6      5       711   53 0.9254571    0.658
7      6       658   57 0.9133739    0.601
8      7       601   34 0.9434276    0.567
9      8       567   37 0.9347443    0.530
10     9       530   29 0.9452830    0.501
11    10       501   16 0.9680639    0.485
12    11       485   19 0.9608247    0.466
13    12       466    0 1.0000000    0.466