R version 4.4.0 RC (2024-04-16 r86457 ucrt) -- "Puppy Cup"
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> options(na.action=na.exclude) # preserve missings
> options(contrasts=c('contr.treatment', 'contr.poly')) #ensure constrast type
> library(survival)
> 
> #
> # The test data set from Turnbull, JASA 1974, 169-73.
> #
> #  status  0=right censored
> #          1=exact
> #          2=left censored
> #
> aeq <- function(x,y, ...) all.equal(as.vector(x), as.vector(y), ...)
> 
> 
> turnbull <- data.frame( time  =c( 1,1,1, 2,2,2, 3,3,3, 4,4,4),
+ 			status=c( 1,0,2, 1,0,2, 1,0,2, 1,0,2),
+ 			  n   =c(12,3,2, 6,2,4, 2,0,2, 3,3,5))
> #
> # Compute the K-M for the Turnbull data
> #      via a slow EM calculation
> #
> 
> emsurv <- function(time, status, wt, verbose=T) {
+     left.cen <- (status==2)
+     if (!any(left.cen)) stop("No left censored data!")
+     if (!any(status==1))stop("Must have some exact death times")
+ 
+     tempy <- Surv(time[!left.cen], status[!left.cen])
+     ww <- wt[!left.cen]
+     tempx <- factor(rep(1, sum(!left.cen)))
+     tfit <- survfit(tempy~tempx, weight=ww)
+     if (verbose)
+        cat("Iteration 0, survival=", format(round(tfit$surv[tfit$n.event>0],3)),
+ 		       "\n")
+ 
+     stimes <- tfit$time[tfit$n.event>0]
+     ltime <- time[left.cen]
+     lwt   <- wt[left.cen]
+     tempx <- factor(rep(1, length(stimes) + sum(!left.cen)))
+     tempy <- Surv(c(time[!left.cen], stimes),
+ 		  c(status[!left.cen], rep(1, length(stimes))))
+     for (iter in 1:4) {
+ 	wt2 <- stimes*0
+ 	ssurv <- tfit$surv[tfit$n.event>0]
+ 	sjump <- diff(c(1, ssurv))
+ 	for (j in 1:(length(ltime))) {
+ 	    k <- sum(ltime[j]>=stimes)   #index of the death time
+ 	    if (k==0)
+ 		stop("Left censored observation before the first death")
+ 	    wt2[1:k] <- wt2[1:k] + lwt[j]*sjump[1:k] /(ssurv[k]-1)
+ 	    }
+ 	tfit <- survfit(tempy~tempx, weight=c(ww, wt2))
+ 	if (verbose) {
+ 	   cat("Iteration", iter, "survival=",
+ 		 format(round(tfit$surv[tfit$n.event>0],3)),  "\n")
+ 	   cat("             weights=", format(round(wt2,3)), "\n")
+ 	   }
+ 	}
+     survfit(tempy ~ tempx, weights=c(ww, wt2), robust=FALSE)
+     }
> 
> temp <-emsurv(turnbull$time, turnbull$status, turnbull$n)
Iteration 0, survival= 0.613 0.383 0.287 0.144 
Iteration 1 survival= 0.549 0.303 0.214 0.094 
             weights= 7.856 3.477 0.828 0.839 
Iteration 2 survival= 0.540 0.296 0.210 0.095 
             weights= 8.228 3.394 0.714 0.664 
Iteration 3 survival= 0.538 0.295 0.210 0.095 
             weights= 8.315 3.356 0.690 0.638 
Iteration 4 survival= 0.538 0.295 0.210 0.095 
             weights= 8.338 3.342 0.685 0.635 
> print(summary(temp))
Call: survfit(formula = tempy ~ tempx, weights = c(ww, wt2), robust = FALSE)

 time n.risk n.event survival std.err lower 95% CI upper 95% CI
    1  44.00   20.34   0.5378  0.0752       0.4089        0.707
    2  20.66    9.34   0.2946  0.0719       0.1827        0.475
    3   9.32    2.68   0.2098  0.0673       0.1119        0.393
    4   6.64    3.64   0.0948  0.0507       0.0333        0.270
> # First check, use the data from Turnbull, JASA 1974, 169-173.
> 
> tdata <- data.frame(time  =c(1,1,1,2,2,2,3,3,3,4,4,4),
+ 		    status=rep(c(1,0,2),4),
+ 		    n     =c(12,3,2,6,2,4,2,0,2,3,3,5))
> 
> tfit <- survfit(Surv(time, time, status, type='interval') ~1, tdata, weight=n)
> all.equal(round(tfit$surv,3), c(.538, .295, .210, .095))
[1] TRUE
> 
> 
> # Second check, compare to a reversed survival curve
> # This is not as simple a test as one might think, because left and right
> #  censored observations are not treated symmetrically by the routine:
> #  time <= y for left and time> y for right (this is to make the routine
> #  correct for the common situation of panel data).
> # To get equivalence, make the left censoreds happen just a little bit
> #  earlier.  The left-continuous/right-continuous shift is also a bother.
> #
> test1 <- data.frame(time=  c(9, 3,1,1,6,6,8),
+                     status=c(1,NA,1,0,1,1,0),
+                     x=     c(0, 2,1,1,1,0,0))
> fit1 <- survfit(Surv(time, status) ~1, test1)
> temp <- ifelse(test1$status==0, 4.99,5) - test1$time
> fit2 <- survfit(Surv(temp, status, type='left') ~1, test1)
> 
> all.equal(round(fit1$surv[1:2],5), round(1-fit2$surv[3:2],5))
[1] TRUE
> 
> rm(tdata, tfit, fit1, temp, fit2)
> #
> # Create a data set similar to the one provided by Al Zinsmeister
> #  It is a hard test case for survfit.turnbull
> #
> time1 <- c(rep(0,100), rep(1,200), 100, 200, 210, 220,
+            rep(365,100), rep(366,5), 731:741)
> 
> time2 <- c((1:100)*3,  10+1:100, rep(365:366, c(60,40)), NA, 500, NA, 450,
+            rep(730,90), rep(NA,10), c(528,571,691,730,731),
+            NA, 1095:1099, NA, 1400, 1200, 772, 1461)
> 
> zfit <- survfit(Surv(time1, time2, type='interval2') ~1)
> 
> #
> # There are 100 intervals of the form (0,x) where x is from 3 to 300,
> #  and 200 more of the form (1,x) where x is from 11 to 366.  These
> #  lead to a mass point in the interval (1,3), which is placed at 2.
> #  The starting estimate has far too little mass placed here, and it takes
> #  the EM a long time to realize that most of the weight for the first 300
> #  subjects goes here.  With acceleration, it takes 16 iterations, without
> #  it takes >40.  (On Al's orginal data, without accel still wasn't there after
> #  165 iters!)
> #
> # The next 4 obs give rise to potential jumps at 100.5, 200.5, 211.5, and
> #  221.  However, the final estimate has no mass at all on any of these.
> #  Assume mass of a,b, and c at 2, 100.5 and 365.5, and consider the 
> #  contributions: 
> #    123 obs that overlap a only
> #    137 obs that overlap a and b
> #     40 obs that overlap a, b, c
> #      1 obs that overlap b, c
> #    108 obs that overlap c   (200, 210,200, 365, and 366 starting points)
> #  For some trial values of a,b,c, compare the loglik to that of (a+b),0,c
> #   First one: a^123 (a+b)^137 (a+b+c)^40 (b+c) c^108
> #   Second:    (a+b)^123 (a+b)^137 (a+b+c)^40 c c^108
> #   Likelhood improves if (1 + b/a)^123 > 1+ b/c, which is true for almost
> #     all a and c.  In particular, at the solution a and c are approx .7 and
> #     .18, respectively.
> #
> # The program can't see this coming, of course, and so iterates towards a
> #  KM with epsilon sized jumps at 100.5, 200.5, and 211.5.  Whether these
> #  intervals should be removed during iteration, as detected, is an open
> #  question for me.
> #
> #
> # True solution: mass points at 2, 365.5, 408, and 756.5, of sizes a, b, c, d
> # Likelihood:    a^260 (a+b)^40 (b+c)^92 (b+c+d)^12 c^5 d^11
> # Solution: a=0.6958, b=0.1674, c=0.1079, d=0.0289
> 
> tfun <- function(x) {
+     if (length(x) ==3) x <- c(x, .03)
+     x <- x/sum(x)  #make probabilities sum to 1
+     loglik <- 260*log(x[1]) + 40*log(x[1]+x[2]) + 92*log(x[2] + x[3]) +
+                        12*log(x[2]+x[3]+x[4]) + 5*log(x[3]) + 11*log(x[4])
+     -loglik  #find the max, not the min
+     }
> 
> nfit <- nlminb(start=c(.7,.15, .1), tfun, lower=0, upper=1)
> nparm <- c(nfit$par, .03)
> nparm <- nparm / sum(nparm)
> zparm <- -diff(c(1, zfit$surv[match(c(2, 365.5, 408, 756.5), zfit$time)]))
> aeq(round(tfun(nparm),4), round(tfun(zparm),4))
[1] TRUE
> # .0001 is the tolerance in survfit.turnbull
> 
> rm(tfun, nfit, nparm, zparm, time1, time2, zfit) 
> 
> proc.time()
   user  system elapsed 
   0.82    0.12    0.95