Setting Up the Hazard and Survival Curve Estimation in the
`Hypothetical' Survival Data Example of Problem 4.7
=================================================== 10/13/03

> Fram4.7 = cbind.data.frame(Entr.age = 
c(58,58,59,60,60,61,61,62,62,62,63,63,64,66,66,67,67,67,68,69,
  69,69,70,70,70,71,72,72,73,73), Exit.age = 
c(60,63,69,62,65,72,69,73,66,65,68,74,71,68,69,70,77,69,72,79,
  72,70,76,71,78,79,76,73,80,74), Dth=c(1,1,0,1,1,0,0,0,1,1,1,
  0,1,1,1,1,1,1,1,0,1,1,0,1,0,0,1,1,0,1))

### This had to be entered by hand: it is not on the book webpage.

## Now we need to create a data-frame consisting of unique 
##   death-times, deaths, and numbers-at-risk for the LEFT-TRUNCATED
##   dataset where each individual is left-truncated at "Entry Age"
##   and dies or is censored at "Exit age". 

## We are treating discrete-time survival here, since we have no
   information about exposure-times within years of age. So we act as
   though entry and exit, including death, can occur only on an
   individual's birthday in a given year of age.

> ages4.7 = sort(unique(Fram4.7[Fram4.7$Dth==1,2]))
  dths4.7 = tapply(rep(1,sum(Fram4.7$Dth==1)),
     Fram4.7[Fram4.7$Dth==1,2],sum)
  atrsk4.7T = numeric(length(ages4.7))
> for(i in 1:length(ages4.7)) atrsk4.7T[i] = sum(Fram4.7[,2] >= 
     ages4.7[i]  & Fram4.7[,1] < ages4.7[i])
## Now, for example, an analogue of the Nelson-Aalen estimator
##  is obtained by cumulatively summing up dths/atrsk, and the
##  standard error formula is essentially as before. 

> Fram4.7T = cbind.data.frame(ages = ages4.7, dths = dths4.7, 
     atrsk = atrsk4.7T, CumHaz = round(cumsum(dths4.7/atrsk4.7T),3), 
     Haz.SE = round(sqrt(cumsum(dths4.7/atrsk4.7T^2)),3))
	### You can verify this by hand !!
> Fram4.7T
   ages dths atrsk CumHaz Haz.SE 
60   60    1     3  0.333  0.333
62   62    1     6  0.500  0.373
63   63    1     8  0.625  0.393
65   65    2    10  0.825  0.418
66   66    1     8  0.950  0.436
68   68    2    12  1.117  0.452
69   69    2    11  1.298  0.470
70   70    2    10  1.498  0.490
71   71    2    11  1.680  0.507
72   72    2    10  1.880  0.526
73   73    1     9  1.991  0.538
74   74    1     9  2.103  0.549
76   76    1     7  2.245  0.568
77   77    1     5  2.445  0.602

### Now we could estimate survival function directly 
###  for the increasing ages as given in Fram4.7T.
### (Cannot use survfit directly because there are 
###  multiple deaths at some times ... 
> Scurv = cumprod(1- Fram4.7T$dths/Fram4.7T$atrsk)
  round(Scurv,3)
   60    62    63    65    66    68    69    70    71
0.667 0.556 0.486 0.389 0.340 0.284 0.232 0.186 0.152 
   72    73    74    76    77
0.121 0.108 0.096 0.082 0.066