Splus Log Handout Related to Stratified Survival Curves,
  Survival Differences, and Cox Models
========================================================

Let's begin by seeing how to call for stratified 
survival curves in Splus. The unifying theme throughout
seems to be that a "formula" used in specifying survift or 
survdiff or coxph can be made stratified through syntax like:

Surv(Time,Dth) ~ Group + strata(Strat)

(I) Survival Curves.  Consider first the Bone Marrow Transplant 
data,downloaded as frame "BMT" from first 5 columns of the 
dataset from Section 1.3 of the book.

> ScurvA <- survfit(Surv(Dtime,Dind) ~ Gp, data=BMT)
  ScurvB <- survfit(Surv(Dtime,Dind) ~ strata(Gp), data=BMT)

### There is no difference between the two objects produced in this
###    way. In both cases, they plot as three separately fitted
###    survival curves which could also have been created with 
###    a for-loop.

(II) Survival Differences. Here let us artificially create "Strata" 
according to whether Relapse (Rtime) proecedes Death (Dtime). 
[This is not an advisable way to analyze the data: ity is done for
illustrative purposes only !]

> SdiffA <- survdiff(Surv(Dtime,Dind) ~ Gp, data=BMT) 
Call:
survdiff(formula = Surv(Dtime, Dind) ~ Gp, data = BMT)

      N Observed Expected (O-E)^2/E (O-E)^2/V 
Gp=1 38       24     21.3     0.337     0.462
Gp=2 54       23     38.8     6.413    12.562
Gp=3 45       34     20.9     8.190    11.162

 Chisq= 15.2  on 2 degrees of freedom, p= 0.000496 
> SdiffB <- survdiff(Surv(Dtime,Dind) ~ Gp + 
         strata(I(Rtime < Dtime)), data=BMT)
Call:
survdiff(formula = Surv(Dtime, Dind) ~ Gp + 
        strata(I(Rtime < Dtime)), data = BMT)

      N Observed Expected (O-E)^2/E (O-E)^2/V 
Gp=1 38       24     22.2     0.149      0.21
Gp=2 54       23     35.7     4.490      8.49
Gp=3 45       34     23.2     5.063      7.36

 Chisq= 10.3  on 2 degrees of freedom, p= 0.00589 

### OK, so these are different now: the point is that the 
###   analysis in SdiffB is stratified according to whether
###   or not  Rtime < Dtime   

> table(BMT$Rtime < BMT$Dtime, BMT$Gp)
       1  2  3 
FALSE 25 45 24
 TRUE 13  9 21

### In the following two "survdiff" calls, we obtain observed
###   and expected within the three groups, for each of the
###   two artificial "strata".

> survdiff(formula = Surv(Dtime, Dind) ~ Gp, data = BMT[
      BMT$Rtime < BMT$Dtime,])
...
      N Observed Expected (O-E)^2/E (O-E)^2/V 
Gp=1 13       13     11.9    0.0937     0.136
Gp=2  9        7     14.8    4.1129     6.801
Gp=3 21       21     14.3    3.1916     5.026

 Chisq= 7.8  on 2 degrees of freedom, p= 0.0205 

> survdiff(formula = Surv(Dtime, Dind) ~ Gp, data = BMT[
      BMT$Rtime == BMT$Dtime,])
      N Observed Expected (O-E)^2/E (O-E)^2/V 
Gp=1 25       11    10.24    0.0569    0.0774
Gp=2 45       16    20.85    1.1279    2.3770
Gp=3 24       13     8.91    1.8729    2.4246

 Chisq= 3.1  on 2 degrees of freedom, p= 0.215 

### Note that the observed and expected, summed across the 
###   two "strata", give the groupwise 
###   observed and expected in  SdiffB.

(III) Cox models.  THIS PART WILL BE ADDED OVER THE NEXT COUPLE OF
CLASSES, AS WE STUDY SINGLE-STRATUM COX MODEL FITTING USING  Coxph
AND THEN GENERALIZE TO MULTIPLE STRATA.
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