Problem I of takehome 

We saw in homework 2 that the use of offset( ) in the survreg()
(also available in coxph() ) can produce an confidence intervel
by appealing to the Wilks theorem
(chi-square limiting distribution of the -2 log likelihood ratio).


In the AFT model, similar thing can be done but more tedious:

Here is an example. 

> library(survival)
> library(rankreg)
> data(myeloma)
> rankaft(x=cbind(myeloma[,3],myeloma[,4]),y=myeloma[,1],delta=myeloma[,2])$beta

We see the Gehan rank estimator of the slope beta in AFT model is:
betag -15.01117 1.317596

So, beta1 hat = -15.01117, beta2 hat = 1.317596.

The estimated model is Y= -15.01117*X1 + 1.317596*X2 + e

How do we find a 90% confidence interval for beta1 ?

There is no such thing like 
offset(-12*myeloma[,3]) in rankaft(), 
we have to do the following two steps:

suppose we want to fix beta1 at -12

step 1. 
> rankaft(x=cbind(myeloma[,4]),y=(myeloma[,1]-(-12*myeloma[,3])),delta=myeloma[,2])$beta

We get:
betag 1.353403

step 2:
> RankRegV(x=cbind(myeloma[,3],myeloma[,4]),y=myeloma[,1],d=myeloma[,2], beta=c(-12,1.353403))

We see (among other things)

chisquare 
           [,1]
[1,]     0.3663783


This means if we do have an offset() command, then the output
test statistics will have a chisquare value of 0.3663783, when
beta1 is fixed at -12.

We can keep repeating the above two steps, with other values replacing -12,
until we get chisquare value of 2.7  (= 1.645^2 ). There should be two
of them and this will be the 90% confidence interval of beta1

This is the two step with offset(-6*myeloma[,3]) and almost got what we needed
chisquare value:

> rankaft(x=cbind(myeloma[,4]),y=(myeloma[,1]-(-6*myeloma[,3])),delta=myeloma[,2])$beta
          xnew
betag 1.324358
betal 2.697085
> RankRegV(x=cbind(myeloma[,3],myeloma[,4]),y=myeloma[,1],d=myeloma[,2], beta=c(-6,1.324358))
$VEF
          [,1]       [,2]
[1,] 141.54615  -31.92112
[2,] -31.92112 6071.29246

$chisquare
         [,1]
[1,] 2.612443

$Pval
          [,1]
[1,] 0.2708415


============================================
OK, I assume you got the above procedure. Now you need to work on 
a different data set:

data(stanford2) 

but with the following: delete the cases that t5 is missing (NA)
and delete the cases where time is smaller than 5.

We are intersted to fit a model where log10(time)=Y
and 2 covariates are: age and t5.

find a 90% confidence interval for age alone using the AFT model
as illustrated above.


Take home problem 2

Use the cancer data set.

> library(survival)
> data(cancer)
> ?cancer

(a) Fit a Cox model relating the survival time (time) of cancer patients
with age, sex, ph.ecog and ph.karno. (Notice the definition of the status).

Write some comments on the findings of your statistical analysis.
(additional model fitting and analysis may be needed depending on your
recommendation).

Estimate the cumulative hazard function based on the 
fitted model in (a) for a hypothetical person
with: age = 50, sex = male, ph.ecog=1, ph.karno= 70.
Plot the curve.