# [R] A model for disease progression

Damon Wischik djw1005 at cam.ac.uk
Tue Jul 29 15:53:51 CEST 2003

```Thank you to the various people who have made suggestions. In particular,
reading the documentation of the addreg package has prompted me to try to
put the question differently. I would be grateful for any comments on the
following.

As I described before, I have a snapshot of a population taken at a
certain time. I am interested in an age-related disease, which progresses
healthy->A->B. (There is no recovery.) For each individual, I know their
age (in years) and the stage of the disease. There are roughly 800 cases,
with ages spanning 40 years.

Suppose I don't distinguish between stages A and B, and all I am
interested in is whether someone has the disease or not. For each
individual, I therefore have a censored observation of a "lifetime"
random variable:
if the individual is age t and is diseased, lifetime is in (0,t].
if the individual is age t and is healthy, lifetime is in (t,inf)

I would like to plot a survival function for this "lifetime" random
variable. According to the documentation (for R1.7.0), the Surv function
does not let me enter left-censored intervals for non-parametric plots.
Are there ways around this? I could simply estimate

Prob(lifetime>t) = fraction of cases of age t who are healthy

and take this as my survival curve, but it produces a noisy plot (in
particular, the curve is not monotone). Is there a good way to get a
better estimate of the survival function?

Once I have a good way to estimate survival functions for this sort of
data, I could estimate the distribution of T1 (the time to reach stage A
or B) and of T2 (the time to reach stage B), and thereby estimate the
distribution of T2-T1 (the time to progress from stage A to stage B) by
some sort of convolution, assuming independence.

Damon.

```