# [R] survdiff

Peter Dalgaard p.dalgaard at biostat.ku.dk
Tue Aug 17 23:37:06 CEST 2004

```"Krista Haanstra" <krista at aha.demon.nl> writes:

> As I am quitte an ignorant user of R, excuse me for any wrongfull usage of
> all the terms.
> My question relates to the statistics behind the survdiff function in the
> package survival.
> My textbook knowledge of the logrank test tells me that if I want to compare
> two survival curves, I have to take the sum of the factors: (O-E)^2/E of
> both groups, which will give me the Chisq.
> If I calculate this by hand, I get a different value than the one R is
> giving me.
> Actually, the (O-E)^2/E that R gives me, those I agree with, but if I then
> take the sum, this is not the chisq R gives.
> Two questions:
> - How is Chisq calculated in R?
> - What does the column (O-E)^2/V mean? What is V, and how does this possibly
> relate to the calculated Chisq?

You really need to read a theory book for this, but here's the basic idea:

V is the theoretical variance of O-E for the first group. If O-E is
approximately normally distributed, as it will be in large samples,
then (O-E)^2/V will be approximately chi-squared distributed on 1 DF.

In *other* models, notably those for contingency tables, the same idea
works out as the familiar sum((O-E)^2/E) formula. That formula has
historically been used for the logrank test too, and it still appears
in some textbooks, but as it turns out, it is not actually correct
(although often quite close).

[To fix ideas, consider testing for a given p in the binomial
distribution, you can either say O=x E=np V=npq and get

chisq = (x-np)^2/npq

or have O = (x, n-x), E = (np, nq) and get

chisq =  (x-np)^2/np + ((n-x) - nq)^2/nq

and a little calculus show that the latter expression is

= (x-np)^2*(1/np + 1/nq) = (x-np)^2 * (p+q)/npq

so the two formulas are one and the same. In this case!]
--
O__  ---- Peter Dalgaard             Blegdamsvej 3
c/ /'_ --- Dept. of Biostatistics     2200 Cph. N
(*) \(*) -- University of Copenhagen   Denmark      Ph: (+45) 35327918
~~~~~~~~~~ - (p.dalgaard at biostat.ku.dk)             FAX: (+45) 35327907

```