# [R] significance in difference of proportions: What problema

(Ted Harding) Ted.Harding at nessie.mcc.ac.uk
Sat Nov 29 11:09:18 CET 2003

```On 28-Nov-03 Torsten Hothorn wrote:
> yes, thats my understanding too. The "enumerative techniques" as
> you call it condition on the data actually observed and determine
> the null distribution of the associated test statistic from the data.
> In contrast, unconditional procedures require some assumptions to the
> underlying data generating process from which the null distribution is
> derived. The appropriate choice depends of the kind of experiment
> under test: In a randomized trial we would like to see all possible
> outcomes of the trial caused by "rerandomization" and the enumerative
> techniques are natural here. When we draw many samples from predefined
> populations, men and women, say, "rerandomization" of gender is of
> course not that easy and we may assume something about the data
> generating process :-)

Nice example, but it depends on how you look at it!

Suppose you have samples of n1 Men and n2 Women and record, for instance,
whether or not each is suffering from a cold (r1 and r2 respectively).
Do M & W differ in their risk of catching cold?

NH: No difference; implies that the R = (r1+r2) colds have selected
a random subset of the N=(n1+n2) individuals as victims; implies
that the n1 Men out of N are a random subset of the R+(N-R)
Colds/NonColds. So you then have a hypergeometric distribution and are
back with an "exact" test. But are we "assuming somthing about the
data generating process" here?

(Of course, in the background lurks the Ogre of Exchangeability,
in that the probability of catching cold may vary from person to
person, whether Man or Woman, but nothing in the information
plus NH suggests any reason to distinguish any arrangement of the
N people from any other; equivalent to a re-randomisation of
gender ... ??).

Best wishes,
Ted.

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Date: 29-Nov-03                                       Time: 10:09:18
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