# [R] OT: A test with dependent samples.

Marc Schwartz marc_schwartz at comcast.net
Tue Feb 10 23:49:00 CET 2009

```on 02/10/2009 03:33 PM Rolf Turner wrote:
>
> I am appealing to the general collective wisdom of this
> list in respect of a statistics (rather than R) question.  This question
> comes to me from a friend who is a veterinary oncologist.  In a study that
> she is writing up there were 73 cats who were treated with a drug called
> piroxicam.  None of the cats were observed to be subject to vomiting prior
> to treatment; 12 of the cats were subject to vomiting after treatment
> commenced.  She wants to be able to say that the treatment had a
> ``significant''
> impact with respect to this unwanted side-effect.
>
> Initially she did a chi-squared test.  (Presumably on the matrix
> matrix(c(73,0,61,12),2,2) --- she didn't give details and I didn't pursue
> this.) I pointed out to her that because of the dependence --- same 73
> cats pre- and post- treatment --- the chi-squared test is inappropriate.
>
> So what *is* appropriate?  There is a dependence structure of some sort,
> but it seems to me to be impossible to estimate.
>
> After mulling it over for a long while (I'm slow!) I decided that a
> non-parametric approach, along the following lines, makes sense:
>
> We have 73 independent pairs of outcomes (a,b) where a or b is 0
> if the cat didn't barf, and is 1 if it did barf.
>
> We actually observe 61 (0,0) pairs and 12 (0,1) pairs.
>
> If there is no effect from the piroxicam, then (0,1) and (1,0) are
> equally likely.  So given that the outcome is in {(0,1),(1,0)} the
> probability of each is 1/2.
>
> Thus we have a sequence of 12 (0,1)-s where (under the null hypothesis)
> the probability of each entry is 1/2.  Hence the probability of this
> sequence is (1/2)^12 = 0.00024.  So the p-value of the (one-sided) test
> is 0.00024.  Hence the result is ``significant'' at the usual levels,
> and my vet friend is happy.
>
> I would very much appreciate comments on my reasoning.  Have I made any
> goof-ups, missed any obvious pit-falls?  Gone down a wrong garden path?
>
> Is there a better approach?
>
> Most importantly (!!!): Is there any literature in which this approach is
> spelled out?  (The journal in which she wishes to publish will almost
> surely
> demand a citation.  They *won't* want to see the reasoning spelled out in
> the paper.)
>
> I would conjecture that this sort of scenario must arise reasonably often
> in medical statistics and the suggested approach (if it is indeed valid
> and sensible) would be ``standard''.  It might even have a name!  But I
> have no idea where to start looking, so I thought I'd ask this wonderfully
> learned list.
>
> Thanks for any input.

Rolf,

I am a little confused, perhaps due to lack of sleep (sick dog with CHF).

Typically in this type of study, essentially looking at the
efficacy/safety profile of a treatment, there are two options.

One does a two arm randomized study, whereby "subjects" are randomized
to one of two treatments. The two treatments may both be "active" or one
may be a placebo. Then a typical two sample comparison of the primary
hypothesis is made. In this setting, you would have a second group of 73
cats who received a comparative treatment (or a placebo) to compare
against the 16.4% observed in this treatment group.

For example, say that patients were undergoing cancer treatment, which
has nausea and vomiting as a side effect. Due to the side effect, it is
common to see a reduction in dosing, which of course reduces treatment
effectiveness. You might want to study a treatment that favorably
reduces that side effect, to enable improved treatment dosing and
patient tolerance.

The other option would be to perform a single sample study, where there
is an a priori hypothesis, based upon prior work, of the expected
incidence of the adverse event or perhaps a "clinically acceptable"
incidence of the adverse event. This would seem to be the scenario
indicated above.

What is lacking is some a priori expectation of the incidence of the
event in question, so that one can show that you have reduced the
incidence from the expected.

50% would not make sense here, though if it did, a single sample
binomial test would be used, presuming a two-sided hypothesis:

> binom.test(12, 73, 0.5)\$p.value
[1] 4.802197e-09

That none of them had vomiting prior to treatment seems to be of little
interest here. You could just as easily argue that there was a
significant increase in the incidence of vomiting from 0% to 16.4% due
to the treatment.

What am I missing?

Regards,

Marc Schwartz

```