[R] null distribution of binom.test p values

[Greg Snow]

Yes that is due to the discreteness of the distribution, consider the following:

> binom.test(39,100,.5)

        Exact binomial test

data:  39 and 100 
number of successes = 39, number of trials = 100, p-value = 0.0352
alternative hypothesis: true probability of success is not equal to 0.5 
95 percent confidence interval:
 0.2940104 0.4926855 
sample estimates:
probability of success 
                  0.39 

> binom.test(40,100,.5)

        Exact binomial test

data:  40 and 100 
number of successes = 40, number of trials = 100, p-value = 0.05689
alternative hypothesis: true probability of success is not equal to 0.5 
95 percent confidence interval:
 0.3032948 0.5027908 
sample estimates:
probability of success 
                   0.4

(you can do the same for 60 and 61)

So notice that the probability of getting 39 or more extreme is 0.0352, but anything less extreme will result in not rejecting the null hypothesis (because the probability of getting a 40 or a 60 (dbinom(40,100,.5)) is about 1% each, so we see a 2% jump there).  So the size/probability of a type I error will generally not be equal to alpha unless n is huge or alpha is chosen to correspond to a jump in the distribution rather than using common round values.

I have seen suggestions that instead of the standard test we use a test that rejects the null for values 39 and more extreme, don't reject the null for 41 and less extreme, and if you see a 40 or 60 then you generate a uniform random number and reject if it is below a certain value (that value chosen to give an overall probability of type I error of 0.05).  This will correctly size the test, but becomes less reproducible (and makes clients nervous when they present their data and you pull out a coin, flip it, and tell them if they have significant results based on your coin flip (or more realistically a die roll)).  I think it is better in this case if you know your final sample size is going to be 100 to explicitly state that alpha will be 0.352 (but then you need to justify why you are not using the common 0.05 to reviewers).

-- 
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.snow at imail.org
801.408.8111


> -----Original Message-----
> From: Chris Wallace [mailto:chris.wallace at cimr.cam.ac.uk]
> Sent: Thursday, January 26, 2012 9:36 AM
> To: Greg Snow
> Cc: r-help at r-project.org
> Subject: Re: [R] null distribution of binom.test p values
> 
> Greg, thanks for the reply.
> 
> Unfortunately, I remain unconvinced!
> 
> I ran a longer simulation, 100,000 reps.  The size of the test is
> consistently too small (see below) and the histogram shows increasing
> bars even within the parts of the histogram with even bar spacing.  See
> https://www-gene.cimr.cam.ac.uk/staff/wallace/hist.png
> 
> y<-sapply(1:100000, function(i,n=100)
>   binom.test(sum(rnorm(n)>0),n,p=0.5,alternative="two")$p.value)
> mean(y<0.01)
> # [1] 0.00584
> mean(y<0.05)
> # [1] 0.03431
> mean(y<0.1)
> # [1] 0.08646
> 
> Can that really be due to the discreteness of the distribution?
> 
> C.
> 
> On 26/01/12 16:08, Greg Snow wrote:
> > I believe that what you are seeing is due to the discrete nature of
> the binomial test.  When I run your code below I see the bar between
> 0.9 and 1.0 is about twice as tall as the bar between 0.0 and 0.1, but
> the bar between 0.8 and 0.9 is not there (height 0), if you average the
> top 2 bars (0.8-0.9 and 0.9-1.0) then the average height is similar to
> that of the lowest bar.  The bar between 0.5 and 0.6 is also 0, if you
> average that one with the next 2 (0.6-0.7 and 0.7-0.8) then they are
> also similar to the bars near 0.
> >
> >
> >