On 2012-08-20 12:24, Tal Galili wrote:
Hello all,
I am trying to understand the different results I am getting from the
following 3 commands:
chisq.test(c(62,50), p = c(0.512,1-0.512), correct = F) # p-value = 0.3788
binom.test(x=62,n=112, p= 0.512) # p-value = 0.3961
2*(1-pbinom(62,112, .512)) # p-value = 0.329
Well, the binom.test was supposed to be "exact" and give the same results
as the pbinom, while the chisq.test relies on the normal asymptotics. So I
would imagine the binom.test should be equal to one of the other two lines,
but it is not.
The same happens for larger numbers as well:
chisq.test(c(1395,1278), p = c(0.512,1-0.512), correct = F)
# chisq.test(c(1395,1278), p = c(0.512,1-0.512), simulate.p.value= T)
binom.test(x=1395,n=2673, p= 0.512)
2*(1-pbinom(1395,2673, .512))
Let's first dispense with the chisq.test value; that uses the Normal
approximation and the calculation is trivial; the other two don't.
Let's do a one-sided test:
binom.test(x=62, n=112, p=.512, alternative="greater")$p.value
# 0.2161936
Compare with:
pbinom(61, 112, .512, lower.tail=FALSE)
# 0.2161936
Why '61' instead of '62'? Because we want to include 62 in the
upper tail probability.
Now it's just a matter of how to calculate the lower tail probability
in binom.test. Hint: it's not just set equal to the upper tail
probability (hence no '2*xxx').
The expected value (under H0) is 112*0.51 = 57.12.
Thus 62 is 4.88 ~ 5 units higher than the EV.
Now calculate the probability of an outcome that is 5 units *less*
(or lower) than the EV:
pbinom(52, 112, .512)
# 0.1799121
Add the two probabilities to get:
0.2161936 + 0.1799121
# 0.3961057
and that's what binom.test() reports.
For details, have a look at the binom.test code.
Peter Ehlers
I'd be happy to know what I might be missing.
Thanks,
Tal
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