I am not sure, but I think R is doing not what it is supposed to do.
Using binom.test() my understanding of the parameter alternative="greater" is the following hypothesis.
$$H_0: p le p_0 quad text{vs.} quad H_1: p > p_0$$
(I mean it says greater and not greater or equal).
P-Value then should be calculated as follows:
$$p(c) = P_{H_0}(Z>c) = 1 – P_{H_0}(Zle c)$$
Here is the equivalent R-Code
binom.test(8, 10, alternative="greater", p=0.5)$p.value #[1] 0.0546875 1-pbinom(8,10,0.5) # 1-P(c<=Z) #[1] 0.01074219 1-pbinom(8-1,10,0.5) # 1-P(c<Z) #[1] 0.0546875 So where is my mistake? Or is R just a little imprecise? And if I am right and R is realy testing the "wrong" (="not exactly what one would expect") hypothesis: What is with the other tests? Does "greater" always mean >= ?
Best Answer
The p-value is defined as the probability of a result at least as extreme as the observed results ($c$, in your notation) and should therefore be $$P_{H_0}(Zgeq c)=1−P_{H_0}(Z<c)=1−P_{H_0}(Zleq c−1).$$
Your error lies in defining the p-value as the probability of a more extreme result than what was observed.
Note that for continuous random variables, $P_{H_0}(Zgeq c)=P_{H_0}(Z> c)$, so that it doesn't matter whether equality is included or not. It is the discreteness of the binomial distribution that causes this to matter.
An effect of this is that a test at significance level $alpha=0.05$ rarely has type I error rate (size) equal to $0.05$. The actual size of the one-sided binomial test for different $pin(0,1)$ and $n$ is shown in the figure below. As you can see, the size oscillates quite a lot. It is however bounded by $alpha$ for all $p$. 