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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: $$p(c) = P_{H_0}(Z>c) = 1 - P_{H_0}(Z\le c)$$

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 >= ?

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    $\begingroup$ 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 $P_{H_0}(Z\geq c)=1-P_{H_0}(Z< c)=1-P_{H_0}(Z\leq c-1)$. Your error lies in defining the p-value as the probability of a more extreme result than what was observed. $\endgroup$ – MånsT Nov 21 '12 at 16:39
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    $\begingroup$ @MansT, how about posting your comment as an answer ... ? $\endgroup$ – Ben Bolker Nov 21 '12 at 17:07
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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 $$P_{H_0}(Z\geq c)=1−P_{H_0}(Z<c)=1−P_{H_0}(Z\leq c−1).$$

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, $P_{H_0}(Z\geq c)=P_{H_0}(Z> c)$, 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 $p\in(0,1)$ 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 $p$. enter image description here

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