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| 1 | +--- |
| 2 | +layout: proof |
| 3 | +mathjax: true |
| 4 | + |
| 5 | +author: "Joram Soch" |
| 6 | +affiliation: "BCCN Berlin" |
| 7 | +e_mail: "joram.soch@bccn-berlin.de" |
| 8 | +date: 2023-12-16 20:01:14 |
| 9 | + |
| 10 | +title: "Binomial test" |
| 11 | +chapter: "Statistical Models" |
| 12 | +section: "Count data" |
| 13 | +topic: "Binomial observations" |
| 14 | +theorem: "Binomial test" |
| 15 | + |
| 16 | +sources: |
| 17 | + - authors: "Wikipedia" |
| 18 | + year: 2023 |
| 19 | + title: "Binomial test" |
| 20 | + in: "Wikipedia, the free encyclopedia" |
| 21 | + pages: "retrieved on 2023-12-16" |
| 22 | + url: "https://en.wikipedia.org/wiki/Binomial_test#Usage" |
| 23 | + - authors: "Wikipedia" |
| 24 | + year: 2023 |
| 25 | + title: "Binomialtest" |
| 26 | + in: "Wikipedia – Die freie Enzyklopädie" |
| 27 | + pages: "retrieved on 2023-12-16" |
| 28 | + url: "https://de.wikipedia.org/wiki/Binomialtest#Signifikanzniveau_und_kritische_Werte" |
| 29 | + |
| 30 | +proof_id: "P429" |
| 31 | +shortcut: "bin-test" |
| 32 | +username: "JoramSoch" |
| 33 | +--- |
| 34 | + |
| 35 | + |
| 36 | +**Theorem:** Let $y$ be the number of successes resulting from $n$ independent trials with unknown success probability $p$, such that $y$ follows a [binomial distribution](/D/bin): |
| 37 | + |
| 38 | +$$ \label{eq:Bin} |
| 39 | +y \sim \mathrm{Bin}(n,p) \; . |
| 40 | +$$ |
| 41 | + |
| 42 | +Then, the [null hypothesis](/D/h0) |
| 43 | + |
| 44 | +$$ \label{eq:bin-test-h0} |
| 45 | +H_0: \; p = p_0 |
| 46 | +$$ |
| 47 | + |
| 48 | +is [rejected](/D/test) at [significance level](/D/alpha) $\alpha$, if |
| 49 | + |
| 50 | +$$ \label{eq:bin-test-rej} |
| 51 | +y \leq c_1 \quad \text{or} \quad y \geq c_2 |
| 52 | +$$ |
| 53 | + |
| 54 | +where $c_1$ is the largest integer value, such that |
| 55 | + |
| 56 | +$$ \label{eq:bin-test-c1} |
| 57 | +\sum_{x=0}^{c_1} \mathrm{Bin}(x; n, p_0) \leq \frac{\alpha}{2} \; , |
| 58 | +$$ |
| 59 | + |
| 60 | +and $c_2$ is the smallest integer value, such that |
| 61 | + |
| 62 | +$$ \label{eq:bin-test-c2} |
| 63 | +\sum_{x=c_2}^{n} \mathrm{Bin}(x; n, p_0) \leq \frac{\alpha}{2} \; , |
| 64 | +$$ |
| 65 | + |
| 66 | +where $\mathrm{Bin}(x; n, p)$ is the [probability mass function of the binomial distribution](/P/bin-pmf): |
| 67 | + |
| 68 | +$$ \label{eq:bin-pmf} |
| 69 | +\mathrm{Bin}(x; n, p) = {n \choose x} \, p^x \, (1-p)^{n-x} \; . |
| 70 | +$$ |
| 71 | + |
| 72 | + |
| 73 | +**Proof:** The [alternative hypothesis](/D/h1) relative to $H_0$ for a [two-sided test](/D/test-tail) is |
| 74 | + |
| 75 | +$$ \label{eq:bin-test-h1} |
| 76 | +H_1: \; p \neq p_0 \; . |
| 77 | +$$ |
| 78 | + |
| 79 | +We can use $y$ as a [test statistic](/D/tstat). Its [sampling distribution](/D/dist-samp) is given by \eqref{eq:Bin}. The [cumulative distribution function](/D/cdf) (CDF) of the test statistic under the null hypothesis is thus equal to the [cumulative distribution function of a binomial distribution](/P/bin-cdf) with [success probability](/D/bin) $p_0$: |
| 80 | + |
| 81 | +$$ \label{eq:y-cdf} |
| 82 | +\mathrm{Pr}(y \leq z \vert H_0) = \sum_{x=0}^{z} \mathrm{Bin}(x; n, p_0) = \sum_{x=0}^{z} {n \choose x} \, p_0^x \, (1-p_0)^{n-x} \; . |
| 83 | +$$ |
| 84 | + |
| 85 | +The [critical value](/D/cval) is the value of $y$, such that the probability of observing this or more extreme values of the test statistic is equal to or smaller than $\alpha$. Since $H_0$ and $H_1$ define a two-tailed test, we need two critical values $y_1$ and $y_2$ that satisfy |
| 86 | + |
| 87 | +$$ \label{eq:y-cvals} |
| 88 | +\begin{split} |
| 89 | +\alpha &\geq \mathrm{Pr}(y \in \left\lbrace 0, \ldots, y_1 \right\rbrace \cup \left\lbrace y_2, \ldots, n \right\rbrace \vert H_0) \\ |
| 90 | +&= \mathrm{Pr}(y \leq y_1 \vert H_0) + \mathrm{Pr}(y \geq y_2 \vert H_0) \\ |
| 91 | +&= \mathrm{Pr}(y \leq y_1 \vert H_0) + (1-\mathrm{Pr}(y \leq (y_2-1) \vert H_0) \; . |
| 92 | +\end{split} |
| 93 | +$$ |
| 94 | + |
| 95 | +Given the test statistic's CDF in \eqref{eq:y-cdf}, this is fulfilled by the values $c_1$ and $c_2$ defined in \eqref{eq:bin-test-c1} and \eqref{eq:bin-test-c2}. Thus, the null hypothesis $H_0$ [can be rejected](/D/cval), if the [observed test statistic](/D/test) is inside the rejection region: |
| 96 | + |
| 97 | +$$ \label{eq:bin-test-rej-qed} |
| 98 | +y \in \left\lbrace 0, \ldots, c_1 \right\rbrace \cup \left\lbrace c_2, \ldots, n \right\rbrace \; . |
| 99 | +$$ |
| 100 | + |
| 101 | +This is equivalent to \eqref{eq:bin-test-rej} and thus completes the proof. |
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