5 Binomial Test

5.1 Binomial Test

The binomial test is used to look for evidence that the proportion of a Binomial distributed random variable may differ from a hypothesized (or previously observed) value.

5.1.1 Test Statistic

The test statistic for a binomial test is the observed frequency of experimental subjects that exhibit the trait of interest.

5.1.2 Definitions

Let \(X\) be a random variable following a binomial distribution with parameters \(n\) and \(\pi\). Let \(x\) be the observed frequency of experimental subjects exhibiting the trait of interest.

5.1.3 Hypotheses

The hypotheses for the Binomial test may take the following forms:

For a two-sided test:

\[\begin{aligned} H_0: \pi = \pi_0 \\ H_a: \pi \neq \pi_0 \end{aligned}\]

For a one-sided test:

\[\begin{aligned} H_0: \pi < \pi_0 \\ H_a: \pi \geq \pi_0 \end{aligned}\]

or

\[\begin{aligned} H_0: \pi > \pi_0 \\ H_a: \pi \leq \pi_0 \end{aligned}\]

5.1.4 Decision Rule

The decision to reject the null hypothesis is made when the observed value of \(x\) lies in the critical region that suggests the probability of that observation is low. We define the critical region as the upper bound we are willing to accept for \(\alpha\), the Type I Error.

In a two-sided test, the upper bound is shared equally in both tails. Due to the discrete nature of the distribution, the total probability in the tails may not equal \(\alpha\). The figures below depict examples of rejection regions for selected values of the Binomial distribution parameters. The decision rule is:

Reject \(H_0\) if \(x < Binomial(\alpha/2, n, \pi_0)\) or \(x > Binomial(1 - \alpha/2, n, \pi_0)\)

The examples displayed use $n = 20$. For the top, middle, and bottom examples, $\pi$ is set at 0.3, 0.5, and 0.75, respectively. Notice that in some cases, the rejection regions for $\alpha = 0.10$ and $\alpha = 0.05$ are identical.

Figure 5.1: The examples displayed use \(n = 20\). For the top, middle, and bottom examples, \(\pi\) is set at 0.3, 0.5, and 0.75, respectively. Notice that in some cases, the rejection regions for \(\alpha = 0.10\) and \(\alpha = 0.05\) are identical.

In the one-sided test, \(\alpha\) is placed in only one tail. The figures below depict examples of rejection regions for selected values of the Binomial distribution parameters. In each case, \(\alpha\) is the area in the tail of the figure. It follows, then, that the decision rule for a lower tailed test is:

Reject \(H_0\) when \(x \leq Binomial(\alpha, n, \pi_0)\)

For an upper tailed test, the decision rule is:

Reject \(H_0\) when \(x \geq Binomial(1 - \alpha, n, \pi_0)\)

The examples displayed use $n = 20$. For the top, middle, and bottom examples, $\pi$ is set at 0.3, 0.5, and 0.75, respectively.

Figure 5.2: The examples displayed use \(n = 20\). For the top, middle, and bottom examples, \(\pi\) is set at 0.3, 0.5, and 0.75, respectively.

5.1.5 Power

The derivations below make use of the following symbols:

  • \(x\): The observed frequency of experimental units exhibiting the trait of interest.
  • \(n\): The total number of experimental units.
  • \(\pi_0\): The proportion of the population that exhibits the trait of interest under the null hypothesis.
  • \(\pi_a\): The proportion of the population that exhibits the trait of interest under the alternative hypothesis.
  • \(\alpha\): The significance level.
  • \(\gamma(\pi)\): The power of the test for the parameter \(\pi\).
  • \(Binomial(\alpha, n, \pi)\): A quantile of the Binomial distribution with a probability \(\alpha\), and parameters \(n\) and \(\pi\).
  • \(C\): The critical region.

Two Sided Test

\[\begin{aligned} \gamma(\pi_a) &= P_{\pi_a}(x \in C) \\ &= P_{\pi_a}(Binomial(\alpha/2, n, \pi_0) \leq Binomial(\alpha / 2, n, \pi_a)) + \\ & \ \ \ \ \ \ \ P_{\pi_a}(Binomial(1 - \alpha / 2, n, \pi_0) \geq Binomial(1 - \alpha / 2, n, \pi_a)) \end{aligned}\]

Left Sided Test

\[\begin{aligned} \gamma(\pi_a) &= P_{\pi_a}(x \in C) \\ &= P_{\pi_a}(Binomial(\alpha, n, \pi_0) \leq Binomial(\alpha, n, \pi_a)) \end{aligned}\]

Right Sided Test

\[\begin{aligned} \gamma(\pi_a) &= P_{\pi_a}(x \in C) \\ &= P_{\pi_a}(Binomial(1 - \alpha, n, \pi_0) \geq Binomial(1 - \alpha, n, \pi_a)) \end{aligned}\]

Since the Binomial distribution is discrete, the power curve has the interesting characteristic of not being monotonic. It is sometimes described as having a “sawtooth” appearance. This behavior means that a larger sample size is not always preferred. For example, in the following figure, a sample size of 10 has better power than a sample size of 12.

Power for a Binomial test with $\pi_0 = .15$ and $\pi_a = 0.25$

Figure 5.3: Power for a Binomial test with \(\pi_0 = .15\) and \(\pi_a = 0.25\)