The Binomial Test

What is the Binomial test?

The Binomial test, sometimes referred to as the Binomial exact test, is a test used in sampling statistics to assess whether a proportion of a binary variable is equal to some hypothesized value. In this article, we explore the key features of this test and walk through an example test.

Sometimes you may also want to test a null hypothesis of the population proportion as being greater than the hypothesized value specifically (or lesser than), rather than different in any direction, in which case you would perform a one tailed significance test, but more commonly the two tailed approach is used.

Note that there is no test statistic generated in a Binomial test as is common in other statistical tests such as theMann-Whitney U Testor theunpaired Student’s t-test, due to the p-value being calculated directly.

When to use the Binomial test

二项测试时使用一个二进制变量of interest (a variable that can take only two possible values e.g. mortality (dead/alive)) is being investigated and you have a hypothesized or expected value with which to compare it to. The test can only be used when sample size is small compared to the population about which you are trying to make an inference.

Changes to the shape of a Binomial distribution at varying values of the proportion of successes, p, and number of trials, n.

The Binomial test is derived from the Binomial distribution, which can be thought of as the distribution that is followed by the number of ‘successes’ or ‘failures’ in a certain number,n, of repeated independent experiments or ‘trials’. In more statistical language, we can say that the distribution relies on the values ofnandp(the probability of any trial being a success), and that these are the parameters of the Binomial distribution. It is useful to note that as the sample size (the value ofn) increases, the distribution becomes more symmetrical and converges to a Normal distribution.

Binomial test assumptions

Assumptions for the Binomial test are as follows, and can be easily remembered using the ‘BINS’ acronym:

• B – the variable of interest should be abinaryoutcome meaning it can take only one of two values (e.g. a coin toss (heads/tails), presence of a disease (yes/no), morality (dead/alive)). This is sometimes also referred to as a dichotomous variable.
• I – observations should beindependent, meaning that one observation should not have any bearing on the probability of another.
• N – the experiment should have a fixed sample size denotedn.
• ——所有独立观测年代hould have thesameprobabilityof having the outcome. This is similar to the independence assumption and can be achieved through random sampling.

Binomial test example

Suppose a population health researcher carries out a small random sample survey to estimate the prevalence (the proportion of a population affected) of herpes simplex virus (HSV), a common viral infection that causes genital and oral herpes. Members of the sample are selected at random with a total of 20 people selected (n=20), are independent from one another and have the same probability of having the outcome, and with a binary outcome of interest (presence of HSV; yes/no).

We can thus conceptualize this as a series of 20 independent trials with the proportion of people with the infection, p, following the Binomial distribution. Suppose in the survey it was found that 6 (30%) of the 20 participants had HSV. The probability of a given survey participant having the disease is therefore p=0.3. Suppose also that a previous survey found the prevalence of HSV to be 20% (this could be from the same population or a comparable population) - the researchers use this as the hypothesized value on which to run the Binomial test for the current survey proportion.

The next step is to run the Binomial test and generate a p-value, which denotes the probability of getting the proportion of people with HSV as extreme or more extreme than what was observed if the truepwas equal to the hypothesized value. Statistical packages such as Stata, SPSS or R Studio can be relied upon to generate the Binomial test p-value, but for illustrative purposes the formula is detailed below. If we havenindependent trials with probability of having HSV beingpwe can calculate the probability of the value being the hypothesized number of HSV cases,r(in this caser=4 as 20% of 20 is 4), using the following formula:

By plugging the values into the Binomial formula we get 0.196, the probability of 6 or fewer HSV cases out of 20 (one tailed test). Since our hypothesis of interest is whether the observed and hypothesized values differ in any direction, we would like to generate a two tailed test, and so we multiply by 2 to get a final p-value of 0.392.

Using a significance level ofα=0.05 we fail to reject the null hypothesis because p > 0.05 and conclude that there is no evidence of a statistically significant difference between the prevalence of HSV in the current survey compared with the previous survey given the sample size.

Elliot McClenaghan is a research fellow in Epidemiology and Medical Statistics at the London School of Hygiene & Tropical Medicine