Binomial Test: An Introduction
The binomial test is a statistical test used to determine whether the observed proportion of successes is significantly different from a hypothesized proportion of successes. The binomial test is a non-parametric approach that can be used when the outcome of interest is binary and the sample size is small. The test can also be used to compare the observed proportion of successes to a hypothetical value, such as 50%. The binomial test is a one-tailed test of significance and is commonly used in the field of medical research (Hu et al., 2020).
The binomial test is based on the binomial distribution, which is a probability distribution that describes the outcome of a single event with two possible outcomes. The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success (p). The binomial test is used to determine whether the observed proportion of successes is significantly different from the hypothesized proportion of successes (p). The null hypothesis for the binomial test is that the observed proportion of successes is equal to the hypothesized proportion of successes (p).
The binomial test is typically conducted using the following steps:
1. Specify the hypothesized proportion of successes (p)
2. Calculate the observed proportion of successes (q)
3. Calculate the test statistic (z)
4. Calculate the p-value
5. Interpret the results
The test statistic for the binomial test is calculated as follows:
z = (q – p) / sqrt(p*(1-p)/n)
where q is the observed proportion of successes, p is the hypothesized proportion of successes, and n is the number of trials.
The p-value for the binomial test is calculated using the cumulative binomial distribution function. The result is interpreted as follows: if the p-value is less than a predetermined significance level (usually 0.05), then the null hypothesis is rejected and the observed proportion of successes is significantly different from the hypothesized proportion of successes.
In conclusion, the binomial test is a non-parametric approach used to determine whether the observed proportion of successes is significantly different from a hypothesized proportion of successes. The test is based on the binomial distribution and is typically conducted using the following steps: specify the hypothesized proportion of successes, calculate the observed proportion of successes, calculate the test statistic, calculate the p-value, and interpret the results.
References
Hu, C., Ma, K., Zhang, L., Zeng, Y., & Li, X. (2020). The binomial test in medical research: an introduction. BMC Medical Research Methodology, 20(1), 42. https://doi.org/10.1186/s12874-020-00952-3