# Single Proportion

This function compares an observed single binomial proportion with an expected proportion.

expected proportion (binomial parameter) = pi

successes = r

observations/trials = n

observed proportion p = r / n

The expected proportion (pi) is the probability of success on each trial, for example pi = 0.5 for coming up heads on the toss of a coin. The sign test is basically a single proportion test based on pi = 0.5. Some authors refer to this method as a "binomial test".

StatsDirect provides an exact confidence interval and an approximate mid-P confidence interval for the single proportion. You are also given exact P and exact mid-P hypothesis tests for the proportion in comparison with an expected proportion, i.e. null hypothesis that p = pi (Armitage and Berry, 1994; Gardner and Altman, 1989).

Assumptions:

• two mutually exclusive outcomes
• random sample

Consider using mid-P values and intervals when you have several similar studies to consider within an overall investigation (Armitage and Berry, 1994; Barnard, 1989).

Technical Validation

The Clopper-Pearson method is used for the exact confidence interval and the Newcombe-Wilson method is used for the mid-P confidence interval (Newcombe, 1998c). Mid-P probabilities are found by subtracting the exact probability for the observed count from the cumulative total; this subtraction is done on each side for the two sided result.

If n is greater than one million then the normal approximation to the binomial distribution is used to calculate the P values, otherwise exact cumulative binomial probabilities are given.

Example

In a trial of two analgesics, X and Y, 100 patients tried each drug for a week. The trial order was randomized. 65 out of 100 preferred drug Y.

To analyse these data in StatsDirect you must select single proportion from the proportions section of the analysis menu. To select a 95% confidence interval just press enter when you are presented with the confidence interval menu. Enter n as 100 and r as 65. Enter the binomial test proportion as 0.5, this is because you would expect 50% of an infinite number of patients to prefer drug Y if there was no difference between X and Y.

For this example:

Total = 100, response = 65

Proportion = 0.65

Exact (Clopper-Pearson) 95% confidence interval = 0.548151 to 0.742706

Using null hypothesis that the population proportion equals .5

Binomial one sided P = .0018

Binomial two sided P = .0035

Approximate (Wilson) 95% mid P confidence interval = 0.552544 to 0.736358

Binomial one sided mid P = .0013

Binomial two sided mid P = .0027

Here we can conclude that the proportion was statistically significantly different from 0.5. With 95% confidence we can state that the true population value for the proportion lies somewhere between 0.55 and 0.74.