# Paired Proportions

This function examines the difference between a pair of binomial proportions.

Two proportions are paired (as opposed to independent) if they share a common feature that affects the outcome. For example, when comparing two laboratory methods (culture media) to detect bacteria in samples of blood, if blood from the same sample is put into both methods, this is the "pairing". Pairs of results from multiple samples can then be compared as a pair of proportions:

 Group/Category A: outcome present outcome absent Group/Category B: outcome present: a b outcome absent: c d

total n = a+b+c+d

responding in both categories = r = a

responding in first category only = s = b

responding in second category only = t = c

proportion 1 = (r + s) / n

proportion 2 = (r + t) / n

proportion difference (delta) = (s - t) / n

Another way of looking at these data is to examine how the grouping into category A depends upon the grouping into category B (see exact test for matched pairs of counts).

StatsDirect gives you exact and exact mid-P hypothesis tests for the equality of the two proportions (i.e. delta = 0) and gives you a confidence interval for the difference between them.

Assumptions:

• two mutually exclusive outcomes
• random sample from one population

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

Technical Validation

Exact methods are used throughout (Armitage and Berry, 1994; Liddell, 1983). The two sided exact P value equates with the exact test for a paired fourfold table (Liddell, 1983). With large numbers an appropriate normal approximation is used in the hypothesis test (note that most asymptotic methods tend to mid-P).

The confidence interval is constructed using Newcombe's refinement of Wilson's score based method, this is close to a mid-P interval (Newcombe, 1998a).

Example

The data below represent a comparison of two media for culturing Mycobacterium tuberculosis. Fifty suspect sputum specimens were plated up on both media and the following results were obtained:

 Medium B: Growth No Growth Medium A: Growth: 20 12 No Growth: 2 16 N = 50

To analyse these data in StatsDirect you must select paired proportions from the proportions section of the analysis menu. Select a 95% confidence interval by pressing enter when you are presented with the confidence interval menu. Enter TOTAL (n) as 50, BOTH (k) as 20, FIRST (r) as 12 and SECOND (s) as 2.

For this example:

Total = 50, both = 20, first only = 12, second only = 2

Proportion 1 = 0.64

Proportion 2 = 0.44

Proportion difference = 0.2

Exact two sided P = .0129

Exact one sided P = .0065

Exact two sided mid P = .0074

Exact one sided mid P = .0037

Score based (Newcombe) 95% confidence interval for the proportion difference:

0.056156 to 0.329207

Here we can conclude that the proportion difference is statistically significantly different from zero. With 95% confidence we can say that the true population value for the proportion difference lies somewhere between 0.06 and 0.33. This leaves us with little doubt that medium A is more effective than medium B for the culture of tubercle bacilli.

Compare these results with the exact test for matched pairs.