Two independent proportions

Menu location: Analysis_Proportions_Two Independent.

This function examines the difference between two independent binomial proportions.

Another way of looking at two proportions is to put the counts/frequencies into a 2 by 2 contingency table and examine the relationship between the grouping into rows and the grouping into columns (see Fisher's exact test and 2 by 2 chi-square test).

	feature present	feature absent
outcome positive	a	b
outcome negative	c	d

r1 = a

r2 = b

n1 = a+c

n2 = b+d

p1 = r1/n1

p2 = r2/n2

proportion difference (delta) = p1-p2

For example, the proportion of students passing a test when taught using method A could be compared with the proportion passing when taught using method B.

StatsDirect provides an hypothesis test for the equality of the two proportions (i.e. delta = 0) and a confidence interval for the difference between the proportions. An exact two sided P value is calculated for the hypothesis test (null hypothesis that there is no difference between the two proportions) using a mid-P approach to Fisher's exact test. The conventional normal approximation is also given for the hypothesis test, you should only use this if the numbers are large and the exact (mid) P is not shown (Armitage and Berry, 1994).

Assumptions:

two mutually exclusive outcomes
two random samples
samples from two independent populations

Technical Validation

The iterative method of Miettinen and Nurminen is used to construct the confidence interval for the difference between the proportions (Mee, 1984; Anbar, 1983; Gart and Nam, 1990; Miettinen and Nurminen, 1985; Newcombe, 1998b). This "near exact" confidence interval will be in close but not in exact agreement with the exact two sided (mid) P value; i.e. just excluding zero just above P = 0.05.

Example

From Armitage and Berry (1994).

Two methods of treatment, A and B, for a particular disease were investigated. Out of 257 patients treated with method A 41 died and out of 244 patients treated with method B 64 died. We want to compare these fatality rates.

To analyse these data in StatsDirect you must select unpaired proportions from the proportions section of the analysis menu. Enter total observations in sample 1 as 257, number responding in sample 1 as 41, total observations in sample 2 as 244 and number responding in sample 2 as 64.

For this example:

Total 1 = 257, response 1 = 41

Proportion 1 = 0.159533

Total 2 = 244, response 2 = 64

Proportion 2 = 0.262295

Proportion difference = -0.102762

Approximate (Miettinen) 95% confidence interval = -0.17432 to -0.031588

Exact two sided (mid) P = .0044

Standard error of proportion difference = 0.03638

Standard normal deviate (z) = -2.824689

Approximate two sided P = .0047

Approximate one sided P = .0024

Here we can conclude that the difference between these two proportions is statistically significantly different from zero. With 95% confidence we can state that the true population fatality rate with treatment B is between 0.03 and 0.17 greater than with treatment A.

P values

confidence intervals