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).




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.



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



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