Sample Size Calculator: Compare Two Proportions – Casagrande, Pike & Smith

Hypothesis:

One sided test:

Two sided test:

Data Input: (Help) (Example)

 

Input

 

Results

α

one sided test
two sided test

 

 

 

β

m

P1

 

 

 

P2

 

N

r

 

 

 

 

Note:

Variables

Descriptions

α

Significance level

1-β

Power of the test

P1

Success proportion in arm 1

P2

Success proportion in arm 2

r

Ratio of arm 2 to arm 1

m

Sample size for arm 1

N

Total sample size for arm 1 and 2

Help Aids Top

 

Application: To calculate the sample sizes needed to detect a difference between two binomial probabilities with specified significance level and power, following hypotheses are usually used:

One sided test:

Two sided test:

It is shown that over fairly wide ranges of parameter values and ratios of sample sizes, the percentage error which results from using the approximation is no greater than 1%.

 

Procedure:

  1. Enter

a)      value of α, the probability of type I error (choose either one-sided test or two-sided test)

b)      value of β, the probability of type II error, or (1-power) of the test

c)   value of P1, proportion of characteristic present in arm 1

c)      value of P2, proportion of characteristic present in arm 2

d)      value of r, ratio of arm 2 to arm 1

  1. Click the button “Calculate” to obtain result sample size for arm 1 m and total sample size N.

 

 

Formula:

             Define be the upper 100(1-p) percentile of the standard normal distribution,

                        m be the required sample size from the first population,

                        rm be the required sample size from the second population,

,  and

                                                                            (*)

            where

           

            Note: (*) is corrected with continuity.

                       

 

Notations:

 

α:               The probability of type I error (significance level) is the probability of rejecting the true null hypothesis.

 

β:               The probability of type II error (1 – power of the test) is the probability of failing to reject the false null hypothesis.

 

 

Examples Top

 

Example 1:

With significance level α=0.05, equal sample size from two proportions (r=1), the probabilityandare considered sufficiently different to warrant rejecting the hypothesis of no difference. Then the required sample size for two arms to achieve an 80% power (β=0.2) can be determined by.

 

 

Reference:     

  1. Casagrande, Pike and Smith (1978) Biometrics 34: 483-486
  2. Fleiss, Tytun and Ury (1980) Biometrics 36: 343-346

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