# Sample Size Calculator: One Sample Proportion

Hypothesis: One-Sided Equivalence

Data Input: (Help) (Example)

 Input Results α β θ N θ0 δ

Note:

 Variables Descriptions α One-sided significance level 1-β Power of the test θ Expected success proportion of sample θ0 Known success proportion δ Equivalence limit N Sample size

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Application: To establish equivalence, you can consider the following hypotheses:

Procedure:

1. Enter

a)      value of α, the probability of type I error

b)      value of β, the probability of type II error

c)      value of θ, a true mean response rate of a test drug

d)     value of θ0, a reference response rate

e)      value of δ, the equivalence limit.

1. Click the button “Calculate” to obtain result sample size N.

Formula:                                                    (*)

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.

θ-θ0:          The difference between the true mean response rates of a test drug (θ) and a reference value (θ0).

δ:               Clinically meaningful difference. The largest change from the reference value (baseline) that is considered to be trivial.

Examples

Example 1: We consider that one brand name drug for osteoporosis on the market has a responder rate of 60%. It is believed that a δ=20% difference in responder rate is of no clinical significance. Hence, the investigator wants to show the study drug is equivalent to the market drug in term of responder rate. If the true responder rate is θ0=65%, by (*), at α=0.05, assuming that the true response rate is θ=60%, then the required sample size for having an 80% power (i.e., β=0.2) is N=92.

Reference:

1. Casagrande, Pike and Smith (1978), Biometrics 34: 483-486.
2. Chow, Shao and Wang, Sample Size Calculations In Clinical Research, Taylor & Francis, NY. (2003) Pages 85-86.
3. Flesis J.L., Statistical Methods for Rates and Proportions (2nd edition). Wiley: New York, 1981.

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