Sample Size Calculator: Confidence Interval – Bristol

Hypothesis:

Data Input: (Help) (Example)

 

Input

 

Results

α

 

 

 

P1

n

P2

 

N

L0

 

 

 

 

Note:

Variables

Descriptions

α

Significance level

P1

Observed proportion of success in arm 1

P2

Observed proportion of success in arm 2

L0

A specific positive value, bound of expected length of 100(1- α)% confidence interval

n

Sample size per arm

N

Total sample size

 

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Application:  The determination of the sample size to control the power and to control the length of the confidence interval, following hypotheses are usually used:

where anddenote the success probabilities of interest.

 

 

Procedure:

  1. Enter

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

b)   value of P1, the observed proportion of success in arm 1

c)   value of P2, the observed proportion of success in arm 2

d)   value of L0, the bound of expected length of 100(1- α)% confidence interval.

  1. Click the button “Calculate” to obtain

     a)    value of n, the sample size per arm

           b)    value of N, the total sample size

               

 

Formula:

Let nL donotes the sample size required to have, a specified positive value.

where is the expected length, , of (1- α) confidence interval of .

where ,is the upper 100(1-p) percentile of the standard normal distribution.

 

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.

 

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Example 1:

 

A clinical trial was planned to compare two active treatments for a certain type of cancer, with respect to their response rates P1 and P2. First consider the hypothesis testing formulation, where it is planned to test againstat 5% significance level (α=0.05). The sample size is chosen such that the power is 0.80 when the smaller response rate is 0.20 (P1=0.20) and the larger response rate is 0.30 (P2=0.30). For these values of P1 and P2, the approximation to the expected length of the 95% confidence interval is . The sample size required to guarantee that the approximation to the expected length of the 95% confidence interval is at most L0=0.2 is 162 patients per treatment (n=162). Total sample size required is 324 (N=324).

 

 

Reference: Bristol, David R. "Sample Sizes For Constructing Confidence Intervals and Testing Hypothesis." Statistics in Medicine, 8 (1988):803-811. Print.

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