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Tests of Total Variablility using a Parallel Design without Replicates

Using the notation of Chow et al., \(\sigma^2_{TR} = \sigma^2_{BR} + \sigma^2_{WR} \) and \(\sigma^2_{TT} = \sigma^2_{BT} + \sigma^2_{WT}\) are the total variance for treatments \(R\) and \(T\) respectively. This calculator performs power and sample size calculatios for the two-sided hypothesis: \[ H_0: \frac{\sigma^2_{TT}}{\sigma^2_{TR}} = 1 \text{ versus } H_a: \frac{\sigma^2_{TT}}{\sigma^2_{TR}} \ne 1 \]

or the one-sided hypthesis:

\[ H_0: \frac{\sigma^2_{TT}}{\sigma^2_{TR}} \ge \delta \text{ versus } H_a: \frac{\sigma^2_{TT}}{\sigma^2_{TR}} \lt \delta \]

or the hypothesis of similarity:

\[ H_0: \frac{\sigma^2_{TT}}{\sigma^2_{TR}} \ge \delta \text{ or } \frac{\sigma^2_{TT}}{\sigma^2_{TR}} \le \frac{1}{\delta} \text{ versus } H_a: \frac{1}{\delta} \lt \frac{\sigma^2_{TT}}{\sigma^2_{TR}} \lt \delta \]

where \(\delta\) is the similarity limit using a parallel design with replicates. The model used to test this hypothesis is explained in more depth here.

Power Calculation Parameters

To input multiple values, seperate them by a comma.

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Power Calculation Explanation

Solve For
The unknown you are interested in solving for.
The sample size used to test the hypothesis.
The \(\alpha\) (Type I error rate) level of the hypothesis test.
The power (1 - Type II error rate) of the hypothesis test.
The total variability for the test treatment, \(T\)
The total variability for the reference treatment, \(R\)
Similarity Limit \(\delta\)
The similarity limit against which the ratio of variabilities is tested.

Calculation Results

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N Alpha Power \(\sigma_{TT}\) \(\sigma_{TR}\) Similarity Limit
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  • Chow, S., Shao, J., & Wang, H. (2003), Sample size calculations in clinical research, New York: Marcel Dekker.