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Tests of Total Variablility using a 2 by 2M Crossover Design

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:

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

where \(\delta\) is the similarity limit using a 2 by 2M crossover design. 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.
N
The sample size used to test the hypothesis.
Alpha
The \(\alpha\) (Type I error rate) level of the hypothesis test.
Power
The power (1 - Type II error rate) of the hypothesis test.
Replications
The number of replications in the study (\(m\))
\(\sigma_{BT}\)
The intra-subject variability for the test treatment, \(T\)
\(\sigma_{WT}\)
The inter-subject variability for the test treatment, \(T\)
\(\sigma_{BR}\)
The intra-subject variability for the reference treatment, \(R\)
\(\sigma_{WR}\)
The inter-subject variability for the reference treatment, \(R\)
Similarity Limit \(\delta\)
The similarity limit against which the ratio of variabilities is tested.
\(\rho\)
An element of the covariance between \(S_{ijT}\) and \(S_{ijR}\), the random effects of the crossover model (details). If this is unknown, leave it 1.

Calculation Results

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N Replications Alpha Power \(\sigma_{BT}\) \(\sigma_{WT}\) \(\sigma_{BR}\) \(\sigma_{WR}\) \(\rho\) Similarity Limit
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References

  • Chow, S., Shao, J., & Wang, H. (2003), Sample size calculations in clinical research, New York: Marcel Dekker.