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# Tests of Total Variablility using a 2 by 2 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 2 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.
$$\sigma_{BT}$$
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

No calculation has been generated yet.

N Alpha Power $$\sigma_{BT}$$ $$\sigma_{WT}$$ $$\sigma_{BR}$$ $$\sigma_{WR}$$ $$\rho$$ Similarity Limit
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