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# Intersubject Variance using a 2 by 2$$m$$ Crossover Design

Using the notation of Chow et al., $$\sigma^2_{BR}$$ and $$\sigma^2_{BT}$$ are the inter-subject variance for treatments $$R$$ and $$T$$ respectively. This calculator performs power and sample size calculatios for the two-sided hypothesis: $H_0: \frac{\sigma^2_{BT}}{\sigma^2_{BR}} = 1 \text{ versus } H_a: \frac{\sigma^2_{BT}}{\sigma^2_{BR}} \ne 1$

or the one-sided hypthesis:

$H_0: \frac{\sigma^2_{BT}}{\sigma^2_{BR}} \ge \delta \text{ versus } H_a: \frac{\sigma^2_{BT}}{\sigma^2_{BR}} \lt \delta$

or the hypothesis of similarity:

$H_0: \frac{\sigma^2_{BT}}{\sigma^2_{BR}} \ge \delta \text{ or } \frac{\sigma^2_{BT}}{\sigma^2_{BR}} \le \frac{1}{\delta} \text{ versus } H_a: \frac{1}{\delta} \lt \frac{\sigma^2_{BT}}{\sigma^2_{BR}} \lt \delta$

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.
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.
Hypothesis
There are three types of hypotheses that can be tested: two-sided, one-sided and equivalence. Tests of equivalence must include a margin if the unknown and null means are equal. Read More

## Calculation Results

No calculation has been generated yet.

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.