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# Carry-Over Effect with Binary Outcomes

Consider a standard two-sequence, two-period crossover design. Let $$\tau_i$$ be the $$i^{th}$$ treatment effect, $$\rho_j$$ be the $$j^{th}$$ period effect and $$\gamma_k$$ be the carry-over effect from the first period in the $$k^{th}$$ sequence.

Generally, it is assumed that:

$\tau_1 + \tau_2 = 0 \text{, } \rho_1 + \rho_2 = 0 \text{, and } \gamma_1 + \gamma_2 = 0$

Following the method of Becker and Balagtas, we can test the last assumption that $$\gamma = \gamma_1 + \gamma_2 = 0$$. $$\hat{\gamma}$$ is asymtotically normal with mean $$\gamma$$ and variance $$\sigma^2_1 n_1^{-1} + \sigma^2_2 n_2^{-1}$$ where

$\sigma^2_i = \text{var} \left( \frac{x_{i1k}}{p_{i1}(1 - p_{i1})} + \frac{x_{i2k}}{p_{i2}(1 - p_{i2})} \right)$

is the variance from the $$i^{th}$$ period.

## 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.
$$\gamma$$
The estimated carry-over effect.
$$\sigma_1$$
The standard deviation from period 1.
$$\sigma_2$$
The standard deviation from period 2.

## Calculation Results

No calculation has been generated yet.

N Alpha Power Gamma $$\sigma_1$$ $$\sigma_2$$
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## Power Graph

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## References

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