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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

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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.