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The two sample non parametric test of location tests the hypothesis that

\[ H_0: \theta=0 \text{ versus } H_a: \theta \neq 0. \]To perform the sample size calculation for this test, we need to estimate three parameters: \(p_1\), \(p_2\), and \(p_3\). If we let \(x_i\) and \(y_j\) be two independant samples with \(i = 1, \dots, n_1\) and \(j = 1, \dots, n_2 \), we can estimate these parameters by:

\[ \hat{p}_1 = \frac{1}{n_1 n_2} \sum_{i=1}^{n_2}\sum_{j=1}^{n_1} I\{y_i \ge x_j \} \] \[ \hat{p}_2 = \frac{1}{n_1 n_2 (n_1 - 1)} \sum_{i=1}^{n_2}\sum_{j_1 \ne j_2} I\{y_i \ge x_{j_1} \text{ and } y_i \ge x_{j_2} \} \] \[ \hat{p}_3 = \frac{1}{n_1 n_2 (n_2 - 1)} \sum_{i_1 \ne i_2}\sum_{j=1}^{n_1} I\{y_{i_1} \ge x_j \text{ and } y_{i_2} \ge x_j \} \]To input multiple values, seperate them by a comma.

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- Solve For
- The unknown you are interested in solving for.
- N
- The sample size used to test the hypothesis.
- Ratio
- The ratio of the sample size in arm 1 to arm 2. The total sample size is minimized for a given power when the ratio is 1 (
*i.e.*the design is balanced). - Alpha
- The \(\alpha\) (Type I error rate) level of the hypothesis test.
- Power
- The power (1 - Type II error rate) of the hypothesis test.
- \(p_1\)
- The value of \(p_1\) whose formula is given above.
- \(p_2\)
- The value of \(p_2\) whose formula is given above.
- \(p_3\)
- The value of \(p_3\) whose formula is given above.

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N (Arm 1) | N (Arm 2) | Alpha | Power | \(p_1\) | \(p_2\) | \(p_3\) |
---|---|---|---|---|---|---|

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