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One Sample Non Parametric Test of Location

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

\[ H_0: \theta=0 \text{ versus } H_a: \theta \neq 0. \]

This hypothesis is commonly tested with the Wilcoxan signed rank test.

To set up the power calculation for this hypothesis, we need three values: \(p_2\), \(p_3\), and \(p_4\). These values are generally estimated from pilot studies.

Let \(z_i\) be defined by the model:

\[ z_i = \theta + \epsilon_i \]

where \(\theta\) is location parameter of interest and \(\epsilon_i\) is the error term whose mean is 0. We can estimate the parameters needed by:

\[ \hat{p}_2 = \frac{1}{n(n - 1)} \sum_{i \ne j} I\{|z_i| \ge |z_j|, z_i > 0 \} \] \[ \hat{p}_3 = \frac{1}{n(n - 1)(n - 2)} \sum_{i \ne j_1 \ne j_2} I \{ |z_i| \ge |z_{j_1}|, |z_i| \ge |z_{j_2}|, z_i > 0 \} \] \[ \hat{p}_4 = \frac{1}{n(n - 1)(n - 2)} \sum_{i \ne j_1 \ne j_2} I\{|z_{j_1}| \ge |z_i| \ge |z_{j_2}|, z_i > 0 \} \]

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.
\(p_2\)
The value of \(p_2\) whose formula is given above.
\(p_3\)
The value of \(p_3\) whose formula is given above.
\(p_4\)
The value of \(p_4\) whose formula is given above.

Calculation Results

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N Alpha Power \(p_2\) \(p_3\) \(p_4\)
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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.