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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

No calculation has been generated yet.

N Alpha Power $$p_2$$ $$p_3$$ $$p_4$$
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