Publication date: 11/10/2021

## Power for a Categorical Effect

This section describes how power for the test for a whole categorical effect is computed. Use the following notation:

X

Model matrix. See The Alias Matrix. Vector of parameters. Least squares estimate of β. Vector of Anticipated Coefficient values. Matrix that defines the test for the categorical effect. The matrix L identifies the values of the parameters in β corresponding to the categorical effect and sets them equal to 0. The null hypothesis for the test of the categorical effect is given by: r

Rank of L. Alternatively, r is the number of levels of the categorical effect minus one.

Note: You can view the design matrix by running Fit Model. Then select Save Columns > Save Coding Table from the red triangle menu for the main report.

The covariance matrix of is given by , where σ2 is the error variance.

The error variance, σ2, is estimated by the MSE, and has n p 1 degrees of freedom, where n is the number of observations and p is the number of terms other than the intercept in the model. If n p 1 = 0, then JMP sets the degrees of freedom for the error to 1. This allows the power to be estimated for parameters in a saturated design.

The test of is given by: Under the null hypothesis, the test statistic F0 has an F distribution on r and n p 1 degrees of freedom.

If the true value of β is , then F0 has a noncentral F distribution with noncentrality parameter given by: To compute the power of the test, first solve for the α-level critical value Fc: Then calculate the power as follows: 