Adjustments for Covariates in Univariate Models :: SAS/STAT(R) 13.1 User's Guide

Adjustments for Covariates in Univariate Models

If you specify covariates in a univariate model (whether continuous or categorical), then two adjustments are made in order to compute approximate power in the presence of the covariates. Let $n_\nu$ denote the number of covariates (counting dummy variables for categorical covariates individually) as specified in the NCOVARIATES= option in the POWER statement. In other words, $n_\nu$ is the total degrees of freedom used by the covariates. The adjustments are as follows:

The error degrees of freedom decrease by $n_\nu$ .
The error standard deviation $\sigma$ shrinks by a factor of $(1 - \rho ^2)^\frac {1}{2}$ (if the CORRXY= option is used to specify the correlation $\rho$ between covariates and response) or $(1 - r)^\frac {1}{2}$ (if the PROPVARREDUCTION= option is used to specify the proportional reduction in total $R^2$ incurred by the covariates). Let $\sigma ^\star$ represent the updated value of $\sigma$ .

As a result of these changes, the power is computed as

$\mr {power} = P\left(F(r_ L, \mr {DF_ E}-n_\nu , \lambda ^\star ) \ge F_{1-\alpha }(r_ L, N-r_ x-n_\nu )\right)$

where $\lambda ^\star$ is calculated using $\sigma ^\star$ rather than $\sigma$ :

$\lambda ^\star = N \left(\mb {L} \bbeta - \btheta _0 \right)’\left(\mb {L} \left(\ddot{\mb {X}}’ \mb {W} \ddot{\mb {X}} \right)^{-1} \mb {L}^\prime \right)^{-1} \left(\mb {L} \bbeta - \btheta _0 \right) (\sigma ^\star )^{-2}$

The GLMPOWER Procedure

Adjustments for Covariates in Univariate Models