Adjustments for Covariates :: SAS/STAT(R) 12.1 User's Guide

Adjustments for Covariates

If you specify covariates in the 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). 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 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 incurred by the covariates). Let $\sigma ^\star$ represent the updated value of $\sigma$ .

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

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

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

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

The GLMPOWER Procedure

Adjustments for Covariates