#### Adjustments to the Variance Estimation

The factor in the computation of the matrix reduces the small sample bias associated with using the estimated function to calculate deviations (Morel, 1989; Hidiroglou, Fuller, and Hickman, 1980). For simple random sampling, this factor contributes to the degrees-of-freedom correction applied to the residual mean square for ordinary least squares in which p parameters are estimated. By default, the procedure uses this adjustment in Taylor series variance estimation. It is equivalent to specifying the VADJUST=DF option in the MODEL statement. If you do not want to use this multiplier in the variance estimation, you can specify the VADJUST=NONE option in the MODEL statement to suppress this factor.

In addition, you can specify the VADJUST=MOREL option to request an adjustment to the variance estimator for the model parameters , introduced by Morel (1989):

where for given nonnegative constants and ,

The adjustment does the following:

• reduces the small sample bias reflected in inflated Type I error rates

• guarantees a positive-definite estimated covariance matrix provided that exists

• is close to zero when the sample size becomes large

In this adjustment, is an estimate of the design effect, which has been bounded below by the positive constant . You can use DEFFBOUND= in the VADJUST=MOREL option in the MODEL statement to specify this lower bound; by default, the procedure uses . The factor converges to zero when the sample size becomes large, and has an upper bound . You can use ADJBOUND= in the VADJUST=MOREL option in the MODEL statement to specify this upper bound; by default, the procedure uses .