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Publication date: 11/10/2021

Image shown hereThe Kackar-Harville Correction

The variance matrix of the fixed effects is always modified to include a Kackar-Harville correction. The variance matrix of the BLUPs, and the covariances between the BLUPs and the fixed effects, are not Kackar-Harville corrected. The rationale for this approach is that corrections for BLUPs can be computationally and memory intensive when the random effects have many levels. In SAS, the Kackar-Harville correction is done for both fixed effects and BLUPs only when the DDFM=KENWARDROGER is set.

For covariance structures that have nonzero second derivatives with respect to the covariance parameters, the Kenward-Roger covariance matrix adjustment includes a second-order term. This term can result in standard error shrinkage. Also, the resulting adjusted covariance matrix can then be indefinite and is not invariant under reparameterization. The first-order Kenward-Roger covariance matrix adjustment eliminates the second derivatives from the calculation. All spatial structures and the AR(1) structure are covariance structures that generally lead to nonzero second derivatives.

Because JMP implements the Kenward-Roger first-order adjustment

Standard errors for linear combinations involving only fixed effects parameters match PROC MIXED DDFM=KENWARDROGER(FIRSTORDER). This presumes that one has taken care to transform between the different parameterizations used by PROC MIXED and JMP.

Standard errors for linear combinations involving only BLUP parameters match PROC MIXED DDFM=SATTERTHWAITE.

Standard errors for linear combinations involving both fixed effects and BLUPS do not match PROC MIXED for any DDFM option if the data are unbalanced. However, these standard errors are between those obtained using the DDFM=SATTERTHWAITE and DDFM=KENWARDROGER options. If the data are balanced, JMP matches SAS regardless of the DDFM option, because the Kackar-Harville correction is null.

Image shown hereDegrees of Freedom

The degrees of freedom for tests involving only linear combinations of fixed effect parameters are calculated using the first-order Kenward-Roger correction. Therefore, the JMP results for these tests match PROC MIXED using the DDFM=KENWARDROGER(FIRSTORDER) option. If there are BLUPs in the linear combination, JMP uses a Satterthwaite approximation to get the degrees of freedom. The results then follow a pattern similar to what is described for standard errors in the preceding paragraph.

For more information about the Kackar-Harville correction and the Kenward-Roger DF approach, see Kenward and Roger (1997). The Satterthwaite method is described in detail in the MIXED Procedure chapter in SAS Institute Inc. (2020d, ch. 83).

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