The GLIMMIX Procedure

Design-Adjusted MBN Estimator

Morel (1989) and Morel, Bokossa, and Neerchal (2003) suggested a bias correction of the classical sandwich estimator that rests on an additive correction of the residual crossproducts and a sample size correction. This estimator is available with the EMPIRICAL=MBN option in the PROC GLIMMIX statement. In the notation of the previous section, the residual-based MBN estimator can be written as

ModifyingAbove bold upper Omega With caret left-parenthesis sigma-summation Underscript i equals 1 Overscript m Endscripts ModifyingAbove bold upper D With caret prime Subscript i Baseline ModifyingAbove bold upper Sigma With caret Subscript i Superscript negative 1 Baseline left-parenthesis c bold e Subscript i Baseline bold e prime Subscript i plus bold upper B Subscript i Baseline right-parenthesis ModifyingAbove bold upper Sigma With caret Subscript i Superscript negative 1 Baseline ModifyingAbove bold upper D With caret Subscript i Baseline right-parenthesis ModifyingAbove bold upper Omega With caret

where

  • c equals left-parenthesis f minus 1 right-parenthesis slash left-parenthesis f minus k right-parenthesis times m slash left-parenthesis m minus 1 right-parenthesis or c = 1 when you specify the EMPIRICAL=MBN(NODF) option

  • f is the sum of the frequencies

  • k equals the rank of bold upper X

  • bold upper B Subscript i Baseline equals delta Subscript m Baseline phi ModifyingAbove bold upper Sigma With caret Subscript i

  • phi equals max left-brace r comma normal t normal r normal a normal c normal e left-parenthesis ModifyingAbove bold upper Omega With caret bold upper M right-parenthesis slash k Superscript asterisk Baseline right-brace

  • bold upper M equals sigma-summation Underscript i equals 1 Overscript m Endscripts ModifyingAbove bold upper D With caret prime Subscript i Baseline ModifyingAbove bold upper Sigma With caret Subscript i Superscript negative 1 Baseline bold e Subscript i Baseline bold e prime Subscript i Baseline ModifyingAbove bold upper Sigma With caret Subscript i Superscript negative 1 Baseline ModifyingAbove bold upper D With caret Subscript i

  • k Superscript asterisk Baseline equals k if m greater-than-or-equal-to k, otherwise k Superscript asterisk equals the number of nonzero singular values of ModifyingAbove bold upper Omega With caret bold upper M

  • delta Subscript m Baseline equals k slash left-parenthesis m minus k right-parenthesis if m greater-than left-parenthesis d plus 1 right-parenthesis k and delta Subscript m Baseline equals 1 slash d otherwise

  • d greater-than-or-equal-to 1 and 0 less-than-or-equal-to r less-than-or-equal-to 1 are parameters supplied with the mbn-options of the EMPIRICAL=MBN(mbn-options) option. The default values are d = 2 and r = 1. When the NODF option is in effect, the factor c is set to 1.

Rearranging terms, the MBN estimator can also be written as an additive adjustment to a sample-size corrected classical sandwich estimator

c times ModifyingAbove bold upper Omega With caret left-parenthesis sigma-summation Underscript i equals 1 Overscript m Endscripts ModifyingAbove bold upper D With caret prime Subscript i Baseline ModifyingAbove bold upper Sigma With caret Subscript i Superscript negative 1 Baseline bold e Subscript i Baseline bold e prime Subscript i Baseline ModifyingAbove bold upper Sigma With caret Subscript i Superscript negative 1 Baseline ModifyingAbove bold upper D With caret Subscript i Baseline right-parenthesis ModifyingAbove bold upper Omega With caret plus delta Subscript m Baseline phi ModifyingAbove bold upper Omega With caret

Because delta Subscript m is of order m Superscript negative 1, the additive adjustment to the classical estimator vanishes as the number of independent sampling units (subjects) increases. The parameter phi is a measure of the design effect (Morel, Bokossa, and Neerchal 2003). Besides good statistical properties in terms of Type I error rates in small-m situations, the MBN estimator also has the desirable property of recovering rank when the number of sampling units is small. If m less-than k, the "meat" piece of the classical sandwich estimator is essentially a sum of rank one matrices. A small number of subjects relative to the rank of bold upper X can result in a loss of rank and subsequent loss of numerator degrees of freedom in tests. The additive MBN adjustment counters the rank exhaustion. You can examine the rank of an adjusted covariance matrix with the COVB(DETAILS) option in the MODEL statement.

When the principle of the MBN estimator is applied to the likelihood-based empirical estimator, you obtain

bold upper H left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis Superscript negative 1 Baseline left-parenthesis sigma-summation Underscript i equals 1 Overscript m Endscripts c bold g Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis bold g Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis prime plus bold upper B Subscript i Baseline right-parenthesis bold upper H left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis Superscript negative 1

where bold upper B Subscript i Baseline equals minus delta Subscript m Baseline phi bold upper H Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis, and bold upper H Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis is the second derivative of the log likelihood for the ith sampling unit (subject) evaluated at the vector of parameter estimates, ModifyingAbove bold-italic alpha With caret. Also, bold g Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis is the first derivative of the log likelihood for the ith sampling unit. This estimator is computed if you request EMPIRICAL=MBN with METHOD=LAPLACE or METHOD=QUAD.

In terms of adjusting the classical likelihood-based estimator (White 1982), the likelihood MBN estimator can be written as

c times bold upper H left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis Superscript negative 1 Baseline left-parenthesis sigma-summation Underscript i equals 1 Overscript m Endscripts bold g Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis bold g Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis prime right-parenthesis bold upper H left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis Superscript negative 1 minus delta Subscript m Baseline phi bold upper H left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis Superscript negative 1

The parameter phi is determined as

  • phi equals max left-brace r comma normal t normal r normal a normal c normal e left-parenthesis minus bold upper H left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis Superscript negative 1 Baseline bold upper M right-parenthesis slash k Superscript asterisk Baseline right-brace

  • bold upper M equals sigma-summation Underscript i equals 1 Overscript m Endscripts bold g Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis bold g Subscript i Baseline left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis prime

  • k Superscript asterisk Baseline equals k if m greater-than-or-equal-to k, otherwise k Superscript asterisk equals the number of nonzero singular values of minus bold upper H left-parenthesis ModifyingAbove bold-italic alpha With caret right-parenthesis Superscript negative 1 Baseline bold upper M

Last updated: December 09, 2022