The GLIMMIX Procedure

Notes on Output Statistics

Table 17 lists the statistics computed with the OUTPUT statement of the GLIMMIX procedure and their default names. This section provides further details about these statistics.

The distinction between prediction and confidence limits in Table 17 stems from the involvement of the predictors of the random effects. If the random-effect solutions (BLUPs, EBES) are involved, then the associated standard error used in computing the limits are standard errors of prediction rather than standard errors of estimation. The prediction limits are not limits for the prediction of a new observation.

The Pearson residuals in Table 17 are "Pearson-type" residuals, because the residuals are standardized by the square root of the marginal or conditional variance of an observation. Traditionally, Pearson residuals in generalized linear models are divided by the square root of the variance function. The GLIMMIX procedure divides by the square root of the variance so that marginal and conditional residuals have similar expressions. In other words, scale and overdispersion parameters are included.

When residuals or predicted values involve only the fixed effects part of the linear predictor (that is, ModifyingAbove eta With caret Subscript m Baseline equals bold x prime ModifyingAbove bold-italic beta With caret), then all model quantities are computed based on this predictor. For example, if the variance by which to standardize a marginal residual involves the variance function, then the variance function is also evaluated at the marginal mean, g Superscript negative 1 Baseline left-parenthesis ModifyingAbove eta With caret Subscript m Baseline right-parenthesis. Thus the residuals p minus ModifyingAbove eta With caret and p Subscript m Baseline minus ModifyingAbove eta With caret Subscript m can also be expressed as left-parenthesis y minus mu right-parenthesis slash partial-differential mu and left-parenthesis y minus mu Subscript m Baseline right-parenthesis slash partial-differential mu Subscript m, respectively, where partial-differential mu is the derivative with respect to the linear predictor. To construct the residual p minus ModifyingAbove eta With caret Subscript m in a GLMM, you can add the value of _ZGAMMA_ to the conditional residual p minus ModifyingAbove eta With caret. (The residual p minus ModifyingAbove eta With caret Subscript m is computed instead of the default marginal residual when you specify the CPSEUDO option in the OUTPUT statement.) If the predictor involves the BLUPs, then all relevant expressions and evaluations involve the conditional mean g Superscript negative 1 Baseline left-parenthesis ModifyingAbove eta With caret right-parenthesis.

The naming convention to add "PA" to quantities not involving the BLUPs is chosen to suggest the concept of a population average. When the link function is nonlinear, these are not truly population-averaged quantities, because g Superscript negative 1 Baseline left-parenthesis bold x prime bold-italic beta right-parenthesis does not equal normal upper E left-bracket upper Y right-bracket in the presence of random effects. For example, if

mu Subscript i Baseline equals g Superscript negative 1 Baseline left-parenthesis bold x prime Subscript i Baseline bold-italic beta plus bold z prime Subscript i Baseline bold-italic gamma Subscript i Baseline right-parenthesis

is the conditional mean for subject i, then

g Superscript negative 1 Baseline left-parenthesis bold x prime Subscript i Baseline ModifyingAbove bold-italic beta With caret right-parenthesis

does not estimate the average response in the population of subjects but the response of the average subject (the subject for which bold-italic gamma Subscript i Baseline equals bold 0). For models with identity link, the average response and the response of the average subject are identical.

The GLIMMIX procedure obtains standard errors on the scale of the mean by the delta method. If the link is a nonlinear function of the linear predictor, these standard errors are only approximate. For example,

normal upper V normal a normal r left-bracket g Superscript negative 1 Baseline left-parenthesis ModifyingAbove eta With caret Subscript m Baseline right-parenthesis right-bracket approaches-the-limit left-parenthesis StartFraction partial-differential g Superscript negative 1 Baseline left-parenthesis t right-parenthesis Over partial-differential t EndFraction Subscript vertical-bar ModifyingAbove eta With caret Sub Subscript m Subscript Baseline right-parenthesis squared normal upper V normal a normal r left-bracket ModifyingAbove eta With caret Subscript m Baseline right-bracket

Confidence limits on the scale of the data are usually computed by applying the inverse link function to the confidence limits on the linked scale. The resulting limits on the data scale have the same coverage probability as the limits on the linked scale, but they are possibly asymmetric.

In generalized logit models, confidence limits on the mean scale are based on symmetric limits about the predicted mean in a category. Suppose that the multinomial response in such a model has J categories. The probability of a response in category i is computed as

ModifyingAbove mu With caret Subscript i Baseline equals StartFraction exp left-brace ModifyingAbove eta With caret Subscript i Baseline right-brace Over sigma-summation Underscript j equals 1 Overscript upper J Endscripts exp left-brace ModifyingAbove eta With caret Subscript i Baseline right-brace EndFraction

The variance of ModifyingAbove mu With caret Subscript i is then approximated as

normal upper V normal a normal r left-bracket ModifyingAbove mu With caret Subscript i Baseline right-bracket approaches-the-limit zeta equals bold-italic upsilon prime Subscript i Baseline normal upper V normal a normal r Start 1 By 4 Matrix 1st Row 1st Column ModifyingAbove eta With caret Subscript 1 Baseline 2nd Column ModifyingAbove eta With caret Subscript 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column ModifyingAbove eta With caret Subscript upper J Baseline EndMatrix bold-italic upsilon Subscript i

where bold-italic upsilon Subscript i is a upper J times 1 vector with kth element

StartLayout 1st Row 1st Column ModifyingAbove mu With caret Subscript i Baseline left-parenthesis 1 minus ModifyingAbove mu With caret Subscript i Baseline right-parenthesis 2nd Column i equals k 2nd Row 1st Column minus ModifyingAbove mu With caret Subscript i Baseline ModifyingAbove mu With caret Subscript k 2nd Column i not-equals k EndLayout

The confidence limits in the generalized logit model are then obtained as

ModifyingAbove mu With caret Subscript i Baseline plus-or-minus t Subscript nu comma alpha slash 2 Baseline StartRoot zeta EndRoot

where t Subscript nu comma alpha slash 2 is the 100 left-parenthesis 1 minus alpha slash 2 right-parenthesisth percentile from a t distribution with nu degrees of freedom. Confidence limits are truncated if they fall outside the left-bracket 0 comma 1 right-bracket interval.

Last updated: December 09, 2022