The GLIMMIX Procedure

requests that a t-type confidence interval be constructed for each of the fixed-effects parameters with confidence level . The value of number must be between 0 and 1; the default is 0.05.

requests that chi-square tests be performed for all specified effects in addition to the F tests. Type III tests are the default; you can produce the Type I and Type II tests by using the HTYPE= option.

requests that t-type confidence limits be constructed for each of the fixed-effects parameter estimates. The confidence level is 0.95 by default; this can be changed with the ALPHA= option.

produces the correlation matrix from the approximate covariance matrix of the fixed-effects parameter estimates.

produces the approximate variance-covariance matrix of the fixed-effects parameter estimates . In a generalized linear mixed model this matrix typically takes the form and can be obtained by sweeping the mixed model equations; see the section Estimated Precision of Estimates. In a model without random effects, it is obtained from the inverse of the observed or expected Hessian matrix. Which Hessian is used in the computation depends on whether the procedure is in scoring mode (see the SCORING= option in the PROC GLIMMIX statement) and whether the EXPHESSIAN option is in effect. Note that if you use EMPIRICAL= or DDFM=KENWARDROGER, the matrix displayed by the COVB option is the empirical (sandwich) estimator or the adjusted estimator, respectively.

The DETAILS suboption of the COVB option enables you to obtain a table of statistics about the covariance matrix of the fixed effects. If an adjusted estimator is used because of the EMPIRICAL= or DDFM=KENWARDROGER option, the GLIMMIX procedure displays statistics for the adjusted and unadjusted estimators as well as statistics comparing them. This enables you to diagnose, for example, changes in rank (because of an insufficient number of subjects for the empirical estimator) and to assess the extent of the covariance adjustment. In addition, the GLIMMIX procedure then displays the unadjusted (=model-based) covariance matrix of the fixed-effects parameter estimates. For more details, see the section Exploring and Comparing Covariance Matrices.

produces the inverse of the approximate covariance matrix of the fixed-effects parameter estimates.

enables you to specify your own denominator degrees of freedom for the fixed effects. The value-list specification is a list of numbers or missing values (.) separated by commas. The degrees of freedom should be listed in the order in which the effects appear in the "Type III Tests of Fixed Effects" table. If you want to retain the default degrees of freedom for a particular effect, use a missing value for its location in the list. For example, the statement assigns 3 denominator degrees of freedom to A and 4.7 to A*B, while those for B remain the same:

model Y = A B A*B / ddf=3,.,4.7;

If you select a degrees-of-freedom method with the DDFM= option, then nonmissing, positive values in value-list override the degrees of freedom for the particular effect. For example, the statement assigns 3 and 6 denominator degrees of freedom in the test of the A main effect and the A*B interaction, respectively:

model Y = A B A*B / ddf=3,.,6  ddfm=Satterth;

The denominator degrees of freedom for the test for the B effect are determined from a Satterthwaite approximation.

Note that the DDF= and DDFM= options determine the degrees of freedom in the "Type I Tests of Fixed Effects," "Type II Tests of Fixed Effects," and "Type III Tests of Fixed Effects" tables. These degrees of freedom are also used in determining the degrees of freedom in tests and confidence intervals from the CONTRAST, ESTIMATE, LSMEANS, and LSMESTIMATE statements. Exceptions from this rule are noted in the documentation for the respective statements.

specifies the method for computing the denominator degrees of freedom for the tests of fixed effects resulting from the MODEL, CONTRAST, ESTIMATE, LSMEANS, and LSMESTIMATE statements.

Table 40.7 table lists syntax aliases for the degrees-of-freedom methods.

The DDFM=BETWITHIN option divides the residual degrees of freedom into between-subject and within-subject portions. PROC GLIMMIX then determines whether a fixed effect changes within any subject. If the GLIMMIX procedure does not process the data by subjects, the DDFM=BETWITHIN option has no effect. See the section Processing by Subjects for details. If so, it assigns within-subject degrees of freedom to the effect; otherwise, it assigns the between-subject degrees of freedom to the effect (see Schluchter and Elashoff 1990). If there are multiple within-subject effects containing classification variables, the within-subject degrees of freedom are partitioned into components corresponding to the subject-by-effect interactions.

One exception to the preceding method is the case where you model only R-side covariation with an unstructured covariance matrix (TYPE=UN option). In this case, all fixed effects are assigned the between-subject degrees of freedom to provide for better small-sample approximations to the relevant sampling distributions. The DDFM=BETWITHIN method is the default for models with only R-side random effects and a SUBJECT= option.

The DDFM=CONTAIN option invokes the containment method to compute denominator degrees of freedom, and this method is the default when the model contains G-side random effects. The containment method is carried out as follows: Denote the fixed effect in question A and search the G-side random effect list for the effects that syntactically contain A. For example, the effect B(A) contains A, but the effect C does not, even if it has the same levels as B(A).

Among the random effects that contain A, compute their rank contributions to the matrix (in order). The denominator degrees of freedom assigned to A is the smallest of these rank contributions. If no effects are found, the denominator df for A is set equal to the residual degrees of freedom, . This choice of degrees of freedom is the same as for the tests performed for balanced split-plot designs and should be adequate for moderately unbalanced designs.

Caution: If you have a matrix with a large number of columns, the overall memory requirements and the computing time after convergence can be substantial for the containment method. If it is too large, you might want to use a different degrees-of-freedom method, such as DDFM=RESIDUAL, DDFM=NONE, or DDFM=BETWITHIN.

DDFM=NONE specifies that no denominator degrees of freedom be applied. PROC GLIMMIX then essentially assumes that infinite degrees of freedom are available in the calculation of -values. The -values for tests are then identical to -values derived from the standard normal distribution. In the case of tests, the -values equal those of chi-square tests determined as follows: if is the observed value of the test with numerator degrees of freedom, then

Regardless of the DDFM= method, you can obtain these chi-square -values with the CHISQ option in the MODEL statement.

The DDFM=RESIDUAL option performs all tests by using the residual degrees of freedom, , where is the sum of the frequencies used. It is the default degrees of freedom method for GLMs and overdispersed GLMs.

The DDFM=KENWARDROGER option applies the (prediction) standard error and degrees-of-freedom correction detailed by Kenward and Roger (1997). This approximation involves inflating the estimated variance-covariance matrix of the fixed and random effects in a manner similar to that of Prasad and Rao (1990), Harville and Jeske (1992), and Kackar and Harville (1984). Satterthwaite-type degrees of freedom are then computed based on this adjustment. By default, the observed information matrix of the covariance parameter estimates is used in the calculations. 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 FIRSTORDER suboption of the DDFM=KENWARDROGER option eliminates the second derivatives from the calculation of the covariance matrix adjustment. For the case of scalar estimable functions, the resulting estimator is referred to as the Prasad-Rao estimator in Harville and Jeske (1992). You can use the COVB(DETAILS) option to diagnose the adjustments made to the covariance matrix of fixed-effects parameter estimates by the GLIMMIX procedure. An application with DDFM=KENWARDROGER is presented in Example 40.8. The following are examples of covariance structures that generally lead to nonzero second derivatives: TYPE=ANTE(1), TYPE=AR(1), TYPE=ARH(1), TYPE=ARMA(1,1), TYPE=CHOL, TYPE=CSH, TYPE=FA0(), TYPE=TOEPH, TYPE=UNR, and all TYPE=SP() structures.

The DDFM=SATTERTHWAITE option performs a general Satterthwaite approximation for the denominator degrees of freedom in a generalized linear mixed model. This method is a generalization of the techniques described in Giesbrecht and Burns (1985), McLean and Sanders (1988), and Fai and Cornelius (1996). The method can also include estimated random effects. The calculations require extra memory to hold matrices that are the size of the mixed model equations, where is the number of covariance parameters. Extra computing time is also required to process these matrices. The Satterthwaite method implemented is intended to produce an accurate F approximation; however, the results can differ from those produced by PROC GLM. Also, the small sample properties of this approximation have not been extensively investigated for the various models available with PROC GLIMMIX. Computational details can be found in the section Satterthwaite Degrees of Freedom Approximation.

When the asymptotic variance matrix of the covariance parameters is found to be singular, a generalized inverse is used. Covariance parameters with zero variance then do not contribute to the degrees of freedom adjustment for DDFM=SATTERTH and DDFM=KENWARDROGER, and a message is written to the log.

specifies the built-in (conditional) probability distribution of the data. If you specify the DIST= option and you do not specify a user-defined link function, a default link function is chosen according to the following table. If you do not specify a distribution, the GLIMMIX procedure defaults to the normal distribution for continuous response variables and to the multinomial distribution for classification or character variables, unless the events/trial syntax is used in the MODEL statement. If you choose the events/trial syntax, the GLIMMIX procedure defaults to the binomial distribution.

Table 40.8 lists the values of the DIST= option and the corresponding default link functions. For the case of generalized linear models with these distributions, you can find expressions for the log-likelihood functions in the section Maximum Likelihood.

Note that the PROC GLIMMIX default link for the gamma or exponential distribution is not the canonical link (the reciprocal link).

The numeric value in the last column of Table 40.8 can be used in combination with DIST=BYOBS. The BYOBS(variable) syntax designates a variable whose value identifies the distribution to which an observation belongs. If the variable is numeric, its values must match values in the last column of Table 40.8. If the variable is not numeric, an observation’s distribution is identified by the first four characters of the distribution’s name in the leftmost column of the table. Distributions whose numeric value is "NA" cannot be used with DIST=BYOBS.

If the variable in BYOBS(variable) is a data set variable, it can also be used in the CLASS statement of the GLIMMIX procedure. For example, this provides a convenient method to model multivariate data jointly while varying fixed-effects components across outcomes. Assume that, for example, for each patient, a count and a continuous outcome were observed; the count data are modeled as Poisson data and the continuous data are modeled as gamma variates. The following statements fit a Poisson and a gamma regression model simultaneously:

proc sort data=yourdata; 
   by patient;
run;
data yourdata; 
   set yourdata; 
   by patient;
   if first.patient then dist='POIS' else dist='GAMM';
run;
proc glimmix data=yourdata;
   class treatment dist;
   model y = dist treatment*dist / dist=byobs(dist);
run;

The two models have separate intercepts and treatment effects. To correlate the outcomes, you can share a random effect between the observations from the same patient:

proc glimmix data=yourdata;
   class treatment dist patient;
   model y = dist treatment*dist / dist=byobs(dist);
   random intercept / subject=patient;      
run;

Or, you could use an R-side correlation structure:

proc glimmix data=yourdata;
   class treatment dist patient;
   model y = dist treatment*dist / dist=byobs(dist);
   random _residual_ / subject=patient type=un;
run;

Although DIST=BYOBS(variable) is used to model multivariate data, you only need a single response variable in PROC GLIMMIX. The responses are in "univariate" form. This allows, for example, different missing value patterns across the responses. It does, however, require that all response variables be numeric.

The default links that are assigned when DIST=BYOBS is in effect correspond to the respective default links in Table 40.8.

When you choose DIST=LOGNORMAL, the GLIMMIX procedure models the logarithm of the response variable as a normal random variable. That is, the mean and variance are estimated on the logarithmic scale, assuming a normal distribution, . This enables you to draw on options that require a distribution in the exponential family—for example, by using a scoring algorithm in a GLM. To convert means and variances for into those of , use the relationships

The DIST=T option models the data as a shifted and scaled central variable. This enables you to model data with heavy-tailed distributions. If denotes the response and has a distribution with degrees of freedom, then PROC GLIMMIX models

In this parameterization, has mean and variance .

By default, . You can supply different degrees of freedom for the variate as in the following statements:

proc glimmix;
   class a b;
   model y = b x b*x / dist=tcentral(9.6);
   random a;
run;

The GLIMMIX procedure does not accept values for the degrees of freedom parameter less than 3.0. If the distribution is used with the DIST=BYOBS(variable) specification, the degrees of freedom are fixed at . For mixed models where parameters are estimated based on linearization, choosing DIST=T instead of DIST=NORMAL affects only the residual variance, which decreases by the factor .

Note that in SAS 9.1, the GLIMMIX procedure modeled . The scale parameter of the parameterizations are related as .

The DIST=BETA option implements the parameterization of the beta distribution in Ferrari and Cribari-Neto (2004). If has a density, so that , this parameterization uses the variance function and .

See the section Maximum Likelihood for the log likelihoods of the distributions fitted by the GLIMMIX procedure.

requests that Type I, Type II, and Type III matrix coefficients be displayed for all specified effects.

requests that Type I matrix coefficients be displayed for all specified effects.

requests that Type II matrix coefficients be displayed for all specified effects.

requests that Type III matrix coefficients be displayed for all specified effects.

indicates the type of hypothesis test to perform on the fixed effects. Valid entries for values in the value-list are 1, 2, and 3, corresponding to Type I, Type II, and Type III tests. The default value is 3. You can specify several types by separating the values with a comma or a space. The ODS table names are "Tests1," "Tests2," and "Tests3" for the Type I, Type II, and Type III tests, respectively.

adds a row to the tables for Type I, II, and III tests corresponding to the overall intercept.

specifies the link function in the generalized linear mixed model. The keywords and their associated built-in link functions are shown in Table 40.9.

For the probit and cumulative probit links, denotes the quantile function of the standard normal distribution. For the other cumulative links, denotes a cumulative category probability. The cumulative and generalized logit link functions are appropriate only for the multinomial distribution. When you choose a cumulative link function, PROC GLIMMIX assumes that the data are ordinal. When you specify LINK=GLOGIT, the GLIMMIX procedure assumes that the data are nominal (not ordered).

The numeric value in the rightmost column of Table 40.9 can be used in conjunction with LINK=BYOBS(variable). This syntax designates a variable whose values identify the link function associated with an observation. If the variable is numeric, its values must match those in the last column of Table 40.9. If the variable is not numeric, an observation’s link function is determined by the first four characters of the link’s name in the first column. Those link functions whose numeric value is "NA" cannot be used with LINK=BYOBS(variable).

You can define your own link function through programming statements. See the section User-Defined Link or Variance Function for more information about how to specify a link function. If a user-defined link function is in effect, the specification in the LINK= option is ignored. If you specify neither the LINK= option nor a user-defined link function, then the default link function is chosen according to Table 40.8.

determines how weights are used in constructing the coefficients for Type I through Type III matrices. The default is LWEIGHT=VAR, and the values of the WEIGHT variable are used in forming crossproduct matrices. If you specify LWEIGHT=FIRO, the weights incorporate the WEIGHT variable as well as the first-order weights of the linearized model. For LWEIGHT=NONE, the matrix coefficients are based on the raw crossproduct matrix, whether a WEIGHT variable is specified or not.

requests that the columns of the matrix are not centered and scaled. By default, the columns of are centered and scaled. Unless the NOCENTER option is in effect, is replaced by during estimation. The columns of are computed as follows:

• In models with an intercept, the intercept column remains the same and the th entry in row of is

• In models without intercept, no centering takes place and the th entry in row of is

The effects of centering and scaling are removed when results are reported. For example, if the covariance matrix of the fixed effects is printed with the COVB option of the MODEL statement, the covariances are reported in terms of the original parameters, not the centered and scaled versions. If you specify the STDCOEF option, fixed-effects parameter estimates and their standard errors are reported in terms of the standardized (scaled and/or centered) coefficients in addition to the usual results in noncentered form.

requests that no intercept be included in the fixed-effects model. An intercept is included by default.

requests estimates of odds ratios and their confidence limits, provided the link function is the logit, cumulative logit, or generalized logit. Odds ratios are produced for the following:

• classification main effects, if they appear in the MODEL statement

• continuous variables in the MODEL statement, unless they appear in an interaction with a classification effect

• continuous variables in the MODEL statement at fixed levels of a classification effect, if the MODEL statement contains an interaction of the two

• continuous variables in the MODEL statement, if they interact with other continuous variables

To compute odds and odds ratios for general estimable functions and least squares means, see the ODDSRATIO option in the LSMEANS statement and the EXP options in the ESTIMATE and LSMESTIMATE statements.

For important details concerning interpretation and computation of odds ratios with the GLIMMIX procedure, see the section Odds and Odds Ratio Estimation.

specifies a variable to be used as an offset for the linear predictor. An offset plays the role of a fixed effect whose coefficient is known to be 1. You can use an offset in a Poisson model, for example, when counts have been obtained in time intervals of different lengths. With a log link function, you can model the counts as Poisson variables with the logarithm of the time interval as the offset variable. The offset variable cannot appear in the CLASS statement or elsewhere in the MODEL or RANDOM statement.

specifies a value for the linear predictor of the reference level in the generalized logit model for nominal data. By default .

requests that a solution for the fixed-effects parameters be produced. Using notation from the section Notation for the Generalized Linear Mixed Model, the fixed-effects parameter estimates are , and their (approximate) estimated standard errors are the square roots of the diagonal elements of . This matrix commonly is of the form in GLMMs. You can output this approximate variance matrix with the COVB option. See the section Details: GLIMMIX Procedure on the construction of in the various models.

Along with the estimates and their approximate standard errors, a t statistic is computed as the estimate divided by its standard error. The degrees of freedom for this t statistic matches the one appearing in the "Type III Tests of Fixed Effects" table under the effect containing the parameter. If DDFM=KENWARDROGER or DDFM=SATTERTHWAITE, the degrees of freedom are computed separately for each fixed-effect estimate, unless you override the value for any specific effect with the DDF=value-list option. The "Pr > |t|" column contains the two-tailed -value corresponding to the t statistic and associated degrees of freedom. You can use the CL option to request confidence intervals for the fixed-effects parameters; they are constructed around the estimate by using a radius of the standard error times a percentage point from the t distribution.

reports solutions for fixed effects in terms of the standardized (scaled and/or centered) coefficients. This option has no effect when the NOCENTER option is specified or in models for multinomial data.

tunes the sensitivity in forming Type III functions. Any element in the estimable function basis with an absolute value less than number is set to 0. The default is 1E8.