MODEL dependent = <fixedeffects> </ options>;
The MODEL statement names a single dependent variable and the fixed effects, which determine the matrix of the mixed model (see the section Parameterization of Mixed Models for details). The specification of effects is the same as in the GLM procedure; however, unlike PROC GLM, you do not specify random effects in the MODEL statement. The MODEL statement is required.
An intercept is included in the fixedeffects model by default. If no fixed effects are specified, only this intercept term is fit. The intercept can be removed by using the NOINT option.
Table 77.8 summarizes the options available in the MODEL statement. These are subsequently discussed in detail in alphabetical order.
Table 77.8: Summary of MODEL Statement Options
Option 
Description 

Model Building 

Excludes fixedeffect intercept from model 

Statistical Computations 

Determines the confidence level () for fixed effects 

Determines the confidence level () for predicted values 

Requests chisquare tests 

Specifies denominator degrees of freedom (list) 

Specifies the method for computing denominator degrees of freedom 

Selects the type of hypothesis test 

Requests influence and casedeletion diagnostics 

Suppresses hypothesis tests for the fixed effects 

Specifies output data set for predicted values and related quantities 

Specifies output data set for predicted means and related quantities 

Adds Pearsontype and studentized residuals to output data sets 

Adds scaled marginal residual to output data sets 

Statistical Output 

Displays confidence limits for fixedeffects parameter estimates 

Displays correlation matrix of fixedeffects parameter estimates 

Displays covariance matrix of fixedeffects parameter estimates 

Displays inverse covariance matrix of fixedeffects parameter estimates 

Displays matrix coefficients 

Adds a row for the intercept to test tables 

Displays fixedeffects parameter estimates (and scale parameter in GLM models) 

Singularity Tolerances 

Tunes sensitivity in computing Cholesky roots 

Tunes singularity criterion for residual variance 

Tunes the sensitivity in sweeping 

Tunes the sensitivity in forming Type 3 functions 
You can specify the following options in the MODEL statement after a slash (/).
requests that a ttype confidence interval be constructed for each of the fixedeffects parameters with confidence level 1 – number. The value of number must be between 0 and 1; the default is 0.05.
requests that a ttype confidence interval be constructed for the predicted values with confidence level 1 – number. The value of number must be between 0 and 1; the default is 0.05.
requests that chisquare tests be performed for all specified effects in addition to the F tests. Type 3 tests are the default; you can produce the Type 1 and Type 2 tests by using the HTYPE= option.
requests that ttype confidence limits be constructed for each of the fixedeffects parameter estimates. The confidence level is 0.95 by default; this can be changed with the ALPHA= option.
has the same effect as the DDFM= CONTAIN option.
produces the approximate correlation matrix of the fixedeffects parameter estimates. The ODS name of this table is CorrB.
produces the approximate variancecovariance matrix of the fixedeffects parameter estimates . By default, this matrix equals and results from sweeping on all but its last pivot and removing the y border. The EMPIRICAL option in the PROC MIXED statement changes this matrix into "empirical sandwich" form. The ODS name of this table is CovB. If the degreesoffreedom method of Kenward and Roger (1997) is in effect (DDFM= KENWARDROGER or DDFM= KENWARDROGER2), the COVB matrix changes because the method entails an adjustment of the variancecovariance matrix of the fixed effects by the method proposed by Prasad and Rao (1990); Harville and Jeske (1992). See also Kackar and Harville (1984).
produces the inverse of the approximate variancecovariance matrix of the fixedeffects parameter estimates. The ODS name of this table is InvCovB.
enables you to specify your own denominator degrees of freedom for
the fixed effects. The valuelist 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 "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 following 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 specify DDFM= SATTERTHWAITE, DDFM= KENWARDROGER, or DDFM= KENWARDROGER2, the DDF= option has no effect.
specifies the method for computing the denominator degrees of freedom for the tests of fixed effects resulting from the MODEL, CONTRAST , ESTIMATE , and LSMEANS statements.
Table 77.9 lists syntax aliases for the degreesoffreedom methods.
Table 77.9: Aliases for DDFM= Option
DDFM= Option 
Alias 

BETWITHIN 
BW 
CONTAIN 
CON 
KENWARDROGER 
KENROG, KR 
KENWARDROGER2 
KENROG2, KR2 
RESIDUAL 
RES 
SATTERTHWAITE 
SATTERTH, SAT 
The DDFM=CONTAIN option invokes the containment method to compute denominator degrees of freedom, and it is the default when you specify a RANDOM
statement. The containment method is carried out as follows: Denote the fixed effect in question A
, and search the RANDOM effect list for the effects that syntactically contain A
. For example, the random effect B
(A
) contains A
, but the random effect C
does not, even if it has the same levels as B
(A
).
Among the random effects that contain A
, compute their rank contribution to the () matrix. The DDF assigned to A
is the smallest of these rank contributions. If no effects are found, the DDF for A
is set equal to the residual degrees of freedom, . This choice of DDF matches the tests performed for balanced splitplot 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 the DDFM=BETWITHIN option.
The DDFM=BETWITHIN option is the default for REPEATED statement specifications (with no RANDOM statements). It is computed by dividing the residual degrees of freedom into betweensubject and withinsubject portions. PROC MIXED then checks whether a fixed effect changes within any subject. If so, it assigns withinsubject degrees of freedom to the effect; otherwise, it assigns the betweensubject degrees of freedom to the effect (see Schluchter and Elashoff 1990). If there are multiple withinsubject effects containing classification variables, the withinsubject degrees of freedom are partitioned into components corresponding to the subjectbyeffect interactions.
One exception to the preceding method is the case where you have specified no RANDOM statements and a REPEATED statement with the TYPE=UN option. In this case, all effects are assigned the betweensubject degrees of freedom to provide for better smallsample approximations to the relevant sampling distributions. DDFM= KENWARDROGER or DDFM= KENWARDROGER2 might be a better option to try for this case.
The DDFM=RESIDUAL option performs all tests by using the residual degrees of freedom, , where n is the number of observations.
The DDFM=SATTERTHWAITE option performs a general Satterthwaite approximation for the denominator degrees of freedom, computed as follows. Suppose is the vector of unknown parameters in , and suppose , where denotes a generalized inverse. Let and be the corresponding estimates.
Consider the onedimensional case, and consider to be a vector defining an estimable linear combination of . The Satterthwaite degrees of freedom for the t statistic
is computed as
where is the gradient of with respect to , evaluated at , and is the asymptotic variancecovariance matrix of obtained from the second derivative matrix of the likelihood equations.
For the multidimensional case, let be an estimable contrast matrix and denote the rank of as q > 1. The Satterthwaite denominator degrees of freedom for the F statistic
are computed by first performing the spectral decomposition , where is an orthogonal matrix of eigenvectors and is a diagonal matrix of eigenvalues, both of dimension . Define to be the mth row of , and let
where is the mth diagonal element of and is the gradient of with respect to , evaluated at . Then let
where the indicator function eliminates terms for which . The degrees of freedom for F are then computed as
provided ; otherwise is set to zero.
This method is a generalization of the techniques described in Giesbrecht and Burns (1985); McLean and Sanders (1988); Fai and Cornelius (1996). The method can also include estimated random effects. In this case, append to and change to be the inverse of the coefficient matrix in the mixed model equations. The calculations require extra memory to hold c matrices that are the size of the mixed model equations, where c is the number of covariance parameters. In the notation of Table 77.29, this is approximately bytes. Extra computing time is also required to process these matrices. The Satterthwaite method implemented here 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 MIXED.
The DDFM=KENWARDROGER option performs the degrees of freedom calculations detailed by Kenward and Roger (1997). This approximation involves inflating the estimated variancecovariance matrix of the fixed and random effects by the method proposed by Prasad and Rao (1990) and Harville and Jeske (1992), see also Kackar and Harville (1984). Satterthwaitetype 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 KenwardRoger covariance matrix adjustment includes a secondorder 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 or LINEAR suboption of the DDFM=KENWARDROGER option eliminates the second derivatives from the calculation of the covariance matrix adjustment. The LINEAR suboption is an alias for FIRSTORDER. For the case of scalar estimable functions, the resulting estimator is referred to as the PrasadRao estimator in Harville and Jeske (1992). 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=CSH , TYPE=FA , TYPE=FA0(q) , TYPE=TOEPH , TYPE=UNR , and all TYPE=SP() structures.
The DDFM=KENWARDROGER2 option specifies an improved approximation of the DDFM=KENWARDROGER method that uses a less biased precision estimator, as proposed by Kenward and Roger (2009). For an intrinsically linear covariance parameterization, this option produces the same precision estimator as that obtained using DDFM=KR(FIRSTORDER).
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 degreesoffreedom adjustment for DDFM=SATTERTHWAITE, DDFM=KENWARDROGER, or DDFM=KENWARDROGER2, and a message is written to the log.
This method changes output in the following tables (listed in Table 77.26): Contrast, CorrB, CovB, Diffs, Estimates, InvCovB, LSMeans, Slices, SolutionF, SolutionR, Tests1–Tests3. The OUTP= and OUTPM= data sets are also affected.
requests that Type 1, Type 2, and Type 3 matrix coefficients be displayed for all specified effects. The ODS name of the table is Coef.
requests that Type 1 matrix coefficients be displayed for all specified effects. The ODS name of the table is Coef.
requests that Type 2 matrix coefficients be displayed for all specified effects. The ODS name of the table is Coef.
requests that Type 3 matrix coefficients be displayed for all specified effects. The ODS name of the table is Coef.
requests that columns of the matrix that consist entirely of zeros not be eliminated from ; otherwise, they are eliminated by default. For a column corresponding to a missing cell to be added to , its particular levels must be present in at least one observation in the analysis data set along with a missing dependent variable. The use of the FULLX option can affect coefficient specifications in the CONTRAST and ESTIMATE statements, as well as covariate coefficients from LSMEANS statements specified with the AT MEANS option.
indicates the type of hypothesis test to perform on the fixed effects. Valid entries for values in the list are 1, 2, and 3; 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 for the Type 1 tests, Tests2 for the Type 2 tests, and Tests3 for the Type 3 tests.
specifies that influence and case deletion diagnostics are to be computed.
The INFLUENCE option computes influence diagnostics by noniterative or iterative methods. The noniterative diagnostics rely on recomputation formulas under the assumption that covariance parameters or their ratios remain fixed. With the possible exception of a profiled residual variance, no covariance parameters are updated. This is the default behavior because of its computational efficiency. However, the impact of an observation on the overall analysis can be underestimated if its effect on covariance parameters is not assessed. Toward this end, iterative methods can be applied to gauge the overall impact of observations and to obtain influence diagnostics for the covariance parameter estimates.
If you specify the INFLUENCE option without further suboptions, PROC MIXED computes singlecase deletion diagnostics and influence statistics for each observation in the data set by updating estimates for the fixedeffects parameter estimates, and also the residual variance, if it is profiled. The EFFECT= , SELECT= , ITER= , SIZE= , and KEEP= suboptions provide additional flexibility in the computation and reporting of influence statistics. Table 77.10 briefly describes important suboptions and their effect on the influence analysis.
Table 77.10: Summary of INFLUENCE Default and Suboptions
Description 
Suboption 

Compute influence diagnostics for individual observations 
Default 
Measure influence of sets of observations chosen according to a classification variable or effect 

Remove pairs of observations and report the results sorted by degree of influence 
SIZE= 2 
Remove triples, quadruples of observations, etc. 

Allow selection of individual observations, observations sharing specific levels of effects, and construction of tuples from specified subsets of observations 

Update fixed effects and covariance parameters by refitting the mixed model, adding up to n iterations 
ITER= n > 0 
Compute influence diagnostics for the covariance parameters 
ITER= n > 0 
Update only fixed effects and the residual variance, if it is profiled 
ITER= 0 
Add the reduceddata estimates to the data set created with ODS OUTPUT 
The modifiers and their default values are discussed in the following paragraphs. The set of computed influence diagnostics varies with the suboptions. The most extensive set of influence diagnostics is obtained when ITER= n with .
You can produce statistical graphics of influence diagnostics when ODS Graphics is enabled. For general information about ODS Graphics, see Chapter 21: Statistical Graphics Using ODS. For specific information about the graphics available in the MIXED procedure, see the section ODS Graphics.
You can specify the following influenceoptions in parentheses:
specifies an effect according to which observations are grouped. Observations sharing the same level of the effect are removed from the analysis as a group. The effect must contain only classification variables, but they need not be contained in the model.
Removing observations can change the rank of the matrix. This is particularly likely to happen when multiple observations are eliminated from the analysis. If the rank of the estimated variancecovariance matrix of changes or its singularity pattern is altered, no influence diagnostics are computed.
specifies that the updated parameter estimates should be written to the
ODS output data set. The values are not displayed in the "Influence" table, but if you use ODS OUTPUT to create a data set
from the listing, the estimates are added to the data set. If ITER=
0, only the fixedeffects estimates are saved. In iterative influence analyses, fixedeffects and covariance parameters are
stored. The p fixedeffects parameter estimates are named Parm1
–Parmp
, and the q covariance parameter estimates are named CovP1
–CovPq
. The order corresponds to that in the "Solution for Fixed Effects" and "Covariance Parameter Estimates" tables. If parameter
updates fail—for example, because of a loss of rank or a nonpositive definite Hessian—missing values are reported.
controls the maximum number of additional iterations PROC MIXED performs to update the fixedeffects and covariance parameter estimates following data point removal. If you specify n > 0, then statistics such as DFFITS, MDFFITS, and the likelihood distances measure the impact of observation(s) on all aspects of the analysis. Typically, the influence will grow compared to values at ITER=0. In models without RANDOM or REPEATED effects, the ITER= option has no effect.
This documentation refers to analyses when n > 0 simply as iterative influence analysis, even if final covariance parameter estimates can be updated in a single step (for example, when METHOD= MIVQUE0 or METHOD= TYPE3). This nomenclature reflects the fact that only if n > 0 are all model parameters updated, which can require additional iterations. If n > 0 and METHOD= REML (default) or METHOD= ML, the procedure updates fixed effects and variancecovariance parameters after removing the selected observations with additional NewtonRaphson iterations, starting from the converged estimates for the entire data. The process stops for each observation or set of observations if the convergence criterion is satisfied or the number of further iterations exceeds n. If n > 0 and METHOD= TYPE1, TYPE2, or TYPE3, ANOVA estimates of the covariance parameters are recomputed in a single step.
Compared to noniterative updates, the computations are more involved. In particular for large data sets or a large number of random effects (or both), iterative updates require considerably more resources. A onestep (ITER=1) or twostep update might be a good compromise. The output includes the number of iterations performed, which is less than n if the iteration converges. If the process does not converge in n iterations, you should be careful in interpreting the results, especially if n is fairly large.
Bounds and other restrictions on the covariance parameters carry over from the fulldata model. Covariance parameters that are not iterated in the model fit to the full data (the NOITER or HOLD= option in the PARMS statement) are likewise not updated in the refit. In certain models, such as randomeffects models, the ratios between the covariance parameters and the residual variance are maintained rather than the actual value of the covariance parameter estimate (see the section Influence Diagnostics).
determines how many observations are retained for display and in the output data set or how many tuples if you specify SIZE= . The output is sorted by an influence statistic as discussed for the SIZE= suboption.
specifies which observations or effect levels are chosen for influence calculations. If the SELECT= suboption is not specified, diagnostics are computed as follows:
for all observations, if EFFECT= or SIZE= are not given
for all levels of the specified effect, if EFFECT= is specified
for all tuples of size k formed from the observations in valuelist, if SIZE=k is specified
When you specify an effect with the EFFECT= option, the values in valuelist represent indices of the levels in the order in which PROC MIXED builds classification effects. Which observations in the data set correspond to this index depends on the order of the variables in the CLASS statement, not the order in which the variables appear in the interaction effect. See the section Parameterization of Mixed Models to understand precisely how the procedure indexes nested and crossed effects and how levels of classification variables are ordered. The actual values of the classification variables involved in the effect are shown in the output so you can determine which observations were removed.
If the EFFECT= suboption is not specified, the SELECT= value list refers to the sequence in which observations are read from the input data set or from the current BY group if there is a BY statement. This indexing is not necessarily the same as the observation numbers in the input data set, for example, if a WHERE clause is specified or during BY processing.
instructs PROC MIXED to remove groups of observations formed as tuples of size n. For example, SIZE=2 specifies all unique pairs of observations. The number of tuples for SIZE=k is and grows quickly with n and k. Using the SIZE= option can result in considerable computing time. The MIXED procedure displays by default only the 50 tuples with the greatest influence. Use the KEEP= option to override this default and to retain a different number of tuples in the listing or ODS output data set. Regardless of the KEEP= specification, all tuples are evaluated and the results are ordered according to an influence statistic. This statistic is the (restricted) likelihood distance as a measure of overall influence if ITER= or when a residual variance is profiled. When likelihood distances are unavailable, the results are ordered by the PRESS statistic.
To reduce computational burden, the SIZE= option can be combined with the SELECT= valuelist modifier. For example, the following statements evaluate all pairs formed from observations 13, 14, 18, 30, 31, and 33 and display the five pairs with the greatest influence:
proc mixed; class a m f; model penetration = a m / influence(size=2 keep=5 select=13,14,18,30,31,33); random f(m); run;
If any observation in a tuple contains missing values or has otherwise not contributed to the analysis, the tuple is not evaluated. This guarantees that the displayed results refer to the same number of observations, so that meaningful statistics are available by which to order the results. If computations fail for a particular tuple—for example, because the matrix changes rank or the matrix is not positive definite—no results are produced. Results are retained when the maximum number of iterative updates is exceeded in iterative influence analyses.
The SIZE= suboption cannot be combined with the EFFECT= suboption. As in the case of the EFFECT= suboption, the statistics being computed are those appropriate for removal of multiple data points, even if SIZE=1.
The ODS name of the "Influence Diagnostics" table is Influence. The variables in this table depend on whether you specify the EFFECT= , SIZE= , or KEEP= suboption and whether covariance parameters are iteratively updated. When ITER= 0 (the default), certain influence diagnostics are meaningful only if the residual variance is profiled. Table 77.11 and Table 77.12 summarize the statistics obtained depending on the model and modifiers. The last column in these tables gives the variable name in the ODS OUTPUT INFLUENCE= data set. Restricted likelihood distances are reported instead of the likelihood distance unless METHOD= ML. See the section Influence Diagnostics for details about the individual statistics.
Table 77.11: Statistics Computed with INFLUENCE Option, Noniterative Analysis (ITER=0)
Suboption 

Statistic 
Variable 

Profiled 
Name 

Default 
Yes 
Observed value 
Observed 
Predicted value 
Predicted 

Marginal residual 
Residual 

Leverage 
Leverage 

PRESS residual 
PRESSRes 

Internally studentized marginal residual 
Student 

Externally studentized marginal residual 
RStudent 

RMSE without deleted observations 
RMSE 

Cook’s D 
CookD 

DFFITS 
DFFITS 

CovRatio 
COVRATIO 

(Restricted) likelihood distance 
RLD, LD 

Default 
No 
Observed value 
Observed 
Predicted value 
Predicted 

Marginal residual 
Residual 

Leverage 
Leverage 

PRESS residual 
PRESSRes 

Internally studentized marginal residual 
Student 

Cook’s D 
CookD 

EFFECT=, 
Yes 
Observations in level (tuple) 
Nobs 
SIZE=, 
PRESS statistic 
PRESS 

or KEEP= 
Cook’s D 
CookD 

MDFFITS 
MDFFITS 

CovRatio 
COVRATIO 

COVTRACE 
COVTRACE 

RMSE without deleted level (tuple) 
RMSE 

(Restricted) likelihood distance 
RLD, LD 

EFFECT=, 
No 
Observations in level (tuple) 
Nobs 
SIZE=, 
PRESS statistic 
PRESS 

or KEEP= 
Cook’s D 
CookD 
Table 77.12: Statistics Computed with INFLUENCE Option, Iterative Analysis (ITER=)
Suboption 
Statistic 
Variable 

Name 

Default 
Number of iterations 
Iter 
Observed value 
Observed 

Predicted value 
Predicted 

Marginal residual 
Residual 

Leverage 
Leverage 

PRESS residual 
PRESSres 

Internally studentized marginal residual 
Student 

Externally studentized marginal residual 
RStudent 

RMSE without deleted obs (if possible) 
RMSE 

Cook’s D 
CookD 

DFFITS 
DFFITS 

CovRatio 
COVRATIO 

Cook’s D CovParms 
CookDCP 

CovRatio CovParms 
COVRATIOCP 

MDFFITS CovParms 
MDFFITSCP 

(Restricted) likelihood distance 
RLD, LD 

EFFECT=, 
Observations in level (tuple) 
Nobs 
SIZE=, 
Number of iterations 
Iter 
or KEEP= 
PRESS statistic 
PRESS 
RMSE without deleted level (tuple) 
RMSE 

Cook’s D 
CookD 

MDFFITS 
MDFFITS 

CovRatio 
COVRATIO 

COVTRACE 
COVTRACE 

Cook’s D CovParms 
CookDCP 

CovRatio CovParms 
COVRATIOCP 

MDFFITS CovParms 
MDFFITSCP 

(Restricted) likelihood distance 
RLD, LD 
adds a row to the tables for Type 1, 2, and 3 tests corresponding to the overall intercept.
requests an estimate for each row of the matrix used to form tests of fixed effects. Components corresponding to Type 3 tests are the default; you can produce the Type 1 and Type 2 component estimates with the HTYPE= option.
Tests of fixed effects involve testing of linear hypotheses of the form . The matrix is constructed from Type 1, 2, or 3 estimable functions. By default the MIXED procedure constructs Type 3 tests. In many situations, the individual rows of the matrix represent contrasts of interest. For example, in a oneway classification model, the Type 3 estimable functions define differences of factorlevel means. In a balanced twoway layout, the rows of correspond to differences of cell means.
For example, suppose factors A
and B
have a and b levels, respectively. The following statements produce (a – 1) one degree of freedom tests for the rows of associated with the Type 1 and Type 3 estimable functions for factor A
, (b – 1) tests for the rows of associated with factor B
, and a single test for the Type 1 and Type 3 coefficients associated with regressor X
:
class A B; model y = A B x / htype=1,3 lcomponents;
The denominator degrees of freedom associated with a row of are the same as those in the corresponding "Tests of Fixed Effects" table, except for DDFM= KENWARDROGER, DDFM= KENWARDROGER2, and DDFM= SATTERTHWAITE. For these degreesoffreedom methods, the denominator degrees of freedom are computed separately for each row of .
The ODS name of the table containing all requested component tests is LComponents. See Example 77.9 for applications of the LCOMPONENTS option.
has the same effect as the DDFM= RESIDUAL option.
requests that no intercept be included in the model. An intercept is included by default.
specifies that no hypothesis tests be performed for the fixed effects.
specifies an output data set containing predicted values and related quantities. This option replaces the P option from SAS 6.
Predicted values are formed by using the rows from () as matrices. Thus, predicted values from the original data are . Their approximate standard errors of prediction are formed from the quadratic form of with defined in the section Statistical Properties. The L95
and U95
variables provide a ttype confidence interval for the predicted values, and they correspond to the L95M
and U95M
variables from the GLM and REG procedures for fixedeffects models. The residuals are the observed minus the predicted values.
Predicted values for data points other than those observed can be obtained by using missing dependent variables in your input
data set.
Specifications that have a REPEATED statement with the SUBJECT= option and missing dependent variables compute predicted values by using empirical best linear unbiased prediction (EBLUP). Using hats to denote estimates, the EBLUP formula is
where represents a hypothetical realization of a missing data vector with associated design matrix . The matrix is the modelbased covariance matrix between and the observed data , and other notation is as presented in the section Mixed Models Theory.
The estimated prediction variance is as follows:
where is the modelbased variance matrix of . For further details, see Henderson (1984) and Harville (1990). This feature can be useful for forecasting time series or for computing spatial predictions.
By default, all variables from the input data set are included in the OUTP= data set. You can select a subset of these variables by using the ID statement.
specifies an output data set containing predicted means and related quantities. This option replaces the PM option from SAS 6.
The output data set is of the same form as that resulting from the OUTP= option, except that the predicted values do not incorporate the EBLUP values . They also do not use the EBLUPs for specifications that have a REPEATED statement with the SUBJECT= option and missing dependent variables. The predicted values are formed as in the OUTPM= data set, and standard errors are quadratic forms in the approximate variancecovariance matrix of as displayed by the COVB option.
By default, all variables from the input data set are included in the OUTPM= data set. You can select a subset of these variables by using the ID statement.
requests that Pearsontype and (internally) studentized residuals be added to the OUTP= and OUTPM= data sets. Studentized residuals are raw residuals standardized by their estimated standard error. When residuals are internally studentized, the data point in question has contributed to the estimation of the covariance parameter estimates on which the standard error of the residual is based. Externally studentized marginal residuals can be computed with the INFLUENCE option. Pearsontype residuals scale the residual by the standard deviation of the response.
The option has no effect unless the OUTP= or OUTPM= option is specified or unless ODS Graphics is enabled. For general information about ODS Graphics, see Chapter 21: Statistical Graphics Using ODS. For specific information about the graphics available in the MIXED procedure, see the section ODS Graphics. For computational details about studentized and Pearson residuals in MIXED, see the section Residual Diagnostics.
tunes the sensitivity in computing Cholesky roots. If a diagonal pivot element is less than D*number as PROC MIXED performs the Cholesky decomposition on a matrix, the associated column is declared to be linearly dependent upon previous columns and is set to . The value D is the original diagonal element of the matrix. The default for number is 1E4 times the machine epsilon; this product is approximately 1E–12 on most computers.
sets the tolerance for which the residual variance is considered to be zero. The default is 1E4 times the machine epsilon; this product is approximately 1E–12 on most computers.
tunes the sensitivity in sweeping. If a diagonal pivot element is less than D*number as PROC MIXED sweeps a matrix, the associated column is declared to be linearly dependent upon previous columns, and the associated parameter is set to 0. The value D is the original diagonal element of the matrix. The default is 1E4 times the machine epsilon; this product is approximately 1E–12 on most computers.
requests that a solution for the fixedeffects parameters be produced. Using notation from the section Mixed Models Theory, the fixedeffects parameter estimates are and their approximate standard errors are the square roots of the diagonal elements of . You can output this approximate variance matrix with the COVB option or modify it with the EMPIRICAL option in the PROC MIXED statement or the DDFM= KENWARDROGER or DDFM= KENWARDROGER2 option in the MODEL statement.
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 "Tests of Fixed Effects" table under the effect containing the parameter. The "Pr > t" column contains the twotailed pvalue corresponding to the t statistic and associated degrees of freedom. You can use the CL option to request confidence intervals for all of the parameters; they are constructed around the estimate by using a radius of the standard error times a percentage point from the t distribution.
requests that responses and marginal residuals be scaled by the
inverse Cholesky root of the marginal variancecovariance matrix. The variables ScaledDep
and ScaledResid
are added to the OUTPM=
data set. These quantities can be important in bootstrapping of data or residuals. Examination of the scaled residuals is
also helpful in diagnosing departures from normality. Notice that the results of this scaling operation can depend on the
order in which the MIXED procedure processes the data.
The VCIRY option has no effect unless you also use the OUTPM= option or unless ODS Graphics is enabled. For general information about ODS Graphics, see Chapter 21: Statistical Graphics Using ODS. For specific information about the graphics available in the MIXED procedure, see the section ODS Graphics.
is an alias for the COVBI option.
is an alias for the COVB option.
tunes the sensitivity in forming Type 3 functions. Any element in the estimable function basis with an absolute value less than number is set to 0. The default is 1E–8.