The RANDOM statement defines the random effects that constitute the vector in the mixed model. You can use this statement to specify traditional variance component models and to specify random coefficients. The random effects can be classification or continuous, and multiple RANDOM statements are possible.
Using notation from the section Linear Mixed Models Theory, the purpose of the RANDOM statement is to define the matrix of the mixed model, the random effects in the vector, and the structure of . The matrix is constructed exactly as the matrix for the fixed effects is constructed, and the matrix is constructed to correspond with the effects that constitute . The structure of is defined by using the TYPE= option.
You can specify INTERCEPT (or INT) as a random effect to indicate the intercept. PROC HPLMIXED does not include the intercept in the RANDOM statement by default as it does in the MODEL statement.
Table 8.5 summarizes important options in the RANDOM statement. All options are subsequently discussed in alphabetical order.
Table 8.5: Summary of Important RANDOM Statement Options
Option 
Description 

Construction of Covariance Structure 

Identifies the subjects in the model 

Specifies the covariance structure 

Statistical Output 

Determines the confidence level () 

Requests confidence limits for predictors of random effects 

Displays solutions of the random effects 
You can specify the following options in the RANDOM statement after a slash (/).
sets the confidence level to be number for each confidence interval of the randomeffects estimates. The value of number must be between 0 and 1; the default is 0.05.
requests that ttype confidence limits be constructed for each of the randomeffect estimates. The confidence level is 0.95 by default; this can be changed with the ALPHA= option.
requests that the solution for the randomeffects parameters be produced. Using notation from the section Linear Mixed Models Theory, these estimates are the empirical best linear unbiased predictors (EBLUPs), . They can be useful for comparing the random effects from different experimental units and can also be treated as residuals in performing diagnostics for your mixed model.
The numbers displayed in the SE Pred column of the "Solution for Random Effects" table are not the standard errors of the displayed in the Estimate column; rather, they are the standard errors of predictions , where is the ith EBLUP and is the ith randomeffect parameter.
identifies the subjects in your mixed model. Complete independence is assumed across subjects; thus, for the RANDOM statement, the SUBJECT= option produces a blockdiagonal structure in with identical blocks. In fact, specifying a subject effect is equivalent to nesting all other effects in the RANDOM statement within the subject effect.
When you specify the SUBJECT= option and a classification random effect, computations are usually much quicker if the levels of the random effect are duplicated within each level of the SUBJECT= effect.
specifies the covariance structure of . Valid values for covariancestructure and their descriptions are listed in Table 8.6. Although a variety of structures are available, most applications call for either TYPE=VC or TYPE=UN . The TYPE=VC (variance components) option is the default structure, and it models a different variance component for each random effect.
The TYPE=UN (unstructured) option is useful for correlated random coefficient models. For example, the following statement specifies a random interceptslope model that has different variances for the intercept and slope and a covariance between them:
random intercept age / type=un subject=person;
You can also use TYPE=FA0(2) here to request a estimate that is constrained to be nonnegative definite.
If you are constructing your own columns of with continuous variables, you can use the TYPE=TOEP (1) structure to group them together to have a common variance component. If you want to have different covariance structures in different parts of , you must use multiple RANDOM statements with different TYPE= options.
Table 8.6: Covariance Structures
Structure 
Description 
Parms 
element 

ANTE(1) 
Antedependence 


AR(1) 
Autoregressive(1) 
2 

ARH(1) 
Heterogeneous AR(1) 


ARMA(1,1) 
Autoregressive moving average(1,1) 
3 

CS 
Compound symmetry 
2 

CSH 
Heterogeneous compound symmetry 


FA(q) 
Factor analytic 


FA0(q) 
No diagonal FA 


FA1(q) 
Equal diagonal FA 


HF 
HuynhFeldt 


SIMPLE 
An alias for VC 
q 
for the kth effect 
TOEP 
Toeplitz 
t 

TOEP(q) 
Banded Toeplitz 
q 

TOEPH 
Heterogeneous TOEP 


TOEPH(q) 
Banded heterogeneous TOEP 


UN 
Unstructured 


UN(q) 
Banded 


UNR 
Unstructured correlation 


UNR(q) 
Banded correlations 


VC 
Variance components 
q 
for the kth effect 
In Table 8.6, the Parms column represents the number of covariance parameters in the structure, t is the overall dimension of the covariance matrix, and equals 1 when A is true and 0 otherwise. For example, 1 equals 1 when and 0 otherwise, and 1 equals 1 when and 0 otherwise. For the TYPE=TOEPH structures, ; for the TYPE=UNR structures, for all i.
Table 8.7 lists some examples of the structures in Table 8.6.
Table 8.7: Covariance Structure Examples
Description 
Structure 
Example 

Variance 
VC (default) 

Compound 
CS 

Unstructured 
UN 

Banded main 
UN(1) 

Firstorder 
AR(1) 

Toeplitz 
TOEP 

Toeplitz with 
TOEP(2) 

Heterogeneous 
ARH(1) 

Firstorder 
ARMA(1,1) 

Heterogeneous 
CSH 

Firstorder 
FA(1) 

HuynhFeldt 
HF 

Firstorder 
ANTE(1) 

Heterogeneous 
TOEPH 

Unstructured 
UNR 

The following list provides some further information about these covariance structures:
specifies the firstorder antedependence structure (Kenward, 1987; Patel, 1991; Macchiavelli and Arnold, 1994). In Table 8.6, is the i variance parameter, and is the k autocorrelation parameter that satisfies .
specifies a firstorder autoregressive structure. PROC HPLMIXED imposes the constraint for stationarity.
specifies a heterogeneous firstorder autoregressive structure. As with TYPE=AR(1), PROC HPLMIXED imposes the constraint for stationarity.
specifies the firstorder autoregressive moving average structure. In Table 8.6, is the autoregressive parameter, models a moving average component, and is the residual variance. In the notation of Fuller (1976, p. 68), and
The example in Table 8.7 and imply that
where and . PROC HPLMIXED imposes the constraints and for stationarity, although the resulting covariance matrix is not positive definite for some values of and in this region. When the estimated value of becomes negative, the computed covariance is multiplied by to account for the negativity.
specifies the compoundsymmetry structure, which has constant variance and constant covariance.
specifies the heterogeneous compoundsymmetry structure. This structure has a different variance parameter for each diagonal element, and it uses the square roots of these parameters in the offdiagonal entries. In Table 8.6, is the i variance parameter, and is the correlation parameter that satisfies .
specifies the factoranalytic structure with q factors (Jennrich and Schluchter, 1986). This structure is of the form , where is a rectangular matrix and is a diagonal matrix with t different parameters. When , the elements of in its upper right corner (that is, the elements in the i row and j column for ) are set to zero to fix the rotation of the structure.
is similar to the FA(q) structure except that no diagonal matrix is included. When (that is, when the number of factors is less than the dimension of the matrix), this structure is nonnegative definite but not of full rank. In this situation, you can use this structure for approximating an unstructured matrix in the RANDOM statement. When , you can use this structure to constrain to be nonnegative definite in the RANDOM statement.
is similar to the TYPE=FA(q) structure except that all of the elements in are constrained to be equal. This offers a useful and more parsimonious alternative to the full factoranalytic structure.
specifies the HuynhFeldt covariance structure (Huynh and Feldt, 1970). This structure is similar to the TYPE=CSH structure in that it has the same number of parameters and heterogeneity along the main diagonal. However, it constructs the offdiagonal elements by taking arithmetic means rather than geometric means.
You can perform a likelihood ratio test of the HuynhFeldt conditions by running PROC HPLMIXED twice, once with TYPE=HF and once with TYPE=UN , and then subtracting their respective values of times the maximized likelihood.
If PROC HPLMIXED does not converge under your HuynhFeldt model, you can specify your own starting values with the PARMS statement. The default MIVQUE(0) starting values can sometimes be poor for this structure. A good choice for starting values is often the parameter estimates that correspond to an initial fit that uses TYPE=CS .
is an alias for TYPE=VC .
specifies a banded Toeplitz structure. This can be viewed as a moving average structure with order equal to . The TYPE=TOEP option is a full Toeplitz matrix, which can be viewed as an autoregressive structure with order equal to the dimension of the matrix. The specification TYPE=TOEP(1) is the same as , where I is an identity matrix, and it can be useful for specifying the same variance component for several effects.
specifies a heterogeneous banded Toeplitz structure. In Table 8.6, is the i variance parameter and is the j correlation parameter that satisfies . If you specify the order parameter q, then PROC HPLMIXED estimates only the first q bands of the matrix, setting all higher bands equal to 0. The option TOEPH(1) is equivalent to both the TYPE=UN (1) and TYPE=UNR (1) options.
specifies a completely general (unstructured) covariance matrix that is parameterized directly in terms of variances and covariances. The variances are constrained to be nonnegative, and the covariances are unconstrained. This structure is not constrained to be nonnegative definite in order to avoid nonlinear constraints. However, you can use the TYPE=FA0 structure if you want this constraint to be imposed by a Cholesky factorization. If you specify the order parameter q, then PROC HPLMIXED estimates only the first q bands of the matrix, setting all higher bands equal to 0.
specifies a completely general (unstructured) covariance matrix that is parameterized in terms of variances and correlations. This structure fits the same model as the TYPE=UN (q) option but with a different parameterization. The i variance parameter is . The parameter is the correlation between the j and k measurements; it satisfies . If you specify the order parameter r, then PROC HPLMIXED estimates only the first q bands of the matrix, setting all higher bands equal to zero.
specifies standard variance components. This is the default structure for both the RANDOM and REPEATED statements. In the RANDOM statement, a distinct variance component is assigned to each effect.
Jennrich and Schluchter (1986) provide general information about the use of covariance structures, and Wolfinger (1996) presents details about many of the heterogeneous structures.