The MIXED Procedure

specifies the first-order antedependence structure (see Kenward 1987; Patel 1991; Macchiavelli and Arnold 1994). In Table 17, is the ith variance parameter, and is the kth autocorrelation parameter satisfying .

specifies a first-order autoregressive structure. PROC MIXED imposes the constraint for stationarity.

specifies a heterogeneous first-order autoregressive structure. As with TYPE=AR(1), PROC MIXED imposes the constraint for stationarity.

specifies the first-order autoregressive moving-average structure. In Table 17, 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 19 and imply that

where and . PROC MIXED imposes the constraints and for stationarity, although for some values of and in this region the resulting covariance matrix is not positive definite. When the estimated value of becomes negative, the computed covariance is multiplied by to account for the negativity.

specifies the compound-symmetry structure, which has constant variance and constant covariance.

specifies the heterogeneous compound-symmetry structure. This structure has a different variance parameter for each diagonal element, and it uses the square roots of these parameters in the off-diagonal entries. In Table 17, is the ith variance parameter, and is the correlation parameter satisfying .

specifies the factor-analytic 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 q > 1, the elements of in its upper-right corner (that is, the elements in the ith row and jth column for j > i) 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 q < t—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 it for approximating an unstructured matrix in the RANDOM statement or for combining with the LOCAL option in the REPEATED statement. When q = t, 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 factor-analytic structure.

specifies the Huynh-Feldt 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 off-diagonal elements by taking arithmetic rather than geometric means.

You can perform a likelihood ratio test of the Huynh-Feldt conditions by running PROC MIXED twice, once with TYPE=HF and once with TYPE=UN, and then subtracting their respective values of –2 times the maximized likelihood.

If PROC MIXED does not converge under your Huynh-Feldt 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 corresponding to an initial fit that uses TYPE=CS.

specifies the general linear covariance structure with q parameters. This structure consists of a linear combination of known matrices that are input with the LDATA= option. This structure is very general, and you need to make sure that the variance matrix is positive definite. By default, PROC MIXED sets the initial values of the parameters to 1. You can use the PARMS statement to specify other initial values.

is an alias for TYPE=LIN(q).

is an alias for TYPE=VC.

specifies the spatial anisotropic exponential structure, where c-list is a list of variables indicating the coordinates. This structure has element equal to

where c is the number of coordinates and is the absolute distance between the kth coordinate () of the ith and jth observations in the input data set. There are 2c + 1 parameters to be estimated: , (), and .

You might want to constrain some of the EXPA parameters to known values. For example, suppose you have three coordinate variables C1, C2, and C3 and you want to constrain the powers to equal 2, as in Sacks et al. (1989). Suppose further that you want to model covariance across the entire input data set and you suspect the and estimates are close to 3, 4, 5, and 1, respectively. Then specify the following statements:

repeated / type=sp(expa)(c1 c2 c3)
   subject=intercept;
parms (3) (4) (5) (2) (2) (2) (1) /
   hold=4,5,6;

specify modification of the isotropic SP(EXP) covariance structure.

specify modification of the isotropic SP(GAU) covariance structure.

specify modification of the isotropic SP(SPH) covariance structure.

These are structures that allow for geometric anisotropy in two dimensions. The coordinates are specified by the variables c1 and c2.

If the spatial process is geometrically anisotropic in , then it is isotropic in the coordinate system

for a properly chosen angle and scaling factor . Elliptical isocorrelation contours are thereby transformed to spherical contours, adding two parameters to the respective isotropic covariance structures. Euclidean distances (see Table 18) are expressed in terms of .

The angle of the clockwise rotation is reported in radians, . The scaling parameter represents the ratio of the range parameters in the direction of the major and minor axis of the correlation contours. In other words, following a rotation of the coordinate system by angle , isotropy is achieved by compressing or magnifying distances in one coordinate by the factor .

Fixing reduces the models to isotropic ones for any angle of rotation. If the scaling parameter is held constant at 1.0, you should also hold constant the angle of rotation, as in the following statements:

repeated / type=sp(expga)(gxc gyc)
           subject=intercept;
parms (6) (1.0) (0.0) (1) / hold=2,3;

If is fixed at any other value than 1.0, the angle of rotation can be estimated. Specifying a starting grid of angles and scaling factors can considerably improve the convergence properties of the optimization algorithm for these models. Only a single random effect with geometrically anisotropic structure is permitted.

specifies covariance structures in the Matérn class of covariance functions (Matérn 1986). Two observations for the same subject (block of ) that are Euclidean distance apart have covariance

where is the modified Bessel function of the second kind of (real) order . The smoothness (continuity) of a stochastic process with covariance function in this class increases with . The Matérn class thus enables data-driven estimation of the smoothness properties. The covariance is identical to the exponential model for (TYPE=SP(EXP)(c-list)), while for the model advocated by Whittle (1954) results. As the model approaches the gaussian covariance structure (TYPE=SP(GAU)(c-list)).

The MATHSW structure represents the Matérn class in the parameterization of Handcock and Stein (1993) and Handcock and Wallis (1994),

Since computation of the function and its derivatives is numerically very intensive, fitting models with Matérn covariance structures can be more time-consuming than with other spatial covariance structures. Good starting values are essential.

specifies the spatial power structures. When the estimated value of becomes negative, the computed covariance is multiplied by to account for the negativity.

specifies a linear exponent autoregressive (LEAR) correlation structure as proposed by Simpson et al. (2010). For two observations with distance metric , the covariance is

where and are the smallest and largest distance between any two observations, is the decay speed, and . See TYPE=SP(EXP) for the computation of the distance from the variables specified in c-list. When the estimated value of becomes negative, the computed covariance is multiplied by to account for the negativity.

For power analysis of repeated measures designs that have a LEAR correlation structure, see the section POWER Statement in Chapter 55, The GLMPOWER Procedure.

Note that TYPE=SP(LEAR) is not supported for GROUP= option in this SAS release.

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 17, is the ith variance parameter and is the jth correlation parameter satisfying . If you specify the order parameter q, then PROC MIXED 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 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 MIXED estimates only the first q bands of the matrix, setting all higher bands equal to 0.

specifies a completely general (unstructured) covariance matrix 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 ith variance parameter is . The parameter is the correlation between the jth and kth measurements; it satisfies . If you specify the order parameter r, then PROC MIXED estimates only the first q bands of the matrix, setting all higher bands equal to zero.

specify direct (Kronecker) product structures designed for multivariate repeated measures (see Galecki 1994). These structures are constructed by taking the Kronecker product of an unstructured matrix (modeling covariance across the multivariate observations) with an additional covariance matrix (modeling covariance across time or another factor). The upper-left value in the second matrix is constrained to equal 1 to identify the model. See the SAS/IML User's Guide for more details about direct products.

To use these structures in the REPEATED statement, you must specify two distinct REPEATED effects, both of which must be included in the CLASS statement. The first effect indicates the multivariate observations, and the second identifies the levels of time or some additional factor. Note that the input data set must still be constructed in "univariate" format; that is, all dependent observations are still listed observation-wise in one single variable. Although this construction provides for general modeling possibilities, it forces you to construct variables indicating both dimensions of the Kronecker product.

For example, suppose your observed data consist of heights and weights of several children measured over several successive years. Your input data set should then contain variables similar to the following:

Your PROC MIXED statements for a Kronecker AR(1) structure across years would then be as follows:

proc mixed;
   class Var Year Child;
   model Y = Var Year Var*Year;
   repeated Var Year / type=un@ar(1)
                       subject=Child;
run;

You should nearly always want to model different means for the multivariate observations; hence the inclusion of Var in the MODEL statement. The preceding mean model consists of cell means for all combinations of VAR and YEAR.

specifies standard variance components and is the default structure for both the RANDOM and REPEATED statements. In the RANDOM statement, a distinct variance component is assigned to each effect. In the REPEATED statement, this structure is usually used only with the GROUP= option to specify a heterogeneous variance model.