The GLIMMIX Procedure

determines the bucket size b. A larger bucket size will result in fewer knots. For k-d trees in more than one dimension, the correspondence between bucket size and number of knots is difficult to determine. It depends on the data configuration and on other suboptions. In the multivariate case, you might need to try out different bucket sizes to obtain the desired number of knots. The default value of number is 4 for univariate trees (a single random effect) and in the multidimensional case.

specifies whether the knots are based on vertices of the tree cells or the centroid. The two possible values of type are VERTEX and CENTER. The default is KNOTTYPE=VERTEX. For multidimensional smoothing, such as smoothing across irregularly shaped spatial domains, the KNOTTYPE=CENTER option is useful to move knot locations away from the bounding hypercube toward the convex hull.

specifies that knot coordinates are the coordinates of the nearest neighbor of either the centroid or vertex of the cell, as determined by the KNOTTYPE= suboption.

displays details about the construction of the k-d tree, such as the cell splits and the split values.

specifies a first-order ante-dependence structure (Kenward, 1987; Patel, 1991) parameterized in terms of variances and correlation parameters. If t ordered random variables have a first-order ante-dependence structure, then each , , is independent of all other , given . This Markovian structure is characterized by its inverse variance matrix, which is tridiagonal. Parameterizing an ANTE(1) structure for a random vector of size t requires 2t – 1 parameters: variances and t – 1 correlation parameters . The covariances among random variables and are then constructed as

PROC GLIMMIX constrains the correlation parameters to satisfy . For variable-order ante-dependence models see Macchiavelli and Arnold (1994).

specifies a first-order autoregressive structure,

The values and are derived for the ith and jth observations, respectively, and are not necessarily the observation numbers. For example, in the following statements the values correspond to the class levels for the time effect of the ith and jth observation within a particular subject:

proc glimmix;
   class time patient;
   model y = x x*x;
   random time / sub=patient type=ar(1);
run;
  

PROC GLIMMIX imposes the constraint for stationarity.

specifies a heterogeneous first-order autoregressive structure,

with . This covariance structure has the same correlation pattern as the TYPE=AR(1) structure, but the variances are allowed to differ.

specifies the first-order autoregressive moving-average structure,

Here, is the autoregressive parameter, models a moving-average component, and is a scale parameter. In the notation of Fuller (1976, p. 68), and

The example in Table 41.19 and imply that

where and . PROC GLIMMIX 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 an unstructured variance-covariance matrix parameterized through its Cholesky root. This parameterization ensures that the resulting variance-covariance matrix is at least positive semidefinite. If all diagonal values are nonzero, it is positive definite. For example, a unstructured covariance matrix can be written as

Without imposing constraints on the three parameters, there is no guarantee that the estimated variance matrix is positive definite. Even if and are nonzero, a large value for can lead to a negative eigenvalue of . The Cholesky root of a positive definite matrix is a lower triangular matrix such that . The Cholesky root of the above matrix can be written as

The elements of the unstructured variance matrix are then simply , , and . Similar operations yield the generalization to covariance matrices of higher orders.

For example, the following statements model the covariance matrix of each subject as an unstructured matrix:

proc glimmix;
   class sub;
   model y = x;
   random _residual_ / subject=sub type=un;
run;
  

The next set of statements accomplishes the same, but the estimated matrix is guaranteed to be nonnegative definite:

proc glimmix;
   class sub;
   model y = x;
   random _residual_ / subject=sub type=chol;
run;
  

The GLIMMIX procedure constrains the diagonal elements of the Cholesky root to be positive. This guarantees a unique solution when the matrix is positive definite.

The optional order parameter determines how many bands below the diagonal are modeled. Elements in the lower triangular portion of in bands higher than q are set to zero. If you consider the resulting covariance matrix , then the order parameter has the effect of zeroing all off-diagonal elements that are at least q positions away from the diagonal.

Because of its good computational and statistical properties, the Cholesky root parameterization is generally recommended over a completely unstructured covariance matrix (TYPE=UN). However, it is computationally slightly more involved.

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

The compound symmetry structure arises naturally with nested random effects, such as when subsampling error is nested within experimental error. The models constructed with the following two sets of GLIMMIX statements have the same marginal variance matrix, provided is positive:

proc glimmix;
   class block A;
   model y = block A;
   random block*A / type=vc;
run;
  

proc glimmix;
   class block A;
   model y = block A;
   random _residual_ / subject=block*A
                       type=cs;
run;
  

In the first case, the block*A random effect models the G-side experimental error. Because the distribution defaults to the normal, the matrix is of form (see Table 41.20), and is the subsampling error variance. The marginal variance for the data from a particular experimental unit is thus . This matrix is of compound symmetric form.

Hierarchical random assignments or selections, such as subsampling or split-plot designs, give rise to compound symmetric covariance structures. This implies exchangeability of the observations on the subunit, leading to constant correlations between the observations. Compound symmetric structures are thus usually not appropriate for processes where correlations decline according to some metric, such as spatial and temporal processes.

Note that R-side compound-symmetry structures do not impose any constraint on . You can thus use an R-side TYPE=CS structure to emulate a variance-component model with unbounded estimate of the variance component.

specifies the heterogeneous compound-symmetry structure, which is an equi-correlation structure but allows for different variances

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 , the elements of in its upper-right corner (that is, the elements in the ith row and jth column for ) are set to zero to fix the rotation of the structure.

specifies a factor-analytic structure with q factors of the form , where is a rectangular matrix and t is the dimension of . When , is a lower triangular matrix. 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 it to approximate an unstructured covariance matrix.

specifies a covariance structure that satisfies the general Huynh-Feldt condition (Huynh and Feldt, 1970). For a random vector with t elements, this structure has positive parameters and covariances

A covariance matrix generally satisfies the Huynh-Feldt condition if it can be written as . The preceding parameterization chooses . Several simpler covariance structures give rise to covariance matrices that also satisfy the Huynh-Feldt condition. For example, TYPE=CS, TYPE=VC, and TYPE=UN(1) are nested within TYPE=HF. You can use the COVTEST statement to test the HF structure against one of these simpler structures. Note also that the HF structure is nested within an unstructured covariance matrix.

The TYPE=HF covariance structure can be sensitive to the choice of starting values and the default MIVQUE(0) starting values can be poor for this structure; you can supply your own starting values with the PARMS statement.

specifies a general linear covariance structure with q parameters. This structure consists of a linear combination of known matrices that you input with the LDATA= option. Suppose that you want to model the covariance of a random vector of length t, and further suppose that are symmetric ) matrices constructed from the information in the LDATA= data set. Then,

where denotes the element in row i, column j of matrix .

Linear structures are very flexible and general. You need to exercise caution to ensure that the variance matrix is positive definite. Note that PROC GLIMMIX does not impose boundary constraints on the parameters of a general linear covariance structure. For example, if classification variable A has 6 levels, the following statements fit a variance component structure for the random effect without boundary constraints:

data ldata;
   retain parm 1 value 1;
   do row=1 to 6; col=row; output; end;
run;

proc glimmix data=MyData;
   class A B;
   model Y = B;
   random A / type=lin(1) ldata=ldata;
run;
  

requests that PROC GLIMMIX form a B-spline basis and fits a penalized B-spline (P-spline, Eilers and Marx 1996) with random spline coefficients. This covariance structure is available only for G-side random effects and only a single continuous random effect can be specified with TYPE=PSPLINE. As for TYPE=RSMOOTH, PROC GLIMMIX forms a modified matrix and fits a mixed model in which the random variables associated with the columns of are independent with a common variance. The matrix is constructed as follows.

Denote as the matrix of B-splines of degree d and denote as the matrix of rth-order differences. For example, for K = 5,

Then, the matrix used in fitting the mixed model is the matrix

The construction of the B-spline knots is controlled with the KNOTMETHOD= EQUAL(m) option and the DEGREE=d suboption of TYPE=PSPLINE. The total number of knots equals the number m of equally spaced interior knots plus d knots at the low end and knots at the high end. The number of columns in the B-spline basis equals K = m + d + 1. By default, the interior knots exclude the minimum and maximum of the random-effect values and are based on m – 1 equally spaced intervals. Suppose and are the smallest and largest random-effect values; then interior knots are placed at

In addition, d evenly spaced exterior knots are placed below and exterior knots are placed above . The exterior knots are evenly spaced and start at times the machine epsilon. For example, based on the defaults d = 3, r = 3, the following statements lead to 26 total knots and 21 columns in , m = 20, K = m + d + 1 = 24, K – r = 21:

proc glimmix;
   model y = x;
   random x / type=pspline knotmethod=equal(20);
run;

Details about the computation and properties of B-splines can be found in de Boor (2001). You can extend or limit the range of the knots with the KNOTMIN= and KNOTMAX= options. Table 41.18 lists some of the parameters that control this covariance type and their relationships.

You can specify the following options for TYPE=PSPLINE:

specifies a radial smoother covariance structure for G-side random effects. This results in an approximate low-rank thin-plate spline where the smoothing parameter is obtained by the estimation method selected with the METHOD= option of the PROC GLIMMIX statement. The smoother is based on the automatic smoother in Ruppert, Wand, and Carroll (2003, Chapter 13.4–13.5), but with a different method of selecting the spline knots. See the section Radial Smoothing Based on Mixed Models for further details about the construction of the smoother and the knot selection.

Radial smoothing is possible in one or more dimensions. A univariate smoother is obtained with a single random effect, while multiple random effects in a RANDOM statement yield a multivariate smoother. Only continuous random effects are permitted with this covariance structure. If denotes the number of continuous random effects in the RANDOM statement, then the covariance structure of the random effects is determined as follows. Suppose that denotes the vector of random effects for the ith observation. Let denote the vector of knot coordinates, , and K is the total number of knots. The Euclidean distance between the knots is computed as

and the distance between knots and effects is computed as

The matrix for the GLMM is constructed as

where the matrix has typical element

and the matrix has typical element

The exponent in these expressions equals , where the optional value m corresponds to the derivative penalized in the thin-plate spline. A larger value of m will yield a smoother fit. The GLIMMIX procedure requires p > 0 and chooses by default m = 2 if and otherwise. The NOLOG option removes the and terms from the computation of the and matrices when is even; this yields invariance under rescaling of the coordinates.

Finally, the components of are assumed to have equal variance . The “smoothing parameter” of the low-rank spline is related to the variance components in the model, . See Ruppert, Wand, and Carroll (2003) for details. If the conditional distribution does not provide a scale parameter , you can add a single R-side residual parameter.

The knot selection is controlled with the KNOTMETHOD= option. The GLIMMIX procedure selects knots automatically based on the vertices of a k-d tree or reads knots from a data set that you supply. See the section Radial Smoothing Based on Mixed Models for further details on radial smoothing in the GLIMMIX procedure and its connection to a mixed model formulation.

is an alias for TYPE=VC.

models an exponential spatial or temporal covariance structure, where the covariance between two observations depends on a distance metric . The c-list contains the names of the numeric variables used as coordinates to determine distance. For a stochastic process in , there are k elements in c-list. If the vectors of coordinates for observations i and j are and , then PROC GLIMMIX computes the Euclidean distance

The covariance between two observations is then

The parameter is not what is commonly referred to as the range parameter in geostatistical applications. The practical range of a (second-order stationary) spatial process is the distance at which the correlations fall below 0.05. For the SP(EXP) structure, this distance is . PROC GLIMMIX constrains to be positive.

models a Gaussian covariance structure,

See TYPE=SP(EXP) for the computation of the distance . The parameter is related to the range of the process as follows. If the practical range is defined as the distance at which the correlations fall below 0.05, then . PROC GLIMMIX constrains to be positive. See TYPE=SP(EXP) for the computation of the distance from the variables specified in c-list.

models a covariance structure in the Matérn class of covariance functions (Matérn, 1986). The covariance is expressed in the parameterization of Handcock and Stein (1993); Handcock and Wallis (1994); it can be written as

The function is the modified Bessel function of the second kind of (real) order . The smoothness (continuity) of a stochastic process with covariance function in the Matérn class increases with . This class thus enables data-driven estimation of the smoothness properties of the process. 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)).

Note that the MIXED procedure offers covariance structures in the Matérn class in two parameterizations, TYPE=SP(MATERN) and TYPE=SP(MATHSW). The TYPE=SP(MAT) in the GLIMMIX procedure is equivalent to TYPE=SP(MATHSW) in the MIXED procedure.

Computation of the function and its derivatives is numerically demanding; fitting models with Matérn covariance structures can be time-consuming. Good starting values are essential.

models a power covariance structure,

where . This is a reparameterization of the exponential structure, TYPE=SP(EXP). Specifically, . 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.

models an anisotropic power covariance structure in k dimensions, provided that the coordinate list c-list has k elements. If denotes the coordinate for the ith observation of the mth variable in c-list, the covariance between two observations is given by

Note that for k = 1, TYPE=SP(POWA) is equivalent to TYPE=SP(POW), which is itself a reparameterization of TYPE=SP(EXP). When the estimated value of becomes negative, the computed covariance is multiplied by to account for the negativity.

models a spherical covariance structure,

The spherical covariance structure has a true range parameter. The covariances between observations are exactly zero when their distance exceeds . See TYPE=SP(EXP) for the computation of the distance from the variables specified in c-list.

models a Toeplitz covariance structure. This structure can be viewed as an autoregressive structure with order equal to the dimension of the matrix,

specifies a banded Toeplitz structure,

This can be viewed as a moving-average structure with order equal to q – 1. The specification TYPE=TOEP(1) is the same as , and it can be useful for specifying the same variance component for several effects.

models a Toeplitz covariance structure. The correlations of this structure are banded as the TOEP or TOEP(q) structures, but the variances are allowed to vary:

The correlation parameters satisfy . If you specify the optional value q, the correlation parameters with are set to zero, creating a banded correlation structure. The specification TYPE=TOEPH(1) results in a diagonal covariance matrix with heterogeneous variances.

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=CHOL structure if you want this constraint to be imposed by a Cholesky factorization. If you specify the order parameter q, then PROC GLIMMIX estimates only the first q bands of the matrix, setting elements in all higher bands equal to 0.

specifies a completely general (unstructured) covariance matrix parameterized in terms of variances and correlations,

where denotes the standard deviation and the correlation is zero when and when , provided the order parameter q is given. 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 ith and jth measurements; it satisfies . If you specify the order parameter q, then PROC GLIMMIX estimates only the first q bands of the matrix, setting all higher bands equal to zero.

specifies standard variance components and is the default structure for both G-side and R-side covariance structures. In a G-side covariance structure, a distinct variance component is assigned to each effect. In an R-side structure TYPE=VC is usually used only to add overdispersion effects or with the GROUP= option to specify a heterogeneous variance model.