Using !VPGROUP to constrain variance components

Introduction

The qualifier

 !VPGROUP f1 [f2] [f3]

facilitates putting (equality) constraints across residual variance parameters. The residual model

 sat(Exp).ar1(Row).ar1v(Col)

defines the spatial residual model where each experiment (Exp) is laid out as a Row/Col grid with autoregressive (ar) correlations across rows and columns (as a direct product) and a common variance (v) within each section.

Sometimes, the grids may be small and grouped in some way and the user wants the parameters to be constrained to be equal across sections/grids within groups.

For example

 residual sat(Exp !VPGROUP Group1 Group2 Group3).ar1(Row).ar1v(Col)

where Groupi (i=1,2,3) are variables that group the data records associated with Exp. This example has 3 spatial parameters for each experiment in the order row correlation, column correlation and variance. ASReml will constain the i-th parameter according to Groupi (i=1,2,3). If only one grouping factor is listed, it will be used for all three parameters.

 residual sat(Exp !VPGROUP mu Field Exp).ar1(Row).ar1v(Col)

allows a common row autocorrelation across all sections, a field specific column autocorrelation and a distinct variance for each section. % If the same grouping is applied to all parameters the specification can be reduced to sat(Exp !VPGROUP Group1).ar1(Row).ar1v(Col)

Simple Relationships Among Variance Structure Parameters: VCC

The same functionality was available, less conveniently but more generally, with the former !VCC mechanism which has been rejigged.

Formal linear relationships between variance structure parameters can be defined by placing the VCC directive after the residual line. Unlike the case of parameter equality, all parameters can be accessed, and the linear relationship is not limited to equality. Since the parameter positions (which are given in the .tsv file) are not easily anticipated, the VCC statement begins by identifying the first model term containing a parameter to be linked to other parameters.

The syntax is
VCC term !LINK relative position [!SCALE scales] [!BLOCKSIZE s]
or
VCC termA [-]termB
where
term is the model term containing the first parameter to be linked,
termA is a model term with a single variance parameter,
termB is another model term with a single variance parameter, which is to be made equal to that for termA, or the same but with opposite sign.
!LINK relative positions      This is a list of parameter positions where 1 is the first parameter in term.
    For equally spaced positions, use ':' to represent positions between
    the second and the last; the increment is taken from the initial interval.
!SCALE scales      Sometimes, the relationship required among linked parameters is not equality.
     Hence, supply the relative size coefficient for each member of the LINK list.
!BLOCKSIZE b      It is provided for the situation where there are g groups of b parameters,
    and we wish to constrain the first parameter of each group numbered (say, s₁s₂ ... s_g
    and then constrain the second parameter, the third, and so on.
    Hence, !BLOCKSIZE allows to constrain the first set of parameters, and
    ASReml will generate the constraints for the other b-1 sets.

Some VCC examples

VCC statement	action
VCC idv(units) !LINK 1 2 !SCALE 1 -1 VCC idv(units) !LINK 1 -2 VCC idv(units) -uni(Check)	These three statements are equivalent and cause the parameter following units ( uni(Check)) in the following code sample, if placed after the residual line, to have the same magnitude but opposite sign as the parameter for units. y ~ mu Cov Ch mv !r, idv(units !INIT 1) uni(Check !INIT -1) residual ar1v(Col):id(Row)
VCC us(Env).idv(blocks) !LINK 2 4 5 7 8 9	For a (4 × 4) US matrix us(Env), the covariances are made equal.

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