RESTRICT
<’label’> constraintspecification <, …, constraintspecification>
<operator <value>> </ option> ;
The RESTRICT statement enables you to specify linear equality or inequality constraints among the parameters of a mixture model. These restrictions are incorporated into the maximum likelihood analysis. The RESTRICT statement is not available for a Bayesian analysis with the FMM procedure.
Following are reasons why you might want to place constraints and restrictions on the model parameters:
to fix a parameter at a particular value
to equate parameters in different components in a mixture
to impose order conditions on the parameters in a model
to specify contrasts among the parameters that the fitted model should honor
A restriction is composed of a lefthand side and a righthand side, separated by an operator. If the operator and righthand side are not specified, the restriction is assumed to be an equality constraint against zero. If the righthand side is not specified, the value is assumed to be zero.
An individual constraintspecification is written in (nearly) the same form as estimable linear functions are specified in the ESTIMATE statement of the GLM, MIXED, or GLIMMIX procedure. The constraintspecification takes the form
modeleffect valuelist < …modeleffect valuelist > <(SCALE= value)>
At least one modeleffect must be specified followed by one or more values in the valuelist. The values in the list correspond to the multipliers of the corresponding parameter that is associated with the position in the model effect. If you specify more values in the valuelist than the modeleffect occupies in the model design matrix, the extra coefficients are ignored.
To specify restrictions for effects in specific components in the model, separate the constraintspecification by commas. The following statements provide an example:
proc fmm; class A; model y/n = A x / k = 2; restrict A 1 0 1; restrict x 2, x 1 >= 0.5; run;
The linear predictors for this twocomponent model can be written as




where is the binary variable associated with the kth level of A
.
The first RESTRICT statement applies only to the first component and specifies that the parameter estimates that are associated
with the first and third level of the A
effect are identical. In terms of the linear predictor, the restriction can be written as
Now suppose that A
has only two levels. Then the FMM procedure ignores the value –1 in the first RESTRICT statement and imposes the restriction
on the fitted model.
The second RESTRICT statement involves parameters in two different components of the model. In terms of the linear predictors, the restriction can be written as
When restrictions are specified explicitly through the RESTRICT statement or implied through the EQUATE=EFFECTS option in the MODEL statement, the FMM procedure lists all restrictions after the model fit in a table of linear constraints and indicates whether a particular constraint is active at the converged solution.
The following operators can be specified to separate the left and righthand sides of the restriction: =
, >
, <
, >=
, <=
.
Some distributions involve scale parameters (the parameter in the expressions of the log likelihood) and you can also use the constraintspecification to involve a component’s scale parameter in a constraint. To this end, assign a value to the keyword SCALE, separated from the model effects and value lists with parentheses. The following statements fit a twocomponent normal model and restrict the component variances to be equal:
proc fmm; model y = / k=2; restrict int 0 (scale 1), int 0 (scale 1); run;
The intercept specification is necessary because each constraintspecification requires at least one model effect. The zero coefficient ensures that the intercepts are not involved in the restriction. Instead, the RESTRICT statement leads to .
You can specify the following option in the RESTRICT statement after a slash (/).