Example 56.9 Examining Individual Test Components

The LCOMPONENTS option in the MODEL statement enables you to perform single-degree-of-freedom tests for individual rows of the matrix. Such tests are useful to identify interaction patterns. In a balanced layout, Type 3 components of associated with A*B interactions correspond to simple contrasts of cell mean differences.

The first example revisits the data from the split-plot design by Stroup (1989a) that was analyzed in Example 56.1. Recall that variables A and B in the following statements represent the whole-plot and subplot factors, respectively:

proc mixed data=sp;
   class a b block;
   model y = a b a*b / LComponents e3;
   random block a*block;
run;

The MIXED procedure constructs a separate matrix for each of the three fixed-effects components. The matrices are displayed in Output 56.9.1. The tests for fixed effects are shown in Output 56.9.2.

If denotes a whole-plot main effect mean, denotes a subplot main effect mean, and denotes a cell mean, the five components shown in Output 56.9.3 correspond to tests of the following:

The first three components are comparisons of marginal means. The fourth component compares the effect of factor B at the first whole-plot level against the effect of B at the third whole-plot level. Finally, the last component tests whether the factor B effect changes between the second and third whole-plot level.

The Type 3 component tests can also be produced with these corresponding ESTIMATE statements:

proc mixed data=sp;
   class a b block ;
   model y = a b a*b;
   random block a*block;
   estimate 'a    1' a 1 0 -1;
   estimate 'a    2' a 0 1 -1;
   estimate 'b    1' b   1 -1;
   estimate 'a*b  1' a*b 1 -1 0  0 -1 1;
   estimate 'a*b  2' a*b 0  0 1 -1 -1 1;
   ods select Estimates;
run;

The results are shown in Output 56.9.4.

A second useful application of the LCOMPONENTS option is in polynomial models, where Type 1 tests are often used to test the entry of model terms sequentially. The SOLUTION option in the MODEL statement displays the regression coefficients that correspond to a Type 3 analysis. That is, the coefficients represent the partial coefficients you would get by adding the regressor variable last in a model containing all other effects, and the tests are identical to those in the "Type 3 Tests of Fixed Effects" table.

Consider the following DATA step and the fit of a third-order polynomial regression model.

data polynomial;
  do x=1 to 20; input y@@; output; end;
  datalines;
1.092   1.758   1.997   3.154   3.880
3.810   4.921   4.573   6.029   6.032
6.291   7.151   7.154   6.469   7.137
6.374   5.860   4.866   4.155   2.711
;

proc mixed data=polynomial;
  model y = x x*x x*x*x / s lcomponents htype=1,3;
run;

The t tests displayed in the "Solution for Fixed Effects" table are Type 3 tests, sometimes referred to as partial tests. They measure the contribution of a regressor in the presence of all other regressor variables in the model.

The Type 3 L components are identical to the tests in the "Solutions for Fixed Effects" table shown in Output 56.9.5. The Type 1 table yields the following:

• sequential (Type 1) tests of regression variables that test the significance of a regressor given all other variables preceding it in the model list

• the regression coefficients for sequential submodels

The estimate of is the regression coefficient in a simple linear regression of Y on X. The estimate of is the partial coefficient for the quadratic term when it is added to a model containing only a linear component. Similarly, the value is the partial coefficient for the cubic term when it is added to a model containing a linear and quadratic component. The last Type 1 component is always identical to the corresponding Type 3 component.