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 The MIXED Procedure

## 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.

Output 56.9.1 Coefficients of Type 3 Estimable Functions
The Mixed Procedure

Type 3 Coefficients for A
Effect A B Row1 Row2
Intercept
A 1   1
A 2     1
A 3   -1 -1
B   1
B   2
A*B 1 1 0.5
A*B 1 2 0.5
A*B 2 1   0.5
A*B 2 2   0.5
A*B 3 1 -0.5 -0.5
A*B 3 2 -0.5 -0.5

Type 3 Coefficients for B
Effect A B Row1
Intercept
A 1
A 2
A 3
B   1 1
B   2 -1
A*B 1 1 0.3333
A*B 1 2 -0.333
A*B 2 1 0.3333
A*B 2 2 -0.333
A*B 3 1 0.3333
A*B 3 2 -0.333

Type 3 Coefficients for A*B
Effect A B Row1 Row2
Intercept
A 1
A 2
A 3
B   1
B   2
A*B 1 1 1
A*B 1 2 -1
A*B 2 1   1
A*B 2 2   -1
A*B 3 1 -1 -1
A*B 3 2 1 1

Output 56.9.2 Type 3 Tests in Split-Plot Example
Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
A 2 6 4.07 0.0764
B 1 9 19.39 0.0017
A*B 2 9 4.02 0.0566

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:

Output 56.9.3 Type 3 L Components Table
L Components of Type 3 Tests of Fixed Effects
Effect L Index Estimate Standard Error DF t Value Pr > |t|
A 1 7.1250 3.1672 6 2.25 0.0655
A 2 8.3750 3.1672 6 2.64 0.0383
B 1 5.5000 1.2491 9 4.40 0.0017
A*B 1 7.7500 3.0596 9 2.53 0.0321
A*B 2 7.2500 3.0596 9 2.37 0.0419

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.

Output 56.9.4 Results from ESTIMATE Statements
The Mixed Procedure

Estimates
Label Estimate Standard Error DF t Value Pr > |t|
a 1 7.1250 3.1672 6 2.25 0.0655
a 2 8.3750 3.1672 6 2.64 0.0383
b 1 5.5000 1.2491 9 4.40 0.0017
a*b 1 7.7500 3.0596 9 2.53 0.0321
a*b 2 7.2500 3.0596 9 2.37 0.0419

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.

Output 56.9.5 Parameter Estimates in Polynomial Model
The Mixed Procedure

Solution for Fixed Effects
Effect Estimate Standard Error DF t Value Pr > |t|
Intercept 0.7837 0.3545 16 2.21 0.0420
x 0.3726 0.1426 16 2.61 0.0189
x*x 0.04756 0.01558 16 3.05 0.0076
x*x*x -0.00306 0.000489 16 -6.27 <.0001

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

Output 56.9.6 Type 1 and Type 3 L Components
L Components of Type 1 Tests of Fixed Effects
Effect L Index Estimate Standard Error DF t Value Pr > |t|
x 1 0.1763 0.01259 16 14.01 <.0001
x*x 1 -0.04886 0.002449 16 -19.95 <.0001
x*x*x 1 -0.00306 0.000489 16 -6.27 <.0001

L Components of Type 3 Tests of Fixed Effects
Effect L Index Estimate Standard Error DF t Value Pr > |t|
x 1 0.3726 0.1426 16 2.61 0.0189
x*x 1 0.04756 0.01558 16 3.05 0.0076
x*x*x 1 -0.00306 0.000489 16 -6.27 <.0001

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.

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