The GLM Procedure

To illustrate the four types of tests and the principles upon which they are based, consider a two-way design with interaction based on the following data:

Invoke PROC GLM and specify all the estimable functions options to examine what the GLM procedure can test. The following statements produce the summary ANOVA table displayed in Figure 42.10.

data example;
   input a b y @@;
   datalines;
1 1 23.5  1 1 23.7  1 2 28.7  2 1  8.9  2 2  5.6
2 2  8.9  3 1 10.3  3 1 12.5  3 2 13.6  3 2 14.6
;

proc glm;
   class a b;
   model y=a b a*b / e e1 e2 e3 e4;
run;

The following sections show the general form of estimable functions and discuss the four standard tests, their properties, and abbreviated output for the two-way crossed example.

Figure 42.11 is the general form of estimable functions for the example. In order to be testable, a hypothesis must be able to fit within the framework displayed here.

If a hypothesis is estimable, the s in the preceding scheme can be set to values that match the hypothesis. All the standard tests in PROC GLM can be shown in the preceding format, with some of the s zeroed and some set to functions of other s.

The following sections show how many of the hypotheses can be tested by comparing the model sum-of-squares regression from one model to a submodel. The notation used is

where SS(A effects) denotes the regression model sum of squares for the model consisting of A effects. This notation is equivalent to the reduction notation defined by Searle (1971) and summarized in Chapter 15: The Four Types of Estimable Functions.

Type I sums of squares (SS), also called sequential sums of squares, are the incremental improvement in error sums of squares as each effect is added to the model. They can be computed by fitting the model in steps and recording the difference in error sum of squares at each step.

Type I sums of squares are displayed by default because they are easy to obtain and can be used in various hand calculations to produce sum of squares values for a series of different models. Nelder (1994) and others have argued that Type I and II sums are essentially the only appropriate ones for testing ANOVA effects; however, see also the discussion of Nelder’s article, especially Rodriguez, Tobias, and Wolfinger (1995) and Searle (1995).

The Type I hypotheses have these properties:

The Type I estimable functions and associated tests for the example are shown in Figure 42.12.

The Type II tests can also be calculated by comparing the error sums of squares (SS) for subset models. The Type II SS are the reduction in error SS due to adding the term after all other terms have been added to the model except terms that contain the effect being tested. An effect is contained in another effect if it can be derived by deleting variables from the latter effect. For example, A and B are both contained in A*B. For this model, the Type II SS are given by the reduced sums of squares as shown in the following table.

Type II SS have these properties:

The Type II estimable functions and associated tests for the example are shown in Figure 42.13.

Type III and Type IV sums of squares (SS), sometimes referred to as partial sums of squares, are considered by many to be the most desirable; see Searle (1987, Section 4.6). Using PROC GLM’s singular parameterization, these SS cannot, in general, be computed by comparing model SS from different models. However, they can sometimes be computed by reduction for methods that reparameterize to full rank, when such a reparameterization effectively imposes Type III linear constraints on the parameters. In PROC GLM, they are computed by constructing a hypothesis matrix and then computing the SS associated with the hypothesis . As long as there are no missing cells in the design, Type III and Type IV SS are the same.

These are properties of Type III and Type IV SS:

The SS are constructed from the general form of estimable functions. Type III and Type IV tests are different only if the design has missing cells. In this case, the Type III tests have an orthogonality property, while the Type IV tests have a balancing property. These properties are discussed in Chapter 15: The Four Types of Estimable Functions. For this example, since the data contain observations for all pairs of levels of A and B, Type IV tests are identical to the Type III tests that are shown in Figure 42.14. (This combines tables from several pages of output.)