The SEVERITY Procedure

You can shorten the specification of a large factorial model by using the bar operator. For example, two ways of writing the model for a full three-way factorial model follow:

scalemodel A B C   A*B A*C B*C   A*B*C;

scalemodel A|B|C;

When you use the bar (|), the right and left sides become effects, and the cross of them becomes an effect. Multiple bars are permitted. The expressions are expanded from left to right, using rules 2–4 from Searle (1971, p. 390).

You can also specify the maximum number of variables involved in any effect that results from bar evaluation by specifying that maximum number, preceded by an at sign (@), at the end of the bar effect. For example, the following specification selects only those effects that contain two or fewer variables:

scalemodel A|B|C@2;

The preceding example is equivalent to the following SCALEMODEL statement:

scalemodel A B C   A*B A*C B*C;

More examples of using the bar and at sign operators follow:

Note: The preceding examples assume the following CLASS statement specification:

class A B C D;

You can simplify the specification of a large model when some of your variables have a common prefix by using the colon (:) operator and the dash (-) operator. The colon operator selects all variables that have a particular prefix, and the dash operator enables you to list variables that are numbered sequentially. For example, if your data set contains the variables X1 through X9, the following SCALEMODEL statements are equivalent:

scalemodel X1 X2 X3 X4 X5 X6 X7 X8 X9;

scalemodel X1-X9;

scalemodel X:;

If your data set contains only the three covariates X1, X2, and X9, then the colon operator selects all three variables:

scalemodel X:;

However, the following specification returns an error because X3 through X8 are not in the data set:

scalemodel X1-X9;

The double dash (- -) operator enables you to select variables that are stored sequentially in the SAS data set, whether or not they have a common prefix. You can use the CONTENTS procedure (see Base SAS Procedures Guide) to determine your variable ordering. For example, if you replace the dash in the preceding SCALEMODEL statement with a double dash, as follows, then all three variables are selected:

scalemodel X1--X9;

If your data set contains the variables A, B, and C, then you can use the double dash operator to select these variables by specifying the following:

scalemodel A--C;

Continuous variables or polynomial terms that involve them can be included in the model as continuous effects. An effect that contains a single continuous variable is referred to as a singleton continuous effect, and an effect that contains an interaction of only continuous variables is referred to as a polynomial continuous effect. The actual values of such terms are included as columns of the relevant model matrices. You can use the bar operator along with a continuous variable to generate polynomial effects. For example, X | X | X expands to X X*X X*X*X, which is a cubic model.

If a classification variable has m levels, the GLM parameterization generates m columns for its main effect in the model matrix. Each column is an indicator variable for a given level. The order of the columns is the sort order of the values of their levels and can be controlled by the ORDER= option in the CLASS statement.

Table 23.10 is an example where denotes the intercept and A and B are classification variables that have two and three levels, respectively.

There are usually more columns for these effects than there are degrees of freedom to estimate them. In other words, the GLM parameterization of main effects is singular.

Often a regression model includes interaction (crossed) effects to account for how the effect of a variable changes along with the values of other variables. In an interaction, the terms are first reordered to correspond to the order of the variables in the CLASS statement. Thus, B*A becomes A*B if A precedes B in the CLASS statement. Then, the GLM parameterization generates columns for all combinations of levels that occur in the data. The order of the columns is such that the rightmost variables in the interaction change faster than the leftmost variables, as illustrated in Table 23.11.

In the matrix in Table 23.11, main-effects columns are not linearly independent of crossed-effects columns. In fact, the column space for the crossed effects contains the space of the main effect.

When your regression model contains many interaction effects, you might be able to code them more parsimoniously by using the bar operator ( | ). The bar operator generates all possible interaction effects. For example, A | B | C expands to A B A*B C A*C B*C A*B*C. To eliminate higher-order interaction effects, use the at sign (@) in conjunction with the bar operator. For example, A | B | C | D@2 expands to A B A*B C A*C B*C D A*D B*D C*D.

Nested effects are generated in the same manner as crossed effects. Hence, the design columns that are generated by the following two statements are the same (but the ordering of the columns is different):

scalemodel A B(A);

scalemodel A A*B;

The nesting operator in PROC SEVERITY is more of a notational convenience than an operation that is distinct from crossing. Nested effects are usually characterized by the property that the nested variables do not appear as main effects. The order of the variables within nesting parentheses is made to correspond to the order of these variables in the CLASS statement. The order of the columns is such that variables outside the parentheses index faster than those inside the parentheses, and the rightmost nested variables index faster than the leftmost variables, as illustrated in Table 23.12.

When a continuous variable nests or crosses with a classification variable, the design columns are constructed by multiplying the continuous values into the design columns for the classification effect, as illustrated in Table 23.13.

Continuous-by-class effects generate the same design columns as continuous-nesting-class effects. Table 23.14 shows the construction of the X*A effect. The two columns for this effect are the same as the columns for the X(A) effect in Table 23.13.

An example that combines all the effects is X1*X2*A*B*C(D E). The continuous list comes first, followed by the crossed list, followed by the nested list in parentheses. PROC SEVERITY might rename effects to correspond to ordering rules. For example, B*A(E D) might be renamed A*B(D E) to satisfy the following:

The sequencing of the parameters that are generated by an effect is determined by the variables whose levels are indexed faster:

For example, suppose a model includes four effects—A, B, C, and D—each of which has two levels, 1 and 2. Assume the CLASS statement is

class A B C D;

Then the order of the parameters for the effect B*A(C D), which is renamed A*B(C D), is

Note that first the crossed effects B and A are sorted in the order in which they appear in the CLASS statement so that A precedes B in the parameter list. Then, for each combination of the nested effects in turn, combinations of A and B appear. The B effect changes fastest because it is rightmost in the cross list. Then A changes next fastest, and D changes next fastest after that. The C effect changes most slowly because it is leftmost in the nested list.