The OPTEX Procedure

Specifying Effects in MODEL Statements

This section discusses how to specify the linear model that you plan to fit with the design. The OPTEX procedure provides for the same general linear models as the GLM procedure, although it does not use the GLM procedure’s over-parameterized technique for generating the design matrix (see the section Static Coding.)

Each term in a model, called an effect, is a variable or combination of variables. To specify effects, you use a special notation involving variables and operators. There are two kinds of variables: classification variables and continuous variables. Classification variables separate observations into groups, and the model depends on them through these groups; on the other hand, the model depends on the actual (or coded) values of continuous variables. There are two primary operators: crossing and nesting. A third operator, the bar operator, simplifies the specification for multiple crossed terms, as in a factorial model. The @ operator, used in combination with the bar operator, further simplifies specification of crossed terms.

When specifying a model, you must list the classification variables in a CLASS statement. Any variables in the model that are not listed in the CLASS statement are assumed to be continuous. Continuous variables must be numeric.

Types of Effects

Five types of effects can be specified in the MODEL statement. Each row of the design matrix is generated by combining values for the independent variables according to effects specified in the MODEL statement. This section discusses how to specify different types of effects and explains how they relate to the columns of the design matrix. In the following, assume that A, B, and C are classification variables and X1, X2, and X3 are continuous variables.

Regressor Effects

Regressor effects are specified by writing continuous variables by themselves.

X1  X2  X3

For regressor effects, the actual values of the variable are used in the design matrix.

Polynomial Effects

Polynomial effects are specified by joining two or more continuous variables with asterisks.

X1*X1  X1*X1*X1  X1*X2  X1*X2*X3  X1*X1*X2

Polynomial effects are also referred to as interactions or crossproducts of continuous variables. When a variable is joined with itself, polynomial effects are referred to as quadratic effects, cubic effects, and so on. In the preceding examples, the first two effects are the quadratic and cubic effects for X1, respectively. The remaining effects are crossproducts.

For polynomial effects, the value used in the design matrix is the product of the values of the constituent variables.

Main Effects

If a classification variable A has k levels, then its main effect has k – 1 degrees of freedom, corresponding to k – 1 independent differences between the mean response at different levels. Main effects are specified by writing classification variables by themselves.

A  B  C

Most designs involve main effects since these correspond to the factors in your experiment. For example, in a factorial experiment for a chemical process, the main effects can be metal type, temperature, pressure, and the level of a catalyst.

For information on how the OPTEX procedure generates the k – 1 columns in the design matrix corresponding to the main effect of a classification variable, see the section Design Coding.

Crossed Effects

Crossed effects (or interactions) are specified by joining class variables with asterisks.

A*B  B*C  A*B*C

The number of degrees of freedom for a crossed effect is the product of the numbers of degrees of freedom for the constituent main effects. The columns in the design matrix corresponding to a crossed effect are formed by the horizontal direct products of the constituent main effects.

Continuous-by-Class Effects

Continuous-by-class effects are specified by joining continuous variables and classification variables with asterisks.

X1*A

The design columns for a continuous-by-class effect are constructed by multiplying the values in the design columns for the continuous variables and the classification variable.

Note that all design matrices start with a column of ones for the assumed intercept term unless you use the NOINT option in the MODEL statement.

Bar and @ Operators

You can shorten the specification of a factorial model by using the bar operator. For example, the following statements show two ways of specifying a full three-way factorial model: 

model a b c a*b a*c b*c a*b*c;
model a|b|c;

When the vertical bar (|) is used, the right- and left-hand sides become effects, and their cross becomes an effect. Multiple bars are permitted. The expressions are expanded from left to right by using rules given by Searle (1971). For example, A|B|C is evaluated as follows:

A | B | C

$\rightarrow $

{ A | B } | C

 

$\rightarrow $

{ A B A*B } | C

 

$\rightarrow $

A  B  A*B  C  A*C  B*C  A*B*C

The bar operator does not cross a variable with itself. To produce a quadratic term, you must specify it directly.

You can also specify the maximum number of variables involved in any effect that results from bar evaluation by putting it at the end of a bar effect, preceded by an @ sign. For example, the specification A|B|C@2 results in only those effects that contain two or fewer variables (in this case A, B, A*B, C, A*C, and B*C).

Examples of Models

Main Effects Model

For a three-factor main effects model with A, B, and C as the factors, the MODEL statement is 

   model a b c;
Factorial Model with Interactions

To specify interactions in a factorial model, join effects with asterisks, as described previously. For example, the following statements show two ways of specifying a complete factorial model, which includes all the interactions: 

   model a b c a*b a*c b*c a*b*c;
   model a|b|c;
Quadratic Model

The following statements show two ways of specifying a model with crossed and quadratic effects (for a central composite design, for example): 

   model x1 x2 x1*x2 x3 x1*x3 x2*x3
         x1*x1 x2*x2 x3*x3;
   model x1|x2|x3@2 x1*x1 x2*x2 x3*x3;