Shared Concepts

Nonsingular Parameterization

When a parameterization of main effects provides the same number of columns for the effects as there are degrees of freedom to estimate them, the parameterization is called nonsingular. A variety of nonsingular parameterizations for classification effects are available for many actions in this book. In most of these actions you use the param subparameter in the class parameter to specify the parameterization.

Consider a model with one classification variable A that has four levels, 1, 2, 5, and 7. Details of the possible choices for the param subparameter follow.

REFERENCE

Three columns are created to indicate group membership of the nonreference levels. For the reference level, all three dummy variables have a value of 0. For example, if the reference level is 7, the design matrix columns for A are as follows.

Reference Coding
Design Matrix
A A1 A2 A5
1 1 0 0
2 0 1 0
5 0 0 1
7 0 0 0

Parameter estimates of classification main effects that use the reference coding scheme estimate the difference in the effect of each nonreference level compared to the effect of the reference level.

EFFECT

Three columns are created to indicate group membership of the nonreference levels. For the reference level, all three dummy variables have a value of –1. For example, if the reference level is 7, the design matrix columns for A are as follows.

Effect Coding
Design Matrix
A A1 A2 A5
1 1 0 0
2 0 1 0
5 0 0 1
7 –1 –1 –1

Parameter estimates of classification main effects that use the effect coding scheme estimate the difference in the effect of each nonreference level compared to the average effect over all four levels.

ORDINAL | THERMOMETER

Three columns are created to indicate group membership of the higher levels of the effect. For the first level of the effect (which for A is 1), all three dummy variables have a value of 0. The design matrix columns for A are as follows.

Ordinal Coding
Design Matrix
A A2 A5 A7
1 0 0 0
2 1 0 0
5 1 1 0
7 1 1 1

The first level of the effect is a control or baseline level. Parameter estimates of classification main effects, using the ordinal coding scheme, estimate the differences between effects of successive levels. When the parameters have the same sign, the effect is monotonic across the levels.

POLYNOMIAL | POLY

Three columns are created. The first represents the linear term left-parenthesis x right-parenthesis, the second represents the quadratic term left-parenthesis x squared right-parenthesis, and the third represents the cubic term left-parenthesis x cubed right-parenthesis, where x is the level value. If the classification levels are not numeric, they are translated into 1, 2, 3, ellipsis according to their sort order. The design matrix columns for A are as follows.

Polynomial Coding
Design Matrix
A APOLY1 APOLY2 APOLY3
1 1 1 1
2 2 4 8
5 5 25 125
7 7 49 343

ORTHEFFECT

The columns are obtained by applying the Gram-Schmidt orthogonalization to the columns for the EFFECT parameterization. The design matrix columns for A are as follows.

Orthogonal Effect Coding
Design Matrix
A AOEFF1 AOEFF2 AOEFF3
1 1.41421 –0.81650 –0.57735
2 0 1.63299 –0.57735
5 0 0 1.73205
7 –1.41421 –0.81649 –0.57735

ORTHORDINAL | ORTHOTHERM

The columns are obtained by applying the Gram-Schmidt orthogonalization to the columns for the ORDINAL parameterization. The design matrix columns for A are as follows.

Orthogonal Ordinal Coding
Design Matrix
A AOORD1 AOORD2 AOORD3
1 –1.73205 0 0
2 0.57735 –1.63299 0
5 0.57735 0.81650 –1.41421
7 0.57735 0.81650 1.41421

ORTHPOLY

The columns are obtained by applying the Gram-Schmidt orthogonalization to the columns for POLY parameterization. The design matrix columns for A are as follows.

Orthogonal Polynomial Coding
Design Matrix
A AOPOLY1 AOPOLY2 AOPOLY5
1 –1.15311 0.90712 –0.92058
2 –0.73380 –0.54041 1.47292
5 0.52414 –1.37034 –0.92058
7 1.36277 1.00363 0.36823

ORTHREF

The columns are obtained by applying the Gram-Schmidt orthogonalization to the columns for the REFERENCE parameterization. The design matrix columns for A are as follows.

Orthogonal Reference Coding
Design Matrix
A AOREF1 AOREF2 AOREF3
1 1.73205 0 0
2 –0.57735 1.63299 0
5 –0.57735 –0.81650 1.41421
7 –0.57735 –0.81650 –1.41421

Last updated: March 05, 2026