Generates regressors for a general linear model.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute]
public class RegressorsForGLM
Visual Basic (Declaration)
<SerializableAttribute> _
Public Class RegressorsForGLM
Visual C++
[SerializableAttribute]
public ref class RegressorsForGLM

Remarks

Class RegressorsForGLM generates regressors for a general linear model from a data matrix. The data matrix can contain classification variables as well as continuous variables. Regressors for effects composed solely of continuous variables are generated as powers and crossproducts. Consider a data matrix containing continuous variables as Columns 3 and 4. The effect indices (3, 3) generate a regressor whose i-th value is the square of the i-th value in Column 3. The effect indices (3, 4) generates a regressor whose i-th value is the product of the i-th value in Column 3 with the i-th value in Column 4.

Regressors for an effect (source of variation) composed of a single classification variable are generated using indicator variables. Let the classification variable A take on values a_1, a_2, \ldots, a_n. From this classification variable, RegressorsForGLM creates n indicator variables. For k = 1, 2, \ldots, n, we have

I_k = \left\{ \begin{array}{rl} 1 & \mbox{if } A = a_k \\ 0 & \mbox{otherwise} \end{array} \right.
For each classification variable, another set of variables is created from the indicator variables. These new variables are called dummy variables. Dummy variables are generated from the indicator variables in one of three manners:
  1. The dummies are the n indicator variables.
  2. The dummies are the first n-1 indicator variables.
  3. The n-1 dummies are defined in terms of the indicator variables so that for balanced data, the usual summation restrictions are imposed on the regression coefficients.
In particular, for dummy method All, the dummy variables are A_k = I_k \: (k = 1, 2, \ldots, n). For dummy method LeaveOutLast, the dummy variables are A_k = I_k \: (k = 1, 2, ..., n - 1). For dummy method SumToZero, the dummy variables are A_k = I_k - I_n \: (k = 1, 2, \ldots, n - 1). The regressors generated for an effect composed of a single-classification variable are the associated dummy variables.

Let m_j be the number of dummies generated for the j-th classification variable. Suppose there are two classification variables A and B with dummies

A_1, A_2, \ldots, A_{m_1}
and
B_1, B_2, \ldots, B_{m_2}

The regressors generated for an effect composed of two classification variables A and B are

\begin{array}{rl} A \otimes B = & (A_1, A_2, \ldots, A_{m_1}) \otimes (B_1, B_2, \ldots, B_{m_2}) \\ = & (A_1 B_1, A_1 B_2, \ldots, A_1 B_{m_2}, A_2, B_1, A_2 B_2, \ldots, \\ = & A_2 B_{m_2}, \ldots, A_{m_1}, B_1, A_{m_1}, B_2, \ldots, A_{m_1} B_{m_2}) \end{array}

More generally, the regressors generated for an effect composed of several classification variables and several continuous variables are given by the Kronecker products of variables, where the order of the variables is specified in SetEffects. Consider a data matrix containing classification variables in Columns 0 and 1 and continuous variables in Columns 2 and 3. Label these four columns A, B, X_1, and X_2. The regressors generated by the effect indices (0, 1, 2, 2, 3) are A \otimes B \otimes X_1 X_1 X_2

.

Remarks

Let the data matrix \mathtt{x} = (A, B, X_1), where A and B are classification variables and X_1 is a continuous variable. The model containing the effects A, B, AB, X_1, A X_1, B X_1, and A B X_1 is specified by setting nClassVariables=2 in the constructor and calling SetEffects(effects), with int effects[][] = { {0}, {1}, {0, 1}, {2}, {0, 2}, {1, 2}, {0, 1, 2} };

For this model, suppose that variable A has two levels, A_1 and A_2, and that variable B has three levels, B_1, B_2, and B_3. For each DummyMethod option, the regressors in their order of appearance in regressors are given below.

DummyMethodRegressors
AllA_1, A_2, B_1, B_2, B_3, A_1 B_1, A_1 B_2, A_1 B_3, A_2 B_1, A_2 B_2, A_2 B_3, X1, A_1 X_1, A_2 X_1, B_1 X_1, B_2 X_1, B_3 X_1, A_1 B_1 X_1, A_1 B_2 X_1, A_1 B_3 X_1, A_2 B_1 X_1, A_2 B_2 X_1, A_2 B_3 X_1
LeaveOutLastA_1, B_1, B_2, A_1 B_1, A_1 B_2, X_1, A_1 X_1, B_1 X_1, B_2 X_1, A_1 B_1 X_1, A_1 B_2 X_1
SumToZeroA_1 - A_2, B_1 - B_3, B_2 - B_3, (A_1 - A_2) (B_1 - B_2), (A_1 - A_2) (B_2 - B_3), X_1, (A_1 - A_2) X_1, (B_1 - B_3) X_1, (B_2 - B_3) X_1, (A_1 - A_2) (B_1 - B_2) X_1, (A_1 - A_2) (B_2 - B_3 )X_1
Within a group of regressors corresponding to an interaction effect, the indicator variables composing the regressors vary most rapidly for the last classification variable, next most rapidly for the next to last classification variable, etc.

By default, RegressorsForGLM internally generates values for effects which correspond to a first order model with nEffects = nContinuousVariables + nClassVariables, where nContinuousVariables is the number of continuous variables and nClassVariables is the number of classification variables. The variables then are used to create the regressor variables. The effects are ordered such that the first effect corresponds to the first column of x, the second effect corresponds to the second column of x, etc. A second order model corresponding to the columns (variables) of x is generated if ModelOrder = 2 is used.

The effects array for a first or second order model can be obtained by first using ModelOrder followed by GetEffects. This array can then be modified and used as the argument to SetEffects. This may be an easier way of setting the effects for an almost linear or quadratic model than creating the effects array from scratch.

There are

\mathtt{nEffects} = \mathtt{nClassVariables} + \mathtt{nContinuousVariables} + \frac{\mathtt{nVar} (\mathtt{nVar} - 1)}{2}
effects, where nVar = nClassVariables+nContinuousVariables. The first nVar effects correspond to the columns of x, such that the first effect corresponds to the first column of x, the second effect corresponds to the second column of x, ..., the nVar-th effect corresponds to the nVar-th column of x (i.e. x[nVar-1]). The next nContinuousVariables effects correspond to squares of the continuous variables. The last \mathtt{nVar} (\mathtt{nVar} - 1) / 2 effects correspond to the two-variable interactions.

  • Let the data matrix \mathtt{x} = (A, B, X_1), where A and B are classification variables and X_1 is a continuous variable. The effects generated and order of appearance is
    A,\: B,\: X_1,\: X_1^2,\: A B,\: A X_1,\: B X_1
  • Let the data matrix \mathtt{x} = (A, X_1, X_2), where A is a classification variable and X_1 and X_2 are continuous variables. The effects generated and order of appearance is
    A,\: X_1,\: X_2,\: X_1^2,\: X_2^2,\: A X_1,\: A X_2,\: X_1 X_2
  • Let the data matrix \mathtt{x} = (X_1, A, X_2), where A is a classification variable and X_1 and X_2 are continuous variables. The effects generated and order of appearance is
    X_1,\: A,\: X_2,\: X_1^2,\: X_2^2,\: X_1 A,\: X_1 X_2,\: A X_2

Higher-order and more complicated models can be specified using SetEffects.

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.RegressorsForGLM

See Also