Generates regressors for a general linear model.

The type double function is imsls_d_regressors_for_glm.

An integer (n_regressors) indicating the number of regressors generated.

Function imsls_f_regressors_for_glm 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 a1, a2, ..., an. From this classification variable, imsls_f_regressors_for_glm creates n indicator variables. For k = 1, 2, ..., n, we have

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:

In particular, for dummy_method = IMSLS_ALL, the dummy variables are Ak = Ik(k = 1, 2, ..., n). For dummy_method = IMSLS_LEAVE_OUT_LAST, the dummy variables are Ak = Ik(k = 1, 2, ..., n − 1). For dummy_method = IMSLS_SUM_TO_ZERO, the dummy variables are Ak = Ik − In(k = 1, 2, ..., n − 1). The regressors generated for an effect composed of a single-classification variable are the associated dummy variables.

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

and

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

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 indices_effects. 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, X1, and X2. The regressors generated by the effect indices (0, 1, 2, 2, 3) are A ⊗ B ⊗ X1X1X2.

Let the data matrix x = (A, B, X1), where A and B are classification variables and X1 is a continuous variable. The model containing the effects A, B, AB, X1, AX1, BX1, and ABX1 is specified as follows (use optional keyword IMSLS_INDICES_EFFECTS):

For this model, suppose that variable A has two levels, A1 and A2, and that variable B has three levels, B1, B2, and B3. For each dummy_method option, the regressors in their order of appearance in regressors are given below.

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, imsls_f_regressors_for_glm internally generates values for n_effects, n_var_effects, and indices_effects, which correspond to a first order model with NEF = n_continuous + n_class. 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 IMSLS_MODEL_ORDER with model_order = 2 is specified.

There are

effects, where NVAR = n_continuous + n_class. 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 n_continuous effects correspond to squares of the continuous variables. The last

effects correspond to the two-variable interactions.

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

In the following example, there are two classification variables, A and B, with two and three values, respectively. Regressors for a one-way model (the default model order) are generated using the IMSLS_ALL dummy method (the default dummy method). The five regressors generated are A1, A2, B1, B2, and B3.

In this example, a two-way analysis of covariance model containing all the interaction terms is fit. First, imsls_f_regressors_for_glm is called to produce a matrix of regressors, regressors, from the data x. Then, regressors is used as the input matrix into imsls_f_regression to produce the final fit. The regressors, generated using dummy_method = IMSLS_LEAVE_OUT_LAST, are the model whose mean function is

where