The CATMOD Procedure

Computational Method

Subsections:

Summary of Basic Dimensions
Notation

The notation used in PROC CATMOD differs slightly from that used in the literature. The following table provides a summary of the basic dimensions and the notation for a contingency table. See the section Computational Formulas for a complete description.

Summary of Basic Dimensions

s	=	number of populations or samples (= number of rows in the underlying contingency table)
r	=	number of response categories (= number of columns in the underlying contingency table)
q	=	number of response functions computed for each population
d	=	number of parameters

Notation

$\mb{j}$	Denotes a column vector of 1s.
$\mb{J}$	Denotes a square matrix of 1s.
$\sum _ k$	Denotes the sum over all the possible values of k.
$n_ i$	Denotes the row sum $\sum _ j n_{ij}$ .
$\mb{DIAG}_ n(\mb{p})$	Denotes the diagonal matrix formed from the first n elements of the vector $\mb{p}$ .
$\mb{DIAG}_ n^{-1}(\mb{p})$	Denotes the inverse of $\mb{DIAG}_ n(\mb{p})$ .
$\mb{DIAG}(\mb{A}_1, \mb{A}_2, \ldots , \mb{A}_ k)$	Denotes a block diagonal matrix with the $\mb{A}$ matrices on the main diagonal.

Input data can be represented by a contingency table, as shown in Table 32.7.

Table 32.7: Input Data Represented by a Contingency Table

	Response
Population	1	2	$\cdots$	r	Total
1	$n_{11}$	$n_{12}$	$\cdots$	$n_{1r}$	$n_1$
2	$n_{21}$	$n_{22}$	$\cdots$	$n_{2r}$	$n_2$
$\vdots$	$\vdots$	$\vdots$	$\ddots$	$\vdots$	$\vdots$
s	$n_{s1}$	$n_{s2}$	$\cdots$	$n_{sr}$	$n_ s$