Computational Method

The CATMOD Procedure

Computational Method

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

j	denotes a column vector of 1s.
J	denotes a square matrix of 1s.
$\sum_k$	is the sum over all the possible values of k.
n_i	denotes the row sum $\sum_j n_{ij}$ .
DIAG_n(p)	is the diagonal matrix formed from the first n elements of the vector p.
DIAG_n^-1(p)	is the inverse of DIAG_n(p).
DIAG(A₁, A₂, ... , A_k)	denotes a block diagonal matrix with the A matrices on the main diagonal.

Input data can be represented by a contingency table, as shown in Table 22.4.

Table 22.4: Input Data Represented by a Contingency Table

	Response
Population	1	2	...	r	Total
1	n₁₁	n₁₂	...	n_1r	n₁
2	n₂₁	n₂₂	...	n_2r	n₂
$\vdots$	$\vdots$	$\vdots$	$\ddots$	$\vdots$	$\vdots$
s	n_s1	n_s2	...	n_sr	n_s

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