The CATMOD Procedure

Computational Method

Subsections:

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 $

$n_{sr}$

$n_ s$