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 The CATMOD Procedure

## Computational Formulas

The following calculations are shown for each population and then for all populations combined.

 Source Formula Dimension Probability Estimates jth response pij = [(nij)/(ni)] 1 ×1 ith population r ×1 all populations sr ×1 Variance of Probability Estimates ith population Vi = [1/(ni)] (DIAG(pi) - pi pi') r ×r all populations V = DIAG(V1, V2, ... , Vs ) sr ×sr Response Functions ith population Fi = F(pi) q ×1 all populations sq ×1 Derivative of Function with Respect to Probability Estimates ith population q ×r all populations H = DIAG(H1, H2, ... , Hs ) sq ×sr Variance of Functions ith population Si = Hi Vi Hi' q ×q all populations S = DIAG(S1, S2, ... , Ss ) sq ×sq Inverse Variance of Functions ith population Si = (Si)-1 q ×q all populations S-1 = DIAG(S1, S2, ... , Ss ) sq ×sq

### Derivative Table for Compound Functions: Y=F(G(p))

In the following table, let G(p) be a vector of functions of p, and let D denote , which is the first derivative matrix of G with respect to p.

 Function Y = F(G) Derivative Multiply matrix Y = A*G A*D Logarithm Y = LOG(G) DIAG-1(G)*D Exponential Y = EXP(G) DIAG(Y)*D Add constant Y = G + A D

### Default Response Functions: Generalized Logits

In the following table, subscripts i for the population are suppressed. Also denote fj = log( [(pj)/(pr)] ) for j = 1, ... , r-1 for each population i = 1, ... , s.

 Inverse of Response Functions for a Population Form of F and Derivative for a Population Covariance Results for a Population

The following calculations are shown for each population and then for all populations combined.
 Source Formula Dimension Design Matrix ith population Xi q ×d all populations sq ×d Crossproduct of Design Matrix ith population Ci = Xi' Si Xi d ×d all populations d ×d Crossproduct of Design Matrix with Function d ×1 Weighted Least-Squares Estimates b = C-1 R = (X' S-1 X)-1 (X' S-1 F) d ×1 Covariance of Weighted Least-Squares Estimates COV(b) = C-1 d ×d Predicted Response Functions sq ×1 Covariance of Predicted Response Functions sq ×sq Residual Chi-Square RSS 1 ×1 Chi-Square for Q = (Lb)' (LC-1 L')-1 (Lb) 1 ×1

### Maximum Likelihood Method

Let C be the Hessian matrix and G be the gradient of the log-likelihood function (both functions of and the parameters ). Let pi* denote the vector containing the first r-1 sample proportions from population i, and let denote the corresponding vector of probability estimates from the current iteration. Starting with the least-squares estimates b0 of (if you use the ML and WLS options; with the ML option alone, the procedure starts with 0), the probabilities are computed, and b is calculated iteratively by the Newton-Raphson method until it converges (see the EPSILON= option). The factor is a step-halving factor that equals one at the start of each iteration. For any iteration in which the likelihood decreases, PROC CATMOD uses a series of subiterations in which is iteratively divided by two. The subiterations continue until the likelihood is greater than that of the previous iteration. If the likelihood has not reached that point after ten subiterations, then convergence is assumed, and a warning message is displayed.

Sometimes, infinite parameters may be present in the model, either because of the presence of one or more zero frequencies or because of a poorly specified model with collinearity among the estimates. If an estimate is tending toward infinity, then PROC CATMOD flags the parameter as infinite and holds the estimate fixed in subsequent iterations. PROC CATMOD regards a parameter to be infinite when two conditions apply:

• The absolute value of its estimate exceeds five divided by the range of the corresponding variable.
• The standard error of its estimate is at least three times greater than the estimate itself.
The estimator of the asymptotic covariance matrix of the maximum likelihood predicted probabilities is given by Imrey, Koch, and Stokes (1981, eq. 2.18).

The following equations summarize the method:

where

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