Computational Formulas

Subsections:

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

Source

Formula

Dimension

Probability Estimates

jth response

ith population

all populations

Variance of Probability Estimates

ith population

all populations

Response Functions

ith population

all populations

Derivative of Function with Respect to Probability Estimates

ith population

all populations

Variance of Functions

ith population

all populations

Inverse Variance of Functions

ith population

all populations

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

In the following table, let be a vector of functions of , and let denote , which is the first derivative matrix of with respect to :

Function

Derivative

Multiply matrix

Logarithm

Exponential

Default Response Functions: Generalized Logits

In the following table, subscripts i for the population are suppressed. Also denote for for each population .

Formula

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

all populations

Crossproduct of Design Matrix

ith population

all populations

In the following table, is the 100pth percentile of the standard normal distribution:

Formula

Dimension

Crossproduct of Design Matrix with Function

Weighted Least Squares Estimates

Covariance of Weighted Least Squares Estimates

Wald Confidence Limits for Parameter Estimates

Predicted Response Functions

Covariance of Predicted Response Functions

Residual Chi-Square

Chi-Square for

Q

Maximum Likelihood Method

Let be the Hessian matrix and be the gradient of the log-likelihood function (both functions of and the parameters ). Let denote the vector containing the first sample proportions from population i, and let denote the corresponding vector of probability estimates from the current iteration. Starting with the least squares estimates of (if you use the ML and WLS options; with the ML option alone, the procedure starts with ), the probabilities are computed, and 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 10 subiterations, then convergence is assumed, and a warning message is displayed.

Sometimes, infinite parameters are 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

Iterative Proportional Fitting

The algorithm used by PROC CATMOD for iterative proportional fitting is described in Bishop, Fienberg, and Holland (1975); Haberman (1972); Agresti (2002). To illustrate the method, consider the observed three-dimensional table for the variables X, Y, and Z, and the following hierarchical model:

The following statements request that PROC CATMOD use IPF to fit the preceding model:

model X*Y*Z = _response_ / ml=ipf;
loglin X|Y|Z@2;


Begin with a table of initial cell estimates . PROC CATMOD produces the initial estimates by setting the structural zero cells to 0 and all other cells to , where n is the total weight of the table and is the total number of cells in the table. Iteratively adjust the estimates at step to the observed marginal tables specified in the model by cycling through the following three-stage process to produce the estimates at step s:

The subscript indicates summation over the missing subscript. The log-likelihood is estimated at each step s by

When the function is less than , the iterations terminate. You can change the comparison value with the EPSILON= option, and you can change the convergence criterion with the CONVCRIT= option. The option CONVCRIT=CELL uses the maximum cell difference

as the criterion while the option CONVCRIT=MARGIN computes the maximum difference of the margins