MODEL Statement

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The CATMOD Procedure

MODEL Statement

MODEL response-effect=< design-effects >< / options > ;

PROC CATMOD requires a MODEL statement. You can specify the following in a MODEL statement:

response-effect

can be either a single variable, a crossed effect with two or more variables joined by asterisks, or _F_. The _F_ specification indicates that the response functions and their estimated covariance matrix are to be read directly into the procedure. The response-effect indicates the dependent variables that determine the response categories (the columns of the underlying contingency table).

design-effects

specify potential sources of variation (such as main effects and interactions) in the model. Thus, these effects determine the number of model parameters, as well as the interpretation of such parameters. In addition, if there is no POPULATION statement, PROC CATMOD uses these variables to determine the populations (the rows of the underlying contingency table). When fitting the model, PROC CATMOD adjusts the independent effects in the model for all other independent effects in the model.

Design-effects can be any of those described in the section "Specification of Effects", or they can be defined by specifying the actual design matrix, enclosed in parentheses (see the "Specifying the Design Matrix Directly" section). In addition, you can use the keyword _RESPONSE_ alone or as part of an effect. Effects cannot be nested within _RESPONSE_, so effects of the form A(_RESPONSE_) are invalid.

For more information, see the "Log-Linear Model Analysis" section and the "Repeated Measures Analysis" section.

Some examples of MODEL statements are

`model r=a b;`	main effects only
*`model r=a b ab;`**	main effects with interaction
`model r=a b(a);`	nested effect
`model r=a\|b;`	complete factorial
`model r=a b(a=1) b(a=2);`	nested-by-value effects
*`model rs=_response_;`**	log-linear model
*`model rs=a _response_(a);`**	nested repeated measurement factor
`model _f_=_response_;`	direct input of the response functions

The relationship between these specifications and the structure of the design matrix X is described in the "Generation of the Design Matrix" section.

The following table summarizes the options available in the MODEL statement.

Task	Options
Specify details of computation
Generates maximum likelihood estimates	ML
Generates weighted least-squares estimates	GLS
	WLS
Omits intercept term from the model	NOINT
Adds a number to each cell frequency	ADDCELL=
Averages main effects across response functions	AVERAGED
Specifies the convergence criterion for maximum likelihood	EPSILON=
Specifies the number of iterations for maximum likelihood	MAXITER=
Request additional computation and tables
Estimated correlation matrix of estimates	CORRB
Covariance matrix of response functions	COV
Estimated covariance matrix of estimates	COVB
Two-way frequency tables	FREQ
One-way frequency tables	ONEWAY
Predicted values	PRED=
	PREDICT
Probability estimates	PROB
Crossproducts matrix	XPX
Title	TITLE=
Suppress output
Design matrix	NODESIGN
Iterations for maximum likelihood	NOITER
Parameter estimates	NOPARM
Population and response profiles	NOPROFILE
_RESPONSE_ matrix	NORESPONSE

The following list describes these options in alphabetical order.

ADDCELL=number

adds number to the frequency count in each cell, where number is any positive number. This option has no effect on maximum likelihood analysis; it is used only for weighted least-squares analysis.

AVERAGED

specifies that dependent variable effects can be modeled and that independent variable main effects are averaged across the response functions in a population. For further information on the effect of using (or not using) the AVERAGED option, see the "Generation of the Design Matrix" section. Direct input of the design matrix or specification of the _RESPONSE_ keyword in the MODEL statement automatically induces an AVERAGED model type.

CORRB

displays the estimated correlation matrix of the parameter estimates.

COV

displays S_i, which is the covariance matrix of the response functions for each population.

COVB

displays the estimated covariance matrix of the parameter estimates.

EPSILON=number

specifies the convergence criterion for the maximum likelihood estimation of the parameters. The iterative estimation process stops when the proportional change in the log likelihood is less than number, or after the number of iterations specified by the MAXITER= option, whichever comes first. By default, EPSILON=1E-8.

FREQ

produces the two-way frequency table for the cross-classification of populations by responses.

MAXITER=number

specifies the maximum number of iterations used for the maximum likelihood estimation of the parameters. By default, MAXITER=20.

ML

computes maximum likelihood estimates. This option is available when generalized logits are used, or for the special case of a single two-level dependent variable where cumulative logits or adjacent category logits are used. For generalized logits (the default response functions), ML is the default estimation method.

NODESIGN

suppresses the display of the design matrix X.

NOINT

suppresses the intercept term in the model.

NOITER

suppresses the display of parameter estimates and other information at each iteration of a maximum likelihood analysis.

NOPARM

suppresses the display of the estimated parameters and the statistics for testing that each parameter is zero.

NOPROFILE

suppresses the display of the population profiles and the response profiles.

NORESPONSE

suppresses the display of the _RESPONSE_ matrix for log-linear models. For further information, see the "Log-Linear Model Design Matrices" section.

ONEWAY

produces a one-way table of frequencies for each variable used in the analysis. This table is useful in determining the order of the observed levels for each variable.

PREDICT

PRED=FREQ | PROB

displays the observed and predicted values of the response functions for each population, together with their standard errors and the residuals (observed - predicted). In addition, if the response functions are the standard ones (generalized logits), then the PRED=FREQ option specifies the computation and display of predicted cell frequencies, while PRED=PROB (or just PREDICT) specifies the computation and display of predicted cell probabilities.

The OUT= data set always contains the predicted probabilities. If the response functions are the generalized logits, the predicted cell probabilities are output unless the option PRED=FREQ is specified, in which case the predicted cell frequencies are output.

PROB

produces the two-way table of probability estimates for the cross-classification of populations by responses. These estimates sum to one across the response categories for each population.

TITLE='title'

displays the title at the top of certain pages of output that correspond to this MODEL statement.

WLS

GLS

computes weighted least-squares estimates. This type of estimation is also called generalized-least-squares estimation. For response functions other than the default (of generalized logits), WLS is the default estimation method.

XPX

displays X'S^-1X, the crossproducts matrix for the normal equations.

Specifying the Design Matrix Directly

If you specify the design matrix directly, adjacent rows of the matrix must be separated by a comma, and the matrix must have q ×s rows, where s is the number of populations and q is the number of response functions per population. The first q rows correspond to the response functions for the first population, the second set of q rows corresponds to the functions for the second population, and so forth. The following is an example using direct specification of the design matrix.

   proc catmod;
      model R=(1 0,
               1 1,
               1 2,
               1 3);
   run;

These statements are appropriate for the case of one population and for R with five levels (generating four response functions), so that 4 ×1 = 4. These statements are also appropriate for a situation with two populations and two response functions per population; giving 2 ×2 = 4 rows of the design matrix. (To induce more than one population, the POPULATION statement is needed.)

When you input the design matrix directly, you also have the option of specifying that any subsets of the parameters be tested for equality to zero. Indicate each subset by specifying the appropriate column numbers of the design matrix, followed by an equal sign and a label (24 characters or less, in quotes) that describes the subset. Adjacent subsets are separated by a comma, and the entire specification is enclosed in parentheses and placed after the design matrix. For example,

   proc catmod;
      population Group Time;
      model R=(1  1  0  0,
               1  1  0  1,
               1  1  0  2,
               1  0  1  0,
               1  0  1  1,
               1  0  1  2,
               1 -1 -1  0,
               1 -1 -1  1,
               1 -1 -1  2) (1  ='Intercept',
                            2 3='Group main effect',
                            4  ='Linear effect of Time');
   run;

The preceding statements are appropriate when Group and Time each have three levels, and R is dichotomous. The POPULATION statement induces nine populations, and q=1 (since R is dichotomous), so q ×s = 1 ×9 = 9.

If you input the design matrix directly but do not specify any subsets of the parameters to be tested, then PROC CATMOD tests the effect of MODEL | MEAN, which represents the significance of the model beyond what is explained by an overall mean. For the previous example, the MODEL | MEAN effect is the same as that obtained by specifying

   (2 3 4='model|mean');

at the end of the MODEL statement.

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