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A: There are many types of models in the area of logistic modeling. Following are some common logistic models. Note that the words logistic and logit are used interchangeably.

The three basic categories of logistic models are the binary, ordinal, and nominal models. They are discussed in the rest of this FAQ along with related, special models. Examples of many of these models can be found in the documentation of the procedures that are mentioned or at the links that are provided. Many of these models are discussed and illustrated in more detail in Stokes et al. (2000) and Allison (1999). All procedures that are mentioned are part of SAS/STAT software unless otherwise indicated.

Binary logistic regression model

Used to model a binary (two-level) response—for example, yes or no.
How to Fit It: This model can be fit by many procedures, including the SAS/STAT procedures LOGISTIC (using asymptotic or exact conditional methods), CATMOD (using weighted least squares or maximum likelihood), PROBIT, GAM, GENMOD, GLIMMIX, SURVEYLOGISTIC, and NLMIXED; SAS/INSIGHT; and the SAS/ETS procedures MDC and QLIM. PROC GAM can fit a more flexible logistic model by using spline or loess smoothers. PROC GLIMMIX and PROC NLMIXED allow the inclusion of random effects in the model. Longitudinal or repeated measures data can be modeled using PROC GENMOD (REPEATED statement), PROC GLIMMIX (RANDOM statement), or PROC CATMOD (REPEATED statement).

Case-control model using matched pairs (or sets)

Used to model binary response data from pairs of subjects or sets of subjects that are matched on certain characteristics. In 1:1 matching, one case is matched to one control. In 1:m or m:n matching, one or more cases are matched to two or more controls. Conditional methods are used to fit this model.
How to Fit It: Use the STRATA statement in PROC LOGISTIC in SAS®9 or later. Exact estimation can also be done by specifying the EXACT statement. With a little additional work, the model can also be fit using the STRATA statement in PROC PHREG.

Bradley-Terry model

Used when subjects are asked to compare many items, two at a time. That is, a subject indicates preference for one item in each of several pairs of items presented. Preference probabilities and ratios of preference probabilities for items can be determined from the model parameters. Conditional methods are used to fit this model.
How to Fit It: Details and an example are given in this sample program.

Ordinal (ordered) logistic regression model

Used to model an ordered response—for example, low, medium, or high. Might also be called the ordinal multinomial logit model. Ordinal logistic models take into account the ordered nature of the response, which can result in simple, more powerful models. Typical response functions that are modeled are cumulative logits, adjacent-category logits, or continuation-ratio logits resulting in ordinal logistic models known as the cumulative logit model, the adjacent-category logit model, and the continuation-ratio logit model. The proportional odds model and the partial proportional odds model are special cases of the cumulative logit model. If the spacing between levels of the ordinal response scale is known, so that numerical scores can reasonably be assigned to the response levels, then a mean response model can be fit.

Cumulative logit model

This is one type of ordinal logistic model that models cumulative logits: CLogiti=log((1-Pr[Y≤i])/Pr[Y≤i]), for Y=1,2,... . Notice that each cumulative logit involves all levels of the response and dichotomizes the response scale. In its full generality, the cumulative logit model has a complete set of parameter estimates for each cumulative logit—that is, multiple intercepts and multiple estimates for each predictor. But such a complex model can result in probabilities that do not accumulate properly. Typically, simpler models that are special cases of the general cumulative logit model are used. Such models are the proportional odds model and the partial proportional odds model.
How to Fit It: PROC CATMOD uses the method of weighted least squares to fit the cumulative logit model when the RESPONSE CLOGITS; statement is specified.

Proportional odds model

This is a cumulative logit model that assumes that the odds of response below a given response level are constant regardless of which level you pick. This model has an intercept for each cumulative logit, but it constrains each continuous predictor to have a single parameter for all logits. A proportional odds model that constrains some predictors to have common parameters and leaves other predictors free to have separate parameters is called a partial proportional odds model.
How to Fit It: PROC LOGISTIC fits the proportional odds model by default when the response has more than two levels. PROC PROBIT can fit the model if you specify DIST=LOGISTIC in the MODEL statement. PROC GENMOD fits it when the DIST=MULT and LINK=CUMLOGIT options are specified in the MODEL statement. It can also be fit in the SAS/ETS procedure QLIM if you specify the DISCRETE(DIST=LOGISTIC) option in the MODEL statement. All four procedures use maximum likelihood estimation.

Adjacent-category logit model

This is a type of ordinal logistic model that models adjacent-category logits: ALogiti=log(Pr[Y=i+1]/Pr[Y=i]), for Y=1,2,... . Notice that each adjacent-category logit contrasts two adjacent response categories rather than involving the entire response scale as with cumulative logits.
How to Fit It: PROC CATMOD uses the method of weighted least squares to fit the adjacent-category logit model when the RESPONSE ALOGITS; statement is specified. The model has a parameter vector for each logit—that is, multiple intercepts and multiple parameters for each predictor. A simpler model with an intercept for each logit but a single parameter vector for the predictors across all logits can be fit by including the _RESPONSE_ keyword in the MODEL statement. This simpler model can also be fit by maximum likelihood in PROC CATMOD because this model can be written as a generalized logit model. See Agresti (2002) for discussion and examples using PROC CATMOD.

Continuation-ratio logit model

This is a type of ordinal logistic model that models continuation-ratio logits: CRLogiti=log(Pr[Y=i+1]/Pr[Y≤i]), for Y=1,2,... .
How to Fit It: This model, which has a parameter vector for each logit, can be fit by weighted least squares (WLS) in PROC CATMOD by using the capabilities in the RESPONSE statement to define custom response functions. It can also be fit by maximum likelihood by taking advantage of the fact that the parameters for each logit can be estimated using a separate binary logistic model. A simpler model with an intercept for each logit but a single parameter vector for the predictors across all logits can be fit by WLS in PROC CATMOD by including the _RESPONSE_ keyword in the MODEL statement. Examples are provided in this FAQ.

Mean response model

This is not really a logistic model because no type of logit response function is modeled. As a result, the model does not guarantee that predicted values correspond to a valid set of probabilities across the response levels. However, it is an alternative to the ordinal logistic models that were discussed earlier when numerical scores can be assigned to the response levels, implying that the spacing among the levels is known. This model is most natural when the ordinal categorical response represents a continuous response that is coarsely measured. Note that this model provides an estimated mean response rather than assigning estimated probabilities to each response level.
How to Fit It: PROC CATMOD fits the mean response model by weighted least squares when the RESPONSE MEANS; statement is specified.

Nominal (unordered) logistic regression model

Used to model a multilevel response with no ordering—for example, eye color with levels brown, green, and blue. Such a response is also called polytomous, polychotomous, or multinomial.

Multinomial logit model

Strictly speaking, the term multinomial indicates only that the response has more than two levels; it does not specify whether they are ordered or unordered. Nevertheless, the term multinomial logit model is often used when the response is a set of unordered choices and refers to the conditional discrete choice model. But the term can also refer to the unconditional ordinal model or to the generalized logit model.

Generalized logit model

Also known as the baseline logit model. It is a type of unconditional, nominal logistic model in which the response functions that are modeled are known as generalized logits or baseline logits.
How to Fit It: This is the default model that is fit by PROC CATMOD when the response has more than two levels. PROC CATMOD can fit the model by using maximum likelihood (the default) or weighted least squares (specify the WLS option). PROC LOGISTIC provides a maximum likelihood fit of the model when you specify the LINK=GLOGIT option in the MODEL statement. Exact estimation of the model is also available.

Discrete choice models

Used to model a response that is the choice of individuals—for example, among transportation modes (car, bus, train, plane). Some or all predictors can be properties of the choices (cost, speed, and so on) rather than properties of the choosers as in unconditional models such as the generalized logit model. McFadden's conditional logit model, the nested logit model, and the mixed logit model are discrete choice models. The generalized logit model is often used as a discrete choice model too when the predictors are all properties of the choosers (subjects) and not of the choices.

McFadden's conditional logit model

Is a discrete choice model in which the predictors are properties of the choices (response levels). Conditional methods are used to fit this model. This model assumes independence from irrelevant alternatives (IIA). More information about this model can be found in this FAQ.
How to Fit It: The discrete choice model can be fit in the SAS Market Research Application, the SAS/ETS procedure MDC (specify TYPE=CLOGIT in the MODEL statement), PROC LOGISTIC with the STRATA statement (SAS®9 or later), and PROC PHREG with the STRATA statement.

Nested logit model

Is a generalization of the conditional logit model that relaxes the IIA assumption to allow for particular patterns of correlation in unobserved utility. It can be used if the set of alternatives that are faced by an individual can be partitioned into subsets such that the IIA property holds within subsets but not across subsets.
How to Fit It: The SAS/ETS procedure MDC (specify TYPE=NESTEDLOGIT in the MODEL statement).

Mixed logit model

Is a generalization of the conditional logit model that can represent very general patterns of substitution among alternatives. In this model, the utility function of each decision maker can be decomposed into a deterministic component (linear combination of observed variables) and a stochastic error component. The choice probability is a mixture of logits. The model for the error component involves random coefficients.
How to Fit It: The SAS/ETS procedure MDC (specify TYPE=MIXEDLOGIT in the MODEL statement).

Special Logistic Models

Logistic model for longitudinal (or repeated measures) data

These models are for a response that is observed more than once on each subject (or item), either at multiple times or under multiple conditions. The response can be binary, ordinal, or nominal. There are three primary types of models: marginal (or population-averaged), subject-specific (includes fixed-effects and random-effects model), and transitional.

Generalized Estimating Equations (GEE)

Available for binary and ordinal responses, the GEE method allows missing values within a subject without losing all data from the subject, and time-varying predictors can appear in the model. The method requires a large number of subjects. For binary responses, a variation on the GEE model that models the association among responses with odds ratios rather than correlations is Alternating Logistic Regression (ALR). The ALR method does not restrict the correlation among the measurements as the usual GEE method does when applied to a binary response. Like the GEE model, ALR provides estimates of the marginal model parameters. But ALR also estimates parameters of the model on the log odds ratios among the measurements. The GEE and ALR models are marginal models. Note that the GEE estimation method is not a maximum likelihood method.
How to Fit It: To fit the GEE model, specify the REPEATED statement in PROC GENMOD. The DIST=BINOMIAL or DIST=MULT option must appear in the MODEL statement to request binary or ordinal multinomial logistic models, respectively. The nominal multinomial model is not available. Use the TYPE= option to specify the correlation structure among the repeated measurements within a subject. To fit the ALR model, specify LOGOR= rather than TYPE= in the REPEATED statement.

Cluster model with variance adjustment

This model is fit by maximum likelihood, but variances are adjusted using the cluster structure of the data. For a small number of clusters, this model with the Morel adjustment (VARADJUST=MOREL) can provide a better fit than the GEE model. It can be used for binary, ordinal, or nominal responses. It is a marginal model.
How to Fit It: In SAS 9.1 or later, use PROC SURVEYLOGISTIC with the CLUSTER statement and optionally use the VARADJUST= option in the MODEL statement.

Weighted Least Squares (WLS)

Available for binary, ordinal, and nominal responses, the WLS method requires complete data for each subject (otherwise the subject is ignored) and does not allow time-varying predictors in the model. It is a marginal model.
How to Fit It: Specify the REPEATED statement in PROC CATMOD.

Fixed-effects logistic model

This model treats each measurement on each subject as a separate observation, and the set of subject coefficients that would appear in an unconditional model are eliminated by conditional methods. This is a conditional, subject-specific model (as opposed to a population-averaged model like the GEE model). See the extensive discussion and examples in P. D. Allison (2005), Fixed Effects Regression Methods for Longitudinal Data Using SAS.
How to Fit It: For binary response data, use the STRATA statement in PROC LOGISTIC in SAS®9 or later. See Allison (2005) concerning fitting a related model for multinomial responses.

Transition models for discrete state space stochastic processes

Discrete time Markov chains can be represented as loglinear models. Predictor variables can be incorporated by fitting a logistic model and treating previous response values as additional predictors.
How to Fit It: Loglinear models can be fit in PROC GENMOD by specifying the cell counts of the table as the response and specifying DIST=POISSON in the MODEL statement. The first-order Markov chain for a process over four discrete times is MODEL COUNT = T1|T2 T2|T3 T3|T4 / DIST=POISSON; The second-order model is MODEL COUNT = T1|T2|T3 T2|T3|T4 / DIST=POISSON; A transitional model incorporating predictor variables is fit as a logistic model, but the input data set should have separate observations for the responses at each time for each subject. Variables representing lags of the response variable can then be used as predictors in the model. Examples are provided in this FAQ.

Generalized Additive Model (GAM) for binary response data

This is a very flexible nonparametric model that relaxes assumptions of linearity and is useful for data exploration. Nonlinearity is allowed via smoothers such as cubic splines, local regression (loess), and bivariate thin-plate splines. It is available only for binary responses.
How to Fit It: In PROC GAM, specify the DIST=BINOMIAL option in the MODEL statement to fit a logistic model.

Logistic model for survey data

Includes models for binary, ordinal, and nominal responses. For proper inferences, the analysis or survey data must incorporate properties of the survey sample design, including stratification, clustering, and unequal weighting.
How to Fit It: Use PROC SURVEYLOGISTIC in SAS 9.1 or later. The RATE= and TOTAL= options in the PROC statement and the CLUSTER, STRATA, and WEIGHT statements allow specification of the sampling design and sampling weights. Note that simply using the WEIGHT statement in PROC LOGISTIC is not sufficient.

Random effects logistic model (as a generalized linear mixed model or nonlinear mixed model)

Allows random effects in a logistic model resulting in a subject-specific model. This is a conditional model that can also be used to model longitudinal or repeated measures data.
How to Fit It: Beginning with SAS 9.1, use PROC GLIMMIX and in the MODEL statement specify DIST=BINOMIAL LINK=LOGIT (for binary logit model), DIST=MULT LINK=CLOGIT (for an ordinal logit model), or DIST=MULT LINK=GLOGIT (for a nominal logit model). Use RANDOM statements to define random effects. The model can also be fit in PROC NLMIXED by using a different methodology that typically limits the number of random effects to one or two. Only binary responses are directly supported (specify BINARY(p) or BINOMIAL(n,p) in the MODEL statement), though multinomial models can be accommodated by defining the multinomial log likelihood (via GENERAL() in the MODEL statement).

Heteroscedastic logistic model

This model allows the dispersion to be modeled as well as the mean. Binary and ordinal responses can be modeled.
How to Fit It: Use the MODEL statement in the SAS/ETS procedure QLIM to specify the model for the mean, and use the HETERO statement to specify the dispersion model. Specify the DISCRETE(DIST=LOGISTIC) option in the MODEL statement to fit a binary or ordinal logistic model (depending on the number of levels that are detected in the response variable).

Multivariate logistic model

Simultaneously models multiple responses, taking into account the correlations among all response functions.
How to Fit It: In PROC CATMOD, specify the RESPONSE LOGITS; statement and multiple response variables in the MODEL statement to fit the model by using weighted least squares estimation. For example, these statements simultaneously model logits that are defined separately on three response variables:
response logits; model x1*x2*x3 = group;

Exact logistic regression

This is a method of testing and estimation rather than a model. It uses conditional methods to obtain exact tests of parameters in binary and nominal logistic models. Exact methods are appropriate for small-sample or sparse data situations that often result in the failure (nonconvergence or separation) of the usual unconditional maximum likelihood estimation method. However, exact methods can take a great deal of time and memory as sample or model sizes increase. For sample sizes too large for the default exact method, a Monte Carlo method is provided.
How to Fit It: Specify the EXACT statement in PROC LOGISTIC. Use the EXACTOPTIONS option in the PROC LOGISTIC statement to select the exact method (METHOD=NETWORKMC requests the Monte Carlo method) and control other aspects of the exact analysis.

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