A categorical variable here refers to a variable that is binary, ordinal, or nominal. Event count data are discrete (categorical) but often treated as continuous variables. When a dependent variable is categorical, the ordinary least squares (OLS) method can no longer produce the best linear unbiased estimator (BLUE); that is, OLS is biased and inefficient. Consequently, researchers have developed various regression models for categorical dependent variables. The nonlinearity of categorical dependent variable models (CDVMs) makes it difficult to fit the models and interpret their results.

In CDVMs, the left-hand side (LHS) variable or dependent variable is neither interval nor ratio, but rather categorical. The level of measurement and data generation process (DGP) of a dependent variable determines the proper type of CDVM. Binary responses (0 or 1) are modeled with binary logit and probit regressions, ordinal responses (1st, 2nd, 3rd, ...) are formulated into (generalized) ordered logit/probit regressions, and nominal responses are analyzed by multinomial logit, conditional logit, or nested logit models depending on specific circumstances. Independent variables on the right-hand side (RHS) may be interval, ratio, or binary (dummy).

The CDVMs adopt the maximum likelihood (ML) estimation method, whereas OLS uses the moment based method. The ML method requires an assumption about probability distribution functions, such as the logistic function and the complementary log-log function. Logit models use the standard logistic probability distribution, while probit models assume the standard normal distribution. This document focuses on logit and probit models only, excluding regression models for event count data (e.g., negative binomial regression model and zero-inflated Poisson regression model). Table 1 summarizes CDVMs in comparison with OLS.

How do logit models differ from probit models? The core difference lies in the distribution of errors (disturbances). In the logit model, errors are assumed to follow the standard logistic distribution, . The errors of the probit model are assumed to follow the standard normal distribution, .

The probability density function (PDF) of the standard normal probability distribution has a higher peak and thinner tails than the standard logistic probability distribution (Figure 1). The standard logistic distribution looks as if someone has weighed down the peak of the standard normal distribution and strained its tails. As a result, the cumulative density function (CDF) of the standard normal distribution is steeper in the middle than the CDF of the standard logistic distribution and quickly approaches zero on the left and one on the right.

The two models, of course, produce different parameter estimates. In binary response models, the estimates of a logit model are roughly pi/sqrt(3) times larger than those of the corresponding probit model. These estimators, however, end up with almost the same standardized impacts of independent variables (Long 1997).

The choice between logit and probit model is more closely related to estimation and familiarity rather than theoretical and interpretive aspects. In general, logit models reach convergence fairly well. Some (multinomial) probit models may take a long time to reach convergence, although the probit works well for bivariate models.

SAS provides several procedures for CDVMs, such as LOGISTIC, PROBIT, GENMOD, QLIM, MDC, PHREG, and CATMOD. Since these procedures support various models, a CDVM can be estimated by multiple procedures. For example, you may run a binary logit model using the LOGISTIC, PROBIT, GENMODE, and QLIM. The LOGISTIC and PROBIT procedures of SAS/STAT have been commonly used, but the QLIM and MDC procedures of SAS/ETS have advantages over other procedures.

The QLIM (Qualitative and LImited dependent variable Model) procedure analyzes various categorical and limited dependent variable regression models such as censored, truncated, and sample-selection models. This QLIM procedure also handles Box-Cox regression and bivariate probit and logit models. The MDC (Multinomial Discrete Choice) procedure can estimate conditional logit, and nested logit models.

Another advantage of using SAS is the Output Delivery System (ODS), which makes it easy to manage SAS output. ODS enables users to redirect the output to HTML (Hypertext Markup Language) and RTF (Rich Text Format) formats. Once SAS output is generated in a HTML document, users can easily handle tables and graphics especially when copying and pasting them into a wordprocessor document.

Unlike SAS, Stata has individualized commands for corresponding CDVMs. For example, the .logit and .probit commands respectively fit the binary logit and probit models, while .mlogit and .nlogit estimate mulitinomial logit and nested logit models. Stata enables users to perform post-hoc analyses such as marginal effects and discrete changes in an easy manner.

The LIMDEP Logit$ and Probit$ commands support a variety of CDVMs that are addressed in Greene's Econometric Analysis (2003). The output format of LIMDEP 9 is slightly different from that of previous version, but key statistics remain unchanged. Note that the nested logit model in LIMDEP is estimated by NLOGIT, a separate package. SPSS supports some CDVMs only and its output is often messy with less useful, if not misleading, information especially for beginners. Table 2 summarizes the procedures and commands for CDVMs.

Stata users may take advantages of user-written modules such as J. Scott Long and Jeremy Freese's SPost. The module allows researchers to conduct follow-up analyses of various CDVMs including event count data models. See section 2.2 for major SPost commands.

In order to install SPost, execute the following commands consecutively. For more details, visit J. Scott Long's Web site at http://www.indiana.edu/~jslsoc/spost_install.htm.

If you want to use Vincent Kang Fu's gologit (1998) and Richard Williams' gologit2 (2005) for the generalized ordered logit model, install their modules.