Introduction to multiple regression for categorical and limited dependent variables

Multiple regression is a versatile and powerful statistical method that can be used to model simultaneously the effects of multiple independent variables on a dependent variable (for example, Cohen & Cohen, 1983; Fox, 1997; Pedhazur, 1997). The simultaneous examination of independent variables makes it possible to estimate their independent and combined effects; to determine more accurately the direction and strength of their effects; to rule out spurious effects; to better understand, predict, and explain a dependent variable; and to control the probability of Type I errors. In addition, with multiple regression it is possible to model main, interacting, or curvilinear effects and to examine the incremental improvement in a model brought about by the addition or deletion of independent variables. Furthermore, multiple regression can accommodate any combination of nominal, ordinal, or interval level independent variables. Finally, multiple regression is useful for the analysis of data collected using diverse research designs, including experimental, quasi-experimental, and nonexperimental designs. Most social work researchers are familiar with the linear regression model, also sometimes referred to as ordinary least squares multiple regression, or simply multiple regression (Cnaan & Cascio, 1999; Gutierrez, Fredricksen, & Soifer, 1999; Harrington, 1999; Icard, Longres, & Spencer, 1999; Miller & Macintosh, 1999; Nash & Bowen, 1999). Binary logistic regression also is used increasingly in social work research to model dichotomous dependent variables (see, for example, Drake, 1996; Rosenthal & Rosenthal, 1991; Smith, Sullivan, & Cohen, 1995; Zuravin & DePanfilis, 1997). These are only two of a large number of available multiple regression models (Fox, 1997; Greene, 2000; Long, 1997). In addition to binary dependent variables, many dependent variables in social work research are multicategorical, ordinal, counted, censored, or assessed using truncated populations. Long (1997) referred to such variables as categorical and limited dependent variables (CLDVs). Most social work researchers are not aware of the numerous multiple regression models for analyzing CLDVs, the considerations involved in the selection of the most appropriate model, or the consequences of using linear regression to model many CLDVs. This lack of knowledge about multiple regression methods for CLDVs is unfortunate because many of the dependent variables of interest in social work are CLDVs that should not be modeled using linear or binary logistic regression. Using a multiple regression model inappropriate for the characteristics of the dependent variable at hand can have a number of undesirable consequences (Breen, 1996; Greene, 2000; Long, 1997). Most notably, perhaps, the effects of independent variables might be over- or underestimated (that is, "biased"). In addition, parameter estimates might be "inefficient" (vary more from sample to sample) or "inconsistent" (have sampling distributions whose variability does not decrease with larger samples--Fox, 1997; Long, 1997; Marriott, 1990; Nunnally & Bernstein, 1994). The purpose of this article is to extend the knowledge of alternative multiple regression models among social work researchers so that they will be in a better position to accurately model important dependent variables of interest to the profession. To this end this article * defines and distinguishes among different types of CLDVs * discusses the similarities and differences between linear and CLDV multiple regression models * provides an overview of multiple regression models for CLDVs * discusses key considerations involved in the selection of the most appropriate model from those available * discusses briefly computer software available for estimating CLDV multiple regression models * directs the reader to reference books in this area * provides examples from the social work literature illustrating the use of each CLDV model discussed, where available. …