UNDERSTANDING MULTIPLE REGRESSIONFrom: Ethington, C. A., Thomas, S. L., & Pike, G. R. (2002). Back to the basics: Regression as it should be. In J. C. Smart (Ed.), Higher education: Handbook of theory and research, Vol. 17. New York: Algora Publishing.Sir Francis Galton (1885) introduced the idea of “regression” to the research community in astudy examining the relationship of fathers’ and sons’ heights. In his study he observed that sonsdo not tend toward their fathers’ heights but instead “regress to” the mean of the population. Hethus formulated the idea of “regression toward mediocrity”, and with the development of themethod of least squares procedures by Carl Friedrich Gauss (Myers, 1990), multiple regressionanalysis using ordinary least squares procedures (OLS) has become one of the most commonstatistical techniques for investigating and modeling relationships among variables.The two predominant uses of multiple regression are for prediction and explanation, and themethodological approach taken in the analyses depends upon the purpose of the estimation of themodel. Suppose, for example, that an institutional research office has been charged withdetermining whether a set of variables (e.g., ability, high school achievement, socioeconomicstatus, interests, motivation) can predict end-of-freshman-year grade point average. If thepurpose is to optimize the prediction and to use such a prediction equation in making admissiondecisions, the goal is the development of the most parsimonious equation with the least errors ofprediction so that the best estimates of yield rates from admission pools can be obtained. The aimis to eliminate superfluous variables, not to test theoretically based hypotheses. The use of theoryis not required in the selection of variables for use in the development of such a predictiveequation, and the parameter estimates are of little importance. Of more importance are economy,availability of data, ease of obtaining needed information, and accuracy of prediction. Variousapproaches may be used to identify the smallest number of variables necessary to produce themost accurate prediction estimates (see Montgomery & Peck, 1992 and Myers, 1990, for apresentation of the development of the regression equation for prediction). Different approachesusing the same data may lead to the retention of different sets of variables, but any approach thatmeets the needs of the researcher and produces accurate estimation is sufficient.While the higher education research literature is replete with regression studies using theprediction terminology, few actually use the methodological approaches involved in thedevelopment of the most efficient prediction equation. Almost all of the regression applicationsin higher education are for explanatory purposes, and it is this approach that we take in ourdevelopment of this chapter, for explanation is the essence of behavioral research. Almost all ofour research questions seek to understand and explain why some phenomenon under study variesfrom person to person, and most generally the phenomena studied in higher education are“multivariate” in nature. That is, the primary focus of a study (the dependent variable) isconceptualized as being related to and influenced by multiple interrelated factors (theindependent variables). Rarely can simple bivariate relationships adequately capture and explainor model reality. The complexity of behavioral science phenomena demands that we study thecovariation among the independent variables as well as that of the dependent variable with theindependent variables. It is this variance and covariance that is the basis of multiple regression.Our search for explanations of variability and attempts to model reality imply that there is atheoretical or conceptual basis for not only the anticipated relationships among variables, but forthe selection of variables studied. In his classic text on the application of multiple regression inbehavioral research, Pedhazur (1982) argues that “methods per se mean little unless they areintegrated within a theoretical context” (p. 3) and goes on to state that “in explanatory researchdata analysis is designed to shed light on theory” (p. 11). This focus on theory in the applicationof multiple regression in behavioral research has been argued for since the 1950s whenregression became commonly used in the social sciences. Ezekiel and Fox (1959) stress thenecessity for “careful logical analysis, and the need both for good theoretical knowledge of thefield in which the problem lies and for thorough technological knowledge of the elementsinvolved in the particular problem” (p. 181). Thus, both good theory and methodologicalcompetence are required for the most effective application of multiple regression.As noted above, the most general application of multiple regression in higher education researchis to explain phenomena, and Kerlinger (1973) states that such explanations are called theories.Kerlinger defines theory as “a set of interrelated constructs (concepts), definitions, andpropositions that present a systematic view of phenomena by specifying relations amongvariables, with the purpose of explaining and predicting the phenomena” (p. 9). Thus, theselection of variables to be used in explaining phenomena and the specific hypotheses to betested are derived from the particular theory in which the research is grounded. The constructsand definitions within the theory lead to the operationalization of the constructs and themeasurement of the variables subsequently used in testing the theoretical hypotheses. Weassume, therefore, that the set of independent variables and the dependent variable used in thestatistical analyses are directly specified by some underlying theory.Using a theory as the basis for the choice of independent variables in multiple regression also hasimplications for how the results of the regression analysis can be discussed. The interpretation ofthe estimated regression coefficients as indices of effects of independent variables on an outcomecan only be applied