Reference Guide on Multiple Regression

Reference Guide on Multiple Regression

Description

Multiple regression analysis is a statistical tool used to understand the relationship between or among two or more variables. Multiple regression involves a variable to be explained—called the dependent variable—and additional explanatory variables that are thought to produce or be associated with changes in the dependent variable. For example, a multiple regression analysis might estimate the effect of the number of years of work on salary. Salary would be the dependent variable to be explained; the years of experience would be the explanatory variable. Multiple regression analysis is sometimes well suited to the analysis of data about competing theories for which there are several possible explanations for the relationships among a number of explanatory variables. Multiple regression typically uses a single dependent variable and several explanatory variables to assess the statistical data pertinent to these theories. In a case alleging sex discrimination in salaries, for example, a multiple regression analysis would examine not only sex, but also other explanatory variables of interest, such as education and experience. The employer-defendant might use multiple regression to argue that salary is a function of the employee’s education and experience, and the employee-plaintiff might argue that salary is also a function of the individual’s sex. Alternatively, in an antitrust cartel damages case, the plaintiff’s expert might utilize multiple regression to evaluate the extent to which the price of a product increased during the period in which the cartel was effective, after accounting for costs and other variables unrelated to the cartel. The defendant’s expert might use multiple regression to suggest that the plaintiff’s expert had omitted a number of price determining variables. More generally, multiple regression may be useful (1) in determining whether a particular effect is present; (2) in measuring the magnitude of a particular effect; and (3) in forecasting what a particular effect would be, but for an intervening event. In a patent infringement case, for example, a multiple regression analysis could be used to determine (1) whether the behavior of the alleged infringer affected the price of the patented product, (2) the size of the effect, and (3) what the price of the product would have been had the alleged infringement not occurred. Over the past several decades, the use of multiple regression analysis in court has grown widely. Regression analysis has been used most frequently in cases of sex and race discrimination antitrust violations, and cases involving class certification (under Rule 23). However, there are a range of other applications, including census undercounts, voting rights, the study of the deterrent effect of the death penalty, rate regulation, and intellectual property. Multiple regression analysis can be a source of valuable scientific testimony in litigation. However, when inappropriately used, regression analysis can confuse important issues while having little, if any, probative value. In EEOC v. Sears, Roebuck & Co., in which Sears was charged with discrimination against women in hiring practices, the Seventh Circuit acknowledged that “[m]ultiple regression analyses, designed to determine the effect of several independent variables on a dependent variable, which in this case is hiring, are an accepted and common method of proving disparate treatment claims.” However, the court affirmed the district court’s findings that the “E.E.O.C.’s regression analyses did not ‘accurately reflect Sears’ complex, nondiscriminatory decision-making processes’” and that the “‘E.E.O.C.’s statistical analyses [were] so flawed that they lack[ed] any persuasive value.’” Serious questions also have been raised about the use of multiple regression analysis in census undercount cases and in death penalty cases. The Supreme Court’s rulings in Daubert and Kumho Tire have encouraged parties to raise questions about the admissibility of multiple regression analyses. Because multiple regression is a well-accepted scientific methodology, courts have frequently admitted testimony based on multiple regression studies, in some cases over the strong objection of one of the parties. However, on some occasions courts have excluded expert testimony because of a failure to utilize a multiple regression methodology. On other occasions, courts have rejected regression studies that did not have an adequate foundation or research design with respect to the issues at hand. In interpreting the results of a multiple regression analysis, it is important to distinguish between correlation and causality. Two variables are correlated—that is, associated with each other—when the events associated with the variables occur more frequently together than one would expect by chance. For example, if higher salaries are associated with a greater number of years of work experience, and lower salaries are associated with fewer years of experience, there is a positive correlation between salary and number of years of work experience. However, if higher salaries are associated with less experience, and lower salaries are associated with more experience, there is a negative correlation between the two variables. A correlation between two variables does not imply that one event causes the second. Therefore, in making causal inferences, it is important to avoid spurious correlation. Spurious correlation arises when two variables are closely related but bear no causal relationship because they are both caused by a third, unexamined variable. For example, there might be a negative correlation between the age of certain skilled employees of a computer company and their salaries. One should not conclude from this correlation that the employer has necessarily discriminated against the employees on the basis of their age. A third, unexamined variable, such as the level of the employees’ technological skills, could explain differences in productivity and, consequently, differences in salary. Or, consider a patent infringement case in which increased sales of an allegedly infringing product are associated with a lower price of the patented product. This correlation would be spurious if the two products have their own noncompetitive market niches and the lower price is the result of a decline in the production costs of the patented product. Pointing to the possibility of a spurious correlation will typically not be enough to dispose of a statistical argument. It may be appropriate to give little weight to such an argument absent a showing that the correlation is relevant. For example, a statistical showing of a relationship between technological skills and worker productivity might be required in the age discrimination example, above. Causality cannot be inferred by data analysis alone; rather, one must infer that a causal relationship exists on the basis of an underlying causal theory that explains the relationship between the two variables. Even when an appropriate theory has been identified, causality can never be inferred directly. One must also look for empirical evidence that there is a causal relationship. Conversely, the fact that two variables are correlated does not guarantee the existence of a relationship; it could be that the model—a characterization of the underlying causal theory—does not reflect the correct interplay among the explanatory variables. In fact, the absence of correlation does not guarantee that a causal relationship does not exist. Lack of correlation could occur if (1) there are insufficient data, (2) the data are measured inaccurately, (3) the data do not allow multiple causal relationships to be sorted out, or (4) the model is specified wrongly because of the omission of a variable or variables that are related to the variable of interest. There is a tension between any attempt to reach conclusions with near certainty and the inherently uncertain nature of multiple regression analysis. In general, the statistical analysis associated with multiple regression allows for the expression of uncertainty in terms of probabilities. The reality that statistical analysis generates probabilities concerning relationships rather than certainty should not be seen in itself as an argument against the use of statistical evidence, or worse, as a reason to not admit that there is uncertainty at all. The only alternative might be to use less reliable anecdotal evidence. This reference guide addresses a number of procedural and methodological issues that are relevant in considering the admissibility of, and weight to be accorded to, the findings of multiple regression analyses. It also suggests some standards of reporting and analysis that an expert presenting multiple regression analyses might be expected to meet. Section II discusses research design—how the multiple regression framework can be used to sort out alternative theories about a case. The guide discusses the importance of choosing the appropriate specification of the multiple regression model and raises the issue of whether multiple regression is appropriate for the case at issue. Section III accepts the regression framework and concentrates on the interpretation of the multiple regression results from both a statistical and a practical point of view. It emphasizes the distinction between regression results that are statistically significant and results that are meaningful to the trier of fact. It also points to the importance of evaluating the robustness of regression analyses, i.e., seeing the extent to which the results are sensitive to changes in the underlying assumptions of the regression model. Section IV briefly discusses the qualifications of experts and suggests a potentially useful role for court-appointed neutral experts. Section V emphasizes procedural aspects associated with use of the data underlying regression analyses. It encourages greater pretrial efforts by the parties to attempt to resolve disputes over statistical studies. Throughout the main body of this guide, hypothetical examples are used as illustrations. Moreover, the basic “mathematics” of multiple regression has been kept to a bare minimum. To achieve that goal, the more formal description of the multiple regression framework has been placed in the Appendix. The Appendix is self-contained and can be read before or after the text. The Appendix also includes further details with respect to the examples used in the body of this guide.

Source Publication

Reference Manual on Scientifc Evidence

Publication Date

2011

Edition

3

Reference Guide on Multiple Regression

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