Bonferroni Adjustments in Tests for Regression Coefficients

University of Northern Colorado A common application of multiple linear regression is to build a model that contains only those predictors that are significantly related to the response. In so doing, tests regarding the unique contribution of individual predictors to the model are often performed. It is not uncommon for practitioners to conduct each of these tests at the nominal α = 0.05 level, without regard to the effect that this practice may have on the overall Type I error rate. This research investigated the utility of making a Bonferroni adjustment when conducting these tests of the partial regression coefficients. Simulated multivariate normal populations with various correlational structures, different numbers of predictors in the model, and differing numbers of “significant” predictors in the model were generated. Ten thousand samples, 5000 each of sizes 50 and 300, were drawn from each population condition and a multiple regression analysis was performed on each sample. In every case, the observed significance levels for the Bonferroni-adjusted tests were controlled below the nominal 0.05 level as expected, and in most cases substantially lower than the observed significance levels for the unadjusted tests. ultiple Linear Regression (MLR) is a popular statistical procedure for investigating the nature of the relationships among several numerical characteristics. Typically, one of the characteristics is identified as the dependent or response variable and the remainder of the characteristics are called independent or predictor variables. Most introductory level statistics texts identify regression analyses as having two uses: 1) to estimate the average response for a sample of individuals having various values for each variable in a set of predictors, and 2) to predict the response for a “new” individual for whom only values of the predictors are measured/observed. In either case, a linear model, based on observed data is used to make the estimation or prediction. In some applications, the researcher knows which variables should be used as predictors in the model and the purpose of the analysis is to predict the value of the response using previous information regarding the nature of the variables’ relationships with each other. Data are collected on the predictor variables and the model is used to predict the value of the response variable for one or more “new” individuals. In other situations, the researcher is interested in determining which, if any, of several numerical characteristics are significantly related to a specific outcome. Data are collected on all the variables of interest—the dependent variable and all the independent variables—and an MLR analysis is performed to build a model that may later be used for prediction, i.e., the researcher determines which of these predictor variables displays a significant unique ability to explain variation in the response variable. While both of these applications of MLR are useful and appropriate, it is the latter situation which is the focus of this research. When the purpose of the regression analysis is to determine which independent variables are unique contributors to the model, it is typical for the researcher to perform separate tests of the partial regression coefficients (i.e., the beta coefficients) for each predictor. Those predictors for which the test of the beta coefficient has a p-value that is less than the specified α-level are deemed to be making a unique contribution to the model and will be retained in the model as a predictor variable. On the other hand, those variables for which the test of the beta coefficient has a p-value that is larger than that specified α level are not identified as useful predictors and may be dropped from the model in the interest of parsimony. It is not an uncommon practice for each of these separate tests to be conducted at the nominal 5% significance level. The purpose of this research is to investigate whether conducting each of these tests at α = 0.05 inflates the overall Type I error rate for the collection of all these tests and if a Bonferroni-type adjustment to the α level for each test would be appropriate to control the overall α-level closer to that nominal level. Making adjustments to the significance level of a statistical test when multiple tests are conducted on the same data is a common statistical practice. Many procedures have been developed for making such adjustments. One of these procedures is the Bonferroni adjustment.