regress

DEALING WITH THE INHERENT POSITIVE BIAS IN MULTIPLE REGRESSION ANALYSIS MARC R. LOCHBAUM, DANIEL M. LANDERS, MIN-QI WANG, WALTER SALAZAR, STEVEN J. PETRUZZELLO, AND JERRY R. THOMAS ARIZONA STATE UNIVERSITY¹ Multiple regression is a commonly used statistical technique in exercise and sport science research. The accuracy of prediction of the independent variables is gauged by the strength of the resultant R² , which indicates the portion of the variance of the dependent variable explained by the independent variables. The purpose of this study was to examine the positive bias that is often associated with the R² , which is often overlooked by researchers. To support this assertion, the use of multiple regression was examined in typical publication outlets for motor behaviorists and sport and exercise psychologists (i.e., Journal of Sport and Exercise Psychology and Research Quarterly for Exercise and Sport). A computer simulation was performed using unrelated random numbers so that any fluctuation in the R² as a function of various subject-to-variable ratios would be considered bias. The results indicated that: (a) incremental decreases in the subject-to-variable ratios resulted in increases in R² ; (b) very small subject-to-variable ratios resulted in large R² s; and (c) larger subject-to-variable ratios resulted in small R² s.
Adjusting the R² for the number of independent variables and the number of subjects used reduced bias, but only for subject-to-variable ratios greater than 10:1. It appears that when using multiple regression to analyze data, it would be prudent for researchers to use at least 10 subjects for every independent variable measured. In addition, research reports submitted to journals should routinely report the adjusted R² so that the reader can more accurately judge how much variance is actually explained when bias is accounted for.

¹ Currently, Min-Qi Wang is at the University of Alabama--Tuscaloosa; Walter Salazar is at the University of Costa Rica; Steven Petruzzello is at the University of Illinois--Champaign-Urbana; and Jerry Thomas is at Iowa State University. Abstract for this presentation is published in the Journal of Sport & Exercise Psychology, 20 (June 1998 Supplement), S95.