Objectives To determine the number of independent variables that can be included in a linear regression model. Study Design and Setting We used a series of Monte Carlo simulations to examine the impact of the number of subjects per variable (SPV) on the accuracy of estimated regression coefficients and standard errors, on the empirical coverage of estimated confidence intervals, and on the accuracy of the estimated R<sup>2</sup> of the fitted model. Results A minimum of approximately two SPV tended to result in estimation of regression coefficients with relative bias of less than 10%. Furthermore, with this minimum number of SPV, the standard errors of the regression coefficients were accurately estimated and estimated confidence intervals had approximately the advertised coverage rates. A much higher number of SPV were necessary to minimize bias in estimating the model R<sup>2</sup>, although adjusted R<sup>2</sup> estimates behaved well. The bias in estimating the model R<sup>2</sup> statistic was inversely proportional to the magnitude of the proportion of variation explained by the population regression model. Conclusion Linear regression models require only two SPV for adequate estimation of regression coefficients, standard errors, and confidence intervals.

Bias, Explained variation, Linear regression, Monte Carlo simulations, Regression, Statistical methods
dx.doi.org/10.1016/j.jclinepi.2014.12.014, hdl.handle.net/1765/82396
Journal of Clinical Epidemiology
Department of Public Health

Austin, P.C, & Steyerberg, E.W. (2015). The number of subjects per variable required in linear regression analyses. Journal of Clinical Epidemiology, 68(6), 627–636. doi:10.1016/j.jclinepi.2014.12.014