title: The number of subjects per variable required in linear regression analyses.
abstract: OBJECTIVE
  To determine the number of independent variables that can be included in a linear regression model.
  
  
  METHODS
  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(2) 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(2), although adjusted R(2) estimates behaved well. The bias in estimating the model R(2) statistic was inversely proportional to the magnitude of the proportion of variation explained by the population regression model.
  
  
  CONCLUSIONS
  Linear regression models require only two SPV for adequate estimation of regression coefficients, standard errors, and confidence intervals.
authors.affiliation: Institute for Clinical Evaluative Sciences, G1 06, 2075 Bayview Avenue, Toronto, Ontario, Canada M4N 3M5; Institute of Health Policy, Management and Evaluation, University of Toronto, 155 College Street, Suite 425 Toronto, ON M5T 3M6, Canada; Schulich Heart Research Program, Sunnybrook Research Institute, 2075 Bayview Avenue, Toronto, ON M4N 3M5, Canada. Electronic address: peter.austin@ices.on.ca.
authors.forename: Peter C
authors.initials: PC
authors.lastname: Austin
authors.affiliation: Department of Public Health, Erasmus MC-University Medical Center Rotterdam, 's-Gravendijkwal 230 3015 CE, Rotterdam, The Netherlands.
authors.forename: Ewout W
authors.initials: EW
authors.lastname: Steyerberg
mesh_terms: D015982:Bias; D016018:Least-Squares Analysis; D016014:Linear Models; D015233:Models, Statistical; D009010:Monte Carlo Method; D012107:Research Design; D018401:Sample Size
keywords: Bias; Explained variation; Linear regression; Monte Carlo simulations; Regression; Statistical methods