title: Nonlinear modeling was applied thoughtfully for risk prediction: the Prostate Biopsy Collaborative Group.
abstract: OBJECTIVE
  We aimed to compare nonlinear modeling methods for handling continuous predictors for reproducibility and transportability of prediction models.
  
  
  METHODS
  We analyzed four cohorts of previously unscreened men who underwent prostate biopsy for diagnosing prostate cancer. Continuous predictors of prostate cancer included prostate-specific antigen and prostate volume. The logistic regression models included linear terms, logarithmic terms, fractional polynomials of degree one or two (FP1 and FP2), or restricted cubic splines (RCS) with three or five knots (RCS3 and RCS5). The resulting models were internally validated by bootstrap resampling and externally validated in the cohorts not used at model development. Performance was assessed with the area under the receiver operating characteristic curve (AUC) and the calibration component of the Brier score (CAL).
  
  
  RESULTS
  At internal validation models with FP2 or RCS5 showed slightly better performance than the other models (typically 0.004 difference in AUC and 0.001 in CAL). At external validation models containing logarithms, FP1, or RCS3 showed better performance (differences 0.01 and 0.002).
  
  
  CONCLUSIONS
  Flexible nonlinear modeling methods led to better model performance at internal validation. However, when application of the model is intended across a wide range of settings, less flexible functions may be more appropriate to maximize external validity.
authors.affiliation: Department of Public Health, Erasmus MC-University Medical Center Rotterdam, Rotterdam, The Netherlands. Electronic address: d.nieboer@erasmusmc.nl.
authors.forename: Daan
authors.initials: D
authors.lastname: Nieboer
authors.affiliation: Department of Public Health, Erasmus MC-University Medical Center Rotterdam, Rotterdam, The Netherlands.
authors.forename: Yvonne
authors.initials: Y
authors.lastname: Vergouwe
authors.affiliation: Department of Urology, Erasmus MC-University Medical Center Rotterdam, Rotterdam, The Netherlands.
authors.forename: Monique J
authors.initials: MJ
authors.lastname: Roobol
authors.affiliation: Department of Mathematics, Technical University Munich, Munich, Germany; University of Texas Health Science Center at San Antonio, San Antonio, TX, USA.
authors.forename: Donna P
authors.initials: DP
authors.lastname: Ankerst
authors.affiliation: Cleveland Clinic Foundation Quantitative Health Sciences Division, Cleveland, OH, USA.
authors.forename: Michael W
authors.initials: MW
authors.lastname: Kattan
authors.affiliation: Department of Epidemiology and Biostatistics, Memorial Sloan-Kettering Cancer Center, New York, USA.
authors.forename: Andrew J
authors.initials: AJ
authors.lastname: Vickers
authors.affiliation: Department of Public Health, Erasmus MC-University Medical Center Rotterdam, Rotterdam, The Netherlands.
authors.forename: Ewout W
authors.initials: EW
authors.lastname: Steyerberg
mesh_terms: D000368:Aged; D001706:Biopsy; D005260:Female; D006801:Humans; D007610:Kallikreins / Q000378:metabolism; D016015:Logistic Models; D008297:Male; D008875:Middle Aged; D015233:Models, Statistical; D017711:Nonlinear Dynamics; D011237:Predictive Value of Tests; D017430:Prostate-Specific Antigen / Q000378:metabolism; D011471:Prostatic Neoplasms / Q000175:diagnosis* / Q000378:metabolism / Q000473:pathology; D012372:ROC Curve; D016032:Randomized Controlled Trials as Topic; D015203:Reproducibility of Results; D012306:Risk
chemical_list: C549870:KLK3 protein, human; D007610:Kallikreins; D017430:Prostate-Specific Antigen
keywords: Calibration; Discrimination; External validation; Internal validation; Nonlinear modeling; Prediction models