title: An easy mathematical proof showed that time-dependent bias inevitably leads to biased effect estimation.
abstract: OBJECTIVE
  Time-dependent bias occurs when future exposure status is analyzed as being known with start of observation. As this bias is common, we sought to determine whether it always leads to biased effect estimation. We also sought to determine the direction of the effect bias.
  
  
  METHODS
  We derived an easy mathematical proof investigating the nature of time-dependent bias. We applied the general mathematical result to data from a prospective cohort study on the incidence of hospital infection in intensive care: Here, we investigated the effect of time-dependent hospital infection status on intensive care unit stay. The nature of time-dependent bias was also illustrated graphically.
  
  
  RESULTS
  Biased effect estimation is a mathematically inevitable consequence of time-dependent bias, because the number of individuals at risk of exposure is distorted over the course of time. In case of a time-dependent exposure that prolongs time until the study endpoint, the prolonging effect will be overestimated.
  
  
  CONCLUSIONS
  Because time-dependent bias inevitably leads to erroneous findings, it is a major concern that it is common in the clinical literature. Time-dependent bias can be avoided by proper hazard-based analyses.
authors.affiliation: Freiburg Centre for Data Analysis and Modelling, Freiburg University, Freiburg, Germany; Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg, Freiburg, Germany. Electronic address: jan.beyersmann@fdm.uni-freiburg.de.
authors.forename: Jan
authors.initials: J
authors.lastname: Beyersmann
authors.affiliation: Institute of Medical Microbiology and Hospital Epidemiology, Hannover Medical School, Germany.
authors.forename: Petra
authors.initials: P
authors.lastname: Gastmeier
authors.affiliation: Freiburg Centre for Data Analysis and Modelling, Freiburg University, Freiburg, Germany; Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg, Freiburg, Germany.
authors.forename: Martin
authors.initials: M
authors.lastname: Wolkewitz
authors.affiliation: Institute of Medical Biometry and Medical Informatics, University Medical Center Freiburg, Freiburg, Germany.
authors.forename: Martin
authors.initials: M
authors.lastname: Schumacher
mesh_terms: D015982:Bias; D003422:Critical Care / Q000706:statistics & numerical data; D003428:Cross Infection / Q000453:epidemiology; D003627:Data Interpretation, Statistical; D005858:Germany / Q000453:epidemiology; D006801:Humans; D007902:Length of Stay / Q000706:statistics & numerical data; D015233:Models, Statistical; D011014:Pneumonia / Q000453:epidemiology; D013997:Time Factors