title: Sample size importantly limits the usefulness of instrumental variable methods, depending on instrument strength and level of confounding.
abstract: OBJECTIVE
  Instrumental variable (IV) analysis is promising for estimation of therapeutic effects from observational data as it can circumvent unmeasured confounding. However, even if IV assumptions hold, IV analyses will not necessarily provide an estimate closer to the true effect than conventional analyses as this depends on the estimates' bias and variance. We investigated how estimates from standard regression (ordinary least squares [OLS]) and IV (two-stage least squares) regression compare on mean squared error (MSE).
  
  
  METHODS
  We derived an equation for approximation of the threshold sample size, above which IV estimates have a smaller MSE than OLS estimates. Next, we performed simulations, varying sample size, instrument strength, and level of unmeasured confounding. IV assumptions were fulfilled by design.
  
  
  RESULTS
  Although biased, OLS estimates were closer on average to the true effect than IV estimates at small sample sizes because of their smaller variance. The threshold sample size above which IV analysis outperforms OLS regression depends on instrument strength and strength of unmeasured confounding but will usually be large given the typical moderate instrument strength in medical research.
  
  
  CONCLUSIONS
  IV methods are of most value in large studies if considerable unmeasured confounding is likely and a strong and plausible instrument is available.
authors.affiliation: Department of Clinical Epidemiology, C7-P, Leiden University Medical Centre, PO Box 9600, 2300 RC Leiden, The Netherlands. Electronic address: a.g.c.boef@lumc.nl.
authors.forename: Anna G C
authors.initials: AG
authors.lastname: Boef
authors.affiliation: Department of Clinical Epidemiology, C7-P, Leiden University Medical Centre, PO Box 9600, 2300 RC Leiden, The Netherlands; Department of Endocrinology and Metabolic Diseases, Leiden University Medical Centre, Leiden, The Netherlands.
authors.forename: Olaf M
authors.initials: OM
authors.lastname: Dekkers
authors.affiliation: Department of Clinical Epidemiology, C7-P, Leiden University Medical Centre, PO Box 9600, 2300 RC Leiden, The Netherlands.
authors.forename: Jan P
authors.initials: JP
authors.lastname: Vandenbroucke
authors.affiliation: Department of Clinical Epidemiology, C7-P, Leiden University Medical Centre, PO Box 9600, 2300 RC Leiden, The Netherlands; Department of Medical Statistics and Bioinformatics, Leiden University Medical Centre, Leiden, The Netherlands.
authors.forename: Saskia
authors.initials: S
authors.lastname: le Cessie
mesh_terms: D015982:Bias; D006801:Humans; D016018:Least-Squares Analysis; D057182:Mendelian Randomization Analysis; D015233:Models, Statistical; D064887:Observational Studies as Topic; D016032:Randomized Controlled Trials as Topic; D012044:Regression Analysis; D012107:Research Design / Q000592:standards*; D018401:Sample Size
keywords: Confounding; Instrumental variable; Observational studies; Simulation study; Therapeutic effects; Variance