title: Simultaneous confidence intervals for ratios with applications to the comparison of several treatments with a control.
abstract: OBJECTIVE
  In this article, we illustrate and compare exact simultaneous confidence sets with various approximate simultaneous confidence intervals for multiple ratios as applied to many-to-one comparisons. Quite different datasets are analyzed to clarify the points.
  
  
  METHODS
  The methods are based on existing probability inequalities (e.g., Bonferroni, Slepian and Sidak), estimation of nuisance parameters and re-sampling techniques. Exact simultaneous confidence sets based on the multivariate t-distribution are constructed and compared with approximate simultaneous confidence intervals.
  
  
  RESULTS
  It is found that the coverage probabilities associated with the various methods of constructing simultaneous confidence intervals (for ratios) in manyto-one comparisons depend on the ratios of the coefficient of variation for the mean of the control group to the coefficient of variation for the mean of the treatments. If the ratios of the coefficients of variations are less than one, the Bonferroni corrected Fieller confidence intervals have almost the same coverage probability as the exact simultaneous confidence sets. Otherwise, the use of Bonferroni intervals leads to conservative results.
  
  
  CONCLUSIONS
  When the ratio of the coefficient of variation for the mean of the control group to the coefficient of variation for the mean of the treatments are greater than one (e.g., in balanced designs with increasing effects), the Bonferroni simultaneous confidence intervals are too conservative. Therefore, we recommend not using Bonferroni for this kind of data. On the other hand, the plug-in method maintains the intended confidence coefficient quite satisfactorily; therefore, it can serve as the best alternative in any case.
authors.affiliation: Bioinformatics Unit, University of Hannover, Hannover, Germany. dilba@bioinf.uni-hannover.de
authors.forename: G
authors.initials: G
authors.lastname: Dilba
authors.forename: F
authors.initials: F
authors.lastname: Bretz
authors.forename: V
authors.initials: V
authors.lastname: Guiard
authors.forename: L A
authors.initials: LA
authors.lastname: Hothorn
mesh_terms: D015746:Abdominal Pain / Q000188:drug therapy; D016001:Confidence Intervals; D005260:Female; D006801:Humans; D007917:Leprostatic Agents / Q000494:pharmacology / Q000627:therapeutic use*; D007918:Leprosy / Q000188:drug therapy; D008297:Male; D016032:Randomized Controlled Trials as Topic / Q000706:statistics & numerical data
chemical_list: D007917:Leprostatic Agents