Bayesian Analysis of Variance (Anova)

  • Ton J. Cleophas
  • Aeilko H. Zwinderman


In studies with two unpaired samples of continuous data as outcome and a binary predictor like treatment modality unpaired t-tests are usually applied. With three or more unpaired samples traditional t-tests are impossible, and analysis of variance (anova) must be applied. Instead of a traditional Anova a Bayesian Anova is possible. A traditional analysis of variance with three treatment modalities as predictor provided a Fisher (F) statistic of 14.110, p-value = 0.0001. A Bayesian analysis of variance of these data did not compute “supporting null versus alternative hypothesis” but, in contrast, the other ways around, and it provided support in favor of the traditional test with a Bayes factor of 389.479. The 95% credible intervals gave an impression about the widths of the likelihood distributions.

95% confidence interval

95% credible interval

Group 0.0



Group 1.0



Group 3.0



The group 0 credible interval was a bit wider than the traditional 95% confidence interval was. The groups 2 and 3 credible intervals were a bit narrower. Obviously, group 0 provided the best likelihood in favor of the alternative hypothesis H1.

The Bayes factor (BF) in this example of 389.479 gave very strong support of the alternative hypothesis, and was compatible with the p-value of <0.0001 as obtained from the traditional one way anova.

Suggested Reading1,2

  1. Statistics applied to clinical studies 5th edition, 2012Google Scholar
  2. Machine learning in medicine a complete overview, 2015Google Scholar
  3. SPSS for starters and 2nd levelers 2nd edition, 2015Google Scholar
  4. Clinical data analysis on a pocket calculator 2nd edition, 2016Google Scholar
  5. Understanding clinical data analysis from published research, 2016Google Scholar
  6. Modern Meta-analysis, 2017Google Scholar
  7. Regression Analysis in Clinical Research, 2018Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ton J. Cleophas
    • 1
  • Aeilko H. Zwinderman
    • 2
  1. 1.Department Medicine Albert Schweitzer HospitalAlbert Schweitzer HospitalSliedrechtThe Netherlands
  2. 2.Department Biostatistics and EpidemiologyAcademic Medical Center Department Biostatistics and EpidemiologyAmsterdamThe Netherlands

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