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Relationship between Omnibus and Post-hoc Tests: An Investigation of performance of the F test in ANOVA
  1. Tian Chen1,
  2. Manfei Xu2,
  3. Justin Tu3,
  4. Hongyue Wang4 and
  5. Xiaohui Niu5
  1. 1Department of Mathematics and Statistics, University of Toledo, OH, USA
  2. 2Shanghai Mental Health Center, Shanghai Jiao Tong University Medical College, Shanghai, China
  3. 3Department of Physical Medicine and Rehabilitation, University of Virginia School of Medicine, Charlottesville, VA, USA
  4. 4Department of Biostatistics and Computational Biology, University of Rochester, NY, USA
  5. 5College of Informatics, Huazhong Agriculture University, Wuhan, China
  1. correspondence: Tian CHEN. Mailing address: 2801 W. Bancroft Street, MS 942, Toledo, OH, USA. Postcode: 43606. E-Mail: tian.chen{at}


Comparison of groups is a common statistical test in many biomedical and psychosocial research studies. When there are more than two groups, one first performs an omnibus test for an overall difference across the groups. If this null is rejected, one then proceeds to the next step of post-hoc pairwise group comparisons to determine sources of difference. Otherwise, one stops and declares no group difference. A common belief is that if the omnibus test is significant, there must exist at least two groups that are significantly different and vice versa. Thus, when the omnibus test is significant, but no post-hoc between-group comparison shows significant difference, one is bewildered at what is going on and wondering how to interpret the results. At the end of the spectrum, when the omnibus test is not significant, one wonders if all post-hoc tests will be non-significant as well so that stopping after a nonsignificant omnibus test will not lead to any missed opportunity of finding group difference. In this report, we investigate this perplexing phenomenon and discuss how to interpret such results.

  • Omnibus test
  • post-hoc test
  • F test
  • Tukey’s test

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