The recommendation on the relative merits and advantages of each of these methods in the NIST Engineering Statistics Handbook page on comparison of these methods are reproduced below: When the number of contrasts to be estimated is small, (about as many as there are factors) Bonferroni is better than Scheffé.
Actually, unless the number of desired contrasts is at least twice the number of factors, Scheffé will always show wider confidence bands than Bonferroni.
What this calculator does: Microsoft Excel can do one-way ANOVA of multiple treatments (columns) nicely. Within Excel, followup of a successful ANOVA with post-hoc Tukey HSD has to be done manually, if you know how to!
This self-contained calculator, with flexibility to vary the number of treatments (columns) to be compared, starts with one-way ANOVA.
This calculator is designed to relieve biomedical scientists from the travails of coding heavy-duty statistical packages: Are you a biomedical or social scientist, who has narrow interest in one-way ANOVA followed automatically by post-hoc Tukey HSD, Scheffé, Bonferroni and Holm methods, but do not have the patience and perseverence to hack code to harness R, Stata, SPSS, SAS or Matlab? It was inspired by the frustration of several biomedical scientists with learning the software setup and coding of these serious statistical packages, almost like operating heavy bulldozer machinery to swat an irritating mosquito.
For code grandmasters, fully working code and setup instructions are provided for replication of the results in the serious academic-research-grade open-source (and hence free) R statistical package.
Among the many reviews of the merits of the Holm method and its uniform superiority over the Bonferroni method, that of Aickin and Gensler (1996) is notable.
The follow-up post-hoc Tukey HSD multiple comparison part of this calculator is based on the formulae and procedures at the NIST Engineering Statistics Handbook page on Tukey's method.The original Bonferroni published paper in Italian dating back to 1936 is hard to find on the web.