Both Scheffe and Tukey are 80-100 years old. Take a look at some modern methods in the attachment. If you wish to stick with 100 year old methods google the tests and look at the computation equations. The one with n1 and n2.in it is your best choice because it handles different sample sizes

Best wishes David Booth

David Eugene Booth , the link you shared mainly proposes Bonferroni's correction (partly as modified by Holm as a step-down procedure), a method from 1936 - so this method is also almost 90 yeras old ;)

• Bonferroni, Carlo E. (1936), "Teoria statistica delle classi e calcolo delle probabilità", Pubbl. D. R. Ist. Super. Di Sci. Econom. E Commerciali di Firenze (in Italian), 8: 1–62, Zbl 0016.41103

Benajmini and Hochberg's paper about the FDR is from 1995, a bit more recent, but the concept of FDR and FWER is quite different and controlling either of them served very different purposes (I have not yet found a single instance where controlling the FDR was resonable and justified better than simply taking the "top-N" candidates from a larger list).

The presentation is also about the multtest package in R. I wonder why. Base R has the function p.adjust() that allows for a variety of adjustments for multiple testing,including Bonferroni(-Holm) and Benjamini-Hochberg.

• https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/p.adjust

The multtest package (its function glht()) has the advantage that is allows for testing any kind of linear hypothesis in models with covariables and that is can be applied to GLMs as wel - something that wasn't discussed at any point in the presentation. So that (mentioning multcomp) was just complicating a simple task.

@Jochen as I am sure you know these were originally set up to use in Bio conductor which did not have such routines at the time before it was integrated into R you see. Further these are new.in the sense that nobody used them in microarray analysis prior to that time. Further these are very commonly used in bioassay today especially when the sample size is very large as in micrroarray experiments. Further I find that this introduction is much easier to follow than the.8 or so methods usually discussed. Best wishes David PS see Steen Knudsen books on Microarrays.

The Wikipedia article on Scheffé's method has a fair description of the difference between these two methods ( https://en.wikipedia.org/wiki/Scheff%C3%A9%27s_method ).

Implementations of Tukey's HSD test in good software can handle unequal sample sizes. (This is usually called Tukey-Kramer.)

As a general approach --- if you are using good statistical software --- is to look at more contemporary and flexible methods, like those implemented for estimated marginal means. Those would usually be called by a function or package called emmeans.

If you don't have these methods available: If you are looking at pairwise comparisons of means, Tukey's HSD is a good one. You may want to look up the assumptions of that test, tho.