Conducting multiple independent-samples $$t$$-tests to compare every pair of group means in an experiment creates a statistical problem. While a single $$t$$-test has a $$5\%$$ chance of mistakenly rejecting a true null hypothesis (a Type I error), conducting several $$t$$-tests simultaneously causes this risk to compound. As more tests are run, the overall probability of mistakenly rejecting at least one true null hypothesis becomes unacceptably high, necessitating specialized statistical procedures to control the error rate.

Multiple Comparisons

The continuous probability distribution of the studentized range of an i.i.d. sample from a normally distributed population.
Differentiating the cumulative distribution function with respect to q gives the probability density function.

Studentized Range Distribution

Single-step multiple comparison procedure and statistical test. It can be used to find means that are significantly different from each other.
Compares all possible pairs of means, and is based on a studentized range distribution

University of Michigan - Ann Arbor

Method for adjusting significance levels in a linear regression analysis to account for multiple comparisons.
Scheffé's method is a single-step multiple comparison procedure which applies to the set of estimates of all possible contrasts among the factor level means, not just the pairwise differences considered by the Tukey–Kramer method.
It works on similar principles as the Working–Hotelling procedure for estimating mean responses in regression, which applies to the set of all possible factor levels.

Scheffe's Method

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, pp. 3-7). New York: springer.

An Introduction to Statistical Learning with Applications in R

1. Analysis of variance (a special case of regression analysis)
2. constructing simultaneous confidence bands for regressions involving basis functions.

What is Scheffe's method useful for?

Tukey-Kramer Method

A method of simultaneous estimation in linear regression models.
One of the first developments in simultaneous inference, it was devised by Working and Hotelling for the simple linear regression model in 1929.
Provides a confidence region for multiple mean responses

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