Multiple comparisons (linear models)

Multiple comparisons make simultaneous inferences about a set of parameters.

When making inferences about more than one parameter (such as comparing many means, or the differences between many means), you must use multiple comparison procedures to make inferences about the parameters of interest. The problem when making multiple comparisons using individual tests such as Student's t-test applied to each comparison is the chance of a type I error increases with the number of comparisons. If you use a 5% significance level with a hypothesis test to decide if two groups are significantly different, there is a 5% probability of observing a significant difference that is simply due to chance (a type I error). If you made 20 such comparisons, the probability that one or more of the comparisons is statistically significant simply due to chance increases to 64%. With 50 comparisons, the chance increases to 92%. Another problem is the dependencies among the parameters of interest also alter the significance level. Therefore, you must use multiple comparison procedures to maintain the simultaneous probability close to the nominal significance level (typically 5%).

Multiple comparison procedures are classified by the strength of inference that can be made and the error rate controlled. A test of homogeneity controls the probability of falsely declaring any pair to be different when in fact all are the same. A stronger level of inference is confident inequalities and confident directions which control the probability of falsely declaring any pair to be different regardless of the values of the others. An even stronger level is a set of simultaneous confidence intervals that guarantees that the simultaneous coverage probability of the intervals is at least 100(1-alpha)% and also have the advantage of quantifying the possible effect size rather than producing just a p-value. A higher strength inference can be used to make an inference of a lesser strength but not vice-versa. Therefore, a confidence interval can be used to perform a confidence directions/inequalities inference, but a test of homogeneity cannot make a confidence direction/inequalities inference.

The most well known multiple comparison procedures, Bonferroni and Šidák, are not multiple comparison procedures per se. Rather they are an inequality useful in producing easy to compute multiple comparison methods of various types. In most scenarios, there are more powerful procedures available. A useful application of Bonferroni inequality is when there are a small number of pre-planned comparisons. In these cases, you can use the standard hypothesis test or confidence interval with the significance level (alpha) set to the Bonferroni inequality (alpha divided by the number of comparisons).

A side effect of maintaining the significance level is a lowering of the power of the test. Different procedures have been developed to maintain the power as high as possible depending on the strength of inference required and the number of comparisons to be made. All contrasts comparisons allow for any possible contrast; all pairs forms the k*(k-1)/2 pairwise contrasts, whereas with best forms k contrasts each with the best of the others, and against control forms k-1 contrasts each against the control group. You should choose the appropriate contrasts of interest before you perform the analysis, if you decide after inspecting the data, then you should only use all contrasts comparison procedures.