Post-hoc analysis

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Post-hoc analysis design and analysis of experiments, refers to looking at the data—after the experiment has concluded—for patterns that were not specified a priori. It is also known as data dredging to evoke the sense that the more one looks the more likely something will be found. More subtly, each time a pattern in the data is considered, a statistical test is effectively performed. This greatly inflates the total number of statistical tests and necessitates the use of multiple testing procedures to compensate. However, this is difficult to do precisely and in fact most results of post-hoc analyses are reported as they are with unadjusted p-values. These p-values must be interpreted in light of the fact that they are a small and selected subset of a potentially large group of p-values. Results of post-hoc analysis should be explicitly labeled as such in reports and publications to avoid misleading readers.

In practice, post-hoc analysis is usually concerned with finding patterns in subgroups of the sample.

In Latin post-hoc means "after this".

[edit] Student Newman-Keuls Post-Hoc ANOVA Analysis

An example of a Post-hoc analysis would be a Student Newman-Keuls Test: "A different approach to evaluating a posteriori pairwise comparisons stems from the work of Student (1927), Newman (1939), and Keuls (1952). The Newman-Keuls procedure is based on a stepwise or layer approach to significance testing. Sample means are ordered from the smallest to the largest. The largest difference, which involves means that are r = p steps apart, is tested first at α level of significance; if significant, means that are $r = p - 1$ steps apart are tested at α level of significance and so on. The Newman-Keuls procedure provides an r-mean significance level equal to α for each group of r ordered means; that is, the probability of falsely rejecting the hypothesis that all means in an ordered group are equal to α. It follows that the concept of error rate applies neither on an experimentwise nor on a per comparison basis--the actual error rate falls somewhere between the two. The Newman-Keuls procedure, like Tukey's procedure, requires equal sample n's.

The critical difference y-hat(Wr), that two means separated by r steps must exceed to be declared significant is, according to the Newman-Keuls procedure,

$\psi - \widehat{W_{r}} = q_{\alpha;p,v} \sqrt{\frac{MSE}{n}} \,$

It should be noted that the Newman-Keuls and Tukey procedures require the same critical difference for the first comparison that is tested. The Tukey procedure uses this critical difference for all of the remaining tests while the Newman-Keuls procedure reduces the size of the critical difference, depending on the number of steps separating the ordered means. As a result, Newman-Keuls test is more powerful than Tukey's test. Remember, however, that Newman-Keuls procedure does not control the experimentwise error rate at α.

Frequently a test of the overall null hypothesis m1 =m2 …= mp is performed with an F statistic in ANOVA rather than with a range statistic. If the F statistic is significant, Shaffer (1979) recommends using the critical difference $y - h a t (W r - 1)$ instead of y-hat(Wr) to evaluate the largest pairwise comparison at the first step of the testing procedure. The testing procedure for all subsequent steps is unchanged. She has shown that the modified procedure leads to greater power at the first step without affecting control of the type I error rate. This makes dissonances, in which the overall null hypothesis is rejected by an F test without rejecting any one of the proper subsets of comparison, less likely." [1]