Type III error

In statistical hypothesis testing, there are various notions of so-called type III errors (or errors of the third kind), and sometimes type IV errors or higher, by analogy with the type I and type II errors of Jerzy Neyman and Egon Pearson. Fundamentally, Type III errors occur when researchers provide the right answer to the wrong question.

Since the paired notions of type I errors (or "false positives") and type II errors (or "false negatives") that were introduced by Neyman and Pearson are now widely used, their choice of terminology ("errors of the first kind" and "errors of the second kind"), has led others to suppose that certain sorts of mistakes that they have identified might be an "error of the third kind", "fourth kind", etc.

None of these proposed categories has been widely accepted. The following is a brief account of some of these proposals.

In systems theory an additional type III error is often defined: type III (δ): asking the wrong question and using the wrong null hypothesis.

Florence Nightingale David (1909–1993) [1] a sometime colleague of both Neyman and Pearson at the University College London, making a humorous aside at the end of her 1947 paper, suggested that, in the case of her own research, perhaps Neyman and Pearson's "two sources of error" could be extended to a third:

I have been concerned here with trying to explain what I believe to be the basic ideas [of my "theory of the conditional power functions"], and to forestall possible criticism that I am falling into error (of the third kind) and am choosing the test falsely to suit the significance of the sample.

In 1948, Frederick Mosteller (1916–2006) argued that a "third kind of error" was required to describe circumstances he had observed, namely:

According to Henry F. Kaiser (1927–1992), in his 1966 paper extended Mosteller's classification such that an error of the third kind entailed an incorrect decision of direction following a rejected two-tailed test of hypothesis. In his discussion (1966, pp. 162–163), Kaiser also speaks of α errors, β errors, and γ errors for type I, type II and type III errors respectively (C.O. Dellomos).

...
Wikipedia