Derangement(s)

Things are deranged if they are out of order. Interestingly, whether we can put them back in order can tell us if we are picking up useful information about them.

Details #

If people are given the numbers 5, 3, 1, 4 and 2 and asked to put them back in ascending order most people will rearrange them into 1, 2, 3, 4 and 5 because they understand that sequence and that it is “ascending order”. As there are 120 ways of ordering five objects the chances of doing that by chance if you didn’t understand “ascending order” is only one in 120: conventionally statistically significant.

That’s not terribly interesting to us but say we have five people we know well and their answers on a purely projective “test”, now if we can map their answers to the people and get all five mapped correctly that has the same implication: we know those people well enough and understand the projective test well enough that that we achieved something with a probability we could have achieved it by chance alone of the same one in 120: p = .00833, better than the traditional criterion of 1 in 20, .05, for rejecting the hypothesis that only chance is operating. This is a nice way to test whether people using purely projective, or idiographic data are extracting meaningful information from them. Getting four or more correct is actually statistically significant regardless of whether it’s all four of four, all five of five, four or six of six, etc.

(With n objects you can never map n – 1 correctly because when you have mapped the first n – 2 correctly the only remaining options are to get the remaining two wrong, and score n – 2 correct, or get them right and score n correctly. n – 1 is always impossible.)

It’s a method dear to me because as far as I know I co-wrote the first paper describing its use as a way of testing information transfer with idiographic “measures”/data:

Evans, Chris, John Hughes, and Julia Houston. 2002. ‘Significance Testing the Validity of Ideographic Methods: A Little Derangement Goes a Long Way’. British Journal of Mathematical and Statistical Psychology 55(2):385–90. doi: 10.1348/000711002760554525. Not open access, contact me if you want a copy.

A clearer explanation and mapping to nomothetic statistical methods is in:

Evans, Chris, Jo-anne Carlyle, and Clara Paz. 2023. ‘Rigorous Idiography: Exploring Subjective and Idiographic Data with Rigorous Methods—The Method of Derangements’. Frontiers in Psychology 13:1007685. doi: 10.3389/fpsyg.2022.1007685. (Open access.)