Complex chi-square

[Nick Cox]

It's the unknown unknowns that get you. (The once famous politician who said this was wrongly mocked for pointing it out.)

Just now I came across "complex chi-square" and had to Google it.

Apparently it means that chi-square is on contingency tables larger than 2 x 2. My first chi-square test must have been about 1971 and I am confident it was already "complex" and certainly not offered as if it were rocket surgery.

Feel free to laugh, but how did I miss this term over nearly 50 years? Is it tribal e.g. some jargon within SPSS?

Am I just missing something utterly standard to many?

Authoritative references welcome.

[William Lisowski]

I believe "complex chi-square" may be more complex (sorry) than you were led to believe, where by "complex" I mean "defined across the complex numbers."

At https://www.quora.com/How-do-I-deal-...e-distribution I get the sense that some generalization of a "chi-square" distribution allows complex degrees of freedom.

In the (likely pirated) PDF found (at the moment) at http://dsp-book.narod.ru/DSPMW/60.PDF we find a discussion of complex random variables and stochastic processes, starting with the complex Gaussian distribution and from thence to the complex cognates of the more familiar derivatives of the Gaussian distribution.

[At this point, you can insert your own joke about what one might use a complex valued distribution for; I am leaning toward life tables for zombies and other imaginary creatures.]

[Andrew Musau]

I think Nick defines it correctly

Apparently it means that chi-square is on contingency tables larger than 2 x 2

Feel free to laugh, but how did I miss this term over nearly 50 years? Is it tribal e.g. some jargon within SPSS?

Not SPSS jargon... more like social science jargon. The following from a 1970 publication provides an insight into the terminology

http://www.jstor.org/stable/pdf/1434469.pdf (p.268)

The fact that dichotomous data can be analyzed using ANOVA techniques should
be important in educational and psychological research. Often investigators spend
much time developing instruments which will yield scores that can be analyzed with
parametric statistical tests. If the data they obtain is dichotomous, they use a chi-
square test. Complex chi-square tests have been developed, but most of them are
difficult to compute.

The term is rarely used today given the increase in computing power. Also from 1972, see

http://journals.sagepub.com/doi/abs/....1972.30.3.743

[Nick Cox]

Thanks very much to both of you.

I have no doubt that William has identified a meaning at higher levels, but it's definitely not what my source is discussing.

I think Andrew's discovery is more about supposedly complex tests than complex chi-square.

My source is a 2017 text.