Why are there two spellings of "heteroskedastic" or "heteroscedastic"?

Question

I frequently see both the spellings "heteroskedastic" and "heteroscedastic", and similarly for "homoscedastic" and "homoskedastic". There seems to be no difference in meaning between the "c" and the "k" variants, simply an orthographic difference related to the Greek etymology of the word.

What are the origins of the two distinct spellings?

Is one usage more common than the other, and do they reflect variation between regions or research fields, or nothing more than authorial (or indeed, editorial) preference?

As an aside, other languages have different policies on latinizing Greek roots to English: I note that in French it is, I think, always "hétéroscédasticité" whereas in German it is always "Heteroskedastizität". So I would not be surprised if authors with English as a second language may have a preference for the English spelling corresponding to their mother tongue's. Perhaps the real test is what Greek statisticians call it when writing in English!

Answer 1

Inside this small and vexed question even smaller questions are struggling to get out.

The most detailed discussion to date appears to be

Alfredo R. Paloyo. 2011. When did we begin to spell “heteros*edasticity” correctly? Ruhr Economic Papers 0300. see here

(a reference I owe to @Andy here in Ten fold chat). I can't do justice to its dense and detailed discussion. What follows is more by nature of an executive summary, modulo a little whimsy.

Modern search facilities make it possible to be confident that homoscedastic(ity) and heteroscedastic(ity) are modern coinages introduced, explicitly or implicitly, by the British statistician Karl Pearson in 1905. (Pearson ranged widely over several disciplines, but in the second half of his life his work was firmly centred on statistics.)

Modifying c to k raises absolutely no statistical issue. The idea is at its simplest that the Greek root being used includes the letter kappa ($\kappa$), whose direct equivalent in English is k, and so that k is the correct spelling.

However, as others have done elsewhere, we note that this suggestion was made particularly by J.H. McCulloch in the journal Econometrica, a journal which failed to follow the same logic by renaming itself Econometrika, nay Ekonometrika. (The roots behind "economics" are also Greek, including the word oikos. Ecologists will want to add that there is a journal Oikos even though, once again, ecology did not call itself oikology.)

Further, it is remarkable that Karl Pearson was no hater of k, as he changed his own name from Carl to Karl and named his own journal Biometrika, in full and conscious recognition of the original Greek words he used when devising that name.

The root question then is purely one of language, and of how faithful it is proper to be to the original words behind a coinage. If you follow up the McCulloch reference, the discussion turns to whether such words came into English directly or via other languages, and so hinges on criteria that may appear to many readers as arbitrary if not arcane. (Note that criteria is another word of Greek origin that escaped the k treatment.) Most language authorities now acknowledge that present spelling can owe much to historical accidents and that any long-established usage eventually can over-turn logic (or more precisely etymology). In total, there is plenty of scope here for scepticism (or skepticism).

In terms of tribal or other preferences, it is my impression that

1. Econometric usage seems to be shifting towards the k form. The McCulloch paper had an effect, indirectly if not directly.

2. British English seems to make more use of c forms over k forms than does American English. The form sceptic is standard in British spelling, for example.

All puns and wordplay here should be considered intentional even when accidental.

Answer 2

There is a tradition in English language to use special letters to indicate that a word is of Greek origin (and as all language "rules", it is not absolutely observed). Most of the times for example, when you see "ph" in an English word, it indicates that it has a Greek origin, as in, say, "photograph" which is the transcription of a Greek word ("phos" means "light" and "graph" is also a Greek root for "write/draw", so "photograph" $\approx$ "a writing/drawing of light").

The same happens with "c" and "k": the use of "k" indicates that the word has a Greek origin. And it does because "Heteroskedasticity" is a composite word: "Hetero + Skedasis" where "Hetero" is a Greek word that indicates "difference" and "Skedasis" means "dispersion". So "Heteroskedasticity = different dispersion", and so different variance, which is what we want to express with the word.

But as I said previously, language "rules" can be flexible, especially for international languages as English (your remark that in "less international" languages like French or German, the spelling appears to be fixed is to the point)-and so people that had to write the word and perhaps were not sure of its spelling, decided to use "c" which is the "natural" choice. Or they normatively thought that words should be "merged" in the language they are used as much as possible.

As for what Greek statisticians do, I guess even the slightest amount of "national pride" (or national chauvinism), would be enough to make them use "k" instead of "c".

Answer 3

In Polish it is "heteroskedastyczność", but sometimes "heteroscedastyczność" is used instead. For examples you can check the book by Andrzej Gałecki and Tomasz Burzykowski, who were born and educated in Poland. They use the "c" form in their book written in English. Notice, however, that the forms used by different authors could just reflect editorial policies of journals and publishers, so may not reflect how authors consider how words should be spelled.

Wikipedia (which uses the "c" form) leads to a paper by McCulloch (1985) who argues that the "k" form is appropriate since the word has Greek origin and is written as $\kappa$, which should be transliterated as "k".

McCulloch, J.H. (1985). On Heteros*edasticity. Econometrica, 53(2), 483.

Answer 4

The missing explanation is that letter 'C' was always pronounced as the modern English 'K' in classical Latin, while K itself was actually a redundant letter. A Greek word with the letter Kappa borrowed into Latin, in Roman times, would have always been spelled with a C. Later, in Vulgar Latin and by extension in French and English, the pronunciation of C became corrupted and was pronounced as a 'S' or a 'CH' when it came before vowels 'E' and 'I'.

Therefore the objectively correct spelling (by Latin standards) would be with a C, and the fact that an alternate K-spelling exists shows that the word is a modernism and doesn't hail from Roman times. By modern English standards, both spellings are equivalent.

What I'm trying to say is that when Pearson first used the spelling "heteroskedasticity", he made a judgement call to intentionally go against the norm (for his own subjective reasons) and spell it with a K (according to Nick Cox's answer, he did the same with "Biometrika"). There is no linguistic motive behind this spelling other than the fact (perhaps) that he knew that these words would have once been pronounced with a 'k' and found it aesthetically more pleasing to spell them so.

Answer 5

This is a very interesting discussion.

One problem with using the c instead of k is that in the modern Italianized Latin pronunciation, the combination ce (as with ci) yields a "soft c" (/ch/ sound); moreover, the combination sce yields the soft /sh/ sound. So the word heteroscedastic would be pronounced /hetero-sheh-das-tic/.

The letter c only makes a hard /k/ sound in front of the vowels a, o, and u. To force a c to make the hard /k/ sound in front of soft vowels (e and i), you need to add a modifier. Different Romance languages have handled this in different ways... In Italian, the convention is to use the otherwise silent h after the c: ce = /cheh/, whereas che = /keh/; and sce = /sheh/, whereas sche = /skeh/. In Spanish, the c is replaced by a qu: que = /keh/. There is a similar practice in French and Portuguese.

For this reason, it would be quite rational to favor the k spelling of heteroskedastic. It pays homage to the original Greek transcription and is arguably more phonetically correct in the Romance languages and by extension in English.