Origin and spelling of (multi)collinear/colinear

Multicollinear and multicolinear both appear to be used in scientific literature, although the former appears to be a lot more popular and is also the name of the tag here. I figured there must be a single spelling of colinear/collinear which is more correct, but that is also spelled both ways.

Two questions about this:

1. Which is more correct? Is there some different meaning implied by the extra L?
2. If not, why is multicollinear used more frequently?

It doesn't seem intuitive to me that the double L version is more popular. I interpreted its definition as 'sharing linearity' with the outcome and hence the 'co-' prefix. What did the statistician who coined the term have to say about this?

• Wasn't sure this was on-topic, but I came across this question: stats.stackexchange.com/q/153526/176202 Nov 29, 2017 at 1:26
• "Multicollinear" is a barbarism in the first place, because the "multi" is completely superfluous. (Statistics never considers the case of a single vector being collinear, because the only possibility is that it's the zero vector, which cannot play any effective role in a linear model.) Looking up dictionary definitions of "collinear" and "colinear" therefore will provide some authoritative insight into their distinctions and origins.
– whuber
Nov 29, 2017 at 17:26
• @whuber I seem to recall having this conversation before. (+1) Of course the multi is redundant as collinearity is collective and common to all lines.
– Carl
Dec 17, 2017 at 15:58

2 Answers

Collinear follows the model of collaborate, collide, &c.: the m of the Latin prefix com- ("together") is assimilated to the initial l of the Latinate stem (cf. commiserate, contemporary, coæval, corrode). It's pronounced with the stress on the second syllable, & therefore an indeterminate vowel or at most a short o.

An excuse for colinear might be that you're treating linear as a native word, following the model of copilot, co-worker, &c.—cases in which the prefix is reduced to co- irrespective of the initial letter of the stem. I imagine people who write it thus also pronounce it with at least a secondary stress on the first syllable, & with a long o.

Pace @Carl I don't think a general predilection among the British for writing double l has much to do with it, though a preference for more traditional word-forms might. Counts of occurrences in published works from Google Ngrams suggest that colinear & its derivatives are disfavoured only slightly less in U.S. than in British English (an odds ratio of 1.4 over 1999 – 2008).

library(ngramr)
#define word list & corpora
words <- "collinear, colinear, collinearity, colinearity, multicollinear, multicolinear, multicollinearity, multicolinearity"
corpora <- c("eng_gb_2012", "eng_us_2012")
# fetch word counts
dd <- ngram(words, corpora, year_start = 1999, smoothing = 0,  count = T, tag = NULL, case_ins = TRUE)
# reduce derivatives
dd$stem <- factor(gsub("multi|ity", "", tolower(dd$Phrase)))
# tabulate
xtabs(Count~stem+Corpus, data=dd) -> tb


What may well be muddying the waters, however, is that there are more recent coinings of the word with different senses from the geometric one "together in a line" (first known use in 1863 according to my dictionary); in these we'd naturally expect the form colinear just because people don't make up Latin words any more. Wikipedia has an article on colinear maps & the on-line Merriam-Webster dictionary gives a second sense of colinear (but not collinear), "having corresponding parts arranged in the same linear order", that finds its use in Genetics & Molecular Biology.

† If you really want to write colinear & anyone's picking on you because of it, ask them if they write complanar.

• I listed the Oxford dictionary spellings as references that explicitly ascribe colinear as the US English spelling but also permits collinear as the spelling as well as the British reference to collinear that only admits one favored spelling.
– Carl
Dec 18, 2017 at 12:52
• @Carl: I'm not contesting that - I just don't see what colinear vs collinear has got to do with the different rules for doubling or not doubling a final l when you add a suffix. It's not as if Americans write colaborate, colapse, colateral, colect, colide, coloquial, colude, ... Dec 18, 2017 at 13:27
• American double "l" usage agrees with British on words that have the stress on the second syllable. Please read through the spelling link provided.
– Carl
Dec 18, 2017 at 13:57
• @Carl: You're misreading IPA there: the mark precedes the stressed syllable. When is schwa ever stressed anyway? Dec 18, 2017 at 14:56
• @Carl: Which spelling is an exception to which rule? I'm afraid I've lost track of what your point is: mine's just that the spelling collinear & pronunciation "co-LIN-ear" have the same explanation as the spelling collaborate & pronunciation "co-LAB-orate"; & that it's a quite separate one from that of the double l in modelling. Dec 18, 2017 at 15:32

Colinear is a U.S. English spelling. In the U.S. "collinear", with two l's is also used. In British English, the collinear spelling would be the accepted form.

Another example of doubled versus single "l" appears in the word modelling (especially British) and modeling (especially U.S.). Multicollinearity versus multicolinearity follows this same pattern.

This general American versus British pattern for spelling with one or two l's occurs for many words. However, "spelling" always has two l's. The divergence probably occurred during the 1800's during one of the U.S. spelling reformations.

The correct spelling is whatever fits the journal's style, and many journals insist on either American or British spellings. However, Canadian journals often accept both spellings.

When to use collinearity and when to use multicollinearity is discussed here. @whuber Indeed, we have discussed this before.

Collinearity was first discussed in the $3^{rd}$ century, and rediscovered 1500 years later.

• I don't see 'model' as quite the same. It's a verb in this context and has the "l" at the end. I hope we can someday get rid of the "ll". In writing I often use a hyphen to try to improve the situation: "co-linear" but I'm not sure it really does. Dec 17, 2017 at 15:28
• @FrankHarrell See the links provided, that will explain better than I can. There was one spelling reform, which originated in the US that the British did not want to follow that apparently got rid of the double letters. BTW, collinearity is more common 1,240,000 on google versus 251,000 for colinearity, so, you are probably fighting city hall.
– Carl
Dec 17, 2017 at 15:43