Talk:Pearson's chi-square test

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--- Link to probability theory

I've tried to improve the 2 cells and Many cells sections but I'm not a great expert -- could someone pls take a look and check them? In particular, we need to explain why (O-E)^2/E ends up being the same as the (O-E)^2/sigma that is used by the chi-square distribution.

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What's wrong with simply chi-square test? Are there more than one that are of encyclopedic interest? --mav

You've got to be kidding!!! There are zillions of them (zillions = at least a dozen or so) that are so different from each other except in sharing a common null distribution that <POV> it is astonishing that anyone could wonder about this </POV>. Well, maybe not astonishing to the layman, but still.... Michael Hardy 00:38 May 14, 2003 (UTC)
[I removed my earlier hasty and ill-considered -- and incorrect -- comment because Michael makes the point better.] Jfitzg

In that case, then perhaps there should be an antry at chi-square test saying that there are lots of them, that the general principle was developed by A and B in century C, that these three tests are the most commonly used although those 7 are sometimes used for purpose X and purpose Y, and that all have in common the idea Z. As a generality, the maths entries on the 'pedia are dense and forbidding to the non-mathematician. This sort of thing helps a lot to make stuff accessible to the general reader - which is what we are here for, isn't it? Tannin 00:47 May 14, 2003 (UTC)

could we have a brief intro in english please?

The current intro is comprehensible. I didn't write it -- it replaces a simpler one I wrote which probably appeared more English but which was not specific enough.Jfitzg

I can comprehend it, sure. But I spent a couple of years studying stats, and even so I don't find it exactly easy reading. If I had happened to take a different minor, there is no way I could read and understand that intro, nor would I expect anyone else without at least some specialist training to be able to do so. I appreciate that the maths people want to get the maths entries as precise and strictly correct as possible, and applaud that urge, but we need to make sure that the casual reader is able to look at an entry and, even if he is unable to understand it in detail (or unwilling to put in the half-hour or so of concentrated effort it might take to grasp the detail), at least he should be able to walk away with a rough idea of what it is all about.

I suggest changing the first para to something like this:

  • Pearson's chi-square test (χ2)—one of a variety of different chi-square tests—is a statistical procedure used with category data to decide if experimental results are statistically significant, or else can reasonably be explained by mere chance. Like most statistical tests, it compares observed frequencies (under some kind of test condition) with expected frequencies: in general, the greater the difference between the two, the less likely it is that the experimental results are simply the result of luck.
  • In more detail, Pearson's chi-square is for testing a null hypothesis that states that relative frequencies of occurrence of several specified mutually exclusive events, at least one of which must occur each time a specified experiment is performed, follow a specified frequency distribution. One of the simplest examples ....... etc.

Tannin 13:03 14 May 2003 (UTC)

THing is, I am a maths person. (I just happen to be allergic to stats). Maybe it's the Chi-square test article which should give an overview of what they are and why they're useful / interesting -- Tarquin 13:10 May 14, 2003 (UTC)

Hmmm ... OK, but Pearson's chi-square is the chi-square in a very real sense. Sure, there are others, but this one is the only one that most people are ever going to use. I think it is a special case. The really obscure ones need less contextualising at the start of the entry. Tannin
After further thought I concluded that an introduction like the one suggested by Tannin would be desirable. I'd suggest starting with a more general definition -- no reference to category data, for example. I don't think it's accurate to say that most statistical tests compare observed and expected frequencies, but perhaps I'm missing something or that wasn't what was intended. Anyway, I had thought of combining the current definition with the less detailed one originally posted. I suppose at some point someone's just going to have to grasp the nettle and change it.Jfitzg
Ahh, I knew it would be better to put it here than straight into the entry - my stats is very rusty, and I'm not surprised to have been caught in an error. I meant that most statistical tests compare observed and expected scores of some kind - not always frequencies, obviously. And I should have said "most statistical testing" (i.e., most frequently performed) as oposed to "most tests" (i.e., largest number of different tests). I mentioned the category data because, for the only-knows-a-little-bit statistician (your average social scientist, let's say), that's the key thing you have to remember: chi-square for category data, f-test or t-test or ANOVA for everything else. And if you can't use one of those, ask a real statistician. :) Tannin
Slip is a better word than error, I think. That's what I call them when I remove mine from contributions I've made, anyway. I'll log back on later (I think it's about time I made some money) and if no one has taken a stab at modifying the beginning I'll have a go and await comments.Jfitzg

I agree that it's too dense; I wrote it hastily. It has the advantage over the earlier version, of being correct. The earlier version spoke of differences between observed and theoretical frequencies, but that is trivial: The observed frequences are nearly always obviously different from the ones specified by the null hypothesis, and that is not what is of interest. What is of interest is whether the unobservable population frequencies differ from the theoretical ones.

I don't think it's a good idea to say that the purpose of the test is to decide whether the data are statistically significant. Statistical significance is of interest only because it indicates that the null hypothesis is false. Whether the null hypothesis is false is what is of interest; interest in statistical significance is secondary and merely a means to an end. The null and alternative hypotheses should be made clear in any statement of the purpose of this test. I'll probably get to this within a few days. Michael Hardy 01:47 15 May 2003 (UTC)

In the meantime I simply shortened the sentences in what you wrote. As I said above, it is thoroughly comprehensible, and if people found it dense it was probably because of the one long sentence. It's only a suggestion -- I didn't make it here first because I wasn't altering the content all that much. Jfitzg

What is the correct name of this test? I see that you have written "chi-square test", but in many references I have seen it written as "chi-squared test". Which version does Pearson use? And while we are here, should one write p-value or P-value?

Which version did Pearson use? Simple: he just used the symbol, as far as I have seen. There's a link on his biography page to a site that has a PDF facsimile of his paper from 1900. Maybe it would be useful to link there from here? As for correct name: "chi-square" and "chi-squared" are used about equally as far as search engine results go. Grotendeels Onschadelijk 03:07, 4 August 2007 (UTC)

I'd like to see some historical perspective. ie, when was it invented? Was this the original chi-square or is it a development of an earlier one? How heavily is it used (my guess is very heavily) and when did it become a standard statistical method in widespread use? 203.164.221.61 02:55, 30 April 2006 (UTC)

Contents

[edit] Intro

I am generally a staunch advocate for practical examples in articles, but shouldn't the first equation introduced be for the general case, not a specific example? I was looking at the limits of the sum (n=1-6) and thinking, "Huh?" until I read that it was an example for a six-sided die. Since the formula is almost completely general, I don't think it's a big deal to change it.

I also think this article could use a bit of general reworking, because it didn't answer the question I had at all (which was how a sample's variance is taken into account).


[edit] Restructuring/Different Emphasis?

I'd understand this topic better coming at it first from the perspective of Fisher's Exact Test, whose justification as a permutation test is intuitively obvious. I would prefer to then be introduced afterwards to the chi-square distribution as an approximation which becomes useful when the permutation test becomes intractable. In other words, I (statistically still fuzzy, like many potential readers) get derailed at the sentence

If the null hypothesis is true... the test statistic will be drawn from a chi-square distribution with one degree of freedom.

because I feel I've got to imediately follow that link before I can understand anything else. If this were postponed till after a discussion of the exact test, I could get the main idea of the chi-squre test from that and only then move on to worry about the approximation that the chi-square distribution supplies.

I know that the article presents it as it is conventionally taught; but I've been reading this Julian Simon book on resampling [1] and becoming increasingly convinced that the conventional presentation is not the best order in which to learn things---since only now am I (generally non-dumbass) finally starting to get this stuff.

Thoughts?

71.127.0.211 16:22, 2 December 2006 (UTC)

(A note on the note: I've reinserted the word "dumbass" which had been inappropriately deleted from my comment and replaced with "expletive deleted" by User:Chris53516. Any speaker of English will explain to you that this is nowhere near an "expletive". And it can hardly be offensive when it's being used in self-deprecation. It would be stylistically inappropriate for most articles, but no one has any business deleting it from a discussion page post. Furthermore, as I understand the policy discussion at Wikipedia_policy/Foul_Language, there's not even a clear policy requiring "expletive deletion" from articles, let alone talk pages). 72.79.228.10 21:31, 29 March 2007 (UTC)



I rather agree, actually. That seems like a more intuitive approach. --Gak 20:41, 8 February 2007 (UTC)

[edit] Reduced Chi-squared?

I'm posting on this page because it seems to be the most active of the chi square test talk pages and I'm not sure where it belongs. I added a reduced chi squared statistic to the goodness of fit page, but I don't know if it should have its own page or not. Is it a subset of the Pearson's chi-square, or independent? I'm afraid I don't even have a statistics textbook handy. --Keflavich 17:17, 9 May 2007 (UTC)

[edit] Interpretation of results

This page should include some more detailed information about interpreting the numeric result of the test. In the intro the article mentions:

"A chi-square probability of 0.05 or less is commonly interpreted by applied workers as justification for rejecting the null hypothesis that the row variable is unrelated (that is, only randomly related) to the column variable."

In the two cells section, the idea that the number of degrees of freedom comes into play when interpreting the result, but this is very vague IMHO. Statistics is a weak point in my mathematics knowledge and i am re-learning a lot of it (i studied this about 15 years ago) so maybe my ignorance makes me think this is more vague than it is...not sure.

Straha 206th 22:56, 17 May 2007 (UTC)

It is vague 216.99.15.253
I agree, this is precisely when I visited the entry. --Belg4mit 14:52, 23 October 2007 (UTC)

[edit] Introduction: Incomprehensible sentence

For a while the introduction of this artice has contained the sentence "The events are assumed to be independent and have the same distribution, and the outcomes of each event must be mutually exclusive." I've read this sentence about 100 times, and each time my confidence level has dropped about 1%. At least there is a confusion of terms, because

  • by the standard terminology events do not have a distribution, although they have a probability, but it can not be intended that they must have the same probability;
  • outcomes are necessarily mutually exclusive, although events may be mutually exclusive if they don't contain the same outcomes.

More seriously, it makes very little sense for events to be independent and mutually exclusive at the same time. For events A and B to be independent, Pr(A and B) = Pr(A)Pr(b) would have to hold. For them to be mutually exclusive, we would have to have Pr(A and B) = 0. So either Pr(A) or Pr(B) would have to be zero.

As far as I can see, this is a big mistake, and really, I can see no reason why this sentence couldn't be replaced by "The events must be mutually exclusive". (as it used to be) Grotendeels Onschadelijk 03:40, 4 August 2007 (UTC)

On further reflection, it occured to me that what is actually meant, if not said, by that sentence, is that it is assumed that the data under consideration is in fact a sample. I think that can be more easily achieved by saying that, and linking to the appropriate page.

I propose to change

to

Note that I have also changed the use of frequency distribution to match the referred page. I am slightly uncertain about the "total probability 1" requirement, but that seems to be assumed everywhere I looked. Grotendeels Onschadelijk 09:34, 9 August 2007 (UTC)