When computing the chi-square test statistics for goodness of fit, almost always integral observed values are compared to fractional expected values. That means there will almost never be a fair chance for the test statistics to attain a value of zero. Thus, it will be biased towards larger values. Unfortunately, I cannot find any sources explicitly addressing this kind of bias. Does somebody know of references (printed or on the web) that are concerned with this bias? Can it simply be neglected in case the most frequently mentioned minimal recommendations on classes' frequencies etc. are fulfilled?
Thanks for any hints... Schorsch
On Wed, 5 Sep 2012 15:16:51 -0700 (PDT), Schorsch_MCMLX
<derstroehleinschor...@googlemail.com> wrote:
>When computing the chi-square test statistics for goodness of fit, almost always integral observed values are compared to fractional expected values. That means there will almost never be a fair chance for the test statistics to attain a value of zero. Thus, it will be biased towards larger values. Unfortunately, I cannot find any sources explicitly addressing this kind of bias. Does somebody know of references (printed or on the web) that are concerned with this bias? Can it simply be neglected in case the most frequently mentioned minimal recommendations on classes' frequencies etc. are fulfilled?
>Thanks for any hints... Schorsch
Any given small table has a limited *set* of p-values that
can be obtained by a particular, fixed procedure. I don't
think I would use the term "bias" for the absence, sometimes,
of computed values of 0, but there are certainly some interesting issues that can be raised.
If you want "exact probabilities" to use the whole range,
so that you see p's all the way from 0 to 1, you can employ an ad-hoc randomization of what is to be
reported. (So far as I know, no one has ever tried to use this theoretical correction.)
The one place that I found a bunch of discussion was in these
"Journal of the Royal Statistical Society" references
Fishers vs 2x2 Pearson. ] Yates, et al. JRSS Series A (1984) 147:426-463. Shuster. JRSS Series A (1985) 148:317-327. Upton. JRSS Series A (1992) 155:395-402.
In the 1984 article, Upton leant strongly against using Fishers' test.
In this article, he announces own conversion, crediting the arguments of Barnard.
"Schorsch_MCMLX" <derstroehleinschor...@googlemail.com> wrote in message
news:a9113788-0aa3-4ca1-9a60-f207042b2254@googlegroups.com...
When computing the chi-square test statistics for goodness of fit, almost always integral observed values are compared to fractional expected values. That means there will almost never be a fair chance for the test statistics to attain a value of zero. Thus, it will be biased towards larger values. Unfortunately, I cannot find any sources explicitly addressing this kind of bias. Does somebody know of references (printed or on the web) that are concerned with this bias? Can it simply be neglected in case the most frequently mentioned minimal recommendations on classes' frequencies etc. are fulfilled?
Thanks for any hints... Schorsch
The answer depends on what you are looking for... (i) theoretical discussion of properties of chi-squared tests in small samples; (ii) practical testing procedures for actual use. The latter is in principle answered in work that is often labeled as "exact tests", and such tests are built into some of the available software packages. A simple place to start is with a Google search for "exact test for goodness-of-fit". One relatively recent accessible paper is at ftp://wuecon195.wustl.edu/opt/ReDIF/RePEc/ets/papers/jann_mgof.pdf ("Multinomial goodness-of-fit: large sample tests with survey design correction and exact tests for small samples ", Ben Jann, 2008), selected at random from initial google output.
