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Line 273: − The chi-square test p-values are computed by comparing the test statistic to the chi-square distribution. It is generally assumed that the cell frequencies should be greater than five so that the statistic's distribution follows chi-square distribution. However, there is no consensus about what minimum cell frequency is necessary or how many expected values need to cross that threshold.+ + + + + + + + + + + + + + + + + + +
Significance of E. Coli Evolution Experiments (view source)
Revision as of 16:33, March 27, 2009
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==Caveat==
The chi-square test p-values are computed by comparing the test statistic to the chi-square distribution. The chi-square statistic diverges from the chi-square distribution unpredictably as n approaches zero and as expected cell frequencies approach zero. However, there is no single consensus about what minimum expected cell frequency is necessary or how many expected values need to cross that threshold. Most authors suggest that the minimum expected cell frequency is five
<ref>http://www.ling.upenn.edu/~clight/chisquared.htm</ref>
<ref>http://www.graphpad.com/www/Book/Choose.htm</ref>
<ref>http://books.google.com/books?id=bmwhcJqq01cC&pg=PA494&lpg=PA494&dq=chi-square+independent+categories&source=bl&ots=I9KXRYU_pe&sig=Ru8Qi3ApGT_ST5yxNSRhySg3l94&hl=en&ei=DufMSczSNurxnQfihPjkCQ&sa=X&oi=book_result&resnum=7&ct=result#PPA494,M1</ref>
<ref>http://www.uwlax.edu/faculty/toribio/math442_spr08/chi_square_gof_test_442.pdf</ref>
<ref>http://academic.reed.edu/psychology/RDDAwebsite/spssguide/chisquare.html</ref>
, although others suggest that some values may be as low as but not lower than one if 80% of cell values are greater than five; or that the minimum value may be as small as 5r/s where r is the number of expected cells with values less than five and s is the total number of expected cells.
<ref>http://faculty.chass.ncsu.edu/garson/PA765/chisq.htm</ref>
<ref>http://www.physics.csbsju.edu/stats/contingency.problem.html</ref>
<ref>Cochran, William G. Some Methods for Strengthening the Common [Chi-Squared] Tests, ''Biometrics'' Vol 10, No. 4 (Dec 1954) pp. 417-451</ref>
<ref>Yarnold, James K. The Minimum Expectation in [Chi-Squared] Goodness of Fit Tests sand the Accuracy of Approximations for the Null Distribution, ''Journal of the American Statistical Association'', Vol. 65, No. 330 (Jun 1970), pp. 864-886</ref>
<ref>http://books.google.com/books?id=3XuUx2OSPIQC&pg=PA153&lpg=PA153&dq=chi+square+conservative+false+null+hypothesis&source=bl&ots=lyErIRWnUH&sig=ov44qNSEik0hXrJnzwkKXhVta08&hl=en&ei=sPfMSYzREcffnQf709nMCQ&sa=X&oi=book_result&resnum=9&ct=result</ref>
While different authors set different lower cutoffs, the data in the Blount experiment, where all expected cells have values less than one, lie below even the most liberal cutoff. For low n and expected cell counts, the chi-square test becomes increasingly conservative and errs on the side of accepting a false null hypothesis.
Separate from the issue of low cell counts, the chi-square test also assumes and requires that the data for all categories are completely independent and mutually exclusive: that no member of the population under study can contribute to more than one cell. The chi-square test is therefore not to be used, for example, to study the status of the same population of patients before and after treatment, or to follow the characteristics of a single group over time. <ref>http://faculty.chass.ncsu.edu/garson/PA765/chisq.htm</ref>
<ref>http://books.google.com/books?id=bmwhcJqq01cC&pg=PA494&lpg=PA494&dq=chi-square+independent+categories&source=bl&ots=I9KXRYU_pe&sig=Ru8Qi3ApGT_ST5yxNSRhySg3l94&hl=en&ei=DufMSczSNurxnQfihPjkCQ&sa=X&oi=book_result&resnum=7&ct=result#PPA494,M1</ref>
<ref>http://davidmlane.com/hyperstat/B155670.html</ref>
The Blount experiment collects data from the same population of bacteria over a long period of time; this does not satisfy the conditions of the chi-square test. If the Blount experiment could be repeated so as to produce large n and large expected cell frequencies, the chi-square test would still be an inappropriate test to apply to the data for this reason.
==References==
==References==
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