[R] Validity of Pearson's Chi-Square for Large Tables

[Gerard M. Keogh]

Hi,
didn't get your name.

For large tables (5 X 5) or bigger the dist of the log of the cross product
ratios tends to normality. there are (nC2)**2/2 of these (200 in a 5X5
table. The chi-sq test for independence fits a main effects loglinear model
to the table and this can be expressed in terms of the cross product ratios
(see Discrete Multivariate Analysis by bishop fineburg and holland 1975).
Problems only arise when there are a lot of zeros as this posits weight in
the log dist at +/- infinity.
This in turn will result in a poor chi-sq approx.

Gerard


                                                                           
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                                                                   Subject 
                                       [R]  Validity of Pearson's          
             03/06/2009 04:32          Chi-Square for Large Tables         
                                                                           
                                                                           
                                                                           
                                                                           
                                                                           
                                                                           





Is Pearson's Chi-Square test for contingency tables asymptotically unbiased
for large tables (large degrees of freedom) regardless of the expected
values in each cell?  The rule of thumb is that Pearson's Chi-square should
not be used when large numbers of cells have expected values < 5.  However,
I compared the results on 4x4 contingency tables for R's chisq.test using
chi-square approximation vs. chisq.test using a large number of monte carlo
simulations, and the results agree within a fairly small error.  This is
true even when every cell of the table has an expected value < 2.  I tried
several tables, but the best example was:

4  1  1  1
1  4  1  1
1  1  4  1
1  1  1  4

As expected, the chi-square approximation appears to be very poor when both
the expected values and degrees of freedom are small.  Is there a good
theoretical reason why the chi-square test seems to perform well on large
contingency tables even with small expected values?  Are the standard rules
of thumb overly simplistic?
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