Likelihood ratio, Pearson Chisquare, Linear-by-Linear Association in crosstabs

[E. Bernardo]

Can some suggest references that differentiate in terms of usage (when to use) the following three tests in the outputs of crosstabs:   Likelihood ratio, Pearson Chisquare, Linear-by-Linear Association in crosstabs?

 

Thanks

[Bruce Weaver]

Eins Bernardo wrote

Can some suggest references that differentiate in terms of usage (when to use) the following three tests in the outputs of crosstabs:   Likelihood ratio, Pearson Chisquare, Linear-by-Linear Association in crosstabs?
 
Thanks

Re the test of linear-by-linear association, see David Howell's nice note.

   http://www.uvm.edu/~dhowell/methods7/Supplements/OrdinalChiSq.html

Re Pearson's chi-square versus the likelihood ratio chi-square, here is a comment from Agresti's 1990 book "Categorical Data Analysis" that may be helpful.

"It is not simple to describe the sample size needed for the chi-squared distribution to approximate well the exact distributions of X^2 and G^2 [i.e., Pearson and likelihood chi-square test statistics].  For a fixed number of cells, X^2 usually converges more quickly than G^2.  The chi-squared approximation is usually poor for G^2 when n/IJ < 5 [where n = the grand total and IJ = rc = the number of cells in the table].  When I or J [i.e., r or c] is large, it can be decent for X^2 for n/IJ as small as 1, if the table does not contain both very small and moderately large expected frequencies." (Agresti, 1990, p. 49)

Finally, when you are partition an overall contingency table into orthogonal components, the likelihood ratio chi-square has the nice property that the chi-square values for the orthogonal components add up exactly to the chi-square value for the overall table.  There are some examples of this in my notes on categorical data.

   http://www.angelfire.com/wv/bwhomedir/notes/categorical.pdf

HTH.

[Rich Ulrich]

Following what Bruce posted,
Agresti's books are great reading and great references.

You may also gather from Agresti that, contrary to what
some people expect, the two versions of the test do *not*
become equivalent for large samples.  They test slightly
different versions of the hypothesis of heterogeneity. 

The overall difference is such that, in an R x K table, one cell with
a very discrepant expectation may out-weigh several moderate
discrepances; or vice-versa.   The Pearson test is more likely to
reject for a single cell that is off, whereas the Likelihood test is
more likely to reject for several cells.  (I think I didn't get that
backwards.)   So, the nature of what you expect can determine
the choice of your test -- And there is a rational explanation for
why the two tests may lead to different decisions, if you are
'rejecting'  at a particular alpha.

--
Rich Ulrich