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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.
--
Bruce Weaver bweaver@lakeheadu.ca http://sites.google.com/a/lakeheadu.ca/bweaver/ "When all else fails, RTFM." PLEASE NOTE THE FOLLOWING: 1. My Hotmail account is not monitored regularly. To send me an e-mail, please use the address shown above. 2. The SPSSX Discussion forum on Nabble is no longer linked to the SPSSX-L listserv administered by UGA (https://listserv.uga.edu/). |
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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 |
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