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To analyze frequencies in a 2x2 table, I ran a statistical test procedure using SPSS. It returned a p less than .05 for the likelihood Ratio but greater than .05 for Chi-Square and Fisher Exact Test (see output below). How do I make sense of one being statistically significant and the other not?

SPSS output

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We could go really down the rabbit hole on this, but I'll try to keep out of the weeds. I imagine that you have a relatively small sample size. This isn't a super big deal, but the $X^2$ and LRT tend to diverge when you have a low sample size. It's fairly well agree upon that when sample size is low, LRT tends to have more statistical power. This article gives a brief overview if you'd like to get into the math.

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  • $\begingroup$ The 1973 paper you cite concludes: "The likelihood-ratio statistic has an expected value in excess of the nominal and yields far too many rejections under the null distribution." That is a much less positive statement than the part of your answer that says the "LRT tends to have more statistical power." So, should we routinely ignore the LRT, even for small samples. Or are there situations where the LRT is to be preferred? Or something else? Also, is the paper's discussion of general goodness of fit tests applicable to 2x2 analyses? $\endgroup$
    – Joel W.
    Dec 17, 2020 at 1:29

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