![]() ![]() I’d have thought the default method would be to collapse categories until each cell had an decent expected value & calculate a p-value from the asymptotic distribution of Pearson’s chi-square. This entry was posted in Miscellaneous Statistics by Andrew. In the chi-squared statistic, all that noise in the empty cells is diluting the signal. Only one of the cells is really bad-that point on the lower-right-but it has a high expected value (it’s one of the largest cells in the table), and when you take the equally-weighted mean square (which is equivalent to weighting the contributions to the chi-square in proportion to the expected count), you get a big total value. The following graph shows the discrepancies ((Observed – Expected)/sqrt(Expected)) which are squared and summed to form the chi-squared statistic: If people really are going around saying their models fit in such situations, it could be causing real problems. All those zeroes and near-zeroes in the data give you a chi-squared test that is so noisy as to be useless. (Perkins, Tygert, and Ward compute the p-value via simulation.) Rejection! That is, nothing going on.īut it turns out that that if you do an equally-weighted mean square test (rather than chi-square, which weights each cell proportional to expected counts), you get a p-value of 0.039. The p-value of the chi-squared test is 0.693. Here’s one, taken from the classic 1992 genetics paper of Guo and Thomspson:Īnd here are the expected frequencies from the Guo and Thompson model: Rather, the problem is that, when there are lots of cells with near-zero expectation, the chi-squared test is mostly noise.Īnd this is not merely a theoretical problem. The problem is not merely that the chi-squared statistic doesn’t have the advertised chi-squared distribution-a reference distribution can always be computed via simulation, either using the posterior predictive distribution or by conditioning on a point estimate of the cell expectations and then making a degrees-of-freedom sort of adjustment. This often leads to serious trouble in practice - even in the absence of round-off errors. If a discrete probability distribution in a model being tested for goodness-of-fit is not close to uniform, then forming the Pearson χ2 statistic can involve division by nearly zero. ![]() William Perkins, Mark Tygert, and Rachel Ward write: ![]()
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