As with a one-dimensional chi-square, the closer the observed and expected values are to one another, the less likely the dimensions are to be independent of one another. The less similar the observed and expected values are, the more likely the two dimensions are independent of one another. The test statistic is computed in the same manner as we did for a goodness-of-fit test (what a nice surprise); expected values are computed and used along with observed values to compute a chi-square value, which is then tested for significance. However, unlike with the one-dimensional test, the expected
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