Q-Statistic

Approximations to Q-Statistic

Usually, the χ2 test is used to make inferences about ratios or percentages, but when the samples are correlated, the use of this test violates the assumption of the independence among the samples compared.

The Q-test is an inferential statistic developed by the statistician William Gemmell Cochran (1909–1980) that arises as an extension of the McNemar test.

The McNemar test examines the significance of the differences between ratios or percentages of two correlated samples. The Q-statistic allows for the evaluation of the null hypothesis of equality between ratios or percentages for more than two matched samples under a permutation model, so it can be simplified to McNemar's test when there are only two samples.

The Q-statistic is also equivalent to the sign test when the samples are small and there are no significant outliers. The sign test was developed for testing the median difference of independent paired observations.

The distribution of the Q-test for small samples has been approximate to the χ2 and F test using a correction for continuity. However, rows containing only 1s or 0s can yield quite different results using the F approximation without affecting the value of the Q-statistic. Although the F test, corrected for continuity, can get a better approximation than the corrected or not χ2 in the same cases, the latter has been taken as the common approximation because it is easier to calculate. Nevertheless, as those computations are run by computers today, the possible application of the corrected F approximation should be considered.

The accuracy of the Q-statistic in small samples depends on the number of conditions as well as the sample size; the χ2 approximation seems good enough with a total of 24 scores or more, deleting those rows with only 1s or 0s. After deleting those rows without variation, if the total product, columns by rows, is less than 24, the exact distribution should be constructed. The distribution of Q-test can be obtained by the method that Patil proposed in 1975 or by Tate and Brown's 1964 r × c tables, where the probabilities vary in dimensions from 3 columns and 12 rows to 6 columns and 4 rows.

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