Twenty Nonparametric Statistics And Their Large Sample Approximations

Abstract

Nonparametric procedures are often more powerful than classical tests for real world data which are rarely normally distributed. However, there are difficulties in using these tests. Computational formulas are scattered throughout the literature, and there is a lack of availability of tables and critical values. The computational formulas for twenty commonly employed nonparametric tests that have large-sample approximations for the critical value are brought together. Because there is no generally agreed upon lower limit for the sample size, Monte Carlo methods were used to determine the smallest sample size that can be used with the respective large-sample approximation. The statistics reviewed include single-population tests, comparisons of two populations, comparisons of several populations, and tests of association.

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“…Secondly, both of the tested samples had to come from the same type of distribution function differing only in scale. To test this assumption, another rank type test was used, namely Wilcoxon rank sum test [Fahoome 2002]. This let the author check whether H 0 stating that two independent samples used in particular Ansari--Bradley's procedure were selected from the same distributions without statistically significant differences in location parameters.…”

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“…The large-sample approximation to the sampling distribution of Page's L statistic yields acceptable Type I error rates for a directional Page test, as long as n > 11 for α = .05, or n > 18 for α = .01, according to Fahoome (2002). An acceptable Type I error rate was defined in Fahoome (2002) as within 10% of the nominal α rate, in reference to Bradley (1978)'s work.…”

“…An acceptable Type I error rate was defined in Fahoome (2002) as within 10% of the nominal α rate, in reference to Bradley (1978)'s work. Page (1963) also suggested that the large-sample chi-square approximation be used under one of three conditions: (1) for m > 20 with any n, (2) for m > 12 and n ≥ 4, or (3) for any m when n ≥ 9.…”

“…Page (1963) also suggested that the large-sample chi-square approximation be used under one of three conditions: (1) for m > 20 with any n, (2) for m > 12 and n ≥ 4, or (3) for any m when n ≥ 9. Because m = 4 and n = 14, the Page test result and its statistical significance level were judged to be acceptable, according to Bradley (1978), Fahoome (2002), and Page (1963).…”

“…Since the standardized L, or z from (5), follows a standard normal distribution (e.g., Fahoome, 2002;Lyerly, 1952), a nondirectional 95% CI for the standardized L can be constructed using (6) …”