The chi-square statistic is used when the dependent variable is at the nominal level and our parametric assumptions of normality and homogeneity of variance cannot be met as discussed in the previous lesson. In this lesson we explore nonparametric tests of significance for ordinal dependent variables.
Second, the statistically more powerful Mann-Whitney U test, tests not only the median, but also the total distribution (central tendancy and distribution). The null hypothesis specifies that there is no difference in the scores of the two populations sampled. Two U values are calculated and the smaller one is selected for checking in a table of critical values. The null hypothesis is reject if the computed U is less than the table value. The U values take into account the number of data elements in each sample (n1 and n2) and the sum of the ranks in each group (R1 and R2). Calculate Ui = n1n2 + ½ni(ni + 1) - Ri for both i=1 and i=2.
When both groups are larger than 20, the sampling distribution approaches the normal distribution and a z-score can be computed from the sampling distribution mean µU = ½n1n2, and standard deviation U = sqrt( n1n2 (n1 + n2 +1)/12). The z score is calculated in the usual way with z = (U - µU)/ U.
A table of Mann-Whitney critcal U values is given below for alpha=0.05 (1-tailed=directional Ha) and selected sample sizes only. More extensive tables are readily available.
Tied ranks generally have minimal effect on both the Mann-Whitney U and Kruskal-Wallis H and a correction factor can be applied. However, results from either test might be questionable when there are an excessive number of tied ranks.
In conclusion we wish to emphasize both the role of nonnormal populations and small sample size in the use of these statistics. Larger samples give better statistical precision, but sometimes a limited population or expense preclude obtaining such. Thus the best statistical test for a given situation is defined more in terms of the research conditions than in terms of parametric vs nonparametric.
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