statistics NONPARAMETRIC TEST Nonparametric tests are used when data is not normally distributed (distribution free). Data contain extreme high or low value Data are highly dispersed Type of nonparametric tests Mann-Whitney U Test Wilcoxon test Kruskal-Wallis Test Friedman's Test Chi-square test The Spearman Rank Correlation Coefficient Nonparametric Regression Analysis sifat bin momin

Mann-Whitney U Test The Mann-Whitney U test is one of the most powerful non-parametric(distribution-free) statistical significance tests. It is a most useful alternative to the parametric t-test (a nonparametric equivalent to t test). It is used to compare two independent groups of sampled data. We can think of this test as comparing the medians of the two groups, since the median is a ‘nonparametric’ measure of the typical result, obtained by ranking the available evidence.

Characteristics This test is an alternative to the independent group t-test, when the assumption of normality or equality of variance is not met. This, like many non-parametric tests, uses the ranks of the data rather than their raw values to calculate the statistic. Since this test does not make a distribution assumption, it is not as powerful as the t-test.

Mann-Whitney U test for small samples (or <20): Let n1 = the number of cases in the smaller of two independent groups n2 = the number of cases in the large. To apply the U test, we first combine the observations from both groups, rank these in order of increasing size. or, equivalently, Where R1= sum of the ranks assigned to group whose samples size is n1. R2= sum of the ranks assigned to group whose samples size is n2.

Mann-Whitney U test for small samples (or <20): Steps for calculating U value First combine the observations from both groups and arrange the data as ascending order. Rank these in order of increasing size. Separate the ranks of two groups after ranking and take summation of ranks of individual groups (R1 sum of rank for group 1 and R2 for group 2) Put n1, n2, R1 and R2 in the formula of U1 and U2. This value is compared to a table of critical values for U based. If U value exceeds the critical value for U at some significance level (usually 0.05) it means that there is evidence to reject the null hypothesis in favor of the alternative hypothesis. The smallest of U1 or U2 is compared to the critical value for the purpose of the test.

Example: Data from two independent groups are given in the table Group A Group B 6.2 5.3 4.8 10.0 12.1 7.3 3.9 4.3 Data (in ascending order, both group A & B): 3.9 4.3 4.8 5.3 6.2 7.3 10.0 12.1 Rank: 1 2 3 4 5 6 7 8 Group A Rank of A Group B Rank of B 3.9 1 4.3 2 4.8 5.3 4 6.2 5 7.3 6 12.1 8 10.0 7 n1 = 4 R1 = 17 n2 = 4 R2 = 19

Example: Data from two independent groups are given in the table Group A Rank of A Group B Rank of B 3.9 1 4.3 2 4.8 5.3 4 6.2 5 7.3 6 12.1 8 10.0 7 n1 = 4 R1 = 17 n2 = 4 R2 = 19 This value is compared to a table of critical values for U based. If U value exceeds the critical value for U at some significance level (usually 0.05) it means that there is evidence to reject the null hypothesis in favor of the alternative hypothesis. The smallest of U1 or U2 is compared to the critical value for the purpose of the test.

Mann-Whitney U test for large samples (or >20): It has been shown that as an increase in size, the sampling distribution of U rapidly approaches the normal distribution, that is, when n1 or n2 >20 we may determine the significance of an observed value of U by When the normal approximation to the sampling distribution of U is used, it does not matter whether U1 or U2 is used for U The value of z will be the same if either (U1 or U2) is used.

Example: Computing the value of U for large samle ( <20) In a study of memory, the students in the Pharmacology class both males and females to write down everything they remembered that was unique to the previous day’s class. We wish to know whether there is any difference between male and female in recalling the items. Females Males Items recalled Rank 70 51 40 29 24 21 20 17 16 15 14 13 11 10 9 8 7 6 3 16.5 18.5 22 23 24.5 26.5 30.5 33 35.5 37.5 39 41 85 72 65 52 50 43 37 31 30 27 19 12 1 2 4 5 28.5 n = 24 = 603 n = 17 = 258

Computing the value of z for large samle ( <20) Since the distribution of observations is skewed (not normal distribution) Mann-Whitney test is selected. The z-score of 2.62 led to rejection of the null hypothesis. Since the mean rank of the females, 25 (603  24), is higher than that of males, 15 (258  17), it may be concluded that female recalled significantly more items than the male did.

Wilcoxon matched pairs signed ranks test This test is an alternative to the paired t-test, when the assumption of normality or equality of variance is not met. This uses the ranks of the data rather than their raw values to calculate the statistic. Ranks to the differences of pairs.

Steps in calculation First obtain the difference between each pair of scores Rank absolute values of these difference (disregard signs) If there are 2 similar values then add ranks of these two values and divide by 2 (3+4)/2=3.5, 3.5 for these 2 ranks. The ranks will be summed according to the sign of differences (plus and minus) The smallest of these is taken as Wilcoxon T test (small value, n<25), for large value (>25) use z test.

Wilcoxon test for small samples (or <25): X Y (X-Y) Rank of R(+) R(-) difference (R ) 18 6 12 9.5 9.5 15 21 -6 5.0 5 26 14 12 9.5 9.5 16 14 2 1.5 1.5 12 10 2 1.5 1.5 11 15 -4 3.0 3 21 11 10 7.5 7.5 19 13 6 5 5 9 3 6 5 5 20 10 10 7.5 7.5 Sum 47 (R+) 8(R-) The smallest of sum of sign (sum of R+ or R-) is taken as Wilcoxon T test. Here T = 8. For n=10 and calculated T=8 is significant compared to table value.

Wilcoxon test for large samples (or >25): In case of large sample size, ranks is practically normally distributed then T is replaced by z test. Calculated z value is compared with table z value to find p value.