1. Nonparametric statistics

2. Four levels of measurement Nominal Ordinal Interval Ratio  Nominal: the lowest level  Ordinal  Interval  Ratio: the highest level

3. Parametric Assumptions The observations must be independent The observations must be drawn from normally distributed populations These populations must have the same variances

4. Classic Parametric Setting For the Parametric test, we assume we are sampling from normally distributed populations with identical variances. From here we test the hypothesis: H(0):μ1= μ2 = μ3 (All distributions are identical) or at least one is not identical

5. Settings for Nonparametric Tests If you suspect the variable is not normally distributed in the population When you have nominal/ordinal data and are interested in comparing distributions and their relationships. If other assumptions of parametric tests are violated (e.g. homogeneity of variance, ordinal Data)‏ Can provide approximations for small sample sizes

6. A Guide to Testing Differences Nature Sample Type NominalOrdinalInterval 2 Independent Samples Chi-square TestMann-Whitney U Test Independent Samples T-test 2 Related Samples -- Wilcoxon Matched Pairs Test Paired Samples T- test >2 Independent Samples Chi-square TestKruskall-Wallis Test One-way ANOVA Correlation test Rank Spearman's Test Pearson Correlation

7. Mann-Whitney U Test Nonparametric alternative to two-sample t-test Actual measurements not used – ranks of the measurements used Data can be ranked from highest to lowest or lowest to highest values Calculate Mann-Whitney U statistic U = n 1 n 2 + n 1 (n 1 +1) – R 1 2

8. Example of Mann-Whitney U test Two tailed null hypothesis that there is no difference between the heights of male and female students (5% significance level)‏ H o : Male and female students are the same height H A : Male and female students are not the same height

9. R 2 = 48R 1 = 30n 2 = 5n 1 = 7 9170 6178 125163180 114165183 103168185 82173188 71175193 Ranks of female heights Ranks of male heights Heights of females (cm)‏ Heights of males (cm)‏ U = n 1 n 2 + n 1 (n 1 +1) – R 1 2 U=(7)(5) + (7)(8) – 30 2 U = 35 + 28 – 30 U = 33 U 0.025,7,5 = U 0.025,5,7 = 6 As 33 > 6, H o is rejected Zar, 1996

10. The Kruskal-Wallis Test This test is used to test whether several populations have the same mean. It is a nonparametric substitute for a one-factor ANOVA.

11. where n j is the number of observations in the j th sample, n is the total number of observations, and R j is the sum of ranks for the j th sample.

12. Kruskal-Wallis Test Example: Test at the 5% level whether average employee performance is the same at 3 firms, using the following standardized test scores for 20 employees. n 3 =7n 2 = 6n 1 = 7 7380 607290 706275 936187 508485 657795 826878 rankscorerankscorerankscore Firm 3Firm 2Firm 1

13. f(  2 )‏ acceptance region crit. reg..05 5.991 From the  2 table, we see that the 5% critical value for a  2 with 2 dof is 5.991. Since our value for K was 6.641, we reject H 0 that the means are the same and accept H 1 that the means are different.

14. We rank all the scores. Then we sum the ranks for each firm. Then we calculate the K statistic. R 3 = 57n 3 =7R 2 = 47n 2 = 6R 1 = 106n 1 = 7 9731380 2608721890 7704621075 19933611787 15015841685 56511772095 14826681278 rankscorerankscorerankscore Firm 3Firm 2Firm 1