B AD 6243: Applied Univariate Statistics Non-Parametric Statistics Professor Laku Chidambaram Price College of Business University of Oklahoma

BAD 6243: Applied Univariate Statistics 2 Using Non-Parametric Statistics Non-normal distribution of data –Tests referred to as “distribution free” (or sometimes “assumption free”) tests Small sample size Measurement issues –Dependent variables are nominal or ordinal Tests are generally less powerful than their parametric counterparts –Intent is not to estimate population parameter per se Involves testing differences and relationships

A Guide to Testing Differences Nature of DV/ 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 k Independent Samples Chi-square TestKruskall-Wallis Test One-way ANOVA k x k Independent Samples Contingency Analysis (Crosstabs) -- Factorial ANOVA

BAD 6243: Applied Univariate Statistics 4 The Chi-Square Distribution The chi-square distribution refers to a family of distributions (derived from the normal distribution) with one parameter, k, the degrees of freedom The distribution is positively skewed but becomes increasingly symmetric as k increases The mean and variance of the chi-square distribution also increase as k increases The mean = k and variance = 2k

BAD 6243: Applied Univariate Statistics 5 The Chi-square Test The Chi-square Test is based on the chi-square distribution It evaluates the goodness-of-fit of the observed frequencies (O) with the expected frequencies (E) in various categories The Chi-square statistic (shown below) helps determine whether differences between the observed and expected frequencies in the sample represent “real” or random differences  2 =  [(O-E) 2 / E]

BAD 6243: Applied Univariate Statistics 6 An Example H 0 : p Marketing = p Management = p Finance = p MIS H 1 : At least one pair is not equal Is there an equal proportion of majors in the PCB? (Case of the k independent samples)

BAD 6243: Applied Univariate Statistics 7 Notes on the Chi-square Test Same approach as before applies when unequal frequencies are expected In the case of the chi-square test for two independent samples, the expected frequency in each cell should be at least 5 In the case of the chi-square test for n independent samples, the expected frequency should not be less than 5 in more than 20% of the cells Where the above situation arises, you should consider combining categories Observations in all cases should be independent

BAD 6243: Applied Univariate Statistics 8 Contingency Analysis (Crosstabs) (Case of the k x k samples) Is there a relationship between gender and when students are absent from classes?

BAD 6243: Applied Univariate Statistics 9 Mann-Whitney U Test Is there a difference in the average rank of PhD admits who matriculated in 2000 vs. 2001? (Case of the 2 independent samples)

BAD 6243: Applied Univariate Statistics 10 Wilcoxon Matched Pairs Test Where are they now: Is there a difference between the ATP rankings of the top ten seeded tennis players in 2000 and 2003? (Case of the 2 related samples)

BAD 6243: Applied Univariate Statistics 11 Kruskall-Wallis Test Is there a difference among the average rankings of National Merit Scholars admitted to schools of business in the state? (Case of the k independent samples)