1 PARAMETRIC VERSUS NONPARAMETRIC STATISTICS Heibatollah Baghi, and Mastee Badii

2 Parametric Assumptions Parametric Statistics involve hypothesis about population parameters (e.g., µ, ρ). Parametric Statistics involve hypothesis about population parameters (e.g., µ, ρ). They require assumptions about the population distribution. For example, the assumptions for t test for independent samples are: They require assumptions about the population distribution. For example, the assumptions for t test for independent samples are: a) Each of the two populations of observations is normally distributed b) The populations of observations are equally variable : that is σ 2 = σ 2. (Assumption of homogeneity of variance )

3 Nonparametric Alternative The parametric assumptions cannot be justified: normal distribution, equal variances, etc. The parametric assumptions cannot be justified: normal distribution, equal variances, etc. The data as gathered are measured on nominal or ordinal data The data as gathered are measured on nominal or ordinal data Sample size is small. Sample size is small.

4 Spearman Rank Correlation The Spearman rank correlation is used when: The Spearman rank correlation is used when: Distribution assumptions required by Pearson r are in question Distribution assumptions required by Pearson r are in question Small sample size Small sample size

5 Example X: The student’s popularity measure X: The student’s popularity measure Y: The student’s average academic achievement Y: The student’s average academic achievement Research questions : Is popularity related to achievement ? Research questions : Is popularity related to achievement ?

6 Test of Association Using Spearman Rank Correlation Because of doubts regarding the distributional assumptions coupled with small sample size, select the Spearman Rank Correlation to answer this question

7 Calculation of Spearman Rank Correlation Spearman rank correlation

8 Calculation of Spearman Rank Correlation Difference between ranks

9 Calculation of Spearman Rank Correlation Number of cases

10 Calculation of Spearman Rank Correlation X: The student’s popularity measure Y: The student’s average academic achievement

11 Calculation of Spearman Rank Correlation

12 Calculation of Spearman Rank Correlation

13 Calculation of Spearman Rank Correlation

14 Calculation of Spearman Rank Correlation

15 Calculation of Spearman Rank Correlation

16 Calculation of Spearman Rank Correlation

17 Test of Significance Calculated r Rank = Calculated r Rank = Critical value for alpha 0.05 for Spearman Rank Correlation with 8 subjects = Critical value for alpha 0.05 for Spearman Rank Correlation with 8 subjects = Calculated r Ranks is less than critical value Calculated r Ranks is less than critical value The relation between Popularity and academic achievement is not statistically significant The relation between Popularity and academic achievement is not statistically significant

18 When to Use Which Test

19 When to Use Which Test

20 When to Use Which Test

21 Take Home Lesson Spearman Rank Correlation can be used on ordinal data