Lesson Inferences Between Two Variables

Objectives Perform Spearman’s rank-correlation test

Vocabulary Rank-correlation test -- nonparametric procedure used to test claims regarding association between two variables. Spearman’s rank-correlation coefficient -- test statistic, r s 6Σd i ² r s = 1 – n(n²- 1)

Association ●Parametric test for correlation:  Assumption of bivariate normal is difficult to verify  Used regression instead to test whether the slope is significantly different from 0 ●Nonparametric case for association:  Compare the relationship between two variables without assuming that they are bivariate normal  Perform a nonparametric test of whether the association is 0

Tale of Two Associations Similar to our previous hypothesis tests, we can have a two-tailed, a left-tailed, or a right- tailed alternate hypothesis –A two-tailed alternative hypothesis corresponds to a test of association –A left-tailed alternative hypothesis corresponds to a test of negative association –A right-tailed alternative hypothesis corresponds to a test of positive association

Test Statistic for Spearman’s Rank- Correlation Test The test statistic will depend on the size of the sample, n, and on the sum of the squared differences (d i ²). 6Σd i ² r s = 1 – n(n²- 1) where d i = the difference in the ranks of the two observations (Y i – X i ) in the ith ordered pair. Spearman’s rank-correlation coefficient, r s, is our test statistic z 0 = r s √n – 1 Small Sample Case: (n ≤ 100) Large Sample Case: (n > 100)

Critical Value for Spearman’s Rank-Correlation Test Left-TailedTwo-TailedRight-Tailed Significanceαα/2α Decision Rule Reject if r s < -CV Reject if r s CV Reject if r s > CV Using α as the level of significance, the critical value(s) is (are) obtained from Table XIII in Appendix A. For a two-tailed test, be sure to divide the level of significance, α, by 2. Small Sample Case: (n ≤ 100) Large Sample Case: (n > 100)

Hypothesis Tests Using Spearman’s Rank-Correlation Test Step 0 Requirements: 1. The data are a random sample of n ordered pairs. 2. Each pair of observations is two measurements taken on the same individual Step 1 Hypotheses: (claim is made regarding relationship between two variables, X and Y) H 0 : see below H 1 : see below Step 2 Ranks: Rank the X-values, and rank the Y-values. Compute the differences between ranks and then square these differences. Compute the sum of the squared differences. Step 3 Level of Significance: (level of significance determines the critical value) Table XIII in Appendix A. (see below) Step 4 Compute Test Statistic: Step 5 Critical Value Comparison: Left-TailedTwo-TailedRight-Tailed Significanceαα/2α H0H0 not associated H1H1 negatively associatedassociatedpositively associated Decision Rule Reject if r s < -CV Reject if r s CV Reject if r s > CV 6Σd i ² r s = 1 – n(n²- 1)

Expectations If X and Y were positively associated, then  Small ranks of X would tend to correspond to small ranks of Y  Large ranks of X would tend to correspond to large ranks of Y  The differences would tend to be small positive and small negative values  The squared differences would tend to be small numbers ●If X and Y were negatively associated, then  Small ranks of X would tend to correspond to large ranks of Y  Large ranks of X would tend to correspond to small ranks of Y  The differences would tend to be large positive and large negative values  The squared differences would tend to be large numbers

Example 1 from 15.6 SDS-RankD-Rankd = X - Yd² AveSum6 Calculations:

Example 1 Continued Hypothesis: H 0 : X and Y are not associated H a : X and Y are associated Test Statistic: 6 Σd i ² 6 (6) 36 r s = = 1 – = = n(n² - 1) 8(64 - 1) 8(63) Critical Value: (from table XIII) Conclusion: Since r s > CV, we reject H 0 ; therefore there is a relationship between club-head speed and distance.

Summary and Homework Summary –The Spearman rank-correlation test is a nonparametric test for testing the association of two variables –This test is a comparison of the ranks of the paired data values –The critical values for small samples are given in tables –The critical values for large samples can be approximated by a calculation with the normal distribution Homework –problems 3, 6, 7, 10 from the CD