Lesson 15 - 6
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Transcript of Lesson 15 - 6
Lesson 15 - 6
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, rs
6Σdi² rs = 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 (di²).
6Σdi² rs = 1 – --------------
n(n²- 1)
where di = the difference in the ranks of the two observations (Yi – Xi) in the ith ordered pair.
Spearman’s rank-correlation coefficient, rs, is our test statistic
z0 = rs √n – 1
Small Sample Case: (n ≤ 100)
Large Sample Case: (n > 100)
Critical Value for Spearman’s Rank-Correlation Test
Left-Tailed Two-Tailed Right-Tailed
Significance α α/2 α
Decision Rule
Reject if rs < -CV
Reject if rs < -CV or rs > CV
Reject if rs > 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 TestStep 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) H0: see below H1: 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-Tailed Two-Tailed Right-Tailed
Significance α α/2 α
H0 not associated not associated not associated
H1 negatively associated associated positively associated
Decision Rule
Reject if rs < -CVReject if
rs < -CV or rs > CVReject if rs > CV
6Σdi² rs = 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
S D S-Rank D-Rank d = X - Y d²
100 257 2.5 1 1.5 2.25
102 264 5 4 1 1
103 274 6 6 0 0
101 266 4 5 -1 1
105 277 7.5 8 -0.5 0.25
100 263 2.5 3 -0.5 0.25
99 258 1 2 -1 1
105 275 7.5 7 0.5 0.25
102 267 Ave Sum 6
Calculations:
Example 1 Continued
• Hypothesis: H0: X and Y are not associated Ha: X and Y are associated
• Test Statistic:
6 Σdi² 6 (6) 36 rs = 1 - ----------- = 1 – ------------- = 1 - -------- = 0.929 n(n² - 1) 8(64 - 1) 8(63)
• Critical Value: 0.738 (from table XIII)
• Conclusion: Since rs > CV, we reject H0; 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