1. Inferences Between Two Variables
Lesson
Inferences Between Two Variables

2. Objectives
Perform Spearman’s rank-correlation test

3. rs = 1 – -------------- n(n²- 1)
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)

4. 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

5. 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

6. Test Statistic for Spearman’s Rank-Correlation Test
Small Sample Case: (n ≤ 100)
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
Large Sample Case: (n > 100)

7. Critical Value for Spearman’s Rank-Correlation Test
Small Sample Case: (n ≤ 100)
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.
Large Sample Case: (n > 100)
Left-Tailed
Two-Tailed
Right-Tailed
Significance α
α/2
Decision Rule
Reject if rs < -CV
Reject if rs < -CV or rs > CV
Reject if rs > CV

8. 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)
H0: see below Ha: 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:
6Σdi²
rs = 1 – n(n²- 1)
Left-Tailed
Two-Tailed
Right-Tailed
Significance α
α/2 H0
not associated Ha
negatively associated
associated
positively associated
Decision Rule
Reject if rs < -CV
Reject if rs < -CV or rs > CV
Reject if rs > CV

9. 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

10. Example 1 from 15.6 Calculations: S D S-Rank D-Rank dif = X - Y dif²
100
257
2.5 1
1.5
2.25
102
264 5 4
103
274 6
101
266 -1
105
277
7.5 8
-0.5
0.25
263 3 99
258 2
275 7
0.5
267
Ave
Sum

11. Example 1 Continued
Hypothesis: H0: X and Y are not associated Ha: X and Y are associated
Test Statistic: (Two-tailed) Σdi² (6) rs = = 1 – = = n(n² - 1) (64 - 1) (63)
Critical Value: (from table XIII)
Conclusion: Since rs > CV, we reject H0; therefore there is a relationship between club-head speed and distance.

12. HyCCI for Example 1
Hyp: H0: Club head speed and distance are not associated Ha: Club head speed and distance are associated
Conditions: Assume an independent random sample Paired data (explanatory, response) from each golfer
Calculations: (Two-tailed) Σdi² (6) rs = = 1 – = = n(n² - 1) (64 - 1) (63) Critical Value: (from table XIII)
Interpretation: Since rs > CV, we reject H0; therefore there is a relationship between club-head speed and distance.

13. Example done in Excel Club-Head Speed Distance Rank - X Rank - Y
difference d²
100
257
2.5 1
1.5
2.25
102
264 5 4
103
274 6
101
266 -1
105
277
7.5 8
-0.5
0.25
263 3 99
258 2
275 7
0.5
n =
sum
rs =
rc =
0.738

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

15. Homework Problem 3 Problem 3 X Y Rank - X Rank - Y difference d2 2 1.44
1.8 8
2.1
3.5 3
0.5
0.25
2.3
-0.5 9
2.6 5
n =
sum
rs =
0.975
rc =
FTR

16. Homework Problem 6 Problem 6 X Y Rank - X Rank - Y difference d2 0.8 1
0.8 1
0.5
2.3 2 3 -1
1.4
1.9
3.5
1.5
2.25
2.5 4
-0.5
0.25
3.9 5
4.6
6.8 6
n =
sum
rs =
0.9
rc =
0.886
Reject

17. Homework Problem 7 Problem 7 BS % Income Rank - X Rank - Y difference
17.4
24289 1
29.8
33749 9
24.6
29043 4
22.3
26100 2
23.7
28831 3
26.8
30758 8 25
29944 6 7 -1
26.4
29340 5
24.7
29372
n =
sum
rs =
0.95
rc =
0.6
Reject

18. Homework Problem 10 Problem 10 Standings Yards Rank - X Rank - Y
difference d2 12
4803 5 3 2 4 30
5215 7 6 1
4459 11
5134 -1
4972 -2 29
5936
4134
n =
sum
rs =
rc =
0.714
Reject