Lesson 15  6

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Transcript of Lesson 15  6
Lesson 15  6
Inferences Between Two Variables
Objectives
• Perform Spearman’s rankcorrelation test
Vocabulary• Rankcorrelation test  nonparametric procedure
used to test claims regarding association between two variables.
• Spearman’s rankcorrelation 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 twotailed, a lefttailed, or a righttailed alternate hypothesis– A twotailed alternative hypothesis corresponds
to a test of association
– A lefttailed alternative hypothesis corresponds to a test of negative association
– A righttailed alternative hypothesis corresponds to a test of positive association
Test Statistic for Spearman’s RankCorrelation 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 rankcorrelation 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 RankCorrelation Test
LeftTailed TwoTailed RightTailed
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 twotailed 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 RankCorrelation 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 Xvalues, and rank the Yvalues. 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: LeftTailed TwoTailed RightTailed
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 SRank DRank 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 clubhead speed and distance.
Example done in Excel
ClubHead Speed Distance Rank  X Rank  Y difference d²
100 257 2.5 1 1.5 2.25102 264 5 4 1 1103 274 6 6 0 0101 266 4 5 1 1105 277 7.5 8 0.5 0.25100 263 2.5 3 0.5 0.2599 258 1 2 1 1105 275 7.5 7 0.5 0.25n = 8
ClubHead
Speed Distance 6 sumClubHead Speed 1 rs = 0.928571Distance 0.938695838 1 rc = 0.738
Rank  X Rank  YRank  X 1Rank  Y 0.927778183 1
Summary and Homework
• Summary– The Spearman rankcorrelation 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
Homework Problem 3
Problem 3
X Y Rank  X Rank  Y difference d22 1.4 1 1 0 04 1.8 2 2 0 08 2.1 3.5 3 0.5 0.258 2.3 3.5 4 0.5 0.259 2.6 5 5 0 0
n = 5 0.5 sum Rank  X Rank  Y rs = 0.975
Rank  X 1 rc = 1Rank  Y 0.974679434 1 FTR
Homework Problem 6
Problem 6
X Y Rank  X Rank  Y difference d20 0.8 1 1 0 0
0.5 2.3 2 3 1 11.4 1.9 3.5 2 1.5 2.251.4 2.5 3.5 4 0.5 0.253.9 5 5 5 0 04.6 6.8 6 6 0 0n = 6
3.5 sumrs = 0.9
Rank  X Rank  Y rc = 0.886Rank  X 1 RejectRank  Y 0.898645105 1
Homework Problem 7Problem 7
BS % Income Rank  X Rank  Y difference d217.4 24289 1 1 0 029.8 33749 9 9 0 024.6 29043 4 4 0 022.3 26100 2 2 0 023.7 28831 3 3 0 026.8 30758 8 8 0 025 29944 6 7 1 1
26.4 29340 7 5 2 424.7 29372 5 6 1 1n = 9
6 sumrs = 0.95
Rank  X Rank  Y rc = 0.6Rank  X 1 RejectRank  Y 0.95 1
Homework Problem 10
Problem 10
Standings Yards Rank  X Rank  Y difference d212 4803 5 3 2 430 5215 7 6 1 17 4459 3 2 1 1
11 5134 4 5 1 12 4972 2 4 2 4
29 5936 6 7 1 11 4134 1 1 0 0
n = 7 12 sum
rs = 0.785714 Rank  X Rank  Y rc = 0.714
Rank  X 1 RejectRank  Y 0.785714286 1