Inferences Between Two Variables

1
Lesson 15 - 6

• Inferences Between Two Variables

2
Objectives

• Perform Spearmans rank-correlation test

3
Vocabulary

• Rank-correlation test -- nonparametric procedure
  used to test claims regarding association between
  two variables.
• Spearmans rank-correlation coefficient -- test
  statistic, rs

6Sdi² 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 Spearmans 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). 6Sdi² rs 1
-------------- n(n²- 1) where di
the difference in the ranks of the two
observations (Yi Xi) in the ith ordered
pair. Spearmans rank-correlation coefficient,
rs, is our test statistic
z0 rs vn 1
Large Sample Case (n gt 100)
7
Critical Value for Spearmans Rank-Correlation
Test
Small Sample Case (n 100)
Using a 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, a, by 2.
Large Sample Case (n gt 100)
Left-Tailed Two-Tailed Right-Tailed
Significance a a/2 a
Decision Rule Reject if rs lt -CV Reject if rs lt -CV or rs gt CV Reject if rs gt CV
8
Hypothesis Tests Using Spearmans
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
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
6Sdi² rs 1 --------------
n(n²- 1)
Left-Tailed Two-Tailed Right-Tailed
Significance a a/2 a
H0 not associated not associated not associated
H1 negatively associated associated positively associated
Decision Rule Reject if rs lt -CV Reject if rs lt -CV or rs gt CV Reject if rs gt 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 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
11
Example 1 Continued

• Hypothesis H0 X and Y are not associated
  Ha X and Y are associated
• Test Statistic 6 Sdi²
  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 gt CV, we reject H0
  therefore there is a relationship between
  club-head speed and distance.

12
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