Transcript Chap. 15: Nonparametric Statistics

© 2000 Prentice-Hall, Inc.
Statistics
Nonparametric Statistics
Chapter 14
14 - 1
Learning Objectives
© 2000 Prentice-Hall, Inc.
1. Distinguish Parametric & Nonparametric
Test Procedures
2. Explain a Variety of Nonparametric Test
Procedures
3. Solve Hypothesis Testing Problems
Using Nonparametric Tests
4. Compute Spearman’s Rank Correlation
14 - 2
© 2000 Prentice-Hall, Inc.
Hypothesis Testing
Procedures
Hypothesis
Testing
Procedures
Parametric
Nonparametric
Wilcoxon
Rank Sum
Test
Z Test
14 - 3
t Test
One-Way
ANOVA
Kruskal-Wallis
H-Test
Many More Tests Exist!
Parametric Test
Procedures
© 2000 Prentice-Hall, Inc.
1. Involve Population Parameters

Example: Population Mean
2. Require Interval Scale or Ratio Scale


Whole Numbers or Fractions
Example: Height in Inches (72, 60.5, 54.7)
3. Have Stringent Assumptions

Example: Normal Distribution
4. Examples: Z Test, t Test, 2 Test
14 - 4
Nonparametric Test
Procedures
© 2000 Prentice-Hall, Inc.
1. Do Not Involve Population Parameters

Example: Probability Distributions, Independence
2. Data Measured on Any Scale



Ratio or Interval
Ordinal
 Example: Good-Better-Best
Nominal
 Example: Male-Female
3. Example: Wilcoxon Rank Sum Test
14 - 5
© 2000 Prentice-Hall, Inc.
Advantages of
Nonparametric Tests
1.
Used With All Scales
2.
Easier to Compute

Developed Originally Before
Wide Computer Use
3.
Make Fewer
Assumptions
4.
Need Not Involve
Population Parameters
5.
Results May Be as Exact
as Parametric
Procedures
14 - 6
© 1984-1994 T/Maker Co.
Disadvantages of
Nonparametric Tests
© 2000 Prentice-Hall, Inc.
1.
May Waste Information


If Data Permit Using
Parametric Procedures
Example: Converting Data
From Ratio to Ordinal Scale
2.
Difficult to Compute by
Hand for Large Samples
3.
Tables Not Widely
Available
14 - 7
© 1984-1994 T/Maker Co.
© 2000 Prentice-Hall, Inc.
Frequently Used
Nonparametric Tests
1. Sign Test
2. Wilcoxon Rank Sum Test
3. Wilcoxon Signed Rank Test
4. Kruskal Wallis H-Test
5. Friedman Fr-Test
14 - 8
© 2000 Prentice-Hall, Inc.
Sign Test
14 - 9
© 2000 Prentice-Hall, Inc.
Frequently Used
Nonparametric Tests
1. Sign Test
2. Wilcoxon Rank Sum Test
3. Wilcoxon Signed Rank Test
4. Kruskal Wallis H-Test
5. Friedman Fr-Test
14 - 10
Sign Test
© 2000 Prentice-Hall, Inc.
1.
2.
3.
4.
Tests One Population Median,  (eta)
Corresponds to t-Test for 1 Mean
Assumes Population Is Continuous
Small Sample Test Statistic: # Sample
Values Above (or Below) Median

Alternative Hypothesis Determines
5. Can Use Normal Approximation If n  10
14 - 11
Sign Test Uses P-Value
to Make Decision
© 2000 Prentice-Hall, Inc.
P(X)
30%
Binomial: n = 8 p = 0.5
20%
.219
10%
0%
.004 .031
0
1
.273
.219
.109
2
.109
3
4
5
6
.031 .004
7
8 X
P-Value Is the Probability of Getting an Observation At
Least as Extreme as We Got. If 7 of 8 Observations
‘Favor’ Ha, Then P-Value = P(x  7) = .031 + .004 = .035.
If  = .05, Then Reject H0 Since P-Value  .
14 - 12
Sign Test Example
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You’re an analyst for ChefBoy-R-Dee. You’ve asked
7 people to rate a new
ravioli on a 5-point Likert
scale (1 = terrible to
5 = excellent. The ratings
are: 2 5 3 4 1 4 5.
At the .05 level, is there
evidence that the median
rating is at least 3?
14 - 13
Sign Test Solution
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H0:
Ha:
=
Test Statistic:
P-Value:
Decision:
Conclusion:
14 - 14
Sign Test Solution
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H0:  = 3
Ha:  < 3
=
Test Statistic:
P-Value:
Decision:
Conclusion:
14 - 15
Sign Test Solution
© 2000 Prentice-Hall, Inc.
H0:  = 3
Ha:  < 3
 = .05
Test Statistic:
P-Value:
Decision:
Conclusion:
14 - 16
Sign Test Solution
© 2000 Prentice-Hall, Inc.
H0:  = 3
Ha:  < 3
 = .05
Test Statistic:
S=2
(Ratings 1 & 2 Are
Less Than  = 3:
2, 5, 3, 4, 1, 4, 5)
14 - 17
P-Value:
Decision:
Conclusion:
Sign Test Solution
© 2000 Prentice-Hall, Inc.
H0:  = 3
Ha:  < 3
 = .05
Test Statistic:
S=2
(Ratings 1 & 2 Are
Less Than  = 3:
2, 5, 3, 4, 1, 4, 5)
14 - 18
P-Value:
P(x  2) = 1 - P(x  1)
= .937
(Binomial Table, n = 7,
p = 0.50)
Decision:
Conclusion:
Sign Test Solution
© 2000 Prentice-Hall, Inc.
H0:  = 3
Ha:  < 3
 = .05
Test Statistic:
S=2
(Ratings 1 & 2 Are
Less Than  = 3:
2, 5, 3, 4, 1, 4, 5)
14 - 19
P-Value:
P(x  2) = 1 - P(x  1)
= .937
(Binomial Table, n = 7,
p = 0.50)
Decision:
Do Not Reject at  = .05
Conclusion:
Sign Test Solution
© 2000 Prentice-Hall, Inc.
H0:  = 3
Ha:  < 3
 = .05
Test Statistic:
S=2
(Ratings 1 & 2 Are
Less Than  = 3:
2, 5, 3, 4, 1, 4, 5)
14 - 20
P-Value:
P(x  2) = 1 - P(x  1)
= .937
(Binomial Table, n = 7,
p = 0.50)
Decision:
Do Not Reject at  = .05
Conclusion:
There Is No Evidence
Median Is Less Than 3
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
14 - 21
© 2000 Prentice-Hall, Inc.
Frequently Used
Nonparametric Tests
1. Sign Test
2. Wilcoxon Rank Sum Test
3. Wilcoxon Signed Rank Test
4. Kruskal Wallis H-Test
5. Friedman Fr-Test
14 - 22
Wilcoxon Rank Sum Test
© 2000 Prentice-Hall, Inc.
1. Tests Two Independent Population
Probability Distributions
2. Corresponds to t-Test for 2 Independent
Means
3. Assumptions


Independent, Random Samples
Populations Are Continuous
4. Can Use Normal Approximation If ni  10
14 - 23
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Procedure
1. Assign Ranks, Ri, to the n1 + n2
Sample Observations



If Unequal Sample Sizes, Let n1 Refer to
Smaller-Sized Sample
Smallest Value = 1
Average Ties
2. Sum the Ranks, Ti, for Each Sample
3. Test Statistic Is TA (Smallest Sample)
14 - 24
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Example
You’re a production planner. You want to
see if the operating rates for 2 factories is
the same. For factory 1, the rates (% of
capacity) are 71, 82, 77, 92, 88. For
factory 2, the rates are 85, 82,
94 & 97. Do the factory rates
have the same probability
distributions at the
.10 level?
14 - 25
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Solution
H0:
Ha:
=
n1 =
n2 =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
 Ranks
14 - 26
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Solution
H0: Identical Distrib.
Ha: Shifted Left or Right
=
n1 =
n2 =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
 Ranks
14 - 27
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Solution
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
 Ranks
14 - 28
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum
Table (Portion)
 = .05 one-tailed;  = .10 two-tailed
n1
3
n2
14 - 29
3
4
5
:
4
5
TL TU TL TU TL TU
..
..
6
7
7
:
..
..
..
:
15 7 17 7 20
17 12 24 13 27
20 13 27 19 36
:
:
:
:
:
Wilcoxon Rank Sum Test
Solution
© 2000 Prentice-Hall, Inc.
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Reject
Do Not
Reject
Reject
13
14 - 30
27  Ranks
Test Statistic:
Decision:
Conclusion:
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
Rank Sum
14 - 31
Factory 2
Rate
Rank
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Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
82
77
92
88
Rank Sum
14 - 32
Factory 2
Rate
Rank
85
82
94
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
77
92
88
Rank Sum
14 - 33
Factory 2
Rate
Rank
85
82
94
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
77
2
92
88
Rank Sum
14 - 34
Factory 2
Rate
Rank
85
82
94
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
3
77
2
92
88
Rank Sum
14 - 35
Factory 2
Rate
Rank
85
82
4
94
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
3 3.5
77
2
92
88
Rank Sum
14 - 36
Factory 2
Rate
Rank
85
82
4 3.5
94
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
3 3.5
77
2
92
88
Rank Sum
14 - 37
Factory 2
Rate
Rank
85
5
82
4 3.5
94
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
3 3.5
77
2
92
88
6
Rank Sum
14 - 38
Factory 2
Rate
Rank
85
5
82
4 3.5
94
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
3 3.5
77
2
92
7
88
6
Rank Sum
14 - 39
Factory 2
Rate
Rank
85
5
82
4 3.5
94
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
3 3.5
77
2
92
7
88
6
Rank Sum
14 - 40
Factory 2
Rate
Rank
85
5
82
4 3.5
94
8
97
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
3 3.5
77
2
92
7
88
6
Rank Sum
14 - 41
Factory 2
Rate
Rank
85
5
82
4 3.5
94
8
97
9
...
...
© 2000 Prentice-Hall, Inc.
Wilcoxon Rank Sum Test
Computation Table
Factory 1
Rate
Rank
71
1
82
3 3.5
77
2
92
7
88
6
Rank Sum
14 - 42
19.5
Factory 2
Rate
Rank
85
5
82
4 3.5
94
8
97
9
...
...
25.5
Wilcoxon Rank Sum Test
Solution
© 2000 Prentice-Hall, Inc.
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Reject
Do Not
Reject
Reject
13
14 - 43
27  Ranks
Test Statistic:
T2 = 5 + 3.5 + 8+ 9 = 25.5
(Smallest Sample)
Decision:
Conclusion:
Wilcoxon Rank Sum Test
Solution
© 2000 Prentice-Hall, Inc.
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Reject
Do Not
Reject
Reject
13
14 - 44
27  Ranks
Test Statistic:
T2 = 5 + 3.5 + 8+ 9 = 25.5
(Smallest Sample)
Decision:
Do Not Reject at  = .10
Conclusion:
Wilcoxon Rank Sum Test
Solution
© 2000 Prentice-Hall, Inc.
H0: Identical Distrib.
Ha: Shifted Left or Right
 = .10
n1 = 4 n2 = 5
Critical Value(s):
Reject
Do Not
Reject
Reject
13
14 - 45
Test Statistic:
T2 = 5 + 3.5 + 8+ 9 = 25.5
(Smallest Sample)
Decision:
Do Not Reject at  = .10
Conclusion:
There Is No Evidence
27  Ranks
Distrib. Are Not Equal
© 2000 Prentice-Hall, Inc.
Wilcoxon Signed Rank Test
14 - 46
© 2000 Prentice-Hall, Inc.
Frequently Used
Nonparametric Tests
1. Sign Test
2. Wilcoxon Rank Sum Test
3. Wilcoxon Signed Rank Test
4. Kruskal Wallis H-Test
5. Friedman Fr-Test
14 - 47
Wilcoxon
Signed Rank Test
© 2000 Prentice-Hall, Inc.
1. Tests Probability Distributions of
2 Related Populations
2. Corresponds to t-test for Dependent
(Paired) Means
3. Assumptions


Random Samples
Both Populations Are Continuous
4. Can Use Normal Approximation If n  25
14 - 48
Signed Rank Test
Procedure
© 2000 Prentice-Hall, Inc.
1.
2.
3.
4.
5.
6.
Obtain Difference Scores, Di = X1i - X2i
Take Absolute Value of Differences, Di
Delete Differences With 0 Value
Assign Ranks, Ri, Where Smallest = 1
Assign Ranks Same Signs as Di
Sum ‘+’ Ranks (T+) & ‘-’ Ranks (T-)


Test Statistic Is T- (One-Tailed Test)
Test Statistic Is Smaller of T- or T+ (2-Tail)
14 - 49
Signed Rank Test
Computation Table
© 2000 Prentice-Hall, Inc.
X1i X2i Di = X1i - X2i |Di|
Ri Sign Sign Ri
X11 X21 D1 = X11 - X21 |D1| R1
±
± R1
X12 X22 D2 = X12 - X22 |D2| R2
±
± R2
X13 X23 D3 = X13 - X23 |D3| R3
±
± R3
:
:
±
± Rn
:
:
:
:
:
X1n X2n Dn = X1n - X2n |Dn| Rn
Total
14 - 50
T+ & T-
© 2000 Prentice-Hall, Inc.
Signed Rank Test
Example
You work in the finance department. Is the new
financial package faster (.05 level)? You collect
the following data entry times:
User
Current
New
Donna
9.98
9.88
Santosha 9.88
9.86
Sam
9.90
9.83
Tamika
9.99
9.80
Brian
9.94
9.87
Jorge
9.84
9.84
14 - 51
© 1984-1994 T/Maker Co.
© 2000 Prentice-Hall, Inc.
Signed Rank Test
Solution
H0:
Ha:
Test Statistic:
=
n’ =
Critical Value(s):
Do Not
Reject
Reject
T0
14 - 52
Decision:
Conclusion:
© 2000 Prentice-Hall, Inc.
Signed Rank Test
Solution
H0: Identical Distrib.
Ha: Current Shifted
Right
=
n’ =
Critical Value(s):
Do Not
Reject
Reject
T0
14 - 53
Test Statistic:
Decision:
Conclusion:
Signed Rank Test
Computation Table
© 2000 Prentice-Hall, Inc.
X1i
X2i
9.98 9.88
9.88 9.86
9.90 9.83
9.99 9.80
9.94 9.87
9.84 9.84
Total
14 - 54
Di
+0.10
+0.02
+0.07
+0.19
+0.07
0.00
|Di|
Ri
0.10 4
0.02 1
0.07 2 2.5
0.19 5
0.07 3 2.5
0.00 ...
Sign Sign Ri
+
+
+
+
+
...
+4
+1
+2.5
+5
+2.5
Discard
T+ = 15, T- = 0
© 2000 Prentice-Hall, Inc.
Signed Rank Test
Solution
H0: Identical Distrib.
Ha: Current Shifted
Right
 = .05
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Do Not
Reject
Reject
T0
14 - 55
Test Statistic:
Decision:
Conclusion:
© 2000 Prentice-Hall, Inc.
Wilcoxon Signed Rank
Table (Portion)
One-Tailed Two-Tailed
 = .05
 = .025
 = .01
 = .005
 = .10
 = .05
 = .02
 = .01
n=5
n=6
1
2
1
n = 7 ..
4
2
0
..
..
..
..
n = 11 n = 12 n = 13
:
14 - 56
:
:
:
Signed Rank Test
Solution
© 2000 Prentice-Hall, Inc.
H0: Identical Distrib.
Ha: Current Shifted
Right
 = .05
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Do Not
Reject
Reject
1
14 - 57
T0
Test Statistic:
Decision:
Conclusion:
Signed Rank Test
Solution
© 2000 Prentice-Hall, Inc.
H0: Identical Distrib.
Ha: Current Shifted
Right
 = .05
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Do Not
Reject
Reject
1
14 - 58
T0
Test Statistic:
Since One-Tailed
Test & Current
Shifted Right, Use T-:
T- = 0
Decision:
Conclusion:
Signed Rank Test
Solution
© 2000 Prentice-Hall, Inc.
H0: Identical Distrib.
Ha: Current Shifted
Right
 = .05
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Do Not
Reject
Reject
1
14 - 59
T0
Test Statistic:
Since One-Tailed
Test & Current
Shifted Right, Use T-:
T- = 0
Decision:
Reject at  = .05
Conclusion:
Signed Rank Test
Solution
© 2000 Prentice-Hall, Inc.
H0: Identical Distrib.
Ha: Current Shifted
Right
 = .05
n’ = 5 (not 6; 1 elim.)
Critical Value(s):
Do Not
Reject
Reject
1
14 - 60
T0
Test Statistic:
Since One-Tailed
Test & Current
Shifted Right, Use T-:
T- = 0
Decision:
Reject at  = .05
Conclusion:
There Is Evidence New
Package Is Faster
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
14 - 61
© 2000 Prentice-Hall, Inc.
Frequently Used
Nonparametric Tests
1. Sign Test
2. Wilcoxon Rank Sum Test
3. Wilcoxon Signed Rank Test
4. Kruskal Wallis H-Test
5. Friedman Fr-Test
14 - 62
Kruskal-Wallis H-Test
© 2000 Prentice-Hall, Inc.
1. Tests the Equality of More Than 2 (p)
Population Probability Distributions
2. Corresponds to ANOVA for More Than
2 Means
3. Used to Analyze Completely
Randomized Experimental Designs
4. Uses 2 Distribution with p - 1 df

If At Least 1 Sample Size nj > 5
14 - 63
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Assumptions
1. Independent, Random Samples
2. At Least 5 Observations Per Sample
3. Continuous Population Probability
Distributions
14 - 64
Kruskal-Wallis H-Test
Procedure
© 2000 Prentice-Hall, Inc.
1. Assign Ranks, Ri , to the n Combined
Observations


Smallest Value = 1; Largest Value = n
Average Ties
2. Sum Ranks for Each Group
14 - 65
Kruskal-Wallis H-Test
Procedure
© 2000 Prentice-Hall, Inc.
1. Assign Ranks, Ri , to the n Combined
Observations


Smallest Value = 1; Largest Value = n
Average Ties
2. Sum Ranks for Each Group
3. Compute Test Statistic
Squared total of
each group
F
R I
12
HG

 3 a
n  1f
J
Hn  an  1f n K
14 - 66
p
2
j
j 1
j
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Example
As production manager,
you want to see if 3 filling
machines have different
filling times. You assign
15 similarly trained &
experienced workers,
5 per machine, to the
machines. At the .05 level,
is there a difference in the
distribution of filling times?
14 - 67
Mach1
25.40
26.31
24.10
23.74
25.10
Mach2
23.40
21.80
23.50
22.75
21.60
Mach3
20.00
22.20
19.75
20.60
20.40
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
H0:
Ha:
=
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
2
0
14 - 68
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
=
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
2
0
14 - 69
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
2
0
14 - 70
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
Test Statistic:
Decision:
 = .05
Conclusion:
0
5.991
14 - 71
2
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
14 - 72
Ranks
Mach1 Mach2 Mach3
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
14 - 73
Ranks
Mach1 Mach2 Mach3
1
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
14 - 74
Ranks
Mach1 Mach2 Mach3
2
1
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
14 - 75
Ranks
Mach1 Mach2 Mach3
2
1
3
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
Raw Data
Mach1 Mach2 Mach3
25.40 23.40 20.00
26.31 21.80 22.20
24.10 23.50 19.75
23.74 22.75 20.60
25.10 21.60 20.40
14 - 76
Ranks
Mach1 Mach2 Mach3
14
9
2
15
6
7
12
10
1
11
8
4
13
5
3
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
Raw Data
Ranks
Mach1 Mach2 Mach3
Mach1 Mach2 Mach3
25.40 23.40 20.00
14
9
2
26.31 21.80 22.20
15
6
7
24.10 23.50 19.75
12
10
1
23.74 22.75 20.60
11
8
4
25.10 21.60 20.40
13
5
3
38
17
Total 65
14 - 77
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
F
R I
12
n  1f
 3a

HG
J
Hn  an  1f n K
F
F
17f II
38f a
65 f a
a
12
16f
 3 a


G
G
J
J
Ha15f a16f H5 5 5 KK
12 I
F
191.6f 48
a

H240K
p
2
j
j 1
j
2
 11.58
14 - 78
2
2
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
Test Statistic:
H = 11.58
Decision:
 = .05
Conclusion:
0
5.991
14 - 79
2
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
 = .05
Test Statistic:
H = 11.58
Decision:
Reject at  = .05
Conclusion:
0
5.991
14 - 80
2
© 2000 Prentice-Hall, Inc.
Kruskal-Wallis H-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
 = .05
0
5.991
14 - 81
2
Test Statistic:
H = 11.58
Decision:
Reject at  = .05
Conclusion:
There Is Evidence Pop.
Distrib. Are Different
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
14 - 82
© 2000 Prentice-Hall, Inc.
Frequently Used
Nonparametric Tests
1. Sign Test
2. Wilcoxon Rank Sum Test
3. Wilcoxon Signed Rank Test
4. Kruskal Wallis H-Test
5. Friedman Fr-Test
14 - 83
Friedman Fr-Test
© 2000 Prentice-Hall, Inc.
1. Tests the Equality of 2 or More (p)
Population Probability Distributions
When Blocking Variable Used
2. Corresponds to Randomized Block F-Test
3. Used to Analyze Randomized Block
Designs
4. Uses 2 Distribution with p - 1 df

If Number of Blocks or Treatments > 5
14 - 84
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Assumptions
1. Independent, Random Samples
2. Measurements Can Be Ranked Within
Blocks
3. Continuous Population Probability
Distributions
14 - 85
Friedman Fr-Test
Procedure
© 2000 Prentice-Hall, Inc.
1. Assign Ranks, Ri , to the Observations
Within Each Block


Smallest Value = 1; Largest Value = nj
Average Ties
2. Sum Ranks Within Each Block
14 - 86
Friedman Fr-Test
Procedure
© 2000 Prentice-Hall, Inc.
1. Assign Ranks, Ri , to the Observations
Within Each Block


Smallest Value = 1; Largest Value = nj
Average Ties
2. Sum Ranks Within Each Block
3. Compute Test Statistic
Squared total of
p
each block
12
Fr 
  R 2j  3  b  p  1
b  p  p  1 j 1
a f
14 - 87
a f
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Example
You’re a research assistant for the NIH. You’re
investigating the effects of plants on human stress.
You record finger
Subj. Live Photo None
temperatures under 3
1
91.4
93.5
96.6
conditions: presence
2
94.9
96.6
90.5
of a live plant, plant
3
97.0
95.8
95.4
photo, nothing. At
4
93.7
96.2
96.7
the .05 level, does
5
96.0
96.6
93.5
finger temperature
depend on experimental condition?
14 - 88
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
H0:
Ha:
=
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
2
0
14 - 89
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
=
df =
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
2
0
14 - 90
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
Test Statistic:
Decision:
Conclusion:
2
0
14 - 91
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
Test Statistic:
Decision:
 = .05
Conclusion:
0
5.991
14 - 92
2
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 93
Plant
Ranks
Photo None
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 94
Plant
1
Ranks
Photo None
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 95
Plant
1
Ranks
Photo None
2
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 96
Plant
1
Ranks
Photo None
2
3
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 97
Plant
1
Ranks
Photo None
2
3
1
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 98
Plant
1
2
Ranks
Photo None
2
3
1
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 99
Plant
1
2
Ranks
Photo None
2
3
3
1
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 100
Plant
1
2
3
1
2
Ranks
Photo None
2
3
3
1
2
1
2
3
3
1
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
Raw Data
Plant Photo None
91.4
93.5
96.6
94.9
96.6
90.5
97.0
95.8
95.4
93.7
96.2
96.7
96.0
96.6
93.5
14 - 101
Plant
1
2
3
1
2
Total 9
Ranks
Photo None
2
3
3
1
2
1
2
3
3
1
12
9
Friedman Fr-Test
Solution
© 2000 Prentice-Hall, Inc.
p
12
2
Fr 
  Rj  3 b  p  1
b  p  p  1 j 1
a f
a f
12

 af
9 a
12f  af
9
5 3 4
afafaf
12 I
F

a
306f 60
H60K
2
 1.2
14 - 102
2
2
afaf
3 5 4
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
Test Statistic:
Fr = 1.2
Decision:
 = .05
Conclusion:
0
5.991
14 - 103
2
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
 = .05
Test Statistic:
Fr = 1.2
Decision:
Do Not Reject at  = .05
Conclusion:
0
5.991
14 - 104
2
© 2000 Prentice-Hall, Inc.
Friedman Fr-Test
Solution
H0: Identical Distrib.
Ha: At Least 2 Differ
 = .05
df = p - 1 = 3 - 1 = 2
Critical Value(s):
 = .05
0
5.991
14 - 105
2
Test Statistic:
Fr = 1.2
Decision:
Do Not Reject at  = .05
Conclusion:
There Is No Evidence
Distrib. Are Different
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Coefficient
14 - 106
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Coefficient
1. Measures Correlation Between Ranks
2. Corresponds to Pearson Product
Moment Correlation Coefficient
3. Values Range from -1 to +1
14 - 107
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Coefficient
1. Measures Correlation Between Ranks
2. Corresponds to Pearson Product
Moment Correlation Coefficient
3. Values Range from -1 to +1
4. Equation (Shortcut)
rs  1 
14 - 108
6 d 2
d i
n n2  1
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Procedure
1. Assign Ranks, Ri , to the Observations
of Each Variable Separately
2. Calculate Differences, di , Between
Each Pair of Ranks
3. Square Differences, di 2, Between Ranks
4. Sum Squared Differences for Each
Variable
5. Use Shortcut Approximation Formula
14 - 109
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Example
You’re a research assistant for the FBI. You’re
investigating the relationship between a person’s
attempts at deception
Subj. Deception Pupil
& % changes in their
1
87
10
pupil size. You ask
2
63
6
subjects a series of
3
95
11
questions, some of
4
50
7
which they must
5
43
0
answer dishonestly.
At the .05 level, what is the correlation coefficient?
14 - 110
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Table
Subj. Decep. R1i Pupil R2i
Total
14 - 111
di
di2
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Table
Subj. Decep. R1i Pupil R2i
1
2
3
4
5
87
63
95
50
43
10
6
11
7
0
Total
14 - 112
di
di2
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Table
Subj. Decep. R1i Pupil R2i
1
2
3
4
5
87
63
95
50
43
4
3
5
2
1
10
6
11
7
0
Total
14 - 113
di
di2
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Table
Subj. Decep. R1i Pupil R2i
1
2
3
4
5
87
63
95
50
43
4
3
5
2
1
10
6
11
7
0
Total
14 - 114
4
2
5
3
1
di
di2
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Table
Subj. Decep. R1i Pupil R2i
1
2
3
4
5
87
63
95
50
43
4
3
5
2
1
10
6
11
7
0
Total
14 - 115
4
2
5
3
1
di
0
1
0
-1
0
di2
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Table
Subj. Decep. R1i Pupil R2i
1
2
3
4
5
87
63
95
50
43
4
3
5
2
1
10
6
11
7
0
Total
14 - 116
4
2
5
3
1
di
di2
0
1
0
-1
0
0
1
0
1
0
2
© 2000 Prentice-Hall, Inc.
Spearman’s Rank
Correlation Solution
n
rs  1 
6 d i2
i 1
2
d i
6af
2
 1
5d
5  1i
n n 1
2
 1  0.10
 0.90
14 - 117
Conclusion
© 2000 Prentice-Hall, Inc.
1. Distinguished Parametric &
Nonparametric Test Procedures
2. Explained a Variety of Nonparametric
Test Procedures
3. Solved Hypothesis Testing Problems
Using Nonparametric Tests
4. Computed Spearman’s Rank Correlation
14 - 118