Transcript Nonparametric Statistics

Nonparametric Statistics
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Nonparametric Statistics
• Many of the nonparametric statistical tests
answer the same sorts of questions as the
parametric tests. With nonparametric tests
the assumptions can be relaxed considerably.
Consequently, nonparametric methods are
used for situations that violate the
assumptions of parametric procedures.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Test for Randomness;
The Runs Test
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Test for Randomness;
The Runs Test
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
n!
number of arrangement A 
n1 !n2 !
The Runs Test (Small Samples)
• Sequence 1: HHHHH TTTTT
Run 1
Run 2
• Sequence 3: T HHH T H T H TT
Run 1
Run 7
• R = number of Runs
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
The Runs Test (Small Samples)
Ho: The sequence was generated in a random manner
Ha: The sequence was not generated in a random manner

P (R  k1 )   .025
2

P (R  k2 )   .025
2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Test for Randomness;
The Runs Test
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Runs Test (Large Samples)
Ho: The sequence was generated in a random manner
Ha: The sequence was not generated in a random manner
R  R
Z
R
R
2n1 n2
1 
n1  n2
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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R 
2n1 n2 (2n1 n 2  n1  n2 )
( n1  n2 ) 2 (n1  n 2  1)
Reject Ho if |Z| > Z/2
Nonparametric Test Central
Tendency: Two Populations
Introduction to Business Statistics, 5e
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(c)2000 South-Western College
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Mann-Whitney U
(Independent Samples)
Ho: The two populations have identical probability distributions
Ha: The two populations differ in location
T1 = sum of the ranks of the observations from the first sample
T2 = sum of the ranks of the observations from the second sample
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Mann-Whitney U
Small Samples
n1 (n1 1)
U1  n1 n 2 
 T1
2
n2 (n 2 1)
U 2  n1 n2 
 T2
2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Mann-Whitney U
Small Samples
Procedure:
1. Assume that n1 < n2 (reverse the samples if necessary)
2. Determine U1 and U2
3. Use the value from Table A.10 to test Ho vs. Ha
Two-Sided Test
Ha: the two populations differ in location
Reject Ho if Table A.10 value for U is < /2, where U is
the minimum of U1 and U2.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Mann-Whitney U
Small Samples
Procedure:
1. Assume that n1 < n2 (reverse the samples if necessary)
2. Determine U1 and U2
3. Use the value from Table A.10 to test Ho vs. Ha
One-Sided Test
Ha: Population 1 is shifted to
the right of population2
Reject Ho if Table A.10
value for U is less than 
where U = U1.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Ha: Population 1 is shifted to
the left of population 2
Reject Ho if Table A.10
value for U is less than 
Where U = U2
Mann-Whitney U
Large Samples
U 2  U 2
Z
U2
U2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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n1 n 2

2
U2 
n1n2 (n1  n2  1)
12
Mann-Whitney U
Large Samples
Ho: The two populations have identical probability
distribution
Determine U2
Two-Sided Test
Ha: The two populations differ in location
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Reject Ho if |Z| > Z /2
Mann-Whitney U
Large Samples
Ho: The two populations have identical probability
distribution
Determine U2
One-Sided Test
Ha: Population 1 is shifted to
the right of population 2
Ha: Population 1 is shifted to
the left of population 2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
Reject Ho if Z > Z
(c)2000 South-Western College
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Reject Ho if Z < - Z
Wilcoxon Signed Rank Test for
Paired Samples
When small samples from suspected
nonnormal populations are used, a
nonparametric technique is required. The
Wilcoxon test is used for such situations
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Wilcoxon Signed Rank Test for
Paired Samples
• Determine the difference for each sample.
• Arrange the absolute value of these
differences in order, assigning a rank to
each.
• T+ = sum of ranks having a positive value
and T- = sum of ranks having a negative
value
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Wilcoxon Signed Rank Test for
Paired Samples (small n)
Ho: The population differences are centered at 0
Two-Sided Test
Ha: the population differences are not centered at 0
Using the two-sided value from Table A.11,
Reject Ho if T  table value,
where T = the minimum of T+ and T-.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Wilcoxon Signed Rank Test for
Paired Samples (small n)
Ho: The population differences are centered at 0
One-Sided Test
Ha:the population differences
are centered at a value >0
Using the one-sided value from
Table A.11,
Reject Ho if T-  table value.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Ha:the population differences
are centered at a value < 0
Using the one-sided value from
Table A.11,
Reject Ho if T-  table value.
Wilcoxon Signed Rank Test for
Paired Samples (Large n)
T   T
Z
 T
 T
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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n(n  1)

4
 T 
n(n  1)(2n 1)
24
Wilcoxon Signed Rank Test for
Paired Samples (Large n)
Ho: The population differences are centered at 0
Two-Sided Test
Ha: the population differences are not centered at 0
Reject Ho if |Z| > Z/2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Wilcoxon Signed Rank Test for
Paired Samples (Large n)
Ho: The population differences are centered at 0
One-Sided Test
Ha:the population differences
are centered at a value >0
Reject Ho if Z > Z
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Ha:the population differences
are centered at a value < 0
Reject Ho if Z < -Z
Wilcoxon Signed Rank Test for
Paired Samples
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Kruskal-Wallis Test
The nonparametric counterpart to the oneway ANOVA test.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Kruskal-Wallis Test
Ho: the k populations have identical probability distributions
Ha: at least two of the populations differ in location
k T2
12
i
KW 

 3(n  1)
n(n  1) i1 n i
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Reject Ho if KW > 2.df
p-value for KW
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Kruskal-Wallis Test
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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The Friedman Test
The nonparametric counterpart to the
randomized block ANOVA test.
k
12
2
FR 
 Ti  3b( k 1)
bk( k 1) i 1
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Reject Ho if FR > 2.df
The Friedman Test
Ho: the k populations have identical probability distributions
Ha: at least two of the populations differ in location
k
12
FR 
 Ti 2  3b( k 1)
bk( k 1) i 1
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Spearman’s Rank Correlation
Spearman’s is the nonparametric
counterpart to the Pearson Correlation
rs 
 R( x)R( y)  [ R( x)][ R( y)]/ n
 R2 ( x)  [ R( x)]2 / n  R( y)] [ R( y)]2 / n
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Spearman’s Rank Correlation
rs 
 R( x)R( y)  [ R( x)][ R( y)]/ n
 R2 ( x)  [ R( x)]2 / n  R( y)] [ R( y)]2 / n
If there are no ties then a shortcut equation may be used.0
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
6d 2
r s  1
n(n2 1)
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
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Spearman’s Rank Correlation
Ho: No association between X and Y exists
Ha: An association between X and Y exists
Two-Sided Test
Determine rs
Use Table A.12 for /2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Reject Ho if |rs| > table value
Spearman’s Rank Correlation
Ho: No association between X and Y exists
One-Sided Test
Ha: A positive association
between X and Y exists
Determine rs
Use Table A.12 for 
Reject Ho if rs > table value
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur
(c)2000 South-Western College
Publishing
Ha: A negative association
between X and Y exists
Determine rs
Use Table A.12 for 
Reject Ho if rs < table value