Nonparametric%20Statistics

1
Nonparametric Statistics
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
2
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
3
Test for RandomnessThe Runs Test
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
4
Test for RandomnessThe Runs Test
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
5
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
6
The Runs Test (Small Samples)
Ho The sequence was generated in a random
manner Ha The sequence was not generated in a
random manner
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
7
Test for RandomnessThe Runs Test
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
8
Runs Test (Large Samples)
Ho The sequence was generated in a random
manner Ha The sequence was not generated in a
random manner
Reject Ho if Z gt Z?/2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
9
Nonparametric Test Central Tendency Two
Populations
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
10
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
11
Mann-Whitney USmall Samples
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
12
Mann-Whitney USmall Samples
Procedure 1. Assume that n1 lt 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 lt ?/2, where U is the minimum of
U1 and U2.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
13
Mann-Whitney USmall Samples
Procedure 1. Assume that n1 lt 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 Ha Population 1 is shifted to
the right of population2 the left of
population 2 Reject Ho if Table A.10 Reject
Ho if Table A.10 value for U is less than ?
value for U is less than ? where U U1.
Where U U2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
14
Mann-Whitney ULarge Samples
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
15
Mann-Whitney ULarge Samples
Ho The two populations have identical
probability distribution Determine
U2 Two-Sided Test Ha The two populations
differ in location Reject Ho if Z gt Z ?/2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
16
Mann-Whitney ULarge Samples
Ho The two populations have identical
probability distribution Determine
U2 One-Sided Test Ha Population 1 is
shifted to Ha Population 1 is shifted to
the right of population 2 the left of
population 2 Reject Ho if Z gt Z?
Reject Ho if Z lt - Z?
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
17
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
18
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
19
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
20
Wilcoxon Signed Rank Test for Paired Samples
(small n)
Ho The population differences are centered at
0 One-Sided Test Hathe population
differences Hathe population differences
are centered at a value gt0 are centered
at a value lt 0 Using the one-sided value from
Using the one-sided value from Table A.11,
Table A.11, Reject Ho if T- ? table value.
Reject Ho if T- ? table value.
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
21
Wilcoxon Signed Rank Test for Paired Samples
(Large n)
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
22
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 gt Z?/2
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
23
Wilcoxon Signed Rank Test for Paired Samples
(Large n)
Ho The population differences are centered at
0 One-Sided Test Hathe population
differences Hathe population differences
are centered at a value gt0 are centered
at a value lt 0 Reject Ho if Z gt Z? Reject
Ho if Z lt -Z?
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
24
Wilcoxon Signed Rank Test for Paired Samples
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
25
Kruskal-Wallis Test

• The nonparametric counterpart to the one-way
  ANOVA test.

Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
26
Kruskal-Wallis Test
Ho the k populations have identical probability
distributions Ha at least two of the populations
differ in location
Reject Ho if KW gt ?2?.df
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
27
p-value for KW
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
28
Kruskal-Wallis Test
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
29
The Friedman Test

• The nonparametric counterpart to the randomized
  block ANOVA test.

Reject Ho if FR gt ?2?.df
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
30
The Friedman Test
Ho the k populations have identical probability
distributions Ha at least two of the populations
differ in location
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
31
Spearmans Rank Correlation

• Spearmans is the nonparametric counterpart to
  the Pearson Correlation

Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
32
Spearmans Rank Correlation
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
33
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
34
Spearmans Rank Correlation
Ho No association between X and Y exists Ha
An association between X and Y exists Two-Side
d Test Determine rs Use Table A.12 for
?/2 Reject Ho if rs gt table value
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing
35
Spearmans Rank Correlation
Ho No association between X and Y
exists One-Sided Test Ha A positive
association Ha A negative association
between X and Y exists between X and Y
exists Determine rs Determine
rs Use Table A.12 for ? Use Table A.12
for ? Reject Ho if rs gt table value
Reject Ho if rs lt table value
Introduction to Business Statistics, 5e
Kvanli/Guynes/Pavur (c)2000 South-Western
College Publishing