Download The Runs Test

CD-ROM Chapter 15
Introduction to
Nonparametric
Statistics
Chapter 15 - Chapter Outcomes
After studying the material in this chapter, you
should be able to:
Recognize
when and how to use the runs test and
testing for randomness.
Know when and how to perform a Mann-Whitney
U test.
Recognize the situations for which the Wilcoxon
signed rank test applies and be able to use it in a
decision-making context.
Perform nonparametric analysis of variance using
the Kruskal-Wallis one-way ANOVA.
Nonparametric Statistics
Nonparametric statistical procedures are
those statistical methods that do not
concern themselves with population
distributions and/or parameters.
The Runs Test
The runs test is a statistical procedure
used to determine whether the pattern of
occurrences of two types of observations
is determined by a random process.
The Runs Test
A run is a succession of occurrences of a
certain type preceded and followed by
occurrences of the alternate type or by no
occurrences at all.
The Runs Test
(Table 15-1)
Sequence
1
2
3
4
5
6
7
8
9
10
Number
0.34561
0.42789
0.36925
0.89563
0.25679
0.92001
0.58345
0.23114
0.12672
0.88569
Code
+
+
+
+
Sequence Number
11
0.67201
12
0.23790
13
0.24509
14
0.01467
15
0.78345
16
0.69112
17
0.46023
18
0.38633
19
0.60914
20
0.95234
Code
+
+
+
+
+
The Runs Test
(Small Sample Example)
H0: Computer-generated numbers are random
between 0.0 and 1.0.
HA: Computer-generated numbers are not random .
--- + - ++ -- ++ --- ++ -- ++
Runs: 1
2 3 4
5
6
7
8
9
10
There are r = 10 runs
From runs table (Appendix K) with n1 = 9
and n2 = 11, the critical value of r is 6
The Runs Test
(Small Sample Example)
Test Statistic:
r = 10 runs
Critical Values from Runs Table:
Possible
Runs:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Reject H0
Do not reject H0
Reject H0
Decision:
Since r = 10, we do not reject the null hypothesis.
Large Sample Runs Test
MEAN AND STANDARD DEVIATION FOR r
2n1n2
r 
1
n1  n2
(2n1n2 )( 2n1n2  n1  n2 )
r 
2
(n1  n2 ) (n1  n2  1)
where:
n1 = Number of occurrences of first type
n2 = Number of occurrences of second type
Large Sample Runs Test
TEST STATISTIC FOR LARGE
SAMPLE RUNS TEST
z
r  r
r
Large Sample Runs Test
(Example 15-2)
Table 15-2
OOOUOOUOUUOOUUOOOOUUOUUOOO
UUUOOOOUUOOUUUOUUOOUUUUU
OOOUOUUOOOUOOOOUUUOUUOOOU
OOUUOUOOUUUOUUOOOOUUUOOO
n1 = 53 “O’s”
n2 = 47 “U’s”
r = 45 runs
Large Sample Runs Test
(Example 15-2)
H0: Yogurt fill amounts are randomly distributed above and below 24-ounce level.
H1: Yogurt fill amounts are not randomly distributed above and below 24-ounce
level.
 = 0.05
Rejection Region
 /2 = 0.025
Rejection Region
 /2 = 0.025
z.025  1.96
z
r  r
r
0
z.025  1.96
45  50.82

 1.174
4.95659
Since z= -1.174 > -1.96 and < 1.96, we do not reject H0,
Mann-Whitney U Test
The Mann Whitney U test can be used to compare
two samples from two populations if the
following assumptions are satisfied:
• The two samples are independent and random.
• The value measured is a continuous variable.
• The measurement scale used is at least
ordinal.
• If they differ, the distributions of the two
populations will differ only with respect to
the central location.
Mann-Whitney U Test
U-STATISTICS
n1 (n1  1)
U1  n1n2 
  R1
2
where:
n2 (n2  1)
U 2  n1n2 
  R2
2
n1 and n2 are the two sample sizes
R1 and R2 = Sum of ranks for samples 1 and 2
Mann-Whitney U Test
- Large Samples -
MEAN AND STANDARD DEVIATION FOR
THE U-STATISTIC
n1n2

2
(n1 )( n2 )( n1  n2  1)

12
where:
n1 and n2 = Sample sizes from populations 1 and 2
Mann-Whitney U Test
- Large Samples -
MANN-WHITNEY U-TEST STATISTIC
z
n1n2
U
2
(n1 )( n2 )( n1  n2  1)
12
Mann-Whitney U Test
(Example 15-4)
H 0 : ~1  ~2  0
H A : ~1  ~2  0
  0.05
Rejection Region
 = 0.05
z  1.645
z
n1n2
U
2

(n1 )( n2 )( n1  n2  1)
12
~1  ~2  0
27,412  29,088
 1.027
(144)( 404)(144  404  1)
12
Since z= -1.027 > -1.645, we do not reject H0,
Wilcoxon Matched-Pairs Test
The Wilcoxon matched pairs signed rank test can be
used in those cases where the following
assumptions are satisfied:
• The differences are measured on a continuous
variable.
• The measurement scale used is at least
interval.
• The distribution of the population differences
is symmetric about their median.
Wilcoxon Matched-Pairs Test
WILCOXON MEAN AND STANDARD
DEVIATION
n(n  1)

4
n(n  1)( 2n  1)

24
where:
n = Number of paired values
Wilcoxon Matched-Pairs Test
WILCOXON TEST STATISTIC
z
n(n  1)
T
4
n(n  1)( 2n  1)
24
Kruskal-Wallis One-Way
Analysis of Variance
Kruskal-Wallis one-way analysis of variance can be
used in one-way analysis of variance if the
variables satisfy the following:
• They have a continuous distribution.
• The data are at least ordinal.
• The samples are independent.
• The samples come from populations whose only
possible difference is that at least one may have
a different central location than the others.
Kruskal-Wallis One-Way
Analysis of Variance
H-STATISTIC
k
2
i
R
12
H
 3( N  1), with df  k  1

N ( N  1) i 1 ni
where:
N = Sum of sample sizes in all samples
k = Number of samples
Ri = Sum of ranks in the ith sample
ni = Size of the ith sample
Kruskal-Wallis One-Way
Analysis of Variance
CORRECTION FOR TIED RANKINGS
g
1
 (t
i 1
3
i
 ti )
N N
3
where:
g = Number of different groups of ties
ti = Number of tied observations in the ith tied
group of scores
N = Total number of observations
Kruskal-Wallis One-Way
Analysis of Variance
H-STATISTIC CORRECTED FOR TIED
RANKINGS
2
i
k
H
R
12
 3( N  1)

N ( N  1) i 1 ni
g
1
 (t
i 1
3
i
 ti )
N N
3
Key Terms
• Kruskal-Wallis OneWay Analysis of
Variance
• Mann-Whitney U Test
• Nonparametric
Statistical Procedure
• Run
• Runs Test
• Wilcoxon Test