CD-ROM Chapter 15 Introduction to Nonparametric Statistics.
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Transcript of CD-ROM Chapter 15 Introduction to Nonparametric Statistics.
CD-ROM Chapter 15CD-ROM Chapter 15
Introduction to Introduction to Nonparametric Nonparametric
StatisticsStatistics
Chapter 15 - Chapter 15 - Chapter Chapter OutcomesOutcomesAfter 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 StatisticsNonparametric Statistics
Nonparametric statistical Nonparametric statistical proceduresprocedures are those statistical methods that do not concern themselves with population distributions and/or parameters.
The Runs TestThe Runs Test
The runs testruns 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 TestThe Runs Test
A runrun 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 TestThe Runs Test(Table 15-1)(Table 15-1)
Sequence Number Code Sequence Number Code1 0.34561 - 11 0.67201 +2 0.42789 - 12 0.23790 -3 0.36925 - 13 0.24509 -4 0.89563 + 14 0.01467 -5 0.25679 - 15 0.78345 +6 0.92001 + 16 0.69112 +7 0.58345 + 17 0.46023 -8 0.23114 - 18 0.38633 -9 0.12672 - 19 0.60914 +
10 0.88569 + 20 0.95234 +
The Runs TestThe Runs Test(Small Sample Example)(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 runsFrom runs table (Appendix K) with n1 = 9 and n2 = 11, the
critical value of r is 6
The Runs TestThe Runs Test(Small Sample Example)(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 HReject H00Reject HReject H00Do not reject HDo not reject H00
Decision:
Since r = 10, we do not reject the null hypothesis.
Large Sample Runs TestLarge Sample Runs Test
MEAN AND STANDARD DEVIATION FOR MEAN AND STANDARD DEVIATION FOR rr
where:n1 = Number of occurrences of first type
n2 = Number of occurrences of second type
12
21
21
nn
nnr
)1()(
)2)(2(
212
21
212121
nnnn
nnnnnnr
Large Sample Runs TestLarge Sample Runs Test
TEST STATISTIC FOR LARGE TEST STATISTIC FOR LARGE SAMPLE RUNS TESTSAMPLE RUNS TEST
r
rrz
Large Sample Runs TestLarge Sample Runs Test(Example 15-2)(Example 15-2)
OOOUOOUOUUOOUUOOOOUUOUUOOO
UUUOOOOUUOOUUUOUUOOUUUUU
OOOUOUUOOOUOOOOUUUOUUOOOU
OOUUOUOOUUUOUUOOOOUUUOOO
Table 15-2
n1 = 53 “O’s” n2 = 47 “U’s”
r = 45 runs
96.1025. z0
Large Sample Runs Test Large Sample Runs Test (Example 15-2)(Example 15-2)
Rejection Region /2 = 0.025
Since z= -1.174 > -1.96 and < 1.96, we do not reject H0,
96.1025. z
Rejection Region /2 = 0.025
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
174.195659.4
82.5045
r
rrz
Mann-Whitney U TestMann-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 TestMann-Whitney U Test
U-STATISTICSU-STATISTICS
where:n1 and n2 are the two sample sizes
R1 and R2 = Sum of ranks for samples
1 and 2
111
211 2
)1(R
nnnnU
222
212 2
)1(R
nnnnU
Mann-Whitney U TestMann-Whitney U Test- Large Samples -- Large Samples -
MEAN AND STANDARD DEVIATION FOR MEAN AND STANDARD DEVIATION FOR THE THE UU-STATISTIC-STATISTIC
where:n1 and n2 = Sample sizes from
populations 1 and 2
221nn
12
)1)()(( 2121
nnnn
Mann-Whitney U TestMann-Whitney U Test- Large Samples -- Large Samples -
MANN-WHITNEY U-TEST STATISTICMANN-WHITNEY U-TEST STATISTIC
12)1)()((
2
2121
21
nnnn
nnU
z
0~~21
Mann-Whitney U TestMann-Whitney U Test(Example 15-4)(Example 15-4)
Since z= -1.027 > -1.645, we do not reject H0,
645.1z
Rejection Region = 0.05
05.0
0~~:
0~~:
21
210
AH
H
027.1
12)1404144)(404)(144(
088,29412,27
12)1)()((
2
2121
21
nnnn
nnU
z
Wilcoxon Matched-Pairs Wilcoxon Matched-Pairs TestTest
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 Wilcoxon Matched-Pairs TestTest
WILCOXON MEAN AND STANDARD WILCOXON MEAN AND STANDARD DEVIATIONDEVIATION
where:n = Number of paired values
4
)1(
nn
24
)12)(1(
nnn
Wilcoxon Matched-Pairs Wilcoxon Matched-Pairs TestTest
WILCOXON TEST STATISTICWILCOXON TEST STATISTIC
24)12)(1(
4)1(
nnn
nnT
z
Kruskal-Wallis One-Way Kruskal-Wallis One-Way Analysis of VarianceAnalysis 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 Kruskal-Wallis One-Way Analysis of VarianceAnalysis of Variance
H-STATISTICH-STATISTIC
where:N = Sum of sample sizes in all samplesk = Number of samplesRi = Sum of ranks in the ith sample
ni = Size of the ith sample
1),1(3)1(
12
1
2
kdfwithNn
R
NNH
k
i i
i
Kruskal-Wallis One-Way Kruskal-Wallis One-Way Analysis of VarianceAnalysis of Variance
CORRECTION FOR TIED RANKINGSCORRECTION FOR TIED RANKINGS
where:g = Number of different groups of tiesti = Number of tied observations in the
ith tied group of scoresN = Total number of observations
NN
ttg
iii
31
3 )(1
Kruskal-Wallis One-Way Kruskal-Wallis One-Way Analysis of VarianceAnalysis of Variance
H-STATISTIC CORRECTED FOR TIED H-STATISTIC CORRECTED FOR TIED RANKINGSRANKINGS
NN
tt
NnR
NNH g
iii
k
i i
i
31
3
1
2
)(1
)1(3)1(
12
Key TermsKey Terms
• Kruskal-Wallis One-Way Analysis of Variance
• Mann-Whitney U Test
• Nonparametric Statistical Procedure
• Run
• Runs Test
• Wilcoxon Test