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© 2005 Thomson/South-Western© 2005 Thomson/South-Western
AK/ECON 3480 M & NAK/ECON 3480 M & NWINTER 2006WINTER 2006
Power Point Presentation Power Point Presentation
Professor Ying KongProfessor Ying Kong
School of Analytic Studies and Information School of Analytic Studies and Information TechnologyTechnology
Atkinson Faculty of Liberal and Professional Atkinson Faculty of Liberal and Professional StudiesStudies
York UniversityYork University
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© 2005 Thomson/South-Western© 2005 Thomson/South-Western
Chapter 19Chapter 19Nonparametric Methods Nonparametric Methods
Sign TestSign Test Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test Kruskal-Wallis TestKruskal-Wallis Test Rank CorrelationRank Correlation
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Nonparametric MethodsNonparametric Methods
Most of the statistical methods referred to as parametric Most of the statistical methods referred to as parametric require the use of require the use of intervalinterval- or - or ratio-scaled dataratio-scaled data..
Nonparametric methods are often the only way Nonparametric methods are often the only way to analyze to analyze nominalnominal or or ordinal dataordinal data and draw and draw statistical conclusions.statistical conclusions.
Nonparametric methods require no assumptions Nonparametric methods require no assumptions about the population probability distributions.about the population probability distributions.
Nonparametric methods are often called Nonparametric methods are often called distribution-free methodsdistribution-free methods..
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Nonparametric MethodsNonparametric Methods
In general, for a statistical method to be In general, for a statistical method to be classified as nonparametric, it must satisfy at classified as nonparametric, it must satisfy at least one of the following conditions.least one of the following conditions.
• The method can be used with nominal data.The method can be used with nominal data.
• The method can be used with ordinal data.The method can be used with ordinal data.
• The method can be used with interval or The method can be used with interval or ratio data when no assumption can be made ratio data when no assumption can be made about the population probability distribution.about the population probability distribution.
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Sign TestSign Test
A common application of the A common application of the sign testsign test involves involves using a sample of using a sample of n n potential customers to identify potential customers to identify a preference for one of two brands of a product.a preference for one of two brands of a product.
The objective is to determine whether there is The objective is to determine whether there is a difference in preference between the two a difference in preference between the two items being compared.items being compared.
To record the preference data, we use a plus sign To record the preference data, we use a plus sign if the individual prefers one brand and a minus if the individual prefers one brand and a minus sign if the individual prefers the other brand.sign if the individual prefers the other brand.
Because the data are recorded as plus and Because the data are recorded as plus and minus signs, this test is called the sign test.minus signs, this test is called the sign test.
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Sign Test: Small-Sample CaseSign Test: Small-Sample Case
The small-sample case for the sign test should The small-sample case for the sign test should be used whenever be used whenever nn << 20. 20.
The hypotheses areThe hypotheses are
a : .50H pa : .50H p
0 : .50H p0 : .50H p
A preference for one brandA preference for one brandover the other exists.over the other exists.
No preference for one brandNo preference for one brandover the other exists.over the other exists.
The number of plus signs is our test statistic.The number of plus signs is our test statistic. Assuming Assuming HH00 is true, the sampling distribution for the is true, the sampling distribution for the
test statistic is a binomial distribution with test statistic is a binomial distribution with pp = .5. = .5.
HH00 is rejected if the is rejected if the pp-value -value << level of significance, level of significance, ..
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Sign Test: Large-Sample CaseSign Test: Large-Sample Case
Using Using HH00: : pp = .5 and = .5 and nn > 20, the sampling > 20, the sampling distribution for the number of plus signs can distribution for the number of plus signs can be approximated by a normal distribution.be approximated by a normal distribution.
When no preference is stated (When no preference is stated (HH00: : pp = .5), the = .5), the sampling distribution will have:sampling distribution will have:
The test statistic is:The test statistic is:
HH00 is rejected if the is rejected if the pp-value -value << level of significance, level of significance, ..
Mean: Mean: = .50 = .50nnStandard Deviation: Standard Deviation: .25n .25n
xz
x
z
((xx is the number is the number of plus signs)of plus signs)
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Example: Ketchup Taste TestExample: Ketchup Taste Test
Sign Test: Large-Sample CaseSign Test: Large-Sample Case
AAAA BBBB
As part of a market research study, aAs part of a market research study, a
sample of 36 consumers were asked to tastesample of 36 consumers were asked to taste
two brands of ketchup and indicate a two brands of ketchup and indicate a
preference. Do the data shown on the nextpreference. Do the data shown on the next
slide indicate a significant difference in theslide indicate a significant difference in the
consumer preferences for the two brands?consumer preferences for the two brands?
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1818 preferred Brand A Ketchuppreferred Brand A Ketchup (+ sign recorded)(+ sign recorded)
1212 preferred Brand B Ketchuppreferred Brand B Ketchup ((__ sign recorded) sign recorded) 6 had no preference6 had no preference
Sign Test: Large-Sample CaseSign Test: Large-Sample Case
Example: Ketchup Taste TestExample: Ketchup Taste Test
AAAA BBBBThe analysis will be based The analysis will be based
onon
a sample size of 18 + 12 = a sample size of 18 + 12 = 30.30.
The analysis will be based The analysis will be based onon
a sample size of 18 + 12 = a sample size of 18 + 12 = 30.30.
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HypothesesHypotheses
a : .50H pa : .50H p
AAAA BBBB
Sign Test: Large-Sample CaseSign Test: Large-Sample Case
0 : .50H p0 : .50H p
A preference for one brand over the other existsA preference for one brand over the other exists
No preference for one brand over the other existsNo preference for one brand over the other exists
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Sampling Distribution for Number of Plus SignsSampling Distribution for Number of Plus Signs
= .5(30) = 15= .5(30) = 15
AAAA BBBB
Sign Test: Large-Sample CaseSign Test: Large-Sample Case
.25 .25(30) 2.74n .25 .25(30) 2.74n
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Rejection RuleRejection Rule AAAA BBBB
Sign Test: Large-Sample CaseSign Test: Large-Sample Case
pp-Value = 2(.5000 - .3643) = .2714-Value = 2(.5000 - .3643) = .2714
pp-Value-Value
z z = ( = (xx – – )/)/ = (18 - 15)/2.74 = 3/2.74 = 1.10 = (18 - 15)/2.74 = 3/2.74 = 1.10
Test StatisticTest Statistic
Using .05 level of significance:Using .05 level of significance:
Reject Reject HH00 if if pp-value -value << .05 .05
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AAAA BBBB
Sign Test: Large-Sample CaseSign Test: Large-Sample Case
ConclusionConclusion
Because the Because the pp-value > -value > , we cannot reject , we cannot reject HH00. There is insufficient evidence in the sample . There is insufficient evidence in the sample to conclude that a difference in preference exists to conclude that a difference in preference exists for the two brands of ketchup. for the two brands of ketchup.
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Hypothesis Test About a MedianHypothesis Test About a Median
We can apply the sign test by:We can apply the sign test by:• Using a plus sign whenever the data in the sample Using a plus sign whenever the data in the sample
are above the hypothesized value of the medianare above the hypothesized value of the median
• Using a minus sign whenever the data in Using a minus sign whenever the data in the sample are below the hypothesized the sample are below the hypothesized value of the medianvalue of the median
• Discarding any data exactly equal to the Discarding any data exactly equal to the hypothesized medianhypothesized median
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Hypothesis Test About a MedianHypothesis Test About a Median
34 years34 yearsHH00: Median Age: Median Age34 years34 yearsHHaa: Median Age: Median Age
Example: Trim Fitness CenterExample: Trim Fitness Center
A hypothesis test is being conductedA hypothesis test is being conducted
about the median age of female membersabout the median age of female members
of the Trimof the Trim Fitness Center. Fitness Center.
In a sample of 40 female members, 25 are olderIn a sample of 40 female members, 25 are older
than 34, 14 are younger than 34, and 1 is 34. than 34, 14 are younger than 34, and 1 is 34. Is thereIs there
sufficient evidence to reject sufficient evidence to reject HH00? Assume ? Assume = .05. = .05.
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pp-Value = 2(.5000 -Value = 2(.5000 .4608) = .0784.4608) = .0784
= .5(39) = 19.5= .5(39) = 19.5
.25 .25(39) 3.12n .25 .25(39) 3.12n
Hypothesis Test About a MedianHypothesis Test About a Median
pp-Value-Value
z z = ( = (xx – – )/)/ = (25 – 19.5)/3.12 = 1.76 = (25 – 19.5)/3.12 = 1.76
Test StatisticTest Statistic
Mean and Standard DeviationMean and Standard Deviation
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Hypothesis Test About a MedianHypothesis Test About a Median
Rejection RuleRejection Rule
ConclusionConclusion
Do not reject Do not reject HH00. The . The pp-value for this two-tail test -value for this two-tail test is .0784. There is insufficient evidence in the is .0784. There is insufficient evidence in the sample to conclude that the median age is sample to conclude that the median age is notnot 34 34 for female members of Trim for female members of Trim Fitness Center.Fitness Center.
Using .05 level of significance:Using .05 level of significance:
Reject Reject HH00 if if pp-value -value << .05 .05
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Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
This test is the nonparametric alternative to This test is the nonparametric alternative to the parametric matched-sample test the parametric matched-sample test presented in Chapter 10.presented in Chapter 10.
The methodology of the parametric matched-The methodology of the parametric matched-sample analysis requires:sample analysis requires:• interval data, andinterval data, and• the assumption that the population of the assumption that the population of
differences between the pairs of differences between the pairs of observations is normally distributed.observations is normally distributed.
If the assumption of normally distributed If the assumption of normally distributed differences is not appropriate, the Wilcoxon differences is not appropriate, the Wilcoxon signed-rank test can be used.signed-rank test can be used.
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Example: Express DeliveriesExample: Express Deliveries
Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
A firm has decided to select oneA firm has decided to select one
of two express delivery services toof two express delivery services to
provide next-day deliveries to itsprovide next-day deliveries to its
district offices.district offices. To test the delivery times of the two services, theTo test the delivery times of the two services, the
firm sends two reports to a sample of 10 district firm sends two reports to a sample of 10 district
offices, with one report carried by one service and theoffices, with one report carried by one service and the
other report carried by the second service. Do the dataother report carried by the second service. Do the data
on the next slide indicate a difference in the twoon the next slide indicate a difference in the two
services?services?
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Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
SeattleSeattleLos AngelesLos Angeles
BostonBostonClevelandClevelandNew YorkNew YorkHoustonHoustonAtlantaAtlantaSt. LouisSt. LouisMilwaukeeMilwaukeeDenverDenver
32 hrs.32 hrs.3030191916161515181814141010 771616
25 hrs.25 hrs.242415151515131315151515 88 991111
District OfficeDistrict Office OverNightOverNight NiteFliteNiteFlite
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Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
Preliminary Steps of the TestPreliminary Steps of the Test• Compute the differences between the Compute the differences between the
paired observations.paired observations.• Discard any differences of zero.Discard any differences of zero.• Rank the absolute value of the differences Rank the absolute value of the differences
from lowest to highest. Tied differences from lowest to highest. Tied differences are assigned the average ranking of their are assigned the average ranking of their positions.positions.• Give the ranks the sign of the original Give the ranks the sign of the original difference in the data.difference in the data.
• Sum the signed ranks.Sum the signed ranks.. . . next we will determine whether the. . . next we will determine whether the
sum is significantly different from zero.sum is significantly different from zero.
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Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
SeattleSeattleLos AngelesLos Angeles
BostonBostonClevelandClevelandNew YorkNew YorkHoustonHoustonAtlantaAtlantaSt. LouisSt. LouisMilwaukeeMilwaukeeDenverDenver
77 66 44 11 22 3311 2222 55
District OfficeDistrict Office Differ.Differ. |Diff.| Rank Sign. Rank |Diff.| Rank Sign. Rank
10109977
1.51.54466
1.51.54488
+10+10+9+9+7+7
+1.5+1.5+4+4+6+61.51.5+4+4+8+8
+44+44
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Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
HypothesesHypothesesHH00: The delivery times of the two services are the : The delivery times of the two services are the
same; neither offers faster service than the other.same; neither offers faster service than the other.
HHaa: Delivery times differ between the two services; : Delivery times differ between the two services;
recommend the one with the smaller times.recommend the one with the smaller times.
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Sampling Distribution of Sampling Distribution of TT for Identical Populations for Identical Populations
TT = 0 = 0
( 1)(2 1) 10(11)(21)19.62
6 6T
n n n
( 1)(2 1) 10(11)(21)19.62
6 6T
n n n
Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
TT
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Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
Rejection RuleRejection Rule
Using .05 level of significance,Using .05 level of significance,
Reject Reject HH00 if if pp-value -value << .05 .05 Test StatisticTest Statistic
pp-Value-Value
zz = ( = (TT - - T T )/)/TT = (44 - 0)/19.62 = 2.24 = (44 - 0)/19.62 = 2.24
pp-Value = 2(.5000 - .4875) = .025-Value = 2(.5000 - .4875) = .025
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ConclusionConclusion
Reject Reject HH00. The . The pp-value for this two-tail -value for this two-tail test is .025. There is sufficient evidence in the test is .025. There is sufficient evidence in the sample to conclude that a difference exists in sample to conclude that a difference exists in the delivery times provided by the two services. the delivery times provided by the two services.
Wilcoxon Signed-Rank TestWilcoxon Signed-Rank Test
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Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
This test is another nonparametric method for This test is another nonparametric method for determining whether there is a difference determining whether there is a difference between two populations.between two populations.
This test, unlike the Wilcoxon signed-rank test, This test, unlike the Wilcoxon signed-rank test, is is notnot based on a matched sample. based on a matched sample.
This test does This test does notnot require interval data or the require interval data or the assumption that both populations are normally assumption that both populations are normally distributed.distributed.
The only requirement is that the measurement The only requirement is that the measurement scale for the data is at least ordinal.scale for the data is at least ordinal.
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Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
HHaa: The two populations are not identical: The two populations are not identicalHH00: The two populations are identical: The two populations are identical
Instead of testing for the difference between the Instead of testing for the difference between the means of two populations, this method tests to means of two populations, this method tests to determine whether the two populations are identical.determine whether the two populations are identical.
The hypotheses are:The hypotheses are:
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Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
Example: Westin FreezersExample: Westin FreezersManufacturer labels indicate theManufacturer labels indicate the
annual energy cost associated withannual energy cost associated with
operating home appliances such asoperating home appliances such as
freezers.freezers.
The energy costs for a sample ofThe energy costs for a sample of
10 Westin freezers and a sample of 1010 Westin freezers and a sample of 10
Easton Freezers are shown on the next slide. Do theEaston Freezers are shown on the next slide. Do the
data indicate, using data indicate, using = .05, that a difference exists in = .05, that a difference exists in
the annual energy costs for the two brands of freezers?the annual energy costs for the two brands of freezers?
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Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
$55.10 $55.10
54.5054.50 53.2053.20 53.0053.00 55.5055.50 54.9054.90 55.8055.80 54.0054.00 54.2054.20 55.2055.20
$56.10 $56.10
54.7054.70 54.4054.40 55.4055.40 54.1054.10 56.0056.00 55.5055.50 55.0055.00 54.3054.30 57.0057.00
Westin FreezersWestin FreezersEaston FreezersEaston Freezers
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HypothesesHypotheses
Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
HHaa: Annual energy costs differ for : Annual energy costs differ for
the twothe two
brands of freezers.brands of freezers.
HH00: Annual energy costs for Westin freezers: Annual energy costs for Westin freezers
and Easton freezers are the same.and Easton freezers are the same.
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Mann-Whitney-Wilcoxon Test:Mann-Whitney-Wilcoxon Test:Large-Sample CaseLarge-Sample Case
First, rank the First, rank the combinedcombined data from the lowest to data from the lowest to
the highest values, with tied values being the highest values, with tied values being assigned the average of the tied rankings.assigned the average of the tied rankings.
Then, compute Then, compute TT, the sum of the ranks for the , the sum of the ranks for the first sample.first sample.
Then, compare the observed value of Then, compare the observed value of TT to the to the sampling distribution of sampling distribution of TT for identical populations. for identical populations. The value of the standardized test statistic The value of the standardized test statistic zz will will provide the basis for deciding whether to reject provide the basis for deciding whether to reject HH00..
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Mann-Whitney-Wilcoxon Test:Mann-Whitney-Wilcoxon Test:Large-Sample CaseLarge-Sample Case
1 2 1 21 ( 1)12T n n n n 1 2 1 21 ( 1)12T n n n n
Approximately normal, providedApproximately normal, provided
nn11 >> 10 and 10 and nn22 >> 10 10
TT = = nn11((nn11 + + nn22 + 1) + 1)
Sampling Distribution of Sampling Distribution of TT for Identical Populations for Identical Populations
• MeanMean
• Standard DeviationStandard Deviation
• Distribution FormDistribution Form
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Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
$55.10 $55.10
54.5054.50 53.2053.20 53.0053.00 55.5055.50 54.9054.90 55.8055.80 54.0054.00 54.2054.20 55.2055.20
$56.10 $56.10
54.7054.70 54.4054.40 55.4055.40 54.1054.10 56.0056.00 55.5055.50 55.0055.00 54.3054.30 57.0057.00
Westin FreezersWestin Freezers Easton FreezersEaston Freezers
Sum of RanksSum of Ranks Sum of RanksSum of Ranks
RankRank RankRank
86.586.5 123.5123.5
1122
121288
15.15.551010
17173355
1313
19199977
141444
181815.15.55111166
2020
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Sampling Distribution of Sampling Distribution of TT for Identical Populations for Identical Populations
TT = ½(10)(21) = 105 = ½(10)(21) = 105
Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
1 2 1 21 ( 1)12
1 (10)(10)(21)12 13.23
T n n n n
1 2 1 21 ( 1)12
1 (10)(10)(21)12 13.23
T n n n n
TT
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Rejection RuleRejection Rule
Using .05 level of significance,Using .05 level of significance,
Reject Reject HH00 if if pp-value -value << .05 .05 Test StatisticTest Statistic
pp-Value-Value
zz = ( = (TT - - T T )/)/TT = (86.5 = (86.5 105)/13.23 = -1.40 105)/13.23 = -1.40
pp-Value = 2(.5000 - .4192) = .1616-Value = 2(.5000 - .4192) = .1616
Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
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Mann-Whitney-Wilcoxon TestMann-Whitney-Wilcoxon Test
ConclusionConclusionDo not reject Do not reject HH00. The . The pp-value > -value > . There is . There is
insufficient evidence in the sample data to conclude insufficient evidence in the sample data to conclude that there is a difference in the annual energy cost that there is a difference in the annual energy cost associated with the two brands of freezers.associated with the two brands of freezers.
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Kruskal-Wallis TestKruskal-Wallis Test
The Mann-Whitney-Wilcoxon test has been The Mann-Whitney-Wilcoxon test has been extended by Kruskal and Wallis for cases of extended by Kruskal and Wallis for cases of three or more populations.three or more populations.
The Kruskal-Wallis test can be used with ordinal The Kruskal-Wallis test can be used with ordinal data as well as with interval or ratio data.data as well as with interval or ratio data.
Also, the Kruskal-Wallis test does not require the Also, the Kruskal-Wallis test does not require the assumption of normally distributed populations.assumption of normally distributed populations.
HHaa: Not all populations are identical: Not all populations are identicalHH00: All populations are identical: All populations are identical
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Test StatisticTest Statistic
Kruskal-Wallis TestKruskal-Wallis Test
2
1
123( 1)
( 1)
ki
TiT T i
RW n
n n n
2
1
123( 1)
( 1)
ki
TiT T i
RW n
n n n
where: where: kk = number of populations = number of populations
nnii = number of items in sample = number of items in sample ii
nnTT = = nnii = total number of items in all samples = total number of items in all samples
RRii = sum of the ranks for sample = sum of the ranks for sample ii
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Kruskal-Wallis TestKruskal-Wallis Test
When the populations are identical, the When the populations are identical, the sampling distribution of the test statistic sampling distribution of the test statistic WW can can be approximated by a chi-square distribution be approximated by a chi-square distribution with with kk – 1 degrees of freedom. – 1 degrees of freedom.
This approximation is acceptable if each of the This approximation is acceptable if each of the sample sizes sample sizes nnii is is >> 5. 5.
The rejection rule is: The rejection rule is: Reject Reject HH00 if if pp-value -value <<
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Rank CorrelationRank Correlation
The Pearson correlation coefficient, The Pearson correlation coefficient, rr, is a measure of , is a measure of the linear association between two variables for the linear association between two variables for which interval or ratio data are available.which interval or ratio data are available.
The The Spearman rank-correlation coefficientSpearman rank-correlation coefficient, , rrs s , , is a measure of association between two is a measure of association between two variables when only ordinal data are available.variables when only ordinal data are available.
Values of Values of rrss can range from –1.0 to +1.0, where can range from –1.0 to +1.0, where
• values near 1.0 indicate a strong positive values near 1.0 indicate a strong positive association between the rankings, andassociation between the rankings, and
• values near -1.0 indicate a strong negative values near -1.0 indicate a strong negative association between the rankings.association between the rankings.
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Rank CorrelationRank Correlation
Spearman Rank-Correlation Coefficient, Spearman Rank-Correlation Coefficient, rrss
2
2
61
( 1)i
s
dr
n n
2
2
61
( 1)i
s
dr
n n
where: where: nn = number of items being ranked = number of items being ranked
xxii = rank of item = rank of item ii with respect to one variable with respect to one variable
yyii = rank of item = rank of item ii with respect to a second variable with respect to a second variable
ddii = = xxii - - yyii
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Test for Significant Rank CorrelationTest for Significant Rank Correlation
0 : 0sH p 0 : 0sH p
a : 0sH p a : 0sH p
We may want to use sample results to make an We may want to use sample results to make an inference about the population rank correlation inference about the population rank correlation ppss..
To do so, we must test the hypotheses:To do so, we must test the hypotheses:
(No rank correlation exists)(No rank correlation exists)
(Rank correlation exists)(Rank correlation exists)
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Rank CorrelationRank Correlation
0sr
0sr
11sr n
1
1sr n
Approximately normal, provided Approximately normal, provided nn >> 10 10
Sampling Distribution ofSampling Distribution of rrss when when ppss = 0 = 0
• MeanMean
• Standard DeviationStandard Deviation
• Distribution FormDistribution Form
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Rank CorrelationRank Correlation
Example: Crennor InvestorsExample: Crennor Investors Crennor Investors provides Crennor Investors provides
a portfolio management servicea portfolio management service
for its clients. Two of Crennor’sfor its clients. Two of Crennor’s
analysts ranked ten investmentsanalysts ranked ten investments
as shown on the next slide. Useas shown on the next slide. Use
rank correlation, with rank correlation, with = .10, to = .10, to
comment on the agreement of the two analysts’comment on the agreement of the two analysts’
rankings.rankings.
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Rank CorrelationRank Correlation
Analyst #2Analyst #2 1 5 6 2 9 7 3 10 4 81 5 6 2 9 7 3 10 4 8
Analyst #1Analyst #1 1 4 9 8 6 3 5 7 2 101 4 9 8 6 3 5 7 2 10
InvestmentInvestment A B C D E F G H I JA B C D E F G H I J
Example: Crennor InvestorsExample: Crennor Investors
0 : 0sH p 0 : 0sH p
a : 0sH p a : 0sH p (No rank correlation exists)(No rank correlation exists)
(Rank correlation exists)(Rank correlation exists)
• Analysts’ RankingsAnalysts’ Rankings
• HypothesesHypotheses
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Rank CorrelationRank Correlation
AABBCCDDEEFFGGHHIIJJ
114499886633557722
1010
11556622997733
10104488
00-1-13366-3-3-4-422-3-3-2-222
001199
363699
161644994444
Sum =Sum = 9292
InvestmentInvestmentAnalyst #1Analyst #1
RankingRankingAnalyst #2Analyst #2
RankingRanking Differ.Differ. (Differ.(Differ.))22
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Sampling Distribution of rs
Assuming No Rank Correlation
Rank CorrelationRank Correlation
1.333
10 1sr
1
.33310 1sr
rr = 0 = 0rrss
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Test StatisticTest Statistic2
2
6 6(92)1 1 0.4424
( 1) 10(100 1)i
s
dr
n n
2
2
6 6(92)1 1 0.4424
( 1) 10(100 1)i
s
dr
n n
Rank CorrelationRank Correlation
zz = ( = (rrss - - r r )/)/rr = (.4424 - 0)/.3333 = 1.33 = (.4424 - 0)/.3333 = 1.33
Rejection RuleRejection Rule
With .10 level of significance:With .10 level of significance:
Reject Reject HH00 if if pp-value -value << .10 .10
pp-Value-Value
pp-Value = 2(.5000 - .4082) = .1836-Value = 2(.5000 - .4082) = .1836
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Do no reject Do no reject HH00. The . The pp-value > -value > . There is . There is
not a significant rank correlation. The two analysts not a significant rank correlation. The two analysts are not showing agreement in their ranking of the are not showing agreement in their ranking of the risk associated with the different investments.risk associated with the different investments.
Rank CorrelationRank Correlation
ConclusionConclusion
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End of Chapter 19End of Chapter 19