1 1 slide nonparametric methods n sign test n wilcoxon signed-rank test n mann-whitney-wilcoxon test...

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Nonparametric 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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Most of the statistical methods referred to as Most of the statistical methods referred to as parametric require the use of parametric require the use of intervalinterval- or - or ratio-ratio-scaled datascaled data..

Nonparametric methods are often the only Nonparametric methods are often the only way to analyze way to analyze nominalnominal or or ordinal dataordinal data and and draw statistical conclusions.draw statistical conclusions.

Nonparametric methods require no Nonparametric methods require no assumptions about the population probability assumptions about the population probability distributions.distributions.

Nonparametric methods are often called Nonparametric methods are often called distribution-free methodsdistribution-free methods..

Nonparametric MethodsNonparametric 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 about the population probability distribution.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 potential customers to identify a preference for one of two brands of a identify a preference for one of two brands of a product.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 To record the preference data, we use a plus sign if the individual prefers one brand and a sign if the individual prefers one brand and a minus sign if the individual prefers the other minus sign if the individual prefers the other brand.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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Example: Peanut Butter Taste TestExample: Peanut Butter Taste Test

Sign Test: Large-Sample CaseSign Test: Large-Sample Case

As part of a market research study, a sample of 36 As part of a market research study, a sample of 36

consumers were asked to taste two brands of peanutconsumers were asked to taste two brands of peanut

butter and indicate a preference. Do the data shownbutter and indicate a preference. Do the data shown

below indicate a significant difference in the consumerbelow indicate a significant difference in the consumer

preferences for the two brands?preferences for the two brands?

18 18 preferred Hoppy Peanut Butter (+ sign recorded) preferred Hoppy Peanut Butter (+ sign recorded)

1212 preferred Pokey Peanut Butter ( preferred Pokey Peanut Butter (__ sign recorded) sign recorded)

66 had no preference had no preference

The analysis is based on a sample size of 18 + 12 = 30.The analysis is based on a sample size of 18 + 12 = 30.

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HypothesesHypotheses

HH00: No preference for one brand over the other : No preference for one brand over the other existsexists

HHaa: A preference for one brand over the other : A preference for one brand over the other

existsexists Sampling DistributionSampling Distribution

2.742.74

Sampling distribution of the number of “+” values if there is no brand preference

Sampling distribution of the number of “+” values if there is no brand preference

= 15 = .5(30)= 15 = .5(30)


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Rejection RuleRejection RuleUsing .05 level of significance,Using .05 level of significance,

Reject Reject HH00 if if zz < -1.96 or < -1.96 or zz > 1.96 > 1.96 Test StatisticTest Statistic

z z = (18 - 15)/2.74 = 3/2.74 = 1.095 = (18 - 15)/2.74 = 3/2.74 = 1.095 ConclusionConclusion

Do not reject Do not reject HH00. There is insufficient . There is insufficient evidence in the sample to conclude that a evidence in the sample to conclude that a difference in preference exists for the two difference in preference exists for the two brands of peanut butter. brands of peanut butter.

Fewer than 10 or more than 20 individuals Fewer than 10 or more than 20 individuals would have to have a preference for a particular would have to have a preference for a particular brand in order for us to reject brand in order for us to reject HH00..

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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


A firm has decided to select one of two express A firm has decided to select one of two express

delivery services to provide next-day deliveries to thedelivery services to provide next-day deliveries to the

district offices.district offices.

To test the delivery times of the two services, the To test the delivery times of the two services, the firmfirm

sends two reports to a sample of 10 district offices, withsends two reports to a sample of 10 district offices, with

one report carried by one service and the other reportone report carried by one service and the other report

carried by the second service.carried by the second service.

Do the data (delivery times in hours) on the nextDo the data (delivery times in hours) on the next

slide indicate a difference in the two services?slide indicate a difference in the two services?

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District OfficeDistrict Office OvernightOvernight NiteFliteNiteFlite

SeattleSeattle 32 hrs. 32 hrs. 25 hrs. 25 hrs.

Los AngelesLos Angeles 3030 2424

BostonBoston 1919 1515

ClevelandCleveland 1616 1515

New YorkNew York 1515 1313

HoustonHouston 1818 1515

AtlantaAtlanta 1414 1515

St. LouisSt. Louis 1010 88

MilwaukeeMilwaukee 77 99

DenverDenver 1616 1111

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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 are from lowest to highest. Tied differences are assigned the average ranking of their 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 sum . . . next we will determine whether the sum is significantly different from zero.is significantly different from zero.

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District Office Differ. |Diff.| Rank District Office Differ. |Diff.| Rank Sign. RankSign. Rank

SeattleSeattle 77 1010 +10+10Los AngelesLos Angeles 66 99 +9+9BostonBoston 44 77 +7+7ClevelandCleveland 11 1.51.5 +1.5+1.5New YorkNew York 22 44 +4+4HoustonHouston 33 66 +6+6AtlantaAtlanta -1-1 1.51.5 -1.5-1.5St. LouisSt. Louis 22 44 +4+4MilwaukeeMilwaukee -2-2 44 -4-4DenverDenver 55 88 +8+8

+44+44

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HH00: The delivery times of the two services are : The delivery times of the two services are the the

same; neither offers faster service than same; neither offers faster service than the other.the other.

HHaa: Delivery times differ between the two : Delivery times differ between the two

services; services;

recommend the one with the recommend the one with the smaller times.smaller times.

Sampling DistributionSampling Distribution

Sampling distribution of T if populations are identical

Sampling distribution of T if populations are identical 19.6219.62

TT = 0 = 0TT


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Rejection RuleRejection Rule

Using .05 level of significance,Using .05 level of significance,


zz = ( = (TT - - T T )/)/TT = (44 - 0)/19.62 = 2.24 = (44 - 0)/19.62 = 2.24 ConclusionConclusion

Reject Reject HH00. There is sufficient evidence in . There is sufficient evidence in the sample to conclude that a difference exists the sample to conclude that a difference exists in the delivery times provided by the two in the delivery times provided by the two services. Recommend using the NiteFlite services. Recommend using the NiteFlite service. service.


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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

Instead of testing for the difference between Instead of testing for the difference between the means of two populations, this method the means of two populations, this method tests to determine whether the two tests to determine whether the two populations are identical.populations are identical.

The hypotheses are:The hypotheses are:

HH00: The two populations are identical: The two populations are identical

HHaa: The two populations are not identical: The two populations are not identical

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Example: Westin FreezersExample: Westin Freezers

Mann-Whitney-Wilcoxon Test (Large-Sample Mann-Whitney-Wilcoxon Test (Large-Sample Case)Case)

Manufacturer labels indicate the annual Manufacturer labels indicate the annual energy cost associated with operating home energy cost associated with operating home appliances such as freezers.appliances such as freezers.

The energy costs for a sample of 10 The energy costs for a sample of 10 Westin freezers and a sample of 10 Brand-X Westin freezers and a sample of 10 Brand-X Freezers are shown on the next slide. Do the Freezers are shown on the next slide. Do the data indicate, using data indicate, using = .05, that a difference = .05, that a difference exists in the annual energy costs associated exists in the annual energy costs associated with the two brands of freezers?with the two brands of freezers?

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Westin Freezers Brand-X Westin Freezers Brand-X FreezersFreezers

$55.10 $55.10 $56.10 $56.10

54.5054.50 54.70 54.70

53.2053.20 54.40 54.40

53.0053.00 55.40 55.40

55.5055.50 54.10 54.10

54.9054.90 56.00 56.00

55.8055.80 55.50 55.50

54.0054.00 55.00 55.00

54.2054.20 54.30 54.30

55.2055.20 57.00 57.00

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• HypothesesHypotheses

HH00: Annual energy costs for Westin : Annual energy costs for Westin freezersfreezers

and Brand-X freezers are the same. and Brand-X freezers are the same.

HHaa: Annual energy costs differ for the: Annual energy costs differ for the

two brands of freezers.two brands of freezers.

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First, rank the First, rank the combinedcombined data from the lowest data from the lowest to 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 for identical populations. The value of the standardized populations. The value of the standardized test statistic test statistic zz will provide the basis for will provide the basis for deciding whether to reject deciding whether to reject HH00..

Mann-Whitney-Wilcoxon Test:Mann-Whitney-Wilcoxon Test:Large-Sample CaseLarge-Sample Case

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Sampling Distribution of Sampling Distribution of TT for Identical for Identical PopulationsPopulations

• MeanMean

TT = = nn11((nn11 + + nn22 + 1) + 1)

• Standard DeviationStandard Deviation

• Distribution FormDistribution Form

Approximately normal, providedApproximately normal, provided

nn11 >> 10 and 10 and nn22 >> 10 10

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

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Westin Freezers Rank Brand-X Freezers Westin Freezers Rank Brand-X Freezers RankRank

$55.10 $55.10 12 12 $56.10 $56.10 19 19

54.5054.50 8 8 54.70 54.70 9 9 53.2053.20 2 2 54.40 54.40 7 7 53.0053.00 1 1 55.40 55.40 14 14 55.5055.50 15.5 15.5 54.10 54.10 4 4 54.9054.90 10 10 56.00 56.00 18 18 55.8055.80 17 17 55.50 55.50 15.5 15.5 54.0054.00 3 3 55.00 55.00 11 11 54.2054.20 5 5 54.30 54.30 6 6 55.2055.20 1313 57.00 57.00 2020Sum of RanksSum of Ranks 86.5 86.5 Sum of RanksSum of Ranks

123.5123.5

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• Sampling DistributionSampling Distribution

13.2313.23



TT = 105 =1/2(10)(21) = 105 =1/2(10)(21)TT

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Rejection RuleRejection Rule

Using .05 level of significance,Using .05 level of significance,


zz = ( = (TT - - T T )/)/TT = (86.5 - 105)/13.23 = -1.40 = (86.5 - 105)/13.23 = -1.40 ConclusionConclusion

Do not reject Do not reject HH00. There is insufficient . There is insufficient evidence in the sample data to conclude that evidence in the sample data to conclude that there is a difference in the annual energy cost 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 can be used to The Mann-Whitney-Wilcoxon test can be used to test whether two populations are identical.test whether two populations are identical.

The MWW test has been extended by Kruskal The MWW test has been extended by Kruskal and Wallis for cases of three or more and Wallis for cases of three or more populations.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.

The hypotheses are:The hypotheses are:

HH00: All populations are identical: All populations are identical

HHaa: Not all populations are identical: Not all populations are identical

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Rank CorrelationRank Correlation

The Pearson correlation coefficient, The Pearson correlation coefficient, rr, is a , is a measure of the linear association between two measure of the linear association between two variables for which interval or ratio data are variables for which interval or ratio data are available.available.

The The Spearman rank-correlation coefficientSpearman rank-correlation coefficient, , rrs s , is , is a measure of association between two variables a measure of association between two variables when only ordinal data are available.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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Spearman Rank-Correlation Coefficient, Spearman Rank-Correlation Coefficient, rrss

where: where: nn = number of items being ranked = number of items being ranked

xxii = rank of item = rank of item ii with respect to with respect to one variableone variable

yyii = rank of item = rank of item ii with respect to a with respect to a second second variable variable

ddii = = xxii - - yyii

2

2

61

( 1)i

s

dr

n n

2

2

61

( 1)i

s

dr

n n

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Test for Significant Rank CorrelationTest for Significant Rank Correlation

We may want to use sample results to make We may want to use sample results to make an inference about the population rank an inference about the population rank correlation correlation ppss..

To do so, we must test the hypotheses:To do so, we must test the hypotheses:

HH00: : ppss = 0 = 0

HHaa: : ppss = 0 = 0

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Sampling Distribution ofSampling Distribution of rrss when when ppss = 0 = 0

• MeanMean

• Standard DeviationStandard Deviation

• Distribution FormDistribution Form

Approximately normal, provided Approximately normal, provided nn >> 1010


0sr

0sr

11sr n

1

1sr n

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Example: Connor InvestorsExample: Connor Investors


Connor Investors provides a portfolio Connor Investors provides a portfolio management service for its clients. Two of management service for its clients. Two of Connor’s analysts rated ten investments from Connor’s analysts rated ten investments from high (6) to low (1) risk as shown below. Use high (6) to low (1) risk as shown below. Use rank correlation, with rank correlation, with = .10, to comment on = .10, to comment on the agreement of the two analysts’ ratings.the agreement of the two analysts’ ratings.

InvestmentInvestment AA BB CC DD EE FF GG HH II JJ

Analyst #1Analyst #1 11 44 99 88 66 33 55 7 7 22 1010

Analyst #2Analyst #2 11 55 66 22 99 77 33 1010 44 8 8

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Analyst #1 Analyst #2 Analyst #1 Analyst #2

InvestmentInvestment RatingRating RatingRating Differ.Differ. ((Differ.)Differ.)22

AA 11 11 00 00BB 44 55 -1-1 11CC 99 66 33 99DD 88 22 66 3636EE 66 99 -3-3 99FF 33 77 -4-4 1616GG 55 33 22 44HH 77 1010 -3-3 99II 22 44 -2-2 44JJ 1010 88 22 44

Sum =Sum =9292


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HH00: : ppss = 0 (No rank correlation = 0 (No rank correlation exists.)exists.)

HHaa: : ppss = 0 (Rank correlation exists.) = 0 (Rank correlation exists.)

Sampling DistributionSampling Distribution

rr = 0 = 0rrss

Sampling distribution of rs under the assumptionof no rank correlation

Sampling distribution of rs under the assumptionof no rank correlation

1.333

10 1sr

1

.33310 1sr

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Rejection RuleRejection RuleUsing .10 level of significance,Using .10 level of significance,


zz = ( = (rrss - - r r )/)/rr = (.4424 - 0)/.3333 = 1.33 = (.4424 - 0)/.3333 = 1.33 ConclusionConclusion

Do no reject Do no reject HH00. There is not a significant . There is not a significant rank correlation. The two analysts are not rank correlation. The two analysts are not showing agreement in their rating of the risk showing agreement in their rating of the risk associated with the different investments.associated with the different investments.

2

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

1 1 slide nonparametric methods n sign test n wilcoxon signed-rank test n mann-whitney-wilcoxon test...

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