nonparametric tests chapter1616 why use nonparametric tests? one-sample runs test wilcox on...

Nonparametric TestsNonparametric TestsC

hapter16161616Why Use Nonparametric Tests?

One-Sample Runs TestWilcox on Signed-Rank Test

Mann-Whitney TestKruskal-Wallis Test for Independent Samples

Friedman Test for Related Samples

Spearman Rank Correlation Test

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

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Why Use Nonparametric Tests?Why Use Nonparametric Tests?

• Parametric hypothesis tests require the Parametric hypothesis tests require the estimation of one or more unknown parameters estimation of one or more unknown parameters (e.g., population mean or variance).(e.g., population mean or variance).

• Often, unrealistic assumptions are made about Often, unrealistic assumptions are made about the normality of the underlying population.the normality of the underlying population.

• Large sample sizes are often required to invoke Large sample sizes are often required to invoke the Central Limit Theorem.the Central Limit Theorem.

Parametric TestsParametric Tests

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• Nonparametric Nonparametric or distribution-free tests or distribution-free tests

- usually focus on the sign or rank of the - usually focus on the sign or rank of the data rather than the exact numerical value.data rather than the exact numerical value.

- do not specify the shape of the parent- do not specify the shape of the parent population.population.

- can often be used in smaller samples.- can often be used in smaller samples.

- can be used for ordinal data.- can be used for ordinal data.

Nonparametric TestsNonparametric Tests

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Advantages and Disadvantages of Advantages and Disadvantages of Nonparametric Tests Nonparametric Tests

Table 16.1

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Some Common Nonparametric TestsSome Common Nonparametric Tests

Figure 16.1

One-Sample Runs TestOne-Sample Runs Test

• The one-sample The one-sample runs testruns test ( (Wald-Wolfowitz Wald-Wolfowitz testtest) detects non-randomness.) detects non-randomness.

• Ask – Is each observation in a sequence of Ask – Is each observation in a sequence of binary events independent of its predecessor?binary events independent of its predecessor?

• A nonrandom pattern suggests that the A nonrandom pattern suggests that the observations are not observations are not independentindependent..

• The hypotheses areThe hypotheses areHH00: Events follow a random pattern: Events follow a random pattern

HH11: Events do not follow a random : Events do not follow a random

patternpattern

Wald-Wolfowitz Runs TestWald-Wolfowitz Runs Test

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• To test the hypothesis, first count the number To test the hypothesis, first count the number of outcomes of each type.of outcomes of each type.

nn11 = number of outcomes of the first type = number of outcomes of the first type

nn22 = number of outcomes of the second = number of outcomes of the second

typetypenn = total sample size = = total sample size = nn11 + + nn22

• A A run run is a series of consecutive outcomes of is a series of consecutive outcomes of the same type, surrounded by a sequence of the same type, surrounded by a sequence of outcomes of the other type. outcomes of the other type.



• For example, consider the following series For example, consider the following series representing 44 defective (representing 44 defective (DD) or acceptable () or acceptable (AA) ) computer chips:computer chips:

DAAAAAAADDDDAAAAAAAADDAAAAAAAADDDDDAAAAAAADDDDAAAAAAAADDAAAAAAAADDDDAAAAAAAAAAAAAAAAAAAA

• The grouped sequences are:The grouped sequences are:

• A run can be a single outcome if it is preceded A run can be a single outcome if it is preceded and followed by outcomes of the other type.and followed by outcomes of the other type.


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• There are 8 runs (There are 8 runs (RR = 8). = 8).nn11 = number of defective chips ( = number of defective chips (DD) = 11) = 11

nn22 = number of acceptable chips ( = number of acceptable chips (AA) = 33) = 33

nn = total sample size = = total sample size = nn11 + + nn22 = = 11 + 33 = 4411 + 33 = 44

• The hypotheses are:The hypotheses are:HH00: Defects follow a random sequence: Defects follow a random sequence

HH11: Defects follow a nonrandom sequence: Defects follow a nonrandom sequence


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• When When nn11 >> 10 and 10 and nn22 >> 10, then the number of 10, then the number of

runs runs RR may be assumed to be normally may be assumed to be normally distributed with mean distributed with mean RR and standard deviation and standard deviation

RR..


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• The test statistic is:The test statistic is:

• For a given level of significance For a given level of significance , find the , find the critical value critical value zz for a two-tailed test. for a two-tailed test.

• Reject the hypothesis of a random pattern if Reject the hypothesis of a random pattern if zz < - < -zz or if or if zz > + > +zz ..


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• Decision rule for large-sample runs tests:Decision rule for large-sample runs tests:


Figure 16.2

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Wilcox on Signed-Rank TestWilcox on Signed-Rank Test

• The The Wilcox on signed-rank Wilcox on signed-rank testtest compares a compares a single sample median with a benchmark using single sample median with a benchmark using only only ranksranks of the data instead of the original of the data instead of the original observations.observations.

• It is used to compare It is used to compare pairedpaired observations. observations.• Advantages are Advantages are

- freedom from the normality - freedom from the normality assumption,assumption,

- robustness to outliers- robustness to outliers- applicability to ordinal data.- applicability to ordinal data.

• The population should be roughly symmetric.The population should be roughly symmetric.

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• To compare the sample To compare the sample median median ((MM) with a ) with a benchmark benchmark median median ((MM00), the hypotheses are:), the hypotheses are:

• When evaluating the difference between When evaluating the difference between paired observations, use the median paired observations, use the median differencedifference ( (MMdd) and zero as the benchmark.) and zero as the benchmark.

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• Calculate the difference between the paired Calculate the difference between the paired observations.observations.

• Rank the differences from smallest to Rank the differences from smallest to largest by absolute value.largest by absolute value.

• Add the ranks of the Add the ranks of the positivepositive differences to differences to obtain the rank sum obtain the rank sum W.W.

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• For small samples, a special table is For small samples, a special table is required to obtain critical values.required to obtain critical values.

• For large samples (For large samples (nn >> 20), the test statistic 20), the test statistic is approximately normal.is approximately normal.

• Use Excel or Appendix C to get a Use Excel or Appendix C to get a pp-value. -value.

• Reject Reject HH00 if if pp-value -value << ..

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Mann-Whitney TestMann-Whitney Test

• The The Mann-WhitneyMann-Whitney test is a nonparametric test is a nonparametric test that compares two populations.test that compares two populations.

• It does not assume normality.It does not assume normality.• It is a test for the equality of It is a test for the equality of mediansmedians, ,

assuming assuming - the populations differ only in centrality,- the populations differ only in centrality,- equal variances- equal variances

• The hypotheses are The hypotheses are HH00: : MM11 = = MM22 (no difference in medians) (no difference in medians)

HH11: : MM11 ≠≠ MM22 (medians differ) (medians differ)

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• Step 1: Sort the Step 1: Sort the combinedcombined samples from samples from lowest to highest.lowest to highest.

• Step 2: Assign a rank to each value.Step 2: Assign a rank to each value.

If values are tied, the average of the ranksIf values are tied, the average of the ranksis assigned to each.is assigned to each.

• Step 3: The ranks are summed for each Step 3: The ranks are summed for each column (e.g., column (e.g., TT11, T, T22))..

• Step 4: The sum of the ranks Step 4: The sum of the ranks TT11 + T + T22 must be must be

equal to equal to nn((nn + 1)/2, where + 1)/2, where nn = = nn11 + + nn22..

Performing the TestPerforming the Test

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• Step 5: Calculate the mean rank sums Step 5: Calculate the mean rank sums TT11 and and TT22..


• Step 6: For large samples (Step 6: For large samples (nn11 << 10, 10, nn22 >> 10), 10),

use a use a zz test. test.

• Step 7: For a given Step 7: For a given , reject , reject HH00 if if

zz < - < -zz or or zz > + > +zz

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• The The Kruskal-WallisKruskal-Wallis (K-W) test compares (K-W) test compares cc independent medians, assuming the independent medians, assuming the populations differ only in centrality. populations differ only in centrality.

• The K-W test is a generalization of the The K-W test is a generalization of the Mann-Whitney test and is analogous to a Mann-Whitney test and is analogous to a one-factor ANOVA (completely randomized one-factor ANOVA (completely randomized model).model).

• Groups can be of different sizes if each Groups can be of different sizes if each group has 5 or more observations.group has 5 or more observations.

• Populations must be of similar shape but Populations must be of similar shape but normality is not a requirement.normality is not a requirement.

Kruskal-Wallis Test Kruskal-Wallis Test for Independent Samplesfor Independent Samples

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• First, combine the First, combine the samples and assign samples and assign a rank to each a rank to each observation in each observation in each group. For example:group. For example:

• When a tie occurs, When a tie occurs, each observation is each observation is assigned the assigned the average of the ranks.average of the ranks.


Table 16.7

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• Next, arrange Next, arrange the data by the data by groups and sum groups and sum the ranks to the ranks to obtain the obtain the TTjj’s. ’s.

• Remember,Remember,TTjj = = nn((nn+1)/2.+1)/2.


Table 16.8

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• The hypotheses to be tested are:The hypotheses to be tested are:HH00: All : All cc population medians are the same population medians are the sameHH11: Not all the population medians are the same: Not all the population medians are the same

• For a completely randomized design with For a completely randomized design with cc groups, the tests statistic isgroups, the tests statistic is

where where nn = = nn11 + + nn2 2 + … + + … + nncc

nnjj = number of observations in group = number of observations in group jjTTjj = sum of ranks for group = sum of ranks for group jj


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• The The HH test statistic follows a chi-square test statistic follows a chi-square distribution with distribution with = = cc – 1 degrees of – 1 degrees of freedom.freedom.

• This is a right-tailed test, so reject This is a right-tailed test, so reject HH00 if if HH > >

22or if or if pp-value -value << ..


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• The The Friedman test Friedman test determines if determines if cc treatments treatments have the same central tendency (have the same central tendency (mediansmedians) ) when there is a second factor with when there is a second factor with rr levels and levels and the populations are assumed to be the same the populations are assumed to be the same except for centrality.except for centrality.

• This test is analogous to a two-factor ANOVA This test is analogous to a two-factor ANOVA without replication (randomized block design) without replication (randomized block design) with one observation per cell.with one observation per cell.

• The groups must be of the same size.The groups must be of the same size.• Treatments should be randomly assigned Treatments should be randomly assigned

within blocks.within blocks.• Data should be at least interval scale.Data should be at least interval scale.

Friedman Test for Related Friedman Test for Related SamplesSamples

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• In addition to the In addition to the cc treatment levels that define treatment levels that define the columns, the Friedman test also specifies the columns, the Friedman test also specifies rr block factor levels to define each row of the block factor levels to define each row of the observation matrix.observation matrix.

• The hypotheses to be tested are:The hypotheses to be tested are:HH00: All : All cc populations have the same median populations have the same median

HH11: Not all the populations have the same median: Not all the populations have the same median

• Unlike the Kruskal-Wallis test, the Friedman Unlike the Kruskal-Wallis test, the Friedman ranks are computed ranks are computed within each blockwithin each block rather rather than within a pooled sample.than within a pooled sample.

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• First, assign a rank to each observation within First, assign a rank to each observation within each each rowrow. For example, within each . For example, within each TrialTrial::

• When a tie occurs, each observation is When a tie occurs, each observation is assigned the average of the ranks.assigned the average of the ranks.


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• Compute the test statistic:Compute the test statistic:

where where rr = the number of blocks (rows) = the number of blocks (rows) cc = the number of treatments = the number of treatments

(columns)(columns) TTjj = the sum of ranks for treatment = the sum of ranks for treatment jj


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• The Friedman test statistic The Friedman test statistic FF, follows a chi-, follows a chi-square distribution with square distribution with = = cc – 1 degrees of – 1 degrees of freedom.freedom.

• Reject Reject HH00 if if FF > > 22or if or if pp-value -value << ..

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Spearman Rank Correlation TestSpearman Rank Correlation Test

• Spearman’s rank correlation Spearman’s rank correlation coefficient coefficient ((Spearman’s rhoSpearman’s rho) is an overall ) is an overall nonparametric test that measures the nonparametric test that measures the strength of the association (if any) between strength of the association (if any) between two variables.two variables.

• This method does not assume interval This method does not assume interval measurement.measurement.

• The sample rank correlation coefficient The sample rank correlation coefficient rrss

ranges from -1 ranges from -1 << rrss << +1. +1.

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• The sign of The sign of rrss indicates whether the relationship is indicates whether the relationship is directdirect – ranks tend to vary in the same – ranks tend to vary in the same

direction, or direction, or inverseinverse – ranks tend to vary in opposite – ranks tend to vary in opposite

directions directions• The magnitude of The magnitude of rrss indicated the degree of indicated the degree of

relationship. Ifrelationship. Ifrrss is near 0 – there is little or no agreement is near 0 – there is little or no agreement

between rankings between rankingsrrss is near +1 – there is strong is near +1 – there is strong direct direct agreementagreementrrss is near -1 – there is strong is near -1 – there is strong inverseinverse

agreementagreement

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• First, rank First, rank each each variable. For variable. For example,example,

• If more than If more than one value is one value is the same, the same, assign the assign the average of average of the ranks.the ranks.


Table 16.11

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• The sums of ranks within each column must The sums of ranks within each column must always be always be nn((nn+1)/2.+1)/2.

• Next, compute the difference in ranks Next, compute the difference in ranks ddii for for

each observation.each observation.• The rank differences should sum to zero.The rank differences should sum to zero.


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• Calculate the sample rank correlation Calculate the sample rank correlation coefficient coefficient rrss..

where where ddii = difference in ranks for case = difference in ranks for case ii nn = sample size = sample size

• For a right-tailed test, the hypotheses to be For a right-tailed test, the hypotheses to be tested aretested areHH00: True rank correlation is zero (: True rank correlation is zero (ss << 0) 0)HH11: True rank correlation is positive (: True rank correlation is positive (ss > 0) > 0)


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• If If nn is large (at least 20 observations), then is large (at least 20 observations), then rrss

may be assumed to follow the Student’s may be assumed to follow the Student’s tt distribution with degrees of freedom distribution with degrees of freedom = = nn - 1 - 1

• Reject Reject HH00 if if tt > > ttor if or if pp-value -value << ..


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Correlation versus CausationCorrelation versus Causation

• Caution: correlation does not prove Caution: correlation does not prove causation.causation.

• Correlations may prove to be “significant” Correlations may prove to be “significant” even when there is no causal relation even when there is no causal relation between the two variables.between the two variables.

• However, causation is not ruled out.However, causation is not ruled out.• Multiple causes may be present.Multiple causes may be present.

Applied Statistics in Applied Statistics in Business & EconomicsBusiness & Economics

End of Chapter 16End of Chapter 16

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