Class8 Non-Parameter Tests

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Non-parametric Tests

Research II MSW PT

Class 8


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Key Terms

Power of a test refers to the probability of rejectinga false null hypothesis (or detect a relationship whenit exists)

Power Efficiency the power of the test relative tothat of its most powerful alternative. For example,if the power efficiency of a certain nonparametric testfor difference of means with sample size 10 is 0.9, itmeans that if interval scale and the normalityassumptions can be made (more powerful), we canuse the t-test with a sample size of 9 to achieve thesame power.


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Choice of nonparametric test

It depends on the level of measurement obtained (nominal,ordinal, or interval), the power of the test, whether samples arerelated or independent, number of samples, availability ofsoftware support (e.g. SPSS)

Related samples are usually referred to match-pair (usingrandomization) samples or before-after samples.

Other cases are usually treated as independent samples. Forinstance, in a survey using random sampling, we have a sub-sample of males and a sub-sample of females. They can beconsidered as independent samples as they are all randomly

selected.


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Level of

measurement

One-sample

test

Two-sample case K-sample case

Related Samples Independent samples Related

samples

Independent

samples

Nominal Binomial McNemar forsignificance of

changes

Fisher exactprobability

Chi-square

Cochran Q(Dichotomous)

Chi-square

Ordinal Kolmogorov

Smirnov

Runs

Sign Wilcoxon

matched-pair

signed-ranks

Mann-Whitney U

Kolmogorov-Smirnov

Wald-Wolfowitz runs

Moses of extremereactions

Friedman

two-way

analysis of

variance

Kendalls W

Kruskal-Wallis

one-way

analysis of

variance

Interval Walsh Randomization


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One-sample case Binomial tests whether the observed

distribution of dichotomous variable (a

variable that has two values only) is the sameas that expected from a given binomialdistribution.

The default value of p is 0.5.You can changethe value of p.

For example, a couple hasgiven birth consecutively 8 baby girls, and

you would like to test if their probability ofgiven birth to baby girls is > 0.6 or >0.7, youcan test the hypothesis by changing thedefault value of p in the SPSS programme.


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Binomial Test

Category N Observed Prop. Test Prop. Exact Sig. (2-tailed)

Group 1 Male 8 1.00 .50 .008Gender

Total 8 1.00


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Chi-square tests whether the

observed distribution is the same as a

certain hypothesized distribution. The default null hypothesis is even

distribution.


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Age

Observed N Expected N Residual

18 - 24 92 55.9 36.1

25 - 34 78 111.8 -33.8

35 - 44 100 167.7 -67.745 - 55 95 111.8 -16.8

56 or above 138 55.9 82.1

Total 503

Test StatisticsAge

Chi-Square(a) 184.003

df 4

Asymp. Sig. .000

a 0 cells (.0%) have expected frequencies less than 5. The minimum expected cell frequency

is 55.9.


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Kolmogorov-Smirnov Compares

the distribution of a variable with a

uniform, normal, Poisson, orexponential distribution,

Null hypothesis: the observed values

were sampled from a distribution ofthat type.


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Runs

A run is defined as a sequence of cases onthe same side of the cut point. (Anuninterrupted course of some state or

condition, for e.g. a run of good luck). You should use the Runs Test procedure

when you want to test the hypothesis thatthe values of a variable are ordered randomly

with respect to a cut point of your choosing(Default cut point: median.


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E.g. If you ask 20 students about how well they understand alecture on a scale ranged from 1 to 5 (and the median in theclass is 3). If you find that, the first 10 students give a valuehigher than 3 and the second 10 give a value lower than 3

(there are only 2 runs). 5445444545 2222112211 For random situation, there should be more runs (but will not be

close to 20, which means they are ordered exactly in analternative fashion; for example a value below 3 will be followedby one higher than it and vice versa). 2,4,1,5,1,4,2,5,1,4,2,4

The Runs Test is often used as a precursor to running tests that

compare the means of two or more groups, including: The Independent-Samples T Test procedure. The One-Way ANOVA procedure. The Two-Independent-Samples Tests procedure. The Tests for Several Independent Samples procedure.


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Note: In this data set, 80 social workers (1) are listed together, and followed by120 non-social workers (2), obviously, the order in not random. Since there aremore non-social workers, the median is still 2. There are only 2 runs, one lowerthan the median (2) and one higher than or equal to it.


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Runs Test

Social Worker

Test Value(a) 2

Cases < Test Value 80

Cases >= Test Value 120Total Cases 200

Number of Runs 2

Z -14.033

Asymp. Sig. (2-tailed) .000

a Median


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Sample cases (RelatedSamples)

McNemar tests whether the changesin proportions are the same for pairs of

dichotomous variables. McNemars testis computed like the usual chi-square

test, but only the two cells in which theclassification dont match are used.

Null hypothesis: People are equallylikely to fall into two contradictoryclassification categories.


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Support New Dawn Project (Before)

Yes No

Yes 19 13Support New Dawn Project (After)No 1 7


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Test Statistics (b)

Support New Dawn Project

(Before) & Support New DawnProject (After)

N 40

Exact Sig. (2-tailed) .002(a)a Binomial distribution used.

b McNemar Test


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Sign test tests whether the numbers ofdifferences (+ve or ve) between twosamples are approximately the same. Eachpair of scores (before and after) arecompared.

When after > before (+ sign), if smaller (-sign). When both are the same, it is a tie.

Sign-test did not use all the informationavailable (the size of difference), but itrequires less assumptions about the sampleand can avoid the influence of the outliers.


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To test the association between thefollowing two perceptions

Social workers help the disadvantagedand Social workers bring hopes to thosein averse situation


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Frequencies

N

Social workers bring hopes

to those in averse situation- Social workers help thedisadvantaged

Negative

Differences(a) 104

PositiveDifferences(b)

71

Ties(c) 322

Total 497a Social workers bring hopes to those in averse situation < Social workers help the disadvantaged

b Social workers bring hopes to those in averse situation > Social workers help the disadvantagedc Social workers bring hopes to those in averse situation = Social workers help the disadvantaged


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Test Statistics(a)

Social workers bring hopes to those in aversesituation - Social workers help the

disadvantaged

Z -2.419

Asymp. Sig.

(2-tailed) .016a Sign Test


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Wilcoxon matched-pairs signed-ranks test Similar to sign test, but take into consideration theranking of the magnitude of the difference among

the pairs of values. (Sign test only considers thedirection of difference but not the magnitude ofdifferences.)

The test requires that the differences (of the truevalues) be a sample from a symmetric distribution

(but not require normality). Its better to run stem-and-leaf plot of the differences.


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Ranks

N

Mean

Rank

Sum of

RanksNegative Ranks 104(a) 88.07 9159.50

Positive Ranks 71(b) 87.89 6240.50

Ties 322(c)

Social workers bring hopes to those inaverse situation - Social workers helpthe disadvantaged

Total497

a Social workers bring hopes to those in averse situation < Social workers help the disadvantagedb Social workers bring hopes to those in averse situation > Social workers help the disadvantagedc Social workers bring hopes to those in averse situation = Social workers help the disadvantaged


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Test Statistics (b)

Social workers bring hopes to those in aversesituation - Social workers help the disadvantaged

Z -2.340(a)

Asymp. Sig.(2-tailed)

.019

a Based on positive ranks.b Wilcoxon Signed Ranks Test


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Two-sample case(independent samples)

Mann-Whitney U similar to Wilcoxon matched-paired signed-ranks test except that the samples areindependent and not paired. Its the most commonlyused alternative to the independent-samples ttest.

Null hypothesis: the population means are the samefor the two groups.

The actual computation of the Mann-Whitney test issimple. You rank the combined data values for thetwo groups. Then you find the average rank in eachgroup.

Requirement: the population variances for the twogroups must be the same, but the shape of thedistribution does not matter.


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Ranks

Sex N Mean Rank Sum of Ranks

Male 229 247.36 56645.50

Female 272 254.06 69105.50

Social Worker

Total 501

Test Statistics (a)

Social Worker

Mann-Whitney U 30310.500

Wilcoxon W 56645.500Z -.628

Asymp. Sig. (2-tailed) .530

a Grouping Variable: Sex


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Kolmogorov-Smirnov Z to test iftwo distributions are different. It isused when there are only a few valuesavailable on the ordinal scale. K-S testis more powerful than M-W U test if the

two distributions differ in terms ofdispersion instead of central tendency.


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Test Statistics (a)

SocialWorker

Absolute .036

Positive .009

Most ExtremeDifferences

Negative -.036

Kolmogorov-Smirnov Z .397

Asymp. Sig. (2-tailed) .998a Grouping Variable: Sex


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Wald-Wolfowitz Run Based on the

number of runs within each group when

the cases are placed in rank order. Moses test of extreme reactions

Tests whether the range (excluding the

lowest 5% and the highest 5%) of anordinal variables is the same in the twogroups.


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K-sample case(Independent samples)

Kruskal-Wallis One-way ANOVAIts more powerful than Chi-square test

when ordinal scale can be assumed. Itis computed exactly like the Mann-Whitney test, except that there are

more groups. The data must beindependent samples from populationswith the same shape (but notnecessarily normal).


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Ranks

Educationlevel N

MeanRank

Primary orlower

105 264.67

Secondary 239 248.41Post-secondary

159 249.03

SocialWorker

Total 503

Test Statistics(a,b)

Social WorkerChi-Square 1.049

df 2

Asymp. Sig. .592a Kruskal Wallis Test

b Grouping Variable: Education level


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K related samples

Friedman two-way ANOVA test

whether the k related samples could

probably have come from the samepopulation with respect to mean rank.


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Ranks

1.88

2.45

1.68

Social Worker

Doctor

Lawyer

Mean Rank

Test Sta t i s t i csa

503

204.2412

.000

N

Chi-Squaredf

Asymp. Sig.

Friedman Testa.

Ranks

1.88

2.45

1.68

Social Worker

Doctor

Lawyer

Mean Rank

Test Stat i s t i c s503

.203

204.241

2

.000

N

Kendall's Wa

Chi-Square

df

Asymp. Sig.

Kendall's Coefficient of Concordancea.


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Cochran Q determines whether it is

likely that the k related samples could

have come from the same populationwith respect to proportion or frequencyofsuccesses in the various samples.

In other words, it only comparesdichotomous variables.

Lets try this in class

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Class8 Non-Parameter Tests