Nonparametric TechniquesNonparametric Techniques

CJ 526 Statistical Analysis in CJ 526 Statistical Analysis in Criminal JusticeCriminal Justice

Parametric v. Nonparametric: Parametric v. Nonparametric: ParametricParametric

1.1. ParametricParametric1.1. Dependent Variable:Dependent Variable:

1.1. Interval/RatioInterval/Ratio

Parametric v. Nonparamteric: Parametric v. Nonparamteric: NonparametricNonparametric

1.1. NonparametricNonparametric1.1. Dependent Variable:Dependent Variable:

1.1. Nominal/ordinalNominal/ordinal

Uses of Nonparametric Uses of Nonparametric TechniquesTechniques

1.1. Dependent Variable:Dependent Variable:1.1. Nominal/ordinalNominal/ordinal

Nominal Level DataNominal Level Data

1.1. One SampleOne Sample1.1. Chi-Square Test of Goodness of Fit, chisquare test Chi-Square Test of Goodness of Fit, chisquare test

of independenceof independence

Ordinal Level DataOrdinal Level Data

1.1. RankingRanking1.1. Less demandingLess demanding

1.1. Easier to use Easier to use

Information Derived From an Ordinal ScaleInformation Derived From an Ordinal Scale

1.1. Provides information about the direction Provides information about the direction of difference between scoresof difference between scores

1.1. Greater than, less thanGreater than, less than

3.3. Do not need absolute measurement to Do not need absolute measurement to obtain ranksobtain ranks

Information Derived From an Ordinal Scale -- Information Derived From an Ordinal Scale -- continuedcontinued

Can always convert scores to ranksCan always convert scores to ranks

Mann-Whitney U TestMann-Whitney U Test

1.1. Nonparametric analogue of an Nonparametric analogue of an Independent t-TestIndependent t-Test

ExampleExample

1.1. A hospital administrator supervisor to A hospital administrator supervisor to know whether gender has an effect on know whether gender has an effect on rank-ordered judgments of leadership rank-ordered judgments of leadership ability.ability.

Example -- continuedExample -- continued

1.1. Number of samples: 2Number of samples: 2

2.2. Nature of samples: independentNature of samples: independent

3.3. σσ Known: no Known: no

4.4. Independent Variable: genderIndependent Variable: gender

Example -- continuedExample -- continued

5.5. Dependent Variable and its Level of Dependent Variable and its Level of Measurement: judgments of leadership abilityMeasurement: judgments of leadership ability

6.6. Target Population: hospital personnelTarget Population: hospital personnel

7.7. Inferential Statistical Technique: Mann WhitneyInferential Statistical Technique: Mann Whitney

Example -- continuedExample -- continued

8.8. HH00::1.1. Gender will have no effect on rank-ordered Gender will have no effect on rank-ordered

leadership abilityleadership ability

9.9. HH11::1.1. Gender will have an effect on rankings of Gender will have an effect on rankings of

leadership abilityleadership ability

10.10. Decision Rule:Decision Rule:1.1. If the p-value of the obtained test statistic is less If the p-value of the obtained test statistic is less

than .05, reject the null hypothesis, two tailed testthan .05, reject the null hypothesis, two tailed test

Example -- continuedExample -- continued

11.11. Obtained Test Statistic: z is usedObtained Test Statistic: z is used1.1. Z = -3.811, p = .0003Z = -3.811, p = .0003

12.12. Decision: reject the null hypothesisDecision: reject the null hypothesis

Results SectionResults Section

The results of the Mann-Whitney U Test The results of the Mann-Whitney U Test involving gender as the independent involving gender as the independent variable and rank-ordered leadership ability variable and rank-ordered leadership ability as the dependent variable were statistically as the dependent variable were statistically significant, z = -3.811, p < .001.significant, z = -3.811, p < .001.

Discussion SectionDiscussion Section

It appears that males were ranked higher in It appears that males were ranked higher in terms of leadership ability than females terms of leadership ability than females

Mann-Whitney U Test and SPSS Mann-Whitney U Test and SPSS for Windowsfor Windows

Statistics, Nonparametric Tests,2 Statistics, Nonparametric Tests,2 Independent SamplesIndependent Samples

Move DV to Test Variable listMove DV to Test Variable list Move IV to Grouping VariableMove IV to Grouping Variable Define GroupsDefine Groups Make sure M-W is checkedMake sure M-W is checked

Interpreting the PrintoutInterpreting the Printout

Mean ranksMean ranks z-value (obtained test statistic)z-value (obtained test statistic) 2-tailed p (p-value)2-tailed p (p-value)

Sample PrintoutSample Printout

Ranks

10 9.45 94.50

10 11.55 115.50

20

Gender of RespondentFemale

Male

Total

Score on Drink IndexN Mean Rank Sum of Ranks

Test Statisticsb

39.500

94.500

-.801

.423

.436a

Mann-Whitney U

Wilcoxon W

Z

Asymp. Sig. (2-tailed)

Exact Sig. [2*(1-tailedSig.)]

Score onDrink Index

Not corrected for ties.a.

Grouping Variable: Gender of Respondentb.

Wilcoxon Matched-Pairs Signed-Ranks TestWilcoxon Matched-Pairs Signed-Ranks Test

1.1. Wilcoxon Matched-Pairs Signed-Ranks Wilcoxon Matched-Pairs Signed-Ranks TestTest

ExampleExample

A social psychologist wants to know A social psychologist wants to know whether males and females matched for whether males and females matched for physical attractiveness will be ranked physical attractiveness will be ranked differently in terms of leadership ability.differently in terms of leadership ability.

Example -- continuedExample -- continued

1.1. Number of Samples: 2Number of Samples: 2

2.2. Nature of Samples: dependent, matchedNature of Samples: dependent, matched

3.3. σσ Known: no Known: no

4.4. Independent Variable: genderIndependent Variable: gender

Example -- continuedExample -- continued

5.5. Dependent Variable and its Level of Dependent Variable and its Level of Measurement: rankings of leadership abilityMeasurement: rankings of leadership ability

6.6. Target Population: general populationTarget Population: general population

7.7. Inferential Statistical Technique:Inferential Statistical Technique:1.1. Wilcoxon Matched-Pairs Signed-Ranks TestWilcoxon Matched-Pairs Signed-Ranks Test

Example -- continuedExample -- continued

8.8. HH00::1.1. Gender will have no effect on rank-ordered Gender will have no effect on rank-ordered

leadership abilityleadership ability

9.9. HH11: : 1.1. Gender will have an effect on rank-ordered Gender will have an effect on rank-ordered

leadership abilityleadership ability

10.10. Decision Rule:Decision Rule:1.1. If the p-value of the obtained test statistic is less If the p-value of the obtained test statistic is less

than .05, reject the null hypothesisthan .05, reject the null hypothesis

Example -- continuedExample -- continued

11.11. Obtained Test Statistic:Obtained Test Statistic:1.1. Z= -2.7724, p = .006Z= -2.7724, p = .006

12.12. Decision:Decision:

Results SectionResults Section

1.1. The results of the Wilcoxon Matched-The results of the Wilcoxon Matched-Pairs Signed-Ranks Test involving gender Pairs Signed-Ranks Test involving gender as the independent variable and rank-as the independent variable and rank-ordered leadership ability as the ordered leadership ability as the dependent variable were statistically dependent variable were statistically significant, z = -2.7724, p < .01.significant, z = -2.7724, p < .01.

DiscussionDiscussion

It appears that when matched on physical It appears that when matched on physical attractiveness, males are ranked higher than attractiveness, males are ranked higher than females on leadership ability.females on leadership ability.

Wilcoxon Test and SPSS for Wilcoxon Test and SPSS for WindowsWindows

Statistics, Nonparametric Tests, 2 Related Statistics, Nonparametric Tests, 2 Related SamplesSamples

Move pair of variablesMove pair of variables Make sure W is checkedMake sure W is checked

Interpreting the PrintoutInterpreting the Printout

Mean ranksMean ranks z (obtained test statistic)z (obtained test statistic) 2-tailed p (p-value)2-tailed p (p-value)

Sample PrintoutSample Printout

Ranks

19a 11.00 209.00

1b 1.00 1.00

0c

20

Negative Ranks

Positive Ranks

Ties

Total

Score on Drug Index -Score on Drink Index

N Mean Rank Sum of Ranks

Score on Drug Index < Score on Drink Indexa.

Score on Drug Index > Score on Drink Indexb.

Score on Drug Index = Score on Drink Indexc.

Test Statisticsb

-3.886a

.000

Z

Asymp. Sig. (2-tailed)

Score onDrug Index -

Score onDrink Index

Based on positive ranks.a.

Wilcoxon Signed Ranks Testb.