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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.