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PARAMETRIC VERSUS PARAMETRIC VERSUS NONPARAMETRIC STATISTICSNONPARAMETRIC STATISTICS

Heibatollah Baghi, and Heibatollah Baghi, and

Mastee BadiiMastee Badii

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Parametric AssumptionsParametric Assumptions

Parametric Statistics involve hypothesis Parametric Statistics involve hypothesis about population parameters (e.g., about population parameters (e.g., µ, µ, ρρ).).

They require assumptions about the They require assumptions about the population distribution. For example, the population distribution. For example, the assumptions for t test for independent assumptions for t test for independent samples are:samples are:a)a) Each of the two populations of observations is Each of the two populations of observations is

normally distributed normally distributed b)b) The populations of observations are equally The populations of observations are equally

variable : that is variable : that is σσ22 = = σσ2.2. (Assumption of (Assumption of homogeneity of variance ) homogeneity of variance )

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Nonparametric AlternativeNonparametric Alternative

The parametric assumptions cannot The parametric assumptions cannot be justified: normal distribution, be justified: normal distribution, equal variances, etc.equal variances, etc.

The data as gathered are measured The data as gathered are measured on nominal or ordinal dataon nominal or ordinal data

Sample size is small.Sample size is small.

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Spearman Rank CorrelationSpearman Rank Correlation

The Spearman rank correlation is The Spearman rank correlation is used when:used when: Distribution assumptions required by Distribution assumptions required by

Pearson r are in question Pearson r are in question Small sample sizeSmall sample size

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ExampleExample

X: The student’s popularity X: The student’s popularity measuremeasure

Y: The student’s average academic Y: The student’s average academic achievement achievement

Research questionsResearch questions : Is : Is popularity related to popularity related to achievement ?achievement ?

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Test of Association Using Test of Association Using Spearman Rank CorrelationSpearman Rank Correlation

Because of doubts regarding the Because of doubts regarding the distributional assumptions coupled with distributional assumptions coupled with small sample size, select the Spearman small sample size, select the Spearman Rank Correlation to answer this questionRank Correlation to answer this question

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Calculation of Spearman Rank Calculation of Spearman Rank CorrelationCorrelation

Spearman rankcorrelation

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Calculation of Spearman Rank Calculation of Spearman Rank CorrelationCorrelation

Differencebetween ranks

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Calculation of Spearman Rank Calculation of Spearman Rank CorrelationCorrelation

Num

ber o

f case

s

1010

Calculation of Spearman Rank Calculation of Spearman Rank CorrelationCorrelation

X: The student’s popularity measureY: The student’s average academic

achievement

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Calculation of Spearman Rank Calculation of Spearman Rank CorrelationCorrelation

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Calculation of Spearman Rank Calculation of Spearman Rank CorrelationCorrelation

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Calculation of Spearman Rank Calculation of Spearman Rank CorrelationCorrelation

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Calculation of Spearman Rank Calculation of Spearman Rank CorrelationCorrelation

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Calculation of Spearman Calculation of Spearman Rank CorrelationRank Correlation

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Calculation of Spearman Calculation of Spearman Rank CorrelationRank Correlation

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Test of SignificanceTest of Significance

Calculated rCalculated rRankRank= -0.26= -0.26 Critical value for alpha 0.05 for Critical value for alpha 0.05 for

Spearman Rank Correlation with 8 Spearman Rank Correlation with 8 subjects = 0.738subjects = 0.738

Calculated rCalculated rRanksRanks is less than critical value is less than critical value The relation between Popularity and The relation between Popularity and

academic achievement is not academic achievement is not statistically significantstatistically significant

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When to Use Which TestWhen to Use Which Test

Nominal Data

Two-group case Chi-square t test

K – group case Chi-square ANOVA

Dependent groupsMcNamer test

Association Chi-square Pearson r

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When to Use Which TestWhen to Use Which Test

Nominal Data

Ordinal Data

Two-group case Chi-squareMann-Whitney U t test

K – group case Chi-squareKruskal-Wallis H ANOVA

Dependent groupsMcNamer test

Sign Test or Wilcoxon

Association Chi-squareSpearman rank Pearson r

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When to Use Which TestWhen to Use Which Test

Nominal Data

Ordinal Data

Interval or Ratio data

Two-group case Chi-squareMann-Whitney U t test

K – group case Chi-squareKruskal-Wallis H ANOVA

Dependent groupsMcNamer test

Sign Test or Wilcoxon

Paired t test

Association Chi-squareSpearman rank Pearson r

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Take Home LessonTake Home Lesson

Spearman Rank Correlation Spearman Rank Correlation can be used on ordinal datacan be used on ordinal data