Parametric versus Nonparametric Statistics – When to use them and which is more

powerful?

Parametric Assumptions

The observations must be independent The observations must be drawn from normally distributed

populations These populations must have the same variances

Nonparametric Assumptions

Observations are independent Variable under study has underlying continuity

Measurement

1. Nominal or Classificatory Scale Gender, ethnic background

2. Ordinal or Ranking Scale Hardness of rocks, beauty, military ranks

3. Interval Scale Celsius or Fahrenheit

4. Ratio Scale Kelvin temperature, speed, height, mass or weight

Nonparametric Methods

There is at least one nonparametric test equivalent to a parametric test

These tests fall into several categories

1. Tests of differences between groups (independent samples)

2. Tests of differences between variables (dependent samples)

3. Tests of relationships between variables

Differences between independent groups

Two samples – compare mean value for some variable of interest

Parametric Nonparametric

t-test for independent samples

Wald-Wolfowitz runs test

Mann-Whitney U test

Kolmogorov-Smirnov two sample test

Mann-Whitney U Test

Nonparametric alternative to two-sample t-test Actual measurements not used – ranks of the

measurements used Data can be ranked from highest to lowest or lowest to

highest values Calculate Mann-Whitney U statistic

U = n1n2 + n1(n1+1) – R1

2

Example of Mann-Whitney U test

Two tailed null hypothesis that there is no difference between the heights of male and female students

Ho: Male and female students are the same height

HA: Male and female students are not the same height

Heights of males (cm)

Heights of females (cm)

Ranks of male heights

Ranks of female heights

193 175 1 7

188 173 2 8

185 168 3 10

183 165 4 11

180 163 5 12

178 6

170 9

n1 = 7 n2 = 5 R1 = 30 R2 = 48

U = n1n2 + n1(n1+1) – R1

2

U=(7)(5) + (7)(8) – 30 2

U = 35 + 28 – 30

U = 33

U’ = n1n2 – U

U’ = (7)(5) – 33

U’ = 2

U 0.05(2),7,5 = U 0.05(2),5,7 = 30

As 33 > 30, Ho is rejected Zar, 1996

Differences between independent groups

Multiple groupsParametric Nonparametric

Analysis of variance (ANOVA/ MANOVA)

Kruskal-Wallis analysis of ranks

Median test

Differences between dependent groups

Compare two variables measured in the same sample

If more than two variables are measured in same sample

Parametric Nonparametric

t-test for dependent samples

Sign test

Wilcoxon’s matched pairs test

Repeated measures ANOVA

Friedman’s two way analysis of variance

Cochran Q

Relationships between variables

Two variables of interest are categorical

Parametric Nonparametric

Correlation coefficient

Spearman R

Kendall Tau

Coefficient Gamma

Chi square

Phi coefficient

Fisher exact test

Kendall coefficient of concordance

Summary Table of Statistical Tests

Level of Measurement

Sample Characteristics Correlation

1 Sample

2 Sample K Sample (i.e., >2)

Independent Dependent Independent Dependent

Categorical or Nominal

Χ2 or bi-nomial

Χ2 Macnarmar’s Χ2

Χ2 Cochran’s Q  

Rank or Ordinal

  Mann Whitney U

Wilcoxin Matched

Pairs Signed Ranks

Kruskal Wallis H

Friendman’s ANOVA

Spearman’s rho

Parametric (Interval &

Ratio)

z test or t test

t test between groups

t test within groups

1 way ANOVA between groups

1 way ANOVA

(within or repeated measure)

Pearson’s r

Factorial (2 way) ANOVA  

 (Plonskey,

2001)

Advantages of Nonparametric Tests

Probability statements obtained from most nonparametric statistics are exact probabilities, regardless of the shape of the population distribution from which the random sample was drawn

If sample sizes as small as N=6 are used, there is no alternative to using a nonparametric test

Siegel, 1956

Advantages of Nonparametric Tests

Treat samples made up of observations from several different populations.

Can treat data which are inherently in ranks as well as data whose seemingly numerical scores have the strength in ranks

They are available to treat data which are classificatory Easier to learn and apply than parametric tests

Siegel, 1956

Criticisms of Nonparametric Procedures

Losing precision/wasteful of data Low power False sense of security Lack of software Testing distributions only Higher-ordered interactions not dealt with

Power of a Test

Statistical power – probability of rejecting the null hypothesis when it is in fact false and should be rejected

– Power of parametric tests – calculated from formula, tables, and graphs based on their underlying distribution

– Power of nonparametric tests – less straightforward; calculated using Monte Carlo simulation methods

شاخص های رابطه در تحلیل های پارامتری (ضریب های همبستگی دومتغیری در تحلیل های پارامتری)

دامنه شاخص مقیاس اندازه گیری ضریب همبستگی

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بزرگتر باشد

یک متغیر فاصله ای و دیگری اسمی دو

سطحی با فرض متصل بودن توزیع

زیربنایی

ضریب همبستگی دو رشته 1.)rb(ای

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هر دو متغیر اسمی دو سطحی با فرض متصل بودن توزیع

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شاخص های اندازه گیری رابطه در تحلیل های ناپارامتری (ضرایب همبستگی در

تحلیل های ناپارامتری)موارد کاربرد

مقیاس اندازه گیری و حدود

ضریب همبستگیتعداد متغیرها

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هر دو متغیر اسمی و چند سطح دارد

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رابطه رفتار ماقبل و مابعد را در یک توالی از

رفتارها مطرح میکند

B و Aمتغیرهای اسمی وچند

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ضریب همبستگی )rs(اسپیرمن

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هر دو متغیر رتبه ای

ضریب رتبه ای تاو )τ(کندال

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مانند رابطه سطح تحصیالت مادران با

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متغیرها در مقیاس ترتیبی

ضریب توافق کندال )U(

2بیشتر از 11