1. Parametric versus Nonparametric Statistics – When to use them and which is more powerful?
Angela Hebel
Department of Natural Sciences
University of Maryland Eastern Shore
April 5, 2002

2. Parametric Assumptions
The observations must be independent
The observations must be drawn from normally distributed populations
These populations must have the same variances
The means of these normal and homoscedastic populations must be linear combinations of effects due to columns and/or rows*

3. Nonparametric Assumptions
Observations are independent
Variable under study has underlying continuity

4. Measurement
What are the 4 levels of measurement discussed in Siegel’s chapter?
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

5. Nonparametric Methods
There is at least one nonparametric test equivalent to a parametric test
These tests fall into several categories
Tests of differences between groups (independent samples)
Tests of differences between variables (dependent samples)
Tests of relationships between variables

6. 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

7. 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

8. 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

9. U = n1n2 + n1(n1+1) – R1 2 U=(7)(5) + (7)(8) – 30 U = 35 + 28 – 30
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
U = – 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

10. Differences between independent groups
Multiple groups
Parametric
Nonparametric
Analysis of variance (ANOVA/ MANOVA)
Kruskal-Wallis analysis of ranks
Median test

11. 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

12. 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

13. Summary Table of Statistical Tests
Level of Measurement
Sample Characteristics
Correlation
1 Sample
2 Sample
K Sample (i.e., >2)
Independent
Dependent
Categorical or Nominal
Χ2 or bi-nomial Χ2
Macnarmar’s 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)

14. 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

15. 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

16. 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

17. 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 (Mumby, 2002)

18. Questions?