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

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
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*

Nonparametric Assumptions
Observations are independent
 Variable under study has underlying
continuity

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
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
KolmogorovSmirnov 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

U = n1n2 + n1(n1+1) – R1
2
U=(7)(5) + (7)(8) – 30
2
U = 35 + 28 – 30
U = 33
U’ = n1n2 – U
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
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
R2 = 48
Differences between independent
groups

Multiple groups
Parametric
Nonparametric
Analysis of Kruskal-Wallis
variance
analysis of
(ANOVA/
ranks
MANOVA)
Median test
Differences between dependent
groups


Compare two variables
measured in the same
sample
Parametric
t-test for
dependent
samples
If more than two
variables are measured in Repeated
same sample
measures
ANOVA
Nonparametric
Sign test
Wilcoxon’s
matched pairs
test
Friedman’s two
way analysis of
variance
Cochran Q
Relationships between variables
Parametric
Nonparametric
Correlation
coefficient
Spearman R
Kendall Tau
Coefficient Gamma

Two variables of
interest are
categorical
Chi square
Phi coefficient
Fisher exact test
Kendall coefficient of
concordance
Summary Table of Statistical Tests
Level of
Measurement
Sample Characteristics
1
Sample
Categorical
or Nominal
Χ2 or
binomial
Rank or
Ordinal
Parametric
(Interval &
Ratio)
z test
or t test
2 Sample
Correlation
K Sample (i.e., >2)
Independent
Dependent
Independent
Dependent
Χ2
Macnarmar’
s Χ2
Χ2
Cochran’s Q
Mann
Whitney U
Wilcoxin
Matched
Pairs Signed
Ranks
Kruskal Wallis
H
Friendman’s
ANOVA
Spearman’s
rho
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 (Mumby, 2002)
Questions?