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

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Title: Parametric versus Nonparametric Statistics When to use them and which is more powerful


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
    Siegels 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

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(n11) 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(n11) 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 gt 30, Ho is rejected
Zar, 1996
10
Differences between independent groups
  • Multiple groups

11
Differences between dependent groups
  • Compare two variables measured in the same sample
  • If more than two variables are measured in same
    sample

12
Relationships between variables
  • Two variables of interest are categorical

13
Summary Table of Statistical Tests
 
(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 N6 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
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