Chapter
9: Nonparametric Techniques
l Parametric
statistics make assumptions about normality and homogeneity of variance of the distribution
l Nonparametric
statistics is referred to as distribution free, and no assumptions are made
about the distribution of scores.
l They are
versatile in that they can deal with ranked scores and categories.
l When
variables do not have precise, interval-type or ratio-type data.
l It could be
categories on a questionaire, or subjective rating instruments
l Numerical
counting of events in Qualitative research can be analyzed using nonparametric
techniques
l Main
Drawback is that Nonparametric are less Powerful than Parametric statistics
Chi
Square: Testing Observed vs. Expected
l Data sorted
into categories, such as sex, age, grade level , treatment groups are nominal
(categorical) data.
l A
researcher may be interested in determining whether the number of cases in each
category is different from what would be expected on the basis of chance,
utilizing some known source of information
Chi
Square
C2
= S[(O
– E)2/E]
O = observed frequency; E = expected frequency
Example: Question of whether a tennis court is jinxed. In several years there were 120 matches lost
on four courts numbered 1, 2, 3, 4. One
would expect that there would be 30 losses per court.
Court Number
1 2 3
4 Total
Observed Losses: O = 24
34 22 40 120
Expected Losses: E = 30 30
30 30 120
O-E -6 +4 -8
+10
(O – E)2 36 16 64
100
(O – E)2/E 1.20 0.53 2.13
3.33
S[(O – E)2/E] =
7.19
df = n – 1 = 4 – 1 = 3
Table A.7 critical value for 3 df is 7.82 at
.05 level
Example: Is Dr. Niceperson too lenient on
her grading practices. Her observed
grades are compared with the department’s prescribed normal curve grade
distribution:
l The X2
= 46.46
l In Table
A.7 for 4 df at the .01 level of significance a score of 13.28 is needed for
significance.
l What is the
interpretation?
l Dr.
Niceperson is giving too many A’s and B’s, and too few D’s and F’s
Contingency
Table
l Often there
are two or more categories and two or more groups
l Questionaires
or attitude inventories
l Example: A
group of athletes and nonathletes take a sportsmanship inventory: “A baseball
player who traps a fly ball between the ground and his glove should tell the
umpire that he did not catch it.”
In table A.7 for
(r-1)(c-1) df or
(2-1)(3-1)=2 df a chi square of 9.21 is
needed at the .01 level of significance.
Restrictions using Chi Square
The observations must be independent
The categories must be mutually exclusive
Should not be used for small samples; the
expected frequency for any cell must not be less than one
A 2X2 contingency table should have a
correction for continuity call the Yates Correction for Continuity:
Corrected X2 = S[(O – E - 0.5)2/E]
Total N in a contingency table should not be
less than 20.
Rank-Order Tests
Mann-Whitney U test – analogous to the
parametric independent t-test
Wilcoxon matched-pairs signed-ranks test –
analogous to the dependent t-test
Kruskal-Wallis ANOVA by ranks – analogous to
the one-way ANOVA
Friedman two-way ANOVA by ranks – analogous
to the repeated-measures ANOVA
Spearman Rank-difference correlation –
analogous to the Pearson product moment correlation r.
Rank-Order Tests
These tests are not the best statistical
tests.
These tests have been very popular and are
frequently found in the literature.
The better procedure to follow is:
l Change all data values to ranks
l Run the same parametric statistical tests
l Calculate the L-statistic as the significant test and
compare to the X2 table
Correlation
Example
l Correlation
between biceps and forearm skinfolds on 157 participants
l Figure 9.1
l Data not
normally distributed
l Data skewed
and kurtotic
l r =
0.28 using ranks
l L = (N-1)r2
= (157-1)(.28) = 12.23
l Compare to
X2 table for 1 df = 10.83 for p=.001; indicating that this r is significant