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Introduction to Nonparametric
Statistics
Nonparametric Tests
• Nonparametric tests are sometimes called distribution-free
tests because they are based on fewer assumptions (e.g., they do
not assume that the outcome is approximately normally
distributed).
• Parametric tests involve specific probability distributions (e.g., the
normal distribution) and the tests involve estimation of the key
parameters of that distribution (e.g., the mean or difference in
means) from the sample data.
• Non-parametric tests are typically focused on the median (rather
than on the mean) and involve fairly straight-forward procedures
like ordering and counting.
• The cost of fewer assumptions is that nonparametric tests are
generally less powerful than their parametric counterparts (i.e.,
when the alternative is true, they may be less likely to reject H0).
• There are situations, particularly in psychological or in market
research studies, where in the basic assumptions underlying the
parametric tests are not valid or one does not have the knowledge
of the distribution of the population parameter being tested.
• The most practical approach to assessing normality involves
investigating the distributional form of the outcome in the sample
using a histogram.
• There are some situations when it is clear that the outcome does
not follow a normal distribution. These include situations:
– when the outcome is an ordinal variable or a rank,
– when there are definite outliers or
– when the outcome has clear limits of detection.
• The following are some of the typical situations for using
nonparametric tests:
i) In a consumer behaviour survey for new package design, the
response are not likely to be normally distributed but clustering
around two extreme positions, with a very few respondents giving
a neutral response to the package design.
ii) Sometimes, the responses to a question are given in terms of
names (nominal data), which cannot be treated as numbers. For
example, if we ask young graduates “in which part of the country
would you like to take up a job and live”, the replies could be
north, north-west, west or south, etc. Nominal data can be
analysed only by nonparametric methods.
iii) In mailed questionnaire method of survey, more often partially
filled missions data and make necessary adjustments to extract
maximum information form the available data.
iv) Nonparametric tests can be used to provide reasonably good
results even for very small samples.
Measurements or oberrvations for use of non-parametric Statistics
• Using an Ordinal Scale
Consider a field demonstration where study participants are asked to rate the
effectiveness of training on the assigned topic of interest. Training rating might
be measured on a 5 point ordinal scale with response options: much worse,
slightly worse, no change, slightly effective, or much effective.
• When the Outcome is a Rank
In some studies, the outcome is a rank. For example, in organoleptic studies
score is often used to assess the quality of the product. These scores generally do
not follow a normal distribution.
• When There Are Outliers
In some studies, the outcome is continuous but subject to outliers or extreme
values.
• Limits of Detection
In some studies, the outcome is a continuous variable that is measured with
some imprecision (e.g., with clear limits of detection). For example, some
instruments or assays cannot measure presence of specific quantities above or
below certain limits. In social sciences some measurement never go below or
above certain values.
Ex. Scientific equipments, Age of farmer, etc.
Test
Parametric
Non Parametric
One Quantitative Response
Variable
One Sample ttest
Sign Test
One Quantitative Response
Variable – Two Values from
Paired Samples
Paired Sample t- Wilcoxon Signed
test
Rank Test
One Quantitative Response
Variable – One Qualitative
Independent Variable with
two groups
Two
Independent
Sample t-test
Wilcoxon Rank
Sum or Mann
Whitney Test
One Quantitative Response
Variable – One Qualitative
Independent Variable with
three or more groups
ANOVA
Kruskall Wallis
Runs Test for Detecting Non-randomness
• The runs test suggested by Bradley(1968) can be used to decide if
a data set is from a random process.
• A run is defined as a series of increasing values or a series of
decreasing values. The number of increasing, or decreasing,
values is the length of the run.
• In a random data set, the probability that the (I+1)th value is
larger or smaller than the Ith value follows a binomial
distribution, which forms the basis of the runs test.
• The first step in the runs test is to count the number of runs in the
data sequence. For example, a series of 20 coin tosses might
produce the following sequence of heads (H) and tails (T).
HHTTHTHHHHTHHTTTTTHH
The number of runs for this series is nine. There are 11 heads and
9 tails in the sequence.
Runs Test
• We will code values above the median as positive and values below
the median as negative. A run is defined as a series of consecutive
positive (or negative) values.
• Hypothesis for runs test is defined as:
H0: the sequence was produced in a random manner
H1: the sequence was not produced in a random manner
•
R  r
Test Statistic:The test statistic is
Z
where

2n1n2
r 
1
n1  n2
r
(2n1n2 )( 2n1n2  n1  n2 )
r 
(n1  n2 ) 2 (n1  n2  1)
R is the observed number of runs
n1 = Number of occurrences of first type(Positive)
n2 = Number of occurrences of second type(Negative)
The runs test rejects the null hypothesis If |Z| > Z1-α/2
Runs Test
• For a large-sample runs test (where n1 > 10 andn2 > 10), the
test statistic is compared to a standard normal table. That is,
at the 5 % significance level, a test statistic with an absolute
value greater than 1.96 indicates non-randomness.
• For a small-sample runs test, there are tables to determine
critical values that depend on values of n1 and n2
The Runs Test
(Small Sample Example)
Sequence
1
2
3
4
5
6
7
8
9
10
Number
0.34561
0.42789
0.36925
0.89563
0.25679
0.92001
0.58345
0.23114
0.12672
0.88569
Code
+
+
+
+
Sequence Number
11
0.67201
12
0.23790
13
0.24509
14
0.01467
15
0.78345
16
0.69112
17
0.46023
18
0.38633
19
0.60914
20
0.95234
Code
+
+
+
+
+
The Runs Test
(Small Sample Example)
H0: Computer-generated numbers are random
between 0.0 and 1.0.
HA: Computer-generated numbers are not random .
--- + - ++ -- ++ --- ++ -- ++
Runs: 1
2 3 4
5
6
7
8
9
10
There are r = 10 runs
From runs table (Appendix K) with n1 = 9
and n2 = 11, the critical value of r is 6
The Runs Test
(Small Sample Example)
Test Statistic:
R= 10 runs
Critical Values from Runs Table:
Possible
Runs:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Reject H0
Do not reject H0
Reject H0
Decision:
Since R = 10, we do not reject the null hypothesis.
Large Sample Runs Test
MEAN AND STANDARD DEVIATION FOR r
2n1n2
r 
1
n1  n2
(2n1n2 )( 2n1n2  n1  n2 )
r 
2
(n1  n2 ) (n1  n2  1)
where:
n1 = Number of occurrences of first type
n2 = Number of occurrences of second type
Large
Sample
Runs
Test
TEST
STATISTIC
FOR LARGE
SAMPLE
RUNS
TEST
z
R  r
r
Large Sample Runs Test
(Example )
OOOUOOUOUUOOUUOOOOUUOUUOOO
UUUOOOOUUOOUUUOUUOOUUUUU
OOOUOUUOOOUOOOOUUUOUUOOOU
OOUUOUOOUUUOUUOOOOUUUOOO
n1 = 53 “O’s”
n2 = 47 “U’s”
r = 45 runs
Large Sample Runs Test
(Example)
H0: Yogurt fill amounts are randomly distributed above and below 24-ounce level.
H1: Yogurt fill amounts are not randomly distributed above and below 24-ounce
level.
 = 0.05
Rejection Region
 /2 = 0.025
Rejection Region
 /2 = 0.025
z.025  1.96
z
R  r
r
0

z.025  1.96
45  50.82
 1.174
4.95659
Since z= -1.174 > -1.96 and < 1.96, we do not reject H0,
Mann-Whitney U Test
The Mann Whitney U test can be used to compare two
samples from two populations if the following
assumptions are satisfied:
• The two samples are independent and random.
• The value measured is a continuous variable.
• The measurement scale used is at least ordinal.
• If they differ, the distributions of the two
populations will differ only with respect to the
central location.
Mann-Whitney U Test
U-STATISTICS
n1 (n1  1)
U1  n1n2 
  R1
2
n2 (n2  1)
U 2  n1n2 
  R2
2
where:
n1 and n2 are the two sample sizes
R1 and R2 = Sum of ranks for samples 1 and 2
Mann-Whitney U Test
- Large Samples MEAN AND STANDARD DEVIATION FOR THE USTATISTIC
n1n2

2
(n1 )( n2 )( n1  n2  1)

12
where:
n1 and n2 = Sample sizes from populations 1 and 2
Mann-Whitney U Test
- Large Samples -
MANN-WHITNEY U-TEST STATISTIC
z
n1n2
U
2
(n1 )( n2 )( n1  n2  1)
12
Mann-Whitney U Test
(Example 15-4)
H 0 : ~1  ~2  0
H A : ~1  ~2  0
  0.05
Rejection Region
 = 0.05
z  1.645
z
n1n2
U
2

(n1 )( n2 )( n1  n2  1)
12
~1  ~2  0
27,412  29,088
 1.027
(144)( 404)(144  404  1)
12
Since z= -1.027 > -1.645, we do not reject H0,
Wilcoxon Matched-Pairs Test
The Wilcoxon matched pairs signed rank test can be
used in those cases where the following assumptions
are satisfied:
• The differences are measured on a continuous
variable.
• The measurement scale used is at least interval.
• The distribution of the population differences is
symmetric about their median.
Wilcoxon Matched-Pairs Test
WILCOXON MEAN AND STANDARD DEVIATION
n(n  1)

4
n(n  1)( 2n  1)

24
where:
n = Number of paired values
Wilcoxon Matched-Pairs Test
WILCOXON TEST STATISTIC
z
n(n  1)
T
4
n(n  1)( 2n  1)
24
Kruskal-Wallis One-Way Analysis
of Variance
Kruskal-Wallis one-way analysis of variance can be used
in one-way analysis of variance if the variables satisfy
the following:
• They have a continuous distribution.
• The data are at least ordinal.
• The samples are independent.
• The samples come from populations whose only
possible difference is that at least one may have a
different central location than the others.
Kruskal-Wallis One-Way Analysis
of Variance
H-STATISTIC
k
2
i
R
12
H
 3( N  1), with df  k  1

N ( N  1) i 1 ni
where:
N = Sum of sample sizes in all samples
k = Number of samples
Ri = Sum of ranks in the ith sample
ni = Size of the ith sample
Kruskal-Wallis One-Way Analysis
of Variance
CORRECTION FOR TIED RANKINGS
g
1
 (t
i 1
3
i
 ti )
N N
3
where:
g = Number of different groups of ties
ti = Number of tied observations in the ith tied
group of scores
N = Total number of observations
Kruskal-Wallis One-Way Analysis
of Variance
H-STATISTIC CORRECTED FOR TIED RANKINGS
2
i
k
H
R
12
 3( N  1)

N ( N  1) i 1 ni
g
1
 (t
i 1
3
i
 ti )
N N
3
Key Terms
• Kruskal-Wallis One-Way
Analysis of Variance
• Mann-Whitney U Test
• Nonparametric
Statistical Procedure
• Run
• Runs Test
• Wilcoxon Test