Nonparametric and Distribution- Free Statistics _contd

Lecture Series on
Biostatistics
No. Bio-Stat_12
Date – 16.09.2008

Nonparametric and
Distribution- Free Statistics
By
Dr. Bijaya Bhusan Nanda,
M. Sc (Gold Medalist) Ph. D. (Stat.)
Topper Orissa Statistics & Economics Services, 1988
bijayabnanda@yahoo.com

CONTENTS
 Introduction
 The Sign Test
 The Wilcoxon Signed- Rank Test for Location
 The Median Test
 The Mann-Whitney Test
 The Kolmogorov- Smirnov Goodness-of-fit Test
 The Kruskal-Wallis one-way analysis of variance
by ranks

Introduction
 Those procedures that test hypotheses that are
not statements about population parameters are
classified as nonparametric.
 Those procedures that make no assumption
about the sampled population are called
distribution free procedures.
 However, both these terms Nonparametric and
distribution free are used interchangeably and
under the common heading ‘Nonparametric’.

Advantages:
 Allow for the testing of hypothesis that are not
statements about population parameter values.
 It may be used when the form of the sampled
population is unknown.
 Computationally easier.
 May be applied when the data being analyzed consists
merely of rankings or classifications.
Disadvantages:
 Application of Nonparametric procedure with data
that can be handled with a parametric procedure
results in a waste of data.
 Application of the Nonparametric tests may be
laborious for large samples.

The Sign Test
 The ‘t’ test is not strictly valid for testing hypothesis
regarding single population mean or mean difference
between pairs of observations unless the relevant
populations are at least approximately normally
distributed.
 A frequently used nonparametric test that doesn’t
depend on the assumptions of ‘t’ test is the sign test.
 This test focuses on the median rather than the mean as
a measure of central tendency or location.
 The only assumption underlying the test is that the
distribution of the variable of the interest is continuous.

Example: In a school for the mentally retarded, 10 girls
selected at random received special instruction in personal
care and grooming. Two weeks after completion of the course
of instruction the girls were interviewed by a nurse and a
social worker who assigned each girl a score based on her
general appearance. The investigators believed that the score
achieved the level of an ordinal scale. The scores are shown in
the table. We wish to know if we can conclude that the
median score of the population from which we assume this
sample to have been drawn is different from 5
Table: General appearance Scores of 10 mentally Retarded Girls
Girl Score Girl Score Girl Score
1 4 5 9 9 6
2 5 6 6 10 6
3 8 7 10
4 8 8 7

Data: The data are given in the table
Assumption: We assume that that the measurements are
taken on a continuous variable.
Hypothesis: H0: The population median is 5
H: The population median is not 5
Let α =0.05
Test Statistic (TS): This will depend upon the HA:
If HA: P(+) > P(-) A sufficiently smaller number of – signs
causes rejection of H0 . TS = ‘K’ =No. of – signs.
If HA: P(+) < P(-) A sufficiently smaller number of + signs
causes rejection of H0. TS = ‘K’=No. of + signs.
If HA: P(+) ≠ P(-) Either a sufficiently smaller number of +
signs or – signs causes rejection of H0. TS= ‘K’ Less frequently
occurring signs of the two. This is the TS in the present
example.

Distribution of Test statistic: If we assign a +sign to those
score that lies above the hypothesized median of 5 and a –
sign to those that fall below. Then the table is
Girl 1 2 3 4 5 6 7 8 9 10
Score - 0 + + + + + + + +
From the table we constitute a set of n independent random
variable from the Bernoulli population with parameter p. If
we let K= the TS, the sampling distribution of K is the
binomial distribution with parameter p=0.5 if H0 is true.
Decision Rule:
For Reject H0 if the ‘p’ value for the computed test statistic
is less than or equal to 0.05.
Calculation of test statistic: Determine the probability of
observing S or fewer –signs when given a sample of size n
and parameter p=0.5.

Decision rule: The decision rule depend on the alternative
hypothesis.
For HA: P(+) ≠ P(-) reject H0 if (given H0 is true) the
probability of obtaining a value of ‘S’ as extreme as or more
extreme than the actually computed is equal to or less than
α/2.
P(k≤ x| n, p) = Σ nCkpkqn-k
For the example
9C0(0.5) (0.5) + 9C1(0.5) (0.5)
0 9-0 1 9-1

= .00195+.01758
= .0195
Here there are fewer minuses so we focus our attention on
minuses rather than pluses.
Statistical decision: If the number of minuses is so small that
the probability of observing this few of fewer is less than
0.025, we will reject the null hypothesis. The probability .
0915 is less than .025 so reject the null hypothesis.
Conclusion: The median score is not 5.
p value: The p value for this test is 2(.0195) = 0.0390

Sign Test-Paired Data:
Conditions
 The data to be analyzed consist of observation in matched pairs
 The measurement scale is weak.
 Sign test may be employed to test the null hypothesis that the
median difference is zero
 Say (Yi, Xi) are pairs of matched scores, i = 1,2,……..n
One of the matched scores, say Yi is subtracted from the other
score Xi. If Yi<Xi, the sign of the difference is + and if Yi>Xi
the sign of the difference is - Then the null hypothesis is
H0: P(+) = P(-) = 0.5 or P (Xi > Yi) = P (Xi<Yi) =0.5
HA: P(+) ≠ P(-) or HA: P(+) > P(-) or HA: P(+) < P(-)

Example:
A dental research team wished to know if teaching people
how to brush their teeth would be beneficial. 12 pairs of
patients seen in a dental clinic were obtained by carefully
matching on such factors as age, sex, intelligence and initial
oral hygienic scores. One member of each pair received
instruction on hoe to brush the teeth and other oral hygienic
matters. Six months later all 24 subjects were examined and
assigned an oral hygienic score by a dental hygienist on
aware of which subjects had received the instruction. A low
score indicates a high level of oral hygienic. The results are
shown in Table.
Pair No 1 2 3 4 5 6 7 8 9 10 11 12
Inst. 1.5 2.0 3.5 3.0 3.5 2.5 2.0 1.5 1.5 2.0 3.0 2.0
(Xi)
Not 2.0 2.0 4.0 2.5 4.0 3.0 3.5 3.0 2.5 2.5 2.5 2.5
inst.
(Yi)

Solution:
Assumption: We assume that the population of difference
between two pairs of scores is a continuous variable.
Hypothesis:
H0: The median of the differences is zero [P(+)≥P(-)]
HA: The median of the differences is negative [P(+)<P(-1)
Let α =0.05.
Test Statistic: The test statistic is the number of +signs.
Distribution of Test Statistic: The sampling distribution of
k is the binomial distribution with parameters n and 0.5
if H0 is true.

Decision rule: Reject H0 if P(S ≤ 2| 11,5)≤ 0.05.
Calculation Of Test Statistic: The score difference have
been obtained for each pair. Then the result is in Table.
Pair 1 2 3 4 5 6 7 8 9 10 11 12
Score - 0 - + - - - - - - + -
The nature of the hypothesis indicates a one sided test so
that all of α = 0.05 is associated with the rejection region,
which consists of all values of S (equal to number of
+signs).
The experiment yielded 1zero , 2pluses and 9 minuses.
When we eliminate 0, the effective sample size is n=11
with 2 pluses and 9minuses. Since a small no of +signs
will cause rejection of the null hypothesis, test statistic is
S=2.

Statistical Decision: The probability of obtaining no more
than 2 pluses out of eleven tries when the null hypothesis
is true is
P(S ≤ 2| 11, 5) = Σ 11CS(.5)S(.5)11-S. This probability is to be .
0327. since .0327 is less than .05, we must reject H0.
Conclusion: Then we conclude that the median difference is
negative and the instruction was beneficial.
p value: The p value is 0.0327

Sample size
 When the sample size is large and p is close to 0.5 the
binomial distribution may be approximated by
normal distribution.
 According to the rule of thumb when both np and nq
are greater than 5, normal approximation is
appropriate.
 Under H0 p = 0.5, a sample size of 12 would satisfy
normal approximation.
 In this situation normal z test can be applied for sign
test.
 The test statistic here

( k ± 0.5 )- 0. 5n
z= 0 . 5 √n

The Wilcoxon Signed- Rank Test for Location
 To test a null hypothesis about a population mean , but
for some reason neither z nor t is an appropriate test
statistic.
 Wilcoxon Signed- Rank Test is a appropriate test which
makes use of the magnitudes of the differences between
measurements and a hypothesized location parameter
rather than just the signs of the differences.
Assumptions:
 The sample is random
 The variable is continuous.
 The population is symmetrically distributed about its
meanµ .
 The measurement scale is at least interval.

Hypothesis:
The hypothesis are :
(a) H0:µ =µ 0 (b) H0:µ≥µ 0 (c ) H0:µ≤µ 0
HA: µ≠µ 0 HA: µ <µ 0 H A: µ > µ 0
Calculation:
1. Subtract the hypothesized mean µ 0 from each
observation xi, to obtain di = xi- µ 0.
2. Rank the usable di from the smallest to the largest
without regard to the sign of di. If two or more | di| are
equal, assign each tied value the mean of the rank
positions the tied values occupy.
3. Assign each rank the sign of the di that yields that
rank.
4. Find T+, the sum of the ranks with +ve sign and T-, the
sum of the ranks with –ve sign.

The Test Statistic:
The Wilcoxon test statistic is either T+ or T-, depending on
the nature of the alternative hypothesis.
 When the alternative hypothesis is two-sided (μ ≠ μ0)

either a sufficiently small value of T+ or a sufficiently
small value of T- will cause us to reject H0:µ =µ 0. The test
statistic then is T+ or T-, whichever is smaller.
 When H :µ≥µ is true we expect our sample to yield a large
0 0
value of T+ . Therefore, when the one-sided alternative
hypothesis state that the true population mean is less than
the hypothesized mean (µ <µ 0 ), a sufficiently small value
of T+ will cause rejection of H0. and T+ is the test statistic.
 When H :µ≤µ is true we expect our sample to yield a large
0 0
value of T- . Therefore, for the one-sided alternative HA:
µ >µ 0 , a sufficiently small value of T- will cause rejection of
H and T- is the test statistic.

Critical Value:
The following are the decision rules for the three
possible alternative hypothesis:
(a) HA: µ≠µ 0. Reject H0 at the α level of significance if the
calculated T is smaller that or equal to the
tabulated T for n and preselected α/2. Alternatively,
we may enter Table K with n and our calculated T is
less than or equal to our Stated level of significance.
If so, we may reject H0.
(b) HA: µ <µ 0. Reject H0 at the α level of significance if the
calculated T+ is less than or equal to the tabulated T
for n and preselected α.
(c) HA: µ >µ 0. Reject H0 at the α level of significance if the
calculated T- is less than or equal to the tabulated T
for n and preselected α.

Example:
Cardiac output(lit/min) was measured by thermodilution
in a simple random sample of 15 postcardiac surgical
patients in the left lateral position. The results were
4.91 4.10 6.74 7.27 7.42 7.50 6.56 4.64
5.98 3.14 3.23 5.80 6.17 5.39 5.77
We wish to know if we can conclude on the basis of these
data that the population mean is different from 5.05.
Solution:
Assumption: We assume that the requirements for the
application of the Wilcoxon signed-ranks test are met.
Hypothesis:H0: µ = 5.05
HA: µ ≠ 5.05 and Let α = 0.05
Test Statistic: The test statistic will be T+ or T-, whichever
is smaller.

Distribution of Test Statistic: Critical values of the test
statistics are given in Table K.
Decision Rule: The decision rule is reject H0 if the
computed value of T is less than or equal to 25, the
critical value for n=15, and α /2 = .0240, the closest
value to .0250 in Table K.
Calculation of test statistic: The calculation of the test
statistic is in the table-

Table: Calculation of the test statistic
Cardiac output di=xi-5.05 Rank of |di| Signed rank of |di|
4.91 -.14 1 -1
4.10 -.95 7 -7
6.74 +1.69 10 +10
7.27 +2.22 13 +13
7.42 +2.37 14 +14
7.50 +2.45 15 +15
6.56 +1.52 9 +9
4.64 _.41 3 -3
5.98 +.93 6 +6
3.14 -1.91 12 -12
3.23 -1.82 11 -11
5.80 +.75 5 +5
6.17 +1.12 8 +8
5.39 +.34 2 +2
5.77 +.72 4 +4

T+ = 86 , T- =34, T = 34.
Statistical Decision: Since 34 is greater than 25, we are
unable to reject H0 .
Conclusion: We conclude that the population mean may
be 5.05
p value: From Table K we see that the p value is p =
2(.0757) = 0.1514.

The Median Test
 A nonparametric procedure that may be used to test the
null hypothesis that two independent samples have been
drawn from populations with equal medians is the
median test.
Example:
Do urban and rural male junior high school students
differ with respect to their level of mental health?
Urban Rural Urban Rural
35 29 25 50
26 50 27 37
27 43 45 34
21 22 46 31
27 42 33
38 47 26
23 42 46
25 32 41

Solution:
Assumption: The assumptions are the samples are selected
independently ,at random from their respective
population and the populations are of the same form,
differing only in location and the variable of interest is
continuous .
Hypothesis: H0:MU =MR and HA: MU ≠ MR.
MU is the median score of the sampled population of
urban students and MR is the median score of the
sampled population of rural students.
Let α = 0.05.
Test Statistic: The test statistic is X2 as computed.
Distribution of test statistic: When H0 is true and the
assumptions are met, X2 with 1 degree of freedom.
Decision Rule: The decision rule is reject H0 if the
computed value of X2 is ≥ 3.841(since α =0.05)

Calculation of test statistic: The first step in calculating the test
statistic is to compute the common median i.e. (33+34)/2 = 33.5.
The result according to for each group the number of
observations falling above and below the common median is
Urban Rural Total
No. of scores 6 8 14
above median
No. of scores 10 4 14
below median
Total 16 12 28

The test statistic is χ2 = 28[(6)(4) – (8)(10)]2 = 2.33
(16)(12)(14)(14)
Statistical decision: Since 2.33< 3.841, the critical value of X2
with α=.05 and 1 degree of freedom, we are unable to reject the
null hypothesis on the basis of these data.
Conclusion: The two samples may have been drawn from
populations with equal median.
p value: The p value is p>0.10 since 2.33 < 2.706.

Handling Values Equal to the Median:
When one or more observed values will be exactly equal to the
common median and, hence, will fall neither above nor below it
and if n1+n2 is odd, at least one value will always be exactly equal
to the median one solution is drop them from the analysis if
n1+n2 is large and there are only a few values that fall at the
combined median or we may dichotomize the scores into those
that exceed the median and those that do not, in which case the
observations that equal to the median will be counted in the
second category.
Median Test extension:
The median test extends logically to the case where it is desired
to test the null hypothesis that k≥3 samples are from populations
with equal medians. For this test a 2×k contingency table may
be constructed by using the frequencies that fall above and
below the median computed from combined samples. If
conditions as to sample size and expected frequencies are met, χ2
may be computed and compared with the critical χ2 with k-1
degrees of freedom.

The Mann-Whitney Test
 If for testing the desired hypothesis, there is available a
procedure that makes use of more of the information
inherent in the data , that procedure should be used if
possible
 Such a nonparametric procedure that can often be used
instead of the median test is called the Mann-Whitney
Test.
Assumption:
 The two sample of size n and m, respectively available
for analysis have been independently and randomly
drawn from their respective populations.
 measurement scale is at least ordinal.
 variable of interest is continuous.
 If the population differ at all, they differ only with
respect to their medians.

Hypothesis:
When these assumptions are met we may test the null
hypothesis that the two populations have equal medians
against either of the three possible alternatives:
(1) the populations don’t have equal medians(two sided
test)
(2) the median of population 1 is larger than the median
of population 2(one sided test)
(3) the median of population 1 is smaller than the
median of population 2(one-sided test).
Example:
A researcher designed an experiment to asses the effects of
prolonged inhalation of cadmium oxide. 15 laboratory
animals served as experimental subjects, while 10 similar
animals served as controls. The variable of interest was
hemoglobin level following the experiment. If we can
conclude that prolonged inhalation of cadmium oxide

Solution:
Assumption: We presume that the assumptions of the
Mann- Whitney test are met.
Table: Hemoglobin determination (grams) for 25
laboratory animals
Exposed animals (X) Unexposed animals (Y)
14.4 17.4
14.2 16.2
13.4 17.1
16.5 17.5
14.1 15.0
16.6 16.0
15.9 16.9
15.6 15.0
14.1 16.3
15.3 16.8
15.7
16.7
13.7
15.3

The null and alternative hypothesis are as follows:
H0: MX≥ MY
HA:MX< MY where MX is the median of a population of
animals exposed to cadmium oxide and MY is the
median of a population of animals not exposed to the
substance. Let α = 0.05.
Test statistic :
1. Two samples are combined
2. All observations are ranked from smallest to largest
keeping track of the sample to which each
observations belongs.
3. Tied observations are assigned a rank equal to the
mean of the rank position for which they are tied.
The test statistic:
T = S - n (n+1)
Where n is the size of sample X and S is corresponding
2
sum of ranks.

Distribution Of test: Critical values from the
distribution of the test statistic are given in Table L
for various levels of α.
Decision Rule: In general, for one-sided tests of the type
illustrated here the decision rule is :
Reject H0: MX≥ MY if the computed t is less than wα is
the critical value of T obtained by entering Table L
with n, the number of X observations, m, the number
of Y observations, and α, the chosen level of
significance.
Calculation of test statistic: S=145, so that
T = 145 - 15 (15+1) = 25
2

Statistical decision:
when we enter Table L with n=15, m= 10, and α =
0.05, we find the critical value of wα to be 45. since
25<45, we reject H0.
Conclusion: We conclude that MX is smaller than MY.
This leads to the conclusion that prolonged inhalation
of cadmium oxide does reduce the hemoglobin level.
p value: since 22<25<30, we have for this test
0.005>p>0.001.

The Kolmogorov- Smirnov Goodness-of-fit Test
 The Kolmogorov’s work concerned with the one-sample
case as discussed and Smirnov’s work deals with the
case involving two samples in which interest centers on
testing the hypothesis that the distributions of the two
parent populations are identical .
Test Statistic:
 Kolmogorov- Smirnov goodness-of-fit test a comparison
between some theoritical cumulative distribution
function, FT(x), and a sample cumulative distribution
function FS(x).
 The difference between the theoretical cumulative
distribution function, FT(x), and the sample cumulative
distribution function FS(x), is measured by the statistic D
which is the greatest vertical distance between F (x) and

 When a two sided test is appropriate, that is when the
hypothesis are
H0: F(x) =FT(x) for all x from -∞ to +∞
HA: F(x) ≠ FT(x) for at least one x the statistic is
D = sup | Fs(x) - Ft(x) |
x
which is read, “ d equals the supremum (greatest) over
all x, of the absolute value of the difference Fs(x) - Ft(x).”
 The null hypothesis is rejected at the α level of
significance if the computed value of D exceed the for 1-
α (two sided) and the sample size n.
Assumption:
The assumptions are
1.Sample is a random sample
2. Hypothesized distribution FT(x) is continuous.

 When values of D are based on a discrete theoretical
distribution, the test is conservative.
 When the test is used with discrete data, then, the true
probability of committing a type-I error is at most equal
to α , the stated level of significance.
Example:
Fasting, blood glucose determinations made on 36 nonobese,
apparently healthy, adult males in table. We have to test if
we may conclude that these data are not from a normally
distributed population with a mean of 80 and a standard
deviation of 6.
Solution:
Assumption: We assume that the sample available is a
simple random sample from a continuous population
distribution.
Hypothesis: H0: F(x) =FT(x) for all x from -∞ to +∞

Table: Fasting blood glucose values(mg/100ml) for 36
Nonobese, apparently healthy, adult males
75 92 80 80 84 72
84 77 81 77 75 81
80 92 72 77 78 76
77 86 77 92 80 78
68 78 92 68 80 81
87 76 80 87 77 86

Distribution of test statistic: Critical value of the test statistic
for selected values of α are given in Table M.
Decision rule: Reject H0 if the computed value of D exceeds
0.221, the critical value of D for n=36 and α =0.05.
Calculation of test statistic: First step is to compute values of
FS(x). Each value of FS(x) is obtained by dividing the
corresponding cumulative frequency by the sample size like
first value of FS(x) = 2/36= 0.0556
Value of FT(x) can be obtained by converting each observed
value of x to a value of the standard normal variable, z. Then

Table: Value of FS(x) Table: Values of FT(x)
X F C.F FS(x) X Z=(x-80)/6 FT(x)
68 2 2 .0556 68 -2.00 .0228
72 2 4 .1111 72 -1.33 .0918
75 2 6 .1667 75 -.83 .2033
76 2 8 .2222 76 -.67 .2514
77 6 14 .3889 77 -.50 .3085
78 3 17 .4722 78 -.33 .3707
80 6 23 .6389 80 .00 .5000
81 3 26 .7222 81 .17 .5675
84 2 28 .7778 84 .67 .7486
86 2 30 .8333 86 1.00 .8413
87 2 32 .8889 87 1.17 .8790
92 4 36 1.000 92 2.00 .9772

Statistical decision: Reference to Table M reveals that a
computed D of 0.1547 is not significant at any reasonable level.
So we are not willing to reject H0.
Conclusion: The sample may have come from the specified
distribution.

The Kruskal-Wallis one-way analysis of variance
by ranks
 When the populations from which the samples are drawn
are not normally distributed with equal variances and
when the data for analysis consists only of ranks, a
nonparametric alternative to the one way analysis of
variance may be used to test the hypothesis of equal
location parameter.
 The best known of these procedures is the Kruskal-
Wallis one way analysis of variance by ranks.
 The application of the test involves the steps
 The n1, n2,…..,nk observations from the k samples are combined
in to a single series of size n and arranged in order of
magnitude from smaller to largest.
 Then the observations are then replaced by ranks from 1 to n
from smallest to the largest value.

The ranks assigned to observations in each of the k
groups are added separately to give k rank sums.
 The test statistic n(n+1) nj
n(n+1) nj
H=
H= Σ R -3(n+1)
Σ Rjj -3(n+1)
12
12 2
2
Is computed.
k = the number of samples
nj= the number of observation in the jth sample
n= the number of observations in all samples
combined
Rj= the sum of the ranks in the jth sample.
 When there are 3 samples and five or fewer
observations in each sample, the significance of the
computed H is determined by consulting Table.
 When there are more than five observations in one or
more of the samples, H is compared with tabulated
2

Example:
The effects of two drugs on reaction time to a certain stimulus
were studied in three samples of experimental animals. Sample-III
served as a control while the animals in sample-I were treated with
drug A and those in sample II were treated with drug B prior to the
application of the stimulus. Table shows the reaction times in seconds
of the animals.
Can we conclude that the three populations
represented by the three samples differ with respect to reaction time?
We can so conclude if we can reject the null hypothesis that the three
populations do not differ in their reaction times.
Table:1 Reaction time in
seconds of 13 experimental anima
l
Sample
I II III
17 8 2
20 7 5
40 9 4
31 8 3
35

Solution:
Assumption: The samples are independent random samples
from their respective populations. The measurement scale
employed is at least ordinal. The distribution of the values in
the sampled population are identical except for the possibility
that one or more of the populations are composed of values that
tend to be larger than the of the other populations.
Hypothesis:
H0: The population centers are equal
HA: At least one of the populations tends to exhibit larger
values than at least one of the other populations.
Let α=0.01
Distribution of test statistic: critical values of H for various
sample sizes and α levels are given in Table N.
Decision rule: The null hypothesis will be rejected if the
computed value of H is very large that the probability of
obtaining a value that large or larger when H0 is true or equal

Calculation of test statistic: When the three samples are
combined in to a single series and ranked , the Table of ranks
shown in Table 2.
Table:2 The data of Table 1
replaced by Ranks
Sample
I II III
9 6.5 1
10 5 4
13 8 3
11 6.5 2
12
R1=55 R2=26 R3=10

From the Table data and equation we obtain
H=
12
13(13+1) [(55)2 + (26)2 + (10)2 ] - 3(3+1)
5 4 4
= 10.68

Statistical decision: Table N shows that when the nj are 5, 4
and 4, the probability of obtaining a value of H ≥ 10.68 is
less than 0.009. The null hypothesis can be rejected at the .
01 level of significance.
Conclusion: We conclude that there is a difference in the
average reaction time among the three populations.
p value: p < 0.009.

Nonparametric and Distribution- Free Statistics _contd

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Nonparametric and Distribution- Free Statistics _contd