Nonparametric test

1. Nonparametric Methods

2. parametric statistical methods：
Methods of estimation and hypothesis testing have
been based on these assumptions that data come
from some underlying distribution whose general
form was assumed.
( normal or binomial distribution)
nonparametric statistical methods:
it does not depend on the form of the underlying
distribution ;in particular, it does not depend on the
variable following a normal distribution

3. nonparametric statistical methods
1、assumptions about the shape of the distribution
are not made.
2、the central limit theorem also seems inapplicable
because of small sample size
3、Ordinal data
Ordinal data can be ordered but do not have specific
numeric values. Thus common arithmetic cannot be
performed on ordinal data in a meaningful way.

4. type of data
1、Cardinal data(measurement data)
2、 Ordinal data
3、 nominal data
different data values can be classified into categories
but the categories have no specific ordering
classifying cause of death blood type

5. Nonparametric Methods
The Wilcoxon Signed-Rank Test
1、Wilcoxon signed-rank test is a nonparametric test that
is analogous to the paired t test (paired sample)
2、Wilcoxon signed-rank is based on the ranks of
the observations rather than on their actual values, as is
the paired t test.

6. Frank Wilcoxon
Background
Rank and sum of rank
outliers

7. X 110 117 119 122 127 133 135 141
Y 120 127 132 140 143 162 177 181
Rank 1 2 3 4 5 6.5 6.5 8 9 10 11 12 13 14 15 16
Sum of ranks: RX： 48.5
Sum of ranks: Ry：87.5
Rank and sum of rank

8. The systolic blood pressure (in mmHg)of 6 patients
before and after treatment with a specific drug.
Example_ 1.0
number before after differences Rank with-out signs
1 132 136 +4 1
2 160 130 -30 5
3 145 128 -17 4
4 114 114 0
5 125 115 -10 2
6 128 117 -11 3
T+=1
T-=14 05
.
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0
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9. Method:
(1)Rank the differences ignoring the signs of the
difference omit zero differences
(2)Replace the original signs
(3)Add up the ranks of the positive differences
Add up the ranks of the negative differences
(4)Call the smaller of these two sums of ranks , T.
(5)Using the table11 for signed ranks test(n<16)
(6)When n>15,using normal approximation method

11. 0
R 0
1 2 3 4 5
R 1 2 3 4 5
1+2 1+3 1+4 1+5 2+3 2+4 2+5 3+4 3+5 4+5
R 3 4 5 6 5 6 7 7 8 9
1+2+3 1+2+4 1+2+5 1+3+4 1+3+5 1+4+5 2+3+4 2+3+5 2+4+5 3+4+5
R 6 7 8 8 9 10 9 10 11 12
1+2+3+4 1+2+3+5 1+2+4+5 1+3+4+5 2+3+4+5
R 10 11 12 13 14
1+2+3+4+5
R 15
2
5
C
0
5
C
1
5
C
3
5
C
4
5
C
5
5
C
The distribution of the sum of positive rank in n=5

12. Rank sum frequency Percent (%)
0 1 3.125
1 1 3.125
2 1 3.125
3 2 6.250
4 2 6.250
5 3 9.375
6 3 9.375
7 3 9.375
8 3 9.375
9 3 9.375
10 3 9.375
11 2 6.250
12 2 6.250
13 1 3.125
14 1 3.125
15 1 3.125
total 32 1.000
Frequency distribution of the sum of rank in n=5
the expected value and
variance of the rank sum
( ) 7.5,
var( ) 440/32 55/4 13.75
E R
R

  

13. the expected value and variance of the rank sum
(when there are no ties)
If the null hypothesis is true
3
1
1
var( ) ( 1)(2 1) / 24 ( ) / 48
g
i i
i
R n n n t t

    

when there are ties

14. Frequency distribution of the sum of rank in n=8

15. Frequency distribution of the sum of rank in n=15
~ ( ( ), ( ))
R N E R VAR R

16. Example_ 2.0 (Dermatology)
Suppose we want to compare the effectiveness of two
ointments (A, B) in reducing excessive redness in people who
cannot otherwise be exposed to sunlight.
Ointment A is randomly applied to either the left or right arm,
and ointment B is applied to the corresponding area on the
other arm.
45 person is then exposed to 10 minutes of sunlight, and the
two arms are compared for degrees of redness.
Suppose the degree of burn can be quantified on a 10-point
scale, with 10 being the worst burn and 1 being no burn at all.

17. compute di = xi − yi
where xi = degree of burn for ointment A
yi = degree of burn for ointment B.
If di is positive, then ointment B is doing better
than ointment A;
if di is negative, then ointment A is doing better
than ointment B.
How can this additional information be used to test
whether the ointments are equally effective?

18. oin_A oin_B d rank oin_A oin_B d rank
2 10 -8 40.0 5 2 3 28.0
1 8 -7 38.0 7 4 3 28.0
2 9 -7 38.0 9 7 2 19.5
1 8 -7 38.0 8 6 2 19.5
2 8 -6 35.5 7 5 2 19.5
3 9 -6 35.5 5 3 2 19.5
4 9 -5 33.5 5 3 2 19.5
2 7 -5 33.5 4 2 2 19.5
2 6 -4 32.0 5 4 1 7.5
2 5 -3 28.0 6 5 1 7.5
3 6 -3 28.0 5 4 1 7.5
4 7 -3 28.0 5 4 1 7.5
2 5 -3 28.0 4 3 1 7.5
5 8 -3 28.0 3 2 1 7.5
2 4 -2 19.5 2 1 1 7.5
1 3 -2 19.5 7 6 1 7.5
2 4 -2 19.5 6 5 1 7.5
5 7 -2 19.5 5 4 1 7.5
5 6 -1 7.5 5 5 0
6 7 -1 7.5 6 6 0
4 5 -1 7.5 5 5 0
3 4 -1 7.5 4 4 0
8 8 0
Difference in degree of redness between ointment A and ointment B arms
After 10 minutes of exposure to sunlight
Rank sum = 248
Rank sum = 572

19. ti

20. The test is based on the sum of the ranks, or
rank sum (R1)
R1=248 the group of people with positive di
R2=572 the group of people with negative di
1 2
1 2
(1 ) 40 41
820
2 2
(1 )
( ) ( ) 410
4
n n
R R
n n
E R E R
  
   
 
  
If the null hypothesis is true

21. Frequency distribution of the sum of rank in n=40 (m=50000)
~ ( ( ), ( ))
R N E R VAR R

22. (Normal Approximation Method for Two-Sided)
R1：the rank sum of the positive differences
ti : refers to the number of differences with the same absolute
value in the ith tied group
g： the number of tied groups

24. Example_ 2.0
05
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0
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：

25. We therefore can conclude that there is a significant
difference between ointments.
with ointment A doing better than ointment B because
the observed rank sum(248) is smaller than the
expected rank sum (410).

27. The Wilcoxon Rank-Sum Test
1、Wilcoxon rank Sum test is a nonparametric
test that is analogous to two independent
samples t test.
2、Wilcoxon rank Sum is based on the ranks of
the observations rather than on their actual
values, as is the t test.

28. Example_ 3.0
A researcher wants to know whether the protein level in the
Food can affect the weight. He randomly selected 22 rats
From animal center and randomly assigned them into two
groups. One group was given high-protein food and another
was fed with low-protein. The weights of each rat at baseline
and 3 months later were measured, respectively. The additive
weight of rats is shown in the following table.

29. Low protein High protein
36 134
118 146
93 104
85 120
107 124
94 161
119 107
87 38
59 113
65 129
65
123
n1＝10 n2＝12
Additive weight of rats after supplemented with low- or high-protein food

30. Low protein group High protein group
Additive weight rank Additive weight rank
36 1.0 134 20.0
118 14.0 146 21.0
93 8.0 104 10.0
85 6.0 120 16.0
107 11.5 124 18.0
94 9.0 161 22.0
119 15.0 107 11.5
87 7.0 38 2.0
59 3.0 113 13.0
65 4.5 129 19.0
65 4.5
123 17.0
n1＝10 1
T ＝79 n2＝12 2
T ＝174
1 2 (1 ) / 2 23 22 / 2 253
T T N N
      
1 1 2 2
1 1
( ) 10 11.5 115, ( ) 12 11.5 138
2 2
N N
E T n E T n
 
         

31. number 1 2 3 4 5 T_1
1 X1 X2 Y1 Y2 Y3 3
2 X1 Y1 X2 Y2 Y3 4
3 X1 Y1 Y2 X2 Y3 5
4 X1 Y1 Y2 Y3 X2 6
5 Y1 X1 Y2 Y3 X2 7
6 Y1 Y2 X1 Y3 X2 8
7 Y1 Y2 Y3 X1 X2 9
8 Y1 X1 X2 Y2 Y3 5
9 Y1 Y2 X1 X2 Y3 7
10 Y1 X1 Y2 X2 Y3 6
rank T_1 count frequency
3 1 0.1
4 1 0.1
5 2 0.2
6 2 0.2
7 2 0.2
8 1 0.1
9 1 0.1
sum 10 1.0
Frequency distribution of the T1 in n1=2， in n2=3
2
5 10
C 

32. 10
1 2 1 1 22
10, 12.T ( 10)C 646646
n n n
   
Frequency distribution of the T1
Using the appendix table 9 (n1<11, n2-n1<11)
T1=79 151

33. Normal Approximation Method
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34. Example_ 4.0 （Ophthalmology）
Different genetic types of the disease retinitis pigmentosa
(视网膜色素变性,RP) are thought to have different rates of
progression, with the dominant form of the disease progressing
the slowest, the recessive form the next slowest, and the sex-
linked form the fastest.
This hypothesis can be tested by comparing the visual
acuity of people who have different genetic types of RP

35. 367
.
35
30
1061
,
16
.
19
25
479
1061
479
2
55
)
55
1
(
2
1
2









R
R
R

36. H0: FD = FSL
H1: FD(x) = FSL(x - Δ)
FD : cumulative distribution function (CDF) of visual
acuity for the dominant group
FSL :cumulative distribution function of visual acuity
for the sex-linked group
Δ : a location shift of the CDF for the sex-linked group
relative to the dominant group.

37. Ranking Procedure for the Wilcoxon Rank-Sum Test
(1) Combine the data from the two groups, and order the values
from lowest to highest or, in the case of visual acuity, from best
(20–20) to worst (20–80).
(2) Assign ranks to the individual values, with the best visual
acuity (20–20) having the lowest rank and the worst visual
acuity (20–80) having the highest ran.
(3) If a group of observations has the same value, then compute
the range of ranks for the group, as was done for the signed-
rank test, and assign the average rank for each observation in
the group.
The Wilcoxon Rank-Sum Test

38. (1) Rank the observations
(2) Compute the rank sum R1 in the first sample
(the choice of sample is arbitrary)
(3)if both n1 and n2 are at least 10,
Normal Approximation Method can be used
(4)If either sample size is less than 10, the normal
approximation is not valid, and a small-sample table of
exact significance levels must be used. (Table 12)
The Wilcoxon Rank-Sum Test

40. small-sample table of exact significance levels
a：n1 = minimum of the two sample sizes.
b：n2 = maximum of the two sample sizes.

41. Example_ 4.0 （Ophthalmology）
H0: FD = FSL
H1: FD(x) = FSL(x - Δ)

42. the visual acuities of the two groups are significantly different.
the dominant group has better visual acuity than the sex linked
group.

43. Problem
Page 346 SPSS SOFTWARE
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9.15 9.16 9.17 9.18