Download Nonparametric tests I

Nonparametric tests I
Back to basics
Lecture Outline
• What is a nonparametric test?
• Rank tests, distribution free tests and
nonparametric tests
• Which type of test to use
MTB > dotplot 'Male' 'Female';
SUBC> same.
.
: .
.
.
.
. . :: :..:::.. :..:: :... .:.. .. .
:
.
.
---+---------+---------+---------+---------+---------+---MALE
..: . : :
: .
.: ::::::.::.:. ::.: : .
: . .
---+---------+---------+---------+---------+---------+---FEMALE
0.32
0.48
0.64
0.80
0.96
1.12
MTB > dotplot 'Male' 'Female';
SUBC> same.
.
: .
.
.
.
. . :: :..:::.. :..:: :... .:.. .. .
:
.
.
---+---------+---------+---------+---------+---------+---MALE
..: . : :
: .
.: ::::::.::.:. ::.: : .
: . .
---+---------+---------+---------+---------+---------+---FEMALE
0.32
0.48
0.64
0.80
0.96
1.12
MTB > desc 'Male' 'Female’
Variable
MALE
FEMALE
Variable
MALE
FEMALE
N
50
50
Mean
0.5908
0.5180
Min
0.2900
0.3200
Median
0.5600
0.4950
Max
1.1300
0.8500
TrMean
0.5770
0.5102
Q1
0.4275
0.4100
StDev
0.1979
0.1315
Q3
0.7150
0.6125
SEMean
0.0280
0.0186
Lecture Outline
• What is a nonparametric test?
– What is a parameter?
– What are examples of non-parametric
tests?
• Rank tests, distribution free tests and
nonparametric tests
• Which type of test to use
Parameters
• are central to inference in GLM and
ANOVA
• and represent assumptions about the
underlying processes
LET
LET
LET
LET
K1=4.7
K2=-2.5
K3=10.4
K4=1.9
#
#
#
#
Group 1 mean minus grand mean
Group 2 mean minus grand mean
The grand mean
Standard deviation of the error
RANDOM 30 'Error'
LET 'Y'=K3+K1*'DUM1'+K2*'DUM2'+K4*'Error'
LET
LET
LET
LET
K1=4.7
K2=-2.5
K3=10.4
K4=1.9
#
#
#
#
Group 1 mean minus grand mean
Group 2 mean minus grand mean
The grand mean
Standard deviation of the error
RANDOM 30 'Error'
LET 'Y'=K3+K1*'DUM1'+K2*'DUM2'+K4*'Error'
Group
1
1
Fitted value = m + 2
2
3
-1-2
Error has Normal Distribution with zero mean and
standard deviation 
LET
LET
LET
LET
K1=4.7
K2=-2.5
K3=10.4
K4=1.9
#
#
#
#
Group 1 mean minus grand mean
Group 2 mean minus grand mean
The grand mean
Standard deviation of the error
RANDOM 30 'Error'
LET 'Y'=K3+K1*'DUM1'+K2*'DUM2'+K4*'Error'
Group
1
1
Fitted value = m + 2
2
3
-1-2
Error has Normal Distribution with zero mean and
standard deviation 
Parameters
• are central to inference in GLM and
ANOVA
• but represent assumptions about the
underlying processes
Parameters
• are central to inference in GLM and
ANOVA
• but represent assumptions about the
underlying processes
• can be done without in some simple
situations
Parameters
• are central to inference in GLM and
ANOVA
• but represent assumptions about the
underlying processes
• can be done without in some simple
situations – BUT HOW?
Rnk
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Wt Sex
0.29
1
0.32
2
0.34
1
0.34
2
0.34
2
0.36
1
0.36
1
0.37
1
0.37
1
0.37
1
0.37
2
0.37
2
0.38
1
0.38
1
0.38
2
0.38
2
0.39
2
0.40
2
0.40
2
0.40
2
0.41
1
0.41
1
0.41
2
0.41
2
0.41
2
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50
0.41
0.42
0.43
0.43
0.43
0.45
0.45
0.45
0.45
0.46
0.47
0.47
0.48
0.48
0.48
0.48
0.49
0.49
0.50
0.50
0.50
0.50
0.50
0.51
0.51
2
1
1
2
2
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
2
2
1
2
51
52
53
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55
56
57
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60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
0.52
0.52
0.52
0.53
0.53
0.55
0.56
0.56
0.56
0.57
0.58
0.58
0.59
0.59
0.59
0.60
0.61
0.61
0.62
0.62
0.62
0.62
0.62
0.63
0.63
1
2
2
2
2
2
1
1
1
1
2
2
1
2
2
1
1
2
1
1
2
2
2
1
2
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
0.65
0.66
0.67
0.67
0.67
0.67
0.68
0.71
0.72
0.73
0.75
0.75
0.77
0.78
0.78
0.78
0.82
0.83
0.85
0.85
0.88
0.98
0.98
1.05
1.13
1
1
1
2
2
2
1
1
2
1
1
1
1
1
2
2
2
1
1
2
1
1
1
1
1
Rnk
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Wt Sex
0.29
1
0.32
2
0.34
1
0.34
2
0.34
2
0.36
1
0.36
1
0.37
1
0.37
1
0.37
1
0.37
2
0.37
2
0.38
1
0.38
1
0.38
2
0.38
2
0.39
2
0.40
2
0.40
2
0.40
2
0.41
1
0.41
1
0.41
2
0.41
2
0.41
2
Remember ties
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50
0.41
0.42
0.43
0.43
0.43
0.45
0.45
0.45
0.45
0.46
0.47
0.47
0.48
0.48
0.48
0.48
0.49
0.49
0.50
0.50
0.50
0.50
0.50
0.51
0.51
2
1
1
2
2
1
2
2
2
2
1
1
1
1
2
2
2
2
1
1
1
2
2
1
2
51
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53
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55
56
57
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71
72
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75
0.52
0.52
0.52
0.53
0.53
0.55
0.56
0.56
0.56
0.57
0.58
0.58
0.59
0.59
0.59
0.60
0.61
0.61
0.62
0.62
0.62
0.62
0.62
0.63
0.63
1
2
2
2
2
2
1
1
1
1
2
2
1
2
2
1
1
2
1
1
2
2
2
1
2
76
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100
0.65
0.66
0.67
0.67
0.67
0.67
0.68
0.71
0.72
0.73
0.75
0.75
0.77
0.78
0.78
0.78
0.82
0.83
0.85
0.85
0.88
0.98
0.98
1.05
1.13
1
1
1
2
2
2
1
1
2
1
1
1
1
1
2
2
2
1
1
2
1
1
1
1
1
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
Mean Rank
70
80
90 100
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
Mean Rank
The ‘Male’ mean rank = 55.26
The ‘Female’ mean rank = 45.74
70
80
90 100
MTB > mann-whitney male female
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
FEMALE
N =
N =
50
50
Median =
Median =
0.5600
0.4950
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
N = 50
Median =
0.5600
FEMALE
N = 50
Median =
0.4950
Point estimate for ETA1-ETA2 is
0.0500
95.0 Percent CI for ETA1-ETA2 is (-0.0100,0.1200)
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
N = 50
Median =
0.5600
FEMALE
N = 50
Median =
0.4950
Point estimate for ETA1-ETA2 is
0.0500
95.0 Percent CI for ETA1-ETA2 is (-0.0100,0.1200)
W = 2763.0
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
N = 50
Median =
0.5600
FEMALE
N = 50
Median =
0.4950
Point estimate for ETA1-ETA2 is
0.0500
95.0 Percent CI for ETA1-ETA2 is (-0.0100,0.1200)
W = 2763.0
Sum of ranks of 2763 corresponds to
a mean rank of 2763/50 = 55.26
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
Mean Rank
The ‘Male’ mean rank = 55.26
The ‘Female’ mean rank = 45.74
70
80
90 100
140
120
100
80
60
40
20
0
0
10
20
30
40
50
60
Mean Rank
The ‘Male’ mean rank = 55.26
The ‘Female’ mean rank = 45.74
70
80
90 100
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
N = 50
Median =
0.5600
FEMALE
N = 50
Median =
0.4950
Point estimate for ETA1-ETA2 is
0.0500
95.0 Percent CI for ETA1-ETA2 is (-0.0100,0.1200)
W = 2763.0
Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.1016
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
N = 50
Median =
0.5600
FEMALE
N = 50
Median =
0.4950
Point estimate for ETA1-ETA2 is
0.0500
95.0 Percent CI for ETA1-ETA2 is (-0.0100,0.1200)
W = 2763.0
Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.1016
The test is significant at 0.1014 (adjusted for ties)
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
N = 50
Median =
0.5600
FEMALE
N = 50
Median =
0.4950
Point estimate for ETA1-ETA2 is
0.0500
95.0 Percent CI for ETA1-ETA2 is (-0.0100,0.1200)
W = 2763.0
Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.1016
The test is significant at 0.1014 (adjusted for ties)
Cannot reject at alpha = 0.05
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
N = 50
Median =
0.5600
FEMALE
N = 50
Median =
0.4950
Point estimate for ETA1-ETA2 is
0.0500
95.0 Percent CI for ETA1-ETA2 is (-0.0100,0.1200)
W = 2763.0
Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.1016
The test is significant at 0.1014 (adjusted for ties)
Cannot reject at alpha = 0.05
MTB > mann-whitney male female
Mann-Whitney Test and CI: MALE, FEMALE
MALE
N = 50
Median =
0.5600
FEMALE
N = 50
Median =
0.4950
Point estimate for ETA1-ETA2 is
0.0500
95.0 Percent CI for ETA1-ETA2 is (-0.0100,0.1200)
W = 2763.0
Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.1016
The test is significant at 0.1014 (adjusted for ties)
Cannot reject at alpha = 0.05
The null hypothesis is better expressed as “the distributions
of male and female weights are the same”.
Parameters
• are central to inference in GLM and
ANOVA
• but represent assumptions about the
underlying processes
• can be done without in some simple
situations
Nonparametric vs Parametric
Nonparametric vs Parametric
• Sign Test
• One-sample t-test
Nonparametric vs Parametric
• Sign Test
• Mann-Whitney Test
• One-sample t-test
• Two-sample t-test
Nonparametric vs Parametric
• Sign Test
• Mann-Whitney Test
• Spearman Rank Test
• One-sample t-test
• Two-sample t-test
• Correlation/Regression
Nonparametric vs Parametric
•
•
•
•
Sign Test
Mann-Whitney Test
Spearman Rank Test
Kruskal-Wallis Test
•
•
•
•
One-sample t-test
Two-sample t-test
Correlation/Regression
One-way ANOVA
Nonparametric vs Parametric
•
•
•
•
•
Sign Test
Mann-Whitney Test
Spearman Rank Test
Kruskal-Wallis Test
Friedman Test
•
•
•
•
•
One-sample t-test
Two-sample t-test
Correlation/Regression
One-way ANOVA
One-way blocked ANOVA
Lecture Outline
• What is a nonparametric test?
• Rank tests, distribution free tests and
nonparametric tests
• Which type of test to use
A rose by any other name..
• Non-parametric tests lack parameters
• Rank tests start by ranking the data
• Distribution-free tests don’t assume a
Normal distribution (or any other)
These are mainly but not completely
overlapping sets of tests (and some
are scale-invariant too).
Lecture Outline
• What is a nonparametric test?
• Rank tests, distribution free tests and
nonparametric tests
• Which type of test to use
Fewer assumptions but...
• still some assumptions (including independence)
• limited range of situations
– no more than 2 x-variables
– can’t mix continuous and categorical x-variables
• provide p-values but estimation is dodgy
• loss of efficiency if parametric assumptions are
upheld
• there is a grand scheme for parametric statistics
(GLM) but a lot of separate strange names for
nonparametrics
When is there a choice?
• when there is a non-parametric test
– fewer than two or three variables
altogether
• and prediction is not required
How to choose:
• If the assumptions of parametric test are
upheld, use it – on grounds of efficiency
• If not upheld, consider fixing the
assumptions (e.g. by transforming the
data, as in the practical)
• If assumptions not fixable, use
nonparametric test
MTB > dotplot 'LogM' 'LogF';
SUBC> same.
.
.
.
.
. ::: :.. . :::.. :..::.:....: : : .
: . .
+---------+---------+---------+---------+---------+-------LogM
.:
. :
. .
. : ::.:: : :. ::.::. ::.:. : . : ..
+---------+---------+---------+---------+---------+-------LogF
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
MTB > dotplot 'LogM' 'LogF';
SUBC> same.
.
.
.
.
. ::: :.. . :::.. :..::.:....: : : .
: . .
+---------+---------+---------+---------+---------+-------LogM
.:
. :
. .
. : ::.:: : :. ::.::. ::.:. : . : ..
+---------+---------+---------+---------+---------+-------LogF
-1.25
-1.00
-0.75
-0.50
-0.25
0.00
MTB > desc 'LogM' 'LogF'
Variable
LogM
LogF
Variable
LogM
LogF
N
50
50
Mean
-0.5786
-0.6878
Min
-1.2379
-1.1394
Median
-0.5798
-0.7032
Max
0.1222
-0.1625
TrMean
-0.5850
-0.6928
Q1
-0.8499
-0.8916
StDev
0.3248
0.2453
Q3
-0.3355
-0.4902
SEMean
0.0459
0.0347
Lecture Outline
• What is a nonparametric test?
• Rank tests, distribution free tests and
nonparametric tests
• Which type of test to use
Last remarks
• Nonparametric tests are an opportunity
to revise the basic ideas of statistical
inference
• They are sometimes useful in biology
• They are often used in biology
• NEXT WEEK: more nonparametrics,
including confidence intervals and
randomisation tests. READ the handout