Applied Nonparametric Statistical Methods

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©2001 CRC Press LLC
Library of Congress Cataloging-in-Publication Data
Sprent, Peter.
Applied nonparametric statistical methods.--3rd ed. I P Sprent, N.C. Smeeton.
p. cm. -- (Cbapman & Hall/CRC texts in statistical science series)
Includes bibliographical references and index.
ISBN 1-58488-145-3 (alk. paper)
1.
Nonparametric statistics. I. Smecton, N.C. II. Title. III. Texts in statistical science.
QA278.8 S74 20W
519.5'4--dc2l
00-055485
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Contents
Preface
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Introducing nonparametric methods
Basic statistics
Samples and populations
Hypothesis tests
Estimation
Ethical issues
Computers and nonparametric methods
Further reading
Exercises
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Centrality inference for single samples
Using measurement data
Inferences about medians based on ranks
The sign test
Transformation of ranks
Asymptotic results
Robustness
Fields of application
Summary
Exercises
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Other single-sample inference
Inferences for dichotomous data
Tests related to the sign test
Matching samples to distributions
Angular data
A runs test for randomness
Summary
Exercises
4.1
4.2
4.3
4.4
4.5
Methods for paired samples
Comparisons in pairs
A less obvious use of the sign test
Power and sample size
Summary
Exercises
2
3
4
5
Methods for two independent samples
5.1 Centrality tests and estimates
5.2 Rank based tests
5.3 The median test
5.4 Normal scores
5.5 Tests for survival data
5.6 Asymptotic approximations
5.7 Power and sample size
5.8 Tests for equality of variance
5.9 Tests for a common distribution
5.10 Fields of application
5.11 Summary
Exercises
6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Three or more samples
Compaarisons with parametric methods
Centrality tests for independent samples
Centrality tests for related samples
More detailed treatment comparisons
Tests for heterogeneity of variance
Some miscellaneous considerations
Summary
Exercises
7.1
7.2
7.3
7.4
7.5
Correlation and concordance
Correlation and bivariate data
Ranked data for several variables
Agreement
Summary
Exercises
8.1
8.2
8.3
8.4
8.5
8.6
Regression
Bivariate linear regression
Multiple regression
Nonparametric regression models
Other multivariate data problems
Summary
Exercises
7
8
9
9.1
9.2
Categorical data
Categories and counts
Nominal attribute categories
9.3
9.4
9.5
9,6
9.7
10
Ordered categorical data
Goodness-of-fit tests for discrete data
Extension of McNemar's test
Summary
Exercises
10.1
10.2
10.3
10.4
10.5
10.6
Association in categorical data
The analysis of association
Some models for contingency tables
Combining and partitioning of tables
Power
Summary
Exercises
11.1
11.2
11.3
11.4
11.5
11.6
Robust estimation
When assumptions break down
Outliers and influence
The bootstrap
M-estimators and other robust estimators
Summary
Exercises
11
Appendix
References
Solutions to odd-numbered exercises
Preface
T h e s e c o n d e d i t i o n o f t h i s b o o k w a s w r i t t e n b y th e fi r s t - n a m e d a u t h o r
t o pr o v i d e a th e n ( 1 9 9 3 ) up - t o - d a t e in t r o d u c t i o n to n o n p a r a m e t r i c a n d
d i s t r i b u t i o n - f r e e m e t h o d s . It t o o k a mi d w a y c o u r s e b e t w e e n a b a r e
d e s c r i p t i o n of te c h n i q u e s a n d a de t a i l e d e x p o s i t i o n of th e th e o r y .
I n d i v i d u a l me t h o d s a n d li n k s be t w e e n th e m w e r e il l u s t r a t e d ma i n l y
b y e x a m p l e s , M a t h e m a t i c s w a s k e p t t o t h e m i n i m u m n e e d e d fo r a
c l e a r u n d e r s t a n d i n g o f s c o p e a n d l i m i t a t i o n s . Th e b o o k w a s d e s i g n e d
t o me e t t h e n e e d s b o t h o f s t a t i s t i c s s t u d e n t s m a k i n g fi r s t c o n t a c t w i t h
t h e s e m e t h o d s a n d o f re s e a r c h w o r k e r s , m a n a g e r s , re s e a r c h a n d
d e v e l o p m e n t s t a f f , c o n s u l t a n t s a n d o t h e r s w o r k i n g in v a r i o u s fi e l d s
w h o ha d a n un d e r s t a n d i n g of ba s i c s t a t i s t i c s a n d wh o , a l t h o u g h th e y
h a d li t t l e p r e v i o u s k n o w l e d g e o f n o n p a r a m e t r i c m e t h o d s , n o w fo u n d
o r th o u g h t t h e y mi g h t fi n d th e m u s e f u l in t h e i r w o r k .
A p o s i t i v e re s p o n s e fr o m re a d e r s a n d re v i e w e r s h a s e n c o u r a g e d u s
t o re t a i n t h e b a s i c fo r m a t w h i l e t a k i n g t h e o p p o r t u n i t y t o i n t r o d u c e
n e w to p i c s a s w e l l a s c h a n g i n g th e e m p h a s i s to re f l e c t b o t h
d e v e l o p m e n t s in c o m p u t i n g a n d n e w a t t i t u d e s to w a r d s da t a a n a l y s i s .
N o n p a r a m e t r i c me t h o d s a r e ba s i c a l l y a n a l y t i c to o l s , bu t da t a
c o l l e c t i o n , a n a l y s e s a n d t h e i r i n t e r p r e t a t i o n a r e i n t e r r e l a t e d . Th i s i s
w h y we ha v e e x p a n d e d th e c o v e r a g e of to p i c s s u c h a s e t h i c a l
c o n s i d e r a t i o n s an d ca l c u l a t i o n of po w e r an d of s a m p l e s i z e s ne e d e d
t o a c h i e v e s t a t e d a i m s . Th e s e m a k e t h e i r m a i n i m p a c t a t t h e p l a n n i n g
s t a g e , bu t a l s o in f l u e n c e th e a n a l y t i c a n d in f e r e n t i a l ph a s e s .
T h e r e h a s b e e n w i d e s p r e a d c r i t i c i s m in re c e n t y e a r s b y ma n y
s t a t i s t i c i a n s of in a p p r o p r i a t e a n d e v e n im p r o p e r us e of s i g n i f i c a n c e
t e s t s a n d t h e re l a t e d c o n c e p t o f P- v a l u e s . H o w e v e r , t h e s e t o o l s h a v e a
p o s i t i v e ro l e w h e n p r o p e r l y u s e d a n d u n d e r s t o o d . To e n c o u r a g e b e t t e r
u s e th e s e c t i o n o n h y p o t h e s i s t e s t i n g i n C h a p t e r I h a s b e e n re w r i t t e n ,
a n d th r o u g h o u t th e bo o k th e r e i s mo r e e m p h a s i s on ho w t h e s e
c o n c e p t s s h o u l d be us e d a n d wa r n i n g s a b o u t po t e n t i a l mi s u s e .
T h e la y o u t o f C h a p t e r s I to 1 0 fo l l o w s t h e b r o a d p a t t e r n o f t h e
c o r r e s p o n d i n g c h a p t e r s in th e s e c o n d e d i t i o n bu t th e r e a r e ma n y
c h a n g e s in or d e r a n d ot h e r a s p e c t s of pr e s e n t a t i o n in c l u d i n g ne w a n d
m o r e de t a i l e d e x a m p l e s . O n e or tw o to p i c s h a v e be e n dr o p p e d or a r e
t r e a t e d in le s s de t a i l , a n d ne w ma t e r i a l ha s be e n in s e r t e d w h e r e
a p p r o p r i a t e . As we l l as c o m m e n t s on e t h i c a l co n s i d e r a t i o n s a n d
d i s c u s s i o n s on po w e r a n d s a m p l e s i z e , th e r e a r e ne w s e c t i o n s on th e
a n a l y s i s of a n g u l a r da t a , th e u s e of c a p t u r e - r e c a p t u r e me t h o d s , th e
m e a s u r e m e n t of a g r e e m e n t be t w e e n ob s e r v e r s a n d s e v e r a l le s s e r
a d d i t i o n s . Ex a m p l e s h a v e b e e n c h o s e n fr o m a w i d e r r a n g e o f
d i s c i p l i n e s . Fo r a fe w mo r e a d v a n c e d to p i c s s u c h a s re g r e s s i o n
s m o o t h i n g te c h n i q u e s a n d M - e s t i m a t i o n we ha v e no t gi v e n de t a i l s of
s p e c i f i c me t h o d s bu t on l y a br o a d ov e r v i e w of e a c h to p i c to e n a b l e
r e a d e r s to j u d g e w h e t h e r i t ma y b e re l e v a n t t o th e i r p a r t i c u l a r n e e d s .
I n s u c h c a s e s re f e r e n c e s a r e g i v e n to s o u r c e s th a t c o n t a i n th e d e t a i l
n e e d e d fo r i m p l e m e n t a t i o n .
C h a p t e r 1 1 h a s b e e n re w r i t t e n to g i v e a n e l e m e n t a r y in t r o d u c t i o n
t o in f l u e n c e fu n c t i o n s , th e n o n p a r a m e t r i c b o o t s t r a p a n d ro b u s t
e s t i m a t i o n g e n e r a l l y , a g a i n w i t h re f e r e n c e s to s o u r c e m a t e r i a l fo r
t h o s e w h o w a n t to m a k e fu l l u s e o f t h e s e id e a s . M a t e r i a l th a t
a p p e a r e d in C h a p t e r 12 of th e s e c o n d e d i t i o n ha s be e n u p d a t e d a n d
i n c o r p o r a t e d a t re l e v a n t p o i n t s in th e te x t .
W e h a v e n o t in c l u d e d ta b l e s fo r b a s i c n o n p a r a m e t r i c p r o c e d u r e s ,
m a i n l y be c a u s e mo r e s a t i s f a c t o r y in f o r m a t i o n is pr o v i d e d by mo d e m
s t a t i s t i c a l s o f t w a r e , ma k i n g ma n y s t a n d a r d ta b l e s in s u f f i c i e n t or
s u p e r f l u o u s fo r s e r i o u s u s e r s o f t h e m e t h o d s . Th o s e w h o n e e d s u c h
t a b l e s be c a u s e th e y ha v e no a c c e s s to s p e c i a l i z e d s o f t w a r e a r e we l l
c a t e r e d fo r b y s t a n d a r d c o l l e c t i o n s o f s t a t i s t i c a l ta b l e s . W e g i v e
r e f e r e n c e s to th e s e th r o u g h o u t th e bo o k a n d a l s o wh e n r e l e v a n t to
s o m e s p e c i a l i z e d t a b l e s . W e h a v e re t a i n e d t h e s e c t i o n o u t l i n i n g
s o l u t i o n s to od d - n u m b e r e d e x e r c i s e s .
W e a r e g r a t e f u l to m a n y re a d e r s o f th e e a r l i e r e d i t i o n s w h o ma d e
c o n s t r u c t i v e c o m m e n t s a b o u t th e c o n t e n t a n d tr e a t m e n t , or s o m e t i m e s
a b o u t t h e l a c k o f t r e a t m e n t , o f p a r t i c u l a r t o p i c s . Th i s i n p u t t r i g g e r e d
m a n y of th e c h a n g e s ma d e in th i s e d i t i o n . O u r s p e c i a l t h a n k s go to
J i m M c G a n r i c k fo r h e l p f u l d i s c u s s i o n s o n p h y s i o l o g i c a l
m e a s u r e m e n t s a n d to Pr o f e s s o r R i c h a r d H u g h e s fo r a d v i c e o n th e
G u i l l a i n - B a r r é s y n d r o m e . W e h a p p i l y re n e w t h e t h a n k s re c o r d e d i n
t h e s e c o n d e d i t i o n t o Ti m o t h y P . D a v i s a n d C h r i s T h e o b a l d w h o
s u p p l i e d u s w i t h d a t a s e t s u s e d in i t i a l l y i n th a t e d i t i o n fo r e x a m p l e s
t h a t w e h a v e re t a i n e d .
P. Sprent
N. C. Smeeton
July 2000

Applied Nonparametric Statistical Methods

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