Applied Nonparametric Statistical Methods

463 Pages·2001·7.091 MB·English

by  Peter Sprent, Nigel C. Smeeton

©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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©2001 CRC Press LLC
Contents
Preface
1Introducing nonparametric methods
1.1 Basic statistics
1.2 Samples and populations
1.3 Hypothesis tests
1.4 Estimation
1.5 Ethical issues
1.6 Computers and nonparametric methods
1.7 Further reading
Exercises
2 Centrality inference for single samples
2.1 Using measurement data
2.2 Inferences about medians based on ranks
2.3 The sign test
2.4 Transformation of ranks
2.5 Asymptotic results
2.6 Robustness
2.7 Fields of application
2.8 Summary
Exercises
3 Other single-sample inference
3.1 Inferences for dichotomous data
3.2 Tests related to the sign test
3.3 Matching samples to distributions
3.4 Angular data
3.5 A runs test for randomness
3.6 Fields of application
3.7 Summary
Exercises
4 Methods for paired samples
4.1 Comparisons in pairs
4.2 A less obvious use of the sign test
4.3 Power and sample size
4.4 Fields of application
4.5 Summary
Exercises

©2001 CRC Press LLC
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 Three or more samples
6.1 Compaarisons with parametric methods
6.2 Centrality tests for independent samples
6.3 Centrality tests for related samples
6.4 More detailed treatment comparisons
6.5 Tests for heterogeneity of variance
6.6 Some miscellaneous considerations
6.7 Fields of application
6.8 Summary
Exercises
7 Correlation and concordance
7.1 Correlation and bivariate data
7.2 Ranked data for several variables
7.3 Agreement
7.4 Fields of application
7.5 Summary
Exercises
8 Regression
8.1 Bivariate linear regression
8.2 Multiple regression
8.3 Nonparametric regression models
8.4 Other multivariate data problems
8.5 Fields of application
8.6 Summary
Exercises
9 Categorical data
9.1 Categories and counts
9.2 Nominal attribute categories

©2001 CRC Press LLC
9.3 Ordered categorical data
9.4 Goodness-of-fit tests for discrete data
9.5 Extension of McNemar's test
9,6 Fields of application
9.7 Summary
Exercises
10 Association in categorical data
10.1 The analysis of association
10.2 Some models for contingency tables
10.3 Combining and partitioning of tables
10.4 Power
10.5 Fields of application
10.6 Summary
Exercises
11 Robust estimation
11.1 When assumptions break down
11.2 Outliers and influence
11.3 The bootstrap
11.4 M-estimators and other robust estimators
11.5 Fields of application
11.6 Summary
Exercises
Appendix
References
Solutions to odd-numbered exercises

©2001 CRC Press LLC
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 t h e f i r s t - n a m e d a u t h o r
t o p r o v i d e a t h e n ( 1 9 9 3 ) u p - t o - d a t e i n t r o d u c t i o n t o 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 . I t t o o k a m i 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 o f t e c h n i q u e s a n d a d e t a i l e d e x p o s i t i o n o f t h e t h e o r y .
I n d i v i d u a l m e t h o d s a n d l i n k s b e t w e e n t h e m w e r e i l l u s t r a t e d m a 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 f o 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 . T h e b o o k w a s d e s i g n e d
t o m e 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 f i 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 r e s e a r c h w o r k e r s , m a n a g e r s , r e 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 i n v a r i o u s f i e l d s
w h o h a d a n u n d e r s t a n d i n g o f b a s i c s t a t i s t i c s a n d w h o , a l t h o u g h t h e y
h a d l i 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 f o u n d
o r t h o u g h t t h e y m i g h t f i n d t h e m u s e f u l i n t h e i r w o r k .
A p o s i t i v e r e s p o n s e f r o m r e a d e r s a n d r e v i e w e r s h a s e n c o u r a g e d u s
t o r e t a i n t h e b a s i c f o 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 t o p i c s a s w e l l a s c h a n g i n g t h e e m p h a s i s t o r e f l e c t b o t h
d e v e l o p m e n t s i n c o m p u t i n g a n d n e w a t t i t u d e s t o w a r d s d a t a a n a l y s i s .
N o n p a r a m e t r i c m e t h o d s a r e b a s i c a l l y a n a l y t i c t o o l s , b u t d a 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 . T h i s i s
w h y w e h a v e e x p a n d e d t h e c o v e r a g e o f t o 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 a n d c a l c u l a t i o n o f p o w e r a n d o f s a m p l e s i z e s n e e d e d
t o a c h i e v e s t a t e d a i m s . T h 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 , b u t a l s o i n f l u e n c e t h e a n a l y t i c a n d i n f e r e n t i a l p h 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 i n r e c e n t y e a r s b y m a n y
s t a t i s t i c i a n s o f i n a p p r o p r i a t e a n d e v e n i m p r o p e r u s e o f s i g n i f i c a n c e
t e s t s a n d t h e r e 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 r o 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 . T o e n c o u r a g e b e t t e r
u s e t h 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 r e w r i t t e n ,
a n d t h r o u g h o u t t h e b o o k t h e r e i s m o r e e m p h a s i s o n h o w t h e s e
c o n c e p t s s h o u l d b e u s e d a n d w a r n i n g s a b o u t p o t e n t i a l m i s u s e .
T h e l a y o u t o f C h a p t e r s I t o 1 0 f o 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 i n t h e s e c o n d e d i t i o n b u t t h e r e a r e m a n y
c h a n g e s i n o r d e r a n d o t h e r a s p e c t s o f p r e s e n t a t i o n i n c l u d i n g n e w a n d
m o r e d e t a i l e d e x a m p l e s . O n e o r t w o t o p i c s h a v e b e e n d r o p p e d o r a r e
t r e a t e d i n l e s s d e t a i l , a n d n e w m a t e r i a l h a s b e e n i n s e r t e d w h e r e
a p p r o p r i a t e . A s w e l l a s c o m m e n t s o n e t h i c a l c o 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 o n p o w e r a n d s a m p l e s i z e , t h e r e a r e n e w s e c t i o n s o n t h e
©2001 CRC Press LLC
a n a l y s i s o f a n g u l a r d a t a , t h e u s e o f c a p t u r e - r e c a p t u r e m e t h o d s , t h e
m e a s u r e m e n t o f a g r e e m e n t b e t w e e n o b s e r v e r s a n d s e v e r a l l e s s e r
a d d i t i o n s . E x a m p l e s h a v e b e e n c h o s e n f r 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 . F o r a f e w m o r e a d v a n c e d t o p i c s s u c h a s r e g r e s s i o n
s m o o t h i n g t e c h n i q u e s a n d M - e s t i m a t i o n w e h a v e n o t g i v e n d e t a i l s o f
s p e c i f i c m e t h o d s b u t o n l y a b r o a d o v e r v i e w o f e a c h t o p i c t o e n a b l e
r e a d e r s t o j u d g e w h e t h e r i t m a y b e r e l e v a n t t o t h 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 r e f e r e n c e s a r e g i v e n t o s o u r c e s t h a t c o n t a i n t h e d e t a i l
n e e d e d f o 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 r e w r i t t e n t o g i v e a n e l e m e n t a r y i n t r o d u c t i o n
t o i n f l u e n c e f u n c t i o n s , t h 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 r o 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 r e f e r e n c e s t o s o u r c e m a t e r i a l f o r
t h o s e w h o w a n t t o m a k e f u l l u s e o f t h e s e i d e a s . M a t e r i a l t h a t
a p p e a r e d i n C h a p t e r 1 2 o f t h e s e c o n d e d i t i o n h a s b e 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 r e l e v a n t p o i n t s i n t h e t e x t .
W e h a v e n o t i n c l u d e d t a b l e s f o 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 b e c a u s e m o r e s a t i s f a c t o r y i n f o r m a t i o n i s p r o v i d e d b y m o d e m
s t a t i s t i c a l s o f t w a r e , m a k i n g m a n y s t a n d a r d t a b l e s i n s u f f i c i e n t o r
s u p e r f l u o u s f o 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 . T h o s e w h o n e e d s u c h
t a b l e s b e c a u s e t h e y h a v e n o a c c e s s t o s p e c i a l i z e d s o f t w a r e a r e w e l l
c a t e r e d f o 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 t a b l e s . W e g i v e
r e f e r e n c e s t o t h e s e t h r o u g h o u t t h e b o o k a n d a l s o w h e n r e l e v a n t t o
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 r e 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 t o o d 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 t o m a n y r e a d e r s o f t h e e a r l i e r e d i t i o n s w h o m a 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 t h e c o n t e n t a n d t r e a t m e n t , o r 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 . T h i s i n p u t t r i g g e r e d
m a n y o f t h e c h a n g e s m a d e i n t h 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 g o t o
J i m M c G a n r i c k f o 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 t o P r o f e s s o r R i c h a r d H u g h e s f o r a d v i c e o n t h 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 r e n e w t h e t h a n k s r e 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 T i 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 i n i t i a l l y i n t h a t e d i t i o n f o r e x a m p l e s
t h a t w e h a v e r e t a i n e d .
P. Sprent
N. C. Smeeton
July 2000

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1
Introducing nonparametric methods
1.1 BASIC STATISTICS
One need know only a little statistics to make sensible use of simple
nonparametric methods. While we assume that many readers will be
familiar with the basic statistical notions met in introductory or
service courses of some 20 hours instruction, nevertheless those
without formal statistical training should be able to use this book in
parallel with one of the many introductory general statistical texts
available in libraries or from most academic bookshops, or one
recommended by a statistician colleague or friend. The choice may
depend upon how familiar one is with mathematical terms and
notation. For example, Essential Statistics (Rees, 1995) adopts a
straightforward approach that should suffice for most purposes, but
some may prefer a more advanced treatment, or an introductory text
that emphasizes applications in particular areas such as medicine,
biology, agriculture, the social sciences and so on. Readers with
considerable experience in general statistics but who are new to
nonparametric methods will be familiar with some background
material we give, but we urge them at least to skim through this to
see whether we depart from conventional treatments. For example,
our approa ch to tes ting and estima tion in Se ctions 1. 3 and 1. 4 differs
in certain aspects from that in some general statistics courses.
In this chapter we survey some concepts relevant to nonpara-
metric methods. It is the methods and not the data that are
nonparametric.
1.1.1 Parametric and nonparametric methods
Statistics students meet families of probability distributions early in
their courses. One of the best known is the normal or Gaussian
family, where individual members are specified by assigning
constant values to two quantities called parameters. These are
usually denoted by µ and σ2 and represent the mean and variance.

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The notation N(µ, σ2) denotes a normal distribution with these
parameters. Normal distributions are often associated with con-
tinuous data such as measurements.
Given a set of independent observations (a concept explained
more fully in Sec iton 1. 2) from a norma ldis rtibution, we often want
to infer something about the unknown parameters. The sample mean
provides a point (i.e. single value) estimate of the parameter µ
(sometimes called, in statistical jargon, the population mean). Here
the well-known t-test is used to measure the strength of the evidence
pr ov id ed by a sa mp le to su pp or t a n a pri or i hy po th es i ze d val ue µ
0
fo r the population mean. More usefully, we may determine what is
called a confidence interval for the ‘true’ population mean. This is
an interval for which we have, in a sense we describe in Section
1.4.1, reasonable confidence that it contains the true but unknown
mean µ. These are examples of parametric inferences.
The normal distribution is strictly relevant only to some types of
continuous scale data such as measurements, but it often works quite
well if the measurement scale is coarse (e.g. for examination marks
recorded to the nearest integer). More importantly, it is useful for
approximations in a wide range of circumstances when applied to
other types of data.
The binomial distribution is relevant to some types of counts.
The family also has tw o paramete rs n, p, whe re n is the tota l number
of obs er v at ion s an d p is th e pro bab il it y tha t a par tic ul ar on e fro m
tw o possible events occurs at any observation. Subject to certain
conditions, the number of occurrences, r, where 0 ≤ r ≤ n, of that
event in the n observations has a binomial distribution, which we
refer to as a B(n, p) distribution.
With a binomial distribution if we observe r occurrences of an
event in a set of n observations, then p^ = r/n is a point estimate of p,
the probability of success at each independent observation. We may
want to assess how strongly sample evidence supports an a priori
hypothesized value p , say, for p or obtain a confidence interval for
0
the value of p for the population.
The binomial distribution is often relevant to counts in dichot-
om ou s out co me si tu at i on s. Fo r exa mp le, the nu mb er of mal e
ch il dr en in a family of size n is often assumed to have a binomial
distribution with p = 1/ , but we see in Section 1.1.3 that this is only
2
a pproximate. The number of ‘s ixe s’ re corded in 10 cas ts of a fa ir die
has a B(10, 1/ ) distribution. The outcome of interest is sometimes
6
called a ‘favourable’ event, but this is hardly appropriate if, for
example, the event is a positive diagnosis of illness.
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Other well-known families include the uniform (or rectangular),
multinomial, Poisson, exponential, double exponential, gamma, beta
and Weibull distributions. This list is not exhaustive and you may
not be, and need not be, familiar with all of them.
It may be reasonable on theoretical grounds or on the basis of past
experience to assume that observations come from a particular
family of distributions. Also experience, backed by theory, suggests
that for many measurements inferences based on the assumption that
observations form a random sample from some normal distribution
may not be misleading even if the normality assumption is incorrect.
A theorem called the central limit theorem justifies these and other
uses of the normal distribution, particularly in what are called
asymptotic approximations. We often refer to these in this book.
Parametric inference is sometimes inappropriate or even
impossible. To assume that samples come from any specified family
of distributions ma y be unrea sonable . For example , we ma y not have
examination marks on, say, a percentage scale for each candidate but
know only the numbers of candidates in banded and ordered grades
des igna ted Grade A, Grade B, Gra de C, etc . Given thes e numbe rs for
two different schools, we may want to know if they indicate a
difference in performance between schools that might be due to
unequal standards of teaching or the ability of one school to attract
more able pupils. The method of inference is then usually
nonparametric. Even when we have precise measurements it may
be irrational to assume a normal distribution because normality
implies certain properties of symmetry and spread. We may be able
to see that a sample is not from a normal distribution simply by
looking at a few characteristics like the sample mean, median,
standard deviation and range of values. For example, if all
observations are either zero or positive and the standard deviation is
appreciably greater than the mean then, unless the sample is very
small, it is unreasonable to assume it comes from a normal
distribution (see Exercise 1.9). There are well-known distributions
that are flatter than the normal (e.g. the continuous uniform, or
rectangular, distribution) or skew (e.g. the exponential and gamma
distributions). In practice we are often able to say little more than
that our sample appears to come from a distribution that is skew, or
very peaked, or very flat, etc. Here nonparametric inference may
again be appropriate. In this latter situation some writers prefer the
term distribution-free to nonparametric.
The terms ‘distribution-free’ and ‘nonparametric’ are sometimes
regarded as synonymous. Indeed, in defining a distribution-free

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