Table Of ContentHandbook of
PARAMETRIC and
NONPARAMETRIC
STATISTICAL PROCEDURES
SECOND EDITION
© 2000 by Chapman & Hall/CRC
Handbook of
PARAMETRIC and
NONPARAMETRIC
STATISTICAL PROCEDURES
SECOND EDITION
David J.Sheskin
Western Connecticut State University
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.
Library of Congress Cataloging-in-Publication Data
Sheskin, David.
Handbook of parametric and nonparametric statistical procedures / by David J.
Sheskin.--2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN l-58488-133-X (alk. paper)
1. Mathematical statistics--Handbooks, manuals, etc. I. Title: Parametric and nonparametric
statistical procedures. II. Title
QA27625 .S54 2000
519.5—dc21 99-051977
CIP
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Preface
Like the first edition, the second edition of the Handbook of Parametric and Nonparametric
Statistical Procedures is designed to provide researchers, teachers, and students with a compre-
hensive reference book in the areas of parametric and nonparametric statistics. The addition of
a large amount of new material (250 pages) makes the Handbook unparalleled in terms of its
coverage of material in the field of statistics. Rather than being directed at a limited audience,
the Handbook is intended for individuals who are involved in a broad spectrum of academic
disciplines encompassing the fields of mathematics/statistics, the social and biological sciences,
business, and education. My philosophy in writing both this and the previous edition was to
create a reference book on parametric and nonparametric statistical procedures that I (as well as
colleagues and students I have spoken with over the years) have always wanted, yet could never
find. To be more specific, my primary goal was to produce a comprehensive reference book on
univariate and bivariate statistical procedures which covers a scope of material that extends far
beyond that which is covered in any single available source. It was essential that the book be
applications oriented, yet at the same time that it address relevant theoretical and practical issues
which are of concern to the sophisticated researcher. In addition, I wanted to write a book that
is accessible to people who have a limited knowledge of statistics, as well as those who are well
versed in the subject. I believe I have achieved these goals, and on the basis of this I believe that
the Handbook of Parametric and Nonparametric Statistical Procedures will continue to
serve as an invaluable resource for people in multiple academic disciplines who conduct research,
are involved in teaching, or are presently in the process of learning statistics.
I am not aware of any applications-oriented book that provides in-depth coverage of as
many statistical procedures as the number that are covered in the Handbook of Parametric and
Nonparametric Statistical Procedures. Inspection of the Table of Contents and Index should
confirm the scope of material covered in the book. A unique feature of the Handbook, which
distinguishes it from other reference books on statistics, is that it provides the reader with a
practical guide that emphasizes application over theory. Although the book will be of practical
value to statistically sophisticated individuals who are involved in research, it is also accessible
to those who lack the theoretical and/or mathematical background required for understanding the
material documented in more conventional statistics reference books. Since a major goal of
the book is to serve as a practical guide, emphasis is placed on decision making with respect
to which test is most appropriate to employ in evaluating a specific design. Within the frame-
work of being user-friendly, clear computational guidelines, accompanied by easy-to-understand
examples, are provided for all procedures.
One should not, however, get the impression that the Handbook of Parametric and Non-
parametric Statistical Procedures is little more than a cookbook. In point of fact, the design
of the Handbook is such that within the framework of each of the statistical procedures which
are covered, in addition to the basic guidelines for decision making and computation, substantial
in-depth discussion is devoted to a broad spectrum of practical and theoretical issues, many of
which are not discussed in conventional statistics books. Inclusion of the latter material ensures
that the Handbook will serve as an invaluable resource for those who are sophisticated as well
as unsophisticated in statistics.
In order to facilitate its usage, most of the procedures contained in the Handbook are
organized within a standardized format. Specifically, for most of the procedures the following
information is provided:
© 2000 by Chapman & Hall/CRC
I. Hypothesis evaluated with test and relevant background information The first part
of this section provides a general statement of the hypothesis evaluated with the test. This is
followed by relevant background information on the test such as the following: a) Information
regarding the experimental design for which the test is appropriate; b) Any assumptions under-
lying the test which, if violated, would compromise its reliability; and c) General information on
other statistical procedures that are related to the test.
II. Example This section presents a description of an experiment, with an accompanying
data set (or in some instances two experiments utilizing the same data set), for which the test will
be employed. All examples employ small sample sizes, as well as integer data consisting of
small numbers, in order to facilitate the reader’s ability to follow the computational procedures
to be described in Section IV.
III. Null versus alternative hypotheses This section contains both a symbolic and verbal
description of the statistical hypotheses evaluated with the test (i.e., the null hypothesis versus
the alternative hypothesis). It also states the form the data will assume when the null hypothesis
is supported, as opposed to when one or more of the possible alternative hypotheses are
supported.
IV. Test computations This section contains a step-by-step description of the procedure
for computing the test statistic. The computational guidelines are clearly outlined in reference
to the data for the example(s) presented in Section II.
V. Interpretation of the test results This section describes the protocol for evaluating the
computed test statistic. Specifically: a) It provides clear guidelines for employing the appropriate
table of critical values to analyze the test statistic; b) Guidelines are provided delineating the
relationship between the tabled critical values and when a researcher should retain the null
hypothesis, as opposed to when the researcher can conclude that one or more of the possible
alternative hypotheses are supported; c) The computed test statistic is interpreted in reference to the
example(s) presented in Section II; and d) In instances where a parametric and nonparametric test
can be used to evaluate the same set of data, the results obtained using both procedures are
compared with one another, and the relative power of both tests is discussed in this section and/or
in Section VI.
VI. Additional analytical procedures for the test and/or related tests Since many of the
tests described in the Handbook have additional analytical procedures associated with them, such
procedures are described in this section. Many of these procedures are commonly employed (such
as comparisons conducted following an analysis of variance), while others are used and/or
discussed less frequently (such as the tie correction employed for the large sample normal
approximation of many nonparametric test statistics). Many of the analytical procedures covered
in Section VI are not discussed (or if so, only discussed briefly) in other books. Some repre-
sentative topics which are covered in Section VI are planned versus unplanned comparison
procedures, measures of association for inferential statistical tests, computation of confidence
intervals, and computation of power. In addition to the aforementioned material, for many of the
tests there is additional discussion of other statistical procedures that are directly related to the test
under discussion. In instances where two or more tests produce equivalent results, examples are
provided which clearly demonstrate the equivalency of the procedures.
VII. Additional discussion of the test Section VII discusses theoretical concepts and
issues, as well as practical and procedural issues that are relevant to a specific test. In some
instances where a subject is accorded brief coverage in the initial material presented on the test,
the reader is alerted to the fact that the subject is discussed in greater depth in Section VII. Many
of the issues discussed in this section are topics that are generally not covered in other books, or if
they are, they are only discussed briefly. Among the topics covered in Section VII is additional
discussion of the relationship between a specific test and other tests that are related to it. Section
© 2000 by Chapman & Hall/CRC
VII also provides bibliographic information on less commonly employed alternative procedures
that can be used to evaluate the same design for which the test under discussion is used.
VIII. Additional examples illustrating the use of the test This section provides
descriptions of one or more additional experiments for which a specific test is applicable. For the
most part, these examples employ the same data set as that in the original example(s) presented
in Section II for that test. By virtue of using standardized data for most of the examples, the
material for a test contained in Section IV (Test computations) and Section V (Interpretation
of the test results) will be applicable to most of the additional examples. Because of this, the
reader is able to focus on common design elements in various experiments which indicate that a
given test is appropriate for use with a specific type of design.
IX. Addendum At the conclusion of the discussion of a number of tests an Addendum has
been included that describes one or more related tests that are not discussed in Section VI. As an
example, the Addendum of the between-subjects factorial analysis of variance contains an
overview and computational guidelines for the factorial analysis of variance for a mixed design
and the within-subjects factorial analysis of variance.
References This section provides the reader with a listing of primary and secondary source
material on each test.
Endnotes At the conclusion of most tests, a detailed endnotes section contains additional
useful information that further clarifies or expands upon material discussed in the main text.
The first edition of the Handbook of Parametric and Nonparametric Statistical Pro-
cedures was comprised of an Introduction followed by 26 chapters, each of which documented
a specific inferential statistical test (as well as related tests) or measure of correlation/association.
The general label Test was used (and is used in this edition) for all procedures described in the
book (i.e., inferential tests as well as measures of correlation/association). In addition to the
Introduction, the second edition of the Handbook contains 32 chapters. A chapter describing in
detail each of the following six tests has been added the second edition: a) The single-sample test
for evaluating population skewness (Test 4); b) The single-sample test for evaluating
population kurtosis (Test 5) (The D’Agostino–Pearson test of normality (Test 5A) is also
described in this chapter); c) The Kolmogorov–Smirnov goodness-of-fit test for a single sample
(Test 7) (The Lilliefors test for normality (Test 7a) is also described in this chapter); d) The
Kolmogorov–Smirnov test for two independent samples (Test 13); e) The Moses test for
equal variability (Test 15); and f) The van der Waerden normal-scores test for k independent
samples (Test 23). In addition to the aforementioned tests, a substantial amount of new material
has been added to tests that were included in the first edition. Chapters/Tests included in the first
edition are noted below, indicating subject matter that has been added to the second edition.
Introduction: Description and computation of the coefficient of variation; extensive
coverage of skewness and kurtosis, including description and computation of the Pearsonian
coefficient of skewness, the (cid:1) and (cid:2) measures of skewness, and the (cid:1) and (cid:2) measures of
(cid:1) (cid:1) (cid:2) (cid:2)
kurtosis.
The chi-square goodness-of-fit test (Test 8): Illustration of the use of the chi-square
goodness-of-fit test for assessing goodness-of-fit for a normal population distribution; discussion
of Cohen’s w index for computing the power of the chi-square goodness-of-fit test; description
of heterogeneity chi-square analysis. Two additional examples have been added to this chapter
to illustrate the new material.
The binomial sign test for a single sample (Test 9): Discussion of Cohen’s g index
for computing the power of the binomial sign test for a single sample; evaluating goodness-of-fit
for a binomial distribution. An Addendum has been added that provides comprehensive
coverage of the following discrete probability distributions: multinomial distribution; negative
© 2000 by Chapman & Hall/CRC
binomial distribution; hypergeometric distribution; Poisson distribution; and matching
distribution. Twelve additional examples have been added to this chapter to illustrate the new
material.
The single-sample runs test (and other tests of randomness) (Test 10): The extension
of the runs test to data with more than two categories is described. The runs test for serial
randomness (Test 10a) has been added to this chapter. There is additional discussion of the
concept of randomness. An Addendum has been added that describes in detail the generation of
pseudorandom numbers — specifically, the following methods are described: the midsquare
method, the midproduct method, and the linear congruential method. The Addendum also
provides detailed coverage of the following alternative tests of randomness: The frequency test
(Test 10b), The gap test (Test 10c), The poker test (Test 10d), The maximum test (Test 10e),
and The mean square successive difference test (Test 10f). One additional example has been
added to this chapter to illustrate the new material. In addition, a standardized data set is evaluated
with four of the aforementioned tests of randomness.
The t test for two independent samples (Test 11): Comprehensive discussion of outliers
(including Test 11e: Procedures for identifying outliers), robust statistical procedures, and
data transformation (description of and computational examples illustrating the square root,
logarithmic, reciprocal, and acrsine transformations); discussion of Hotelling’s T2.
The Mann–Whitney U test (Test 12): An Addendum has been added that provides
comprehensive coverage of computer-intensive/data-driven statistical procedures/resampling
statistics. The following topics are discussed in the Addendum: Randomization and permu-
tation tests (including The randomization test for two independent samples (Test 12a), The
bootstrap (Test 12b), and The jackknife (Test 12c)). Two additional examples have been added
to this chapter to illustrate the new material.
The chi-square test for r × c tables (Test 16): Discussion of Cohen’s w and h indices for
computing the power of the chi-square test for r × c tables and the (cid:1) test for two independent
proportions (Test16d); heterogeneity chi-square analysis for 2 × 2 contingency tables;
expanded coverage of the odds ratio (Test 16j) (including discussion of the concept of relative
risk, test of significance for an odds ratio (Test 16j-a), and computation of a confidence interval
for an odds ratio); discussion of Simpson’s Paradox; analysis of multidimensional contingency
tables. Three additional examples have been added to this chapter to illustrate the new material.
The McNemar test (Test 20): An Addendum has been that which describes The Bowker
test of symmetry (Test 20a). One additional example has been added to this chapter to illustrate
the new material.
The single-factor between-subjects analysis of variance (Test 21): Discussion of Cohen’s
f index employed in computing the power and magnitude of treatment effect for the single-factor
between-subjects analysis of variance; discussion of multivariate analysis of variance
(MANOVA). An Addendum has been added that provides comprehensive coverage of the single-
factor between-subjects analysis of covariance (Test 21j). One additional example has been
added to this chapter to illustrate the new material.
The Kruskal–Wallis one-way analysis of variance by ranks (Test 22): Discussion of an
alternative pairwise multiple comparison procedure.
The single-factor within-subjects analysis of variance (Test 24): Revised equations for
computing the omega squared statistic for magnitude of treatment effect; discussion of Cohen’s
f index employed in computing the power and magnitude of treatment effect for the of the single-
factor within-subjects analysis of variance; discussion of the Latin square design.
The single-factor between-subjects factorial analysis of variance (Test 27): Revised
equations for computing the omega squared statistic for magnitude of treatment effect; discussion
© 2000 by Chapman & Hall/CRC
of Cohen’s f index employed in computing the power and magnitude of treatment effect for the
single-factor between-subjects factorial analysis of variance.
The Pearson product-moment correlation coefficient (Test 28): The following material
has been added to the Addendum: Nonmathematical descriptions of the following multivariate
procedures: Factor analysis, canonical correlation, discriminant analysis, and logistic re-
gression; meta-analysis and related topics. (This section contains a comprehensive discussion
of meta-analysis and includes a description of the following meta-analytic procedures: Test 28n:
Procedure for comparing k studies with respect to significance level; Test 28o: The Stouffer
procedure for obtaining a combined significance level for k studies; Test 28p: Procedure for
comparing k studies with respect to effect size; Test 28q: Procedure for obtaining a
combined effect size for k studies. This section also discusses Jacob Cohen’s indices for the
power computation of various tests, and the controversy over the conventional significance test
based hypothesis testing model versus the minimum-effect hypothesis testing model.) One
additional example has been added to this chapter to illustrate the new material.
In addition to the aforementioned topics, the second edition provides expanded information
on the asymptotic relative efficiency of nonparametric statistical procedures. All in all, 25 new
tests have been added to the second edition along with 32 additional examples to illustrate the new
material.
Although it is not a prerequisite, the Handbook of Parametric and Nonparametric
Statistical Procedures is designed to be used by those who have a basic familiarity with de-
scriptive statistics and experimental design. Prior familiarity with the latter subject matter will
facilitate one’s ability to use the book efficiently. In order to insure that the reader has familiarity
with these topics, an Introduction has been included which provides a general overview of
descriptive statistics and experimental design. Following the Introduction, the reader is provided
with guidelines and decision tables for selecting the appropriate statistical test for evaluating a
specific experimental design. The Handbook of Parametric and Nonparametric Statistical
Procedures can be used as a reference book or it can be employed as a textbook in undergraduate
and graduate courses that are designed to cover a broad spectrum of parametric and/or non-
parametric statistical procedures.
The author would like to express his gratitude to a number of people who helped make this
book a reality. First, I would like to thank Tim Pletscher of CRC Press for his confidence in and
support of the first edition of the Handbook. Special thanks are due to Bob Stern, the mathematics
editor at CRC Press, who suggested a second edition. Without his efforts and encouragement this
book would not have become a reality. Sylvia Wood of CRC Press deserves thanks for overseeing
the production of the final product. I am also indebted to Glena Ames who did an excellent job
preparing the copy-ready manuscript. Finally, I must express my appreciation to my wife Vicki
and daughter Emily, who both endured and tolerated the difficulties associated with a project of
this magnitude.
David Sheskin
© 2000 by Chapman & Hall/CRC
To Vicki and Emily
© 2000 by Chapman & Hall/CRC
Table of Contents
with Summary of Topics
Introduction
Descriptive versus inferential statistics
Statistic versus parameter
Levels of measurement
Continuous versus discrete variables
Measures of central tendency (mode, median, and mean)
Measures of variability (range; quantiles, percentiles, quartiles, and deciles;
variance and standard deviation; the coefficient of variation)
Measures of skewness and kurtosis
The normal distribution
Hypothesis testing
Estimation in inferential statistics
Basic concepts and terminology employed in experimental design
Correlational research
Parametric versus nonparametric inferential statistical tests
Selection of the appropriate statistical procedure
Outline of Inferential Statistical Tests and Measures of
Correlation/Association
Guidelines and Decision Tables for Selecting the Appropriate
Statistical Procedure
Inferential Statistical Tests Employed with a Single Sample
Test 1. The Single-Sample z Test
I. Hypothesis Evaluated with Test and Relevant Background Information
II. Example
III. Null versus Alternative Hypotheses
IV. Test Computations
V. Interpretation of the Test Results
VI. Additional Analytical Procedures for the Single-Sample z Test and/or
Related Tests
VII. Additional Discussion of the Single-Sample z Test
1. The interpretation of a negative z value
2. The standard error of the population mean and graphical
representation of the results of the single-sample z test
© 2000 by Chapman & Hall/CRC
