Multivariate Analysis 6th Edition Afifi

Download and Read online, DOWNLOAD EBOOK, [PDF EBOOK EPUB ], Ebooks
download, Read Ebook EPUB/KINDE, Download Book Format PDF
Practical multivariate analysis 6th Edition Afifi
OR CLICK LINK
https://textbookfull.com/product/practical-
multivariate-analysis-6th-edition-afifi/
Read with Our Free App Audiobook Free Format PFD EBook, Ebooks dowload PDF
with Andible trial, Real book, online, KINDLE , Download[PDF] and Read and Read
Read book Format PDF Ebook, Dowload online, Read book Format PDF Ebook,
[PDF] and Real ONLINE Dowload [PDF] and Real ONLINE
More products digital (pdf, epub, mobi) instant
download maybe you interests ...
Practical multivariate analysis 6th Edition Afifi
https://textbookfull.com/product/practical-multivariate-
analysis-6th-edition-afifi-2/
Multivariate Data Analysis Joseph F. Hair
https://textbookfull.com/product/multivariate-data-analysis-
joseph-f-hair/
Multivariate Analysis for the Behavioral Sciences Kimmo

Vehkalahti
https://textbookfull.com/product/multivariate-analysis-for-the-
behavioral-sciences-kimmo-vehkalahti/
Multivariate analysis for the behavioral sciences 2nd

Edition Brian S. Everitt
https://textbookfull.com/product/multivariate-analysis-for-the-
behavioral-sciences-2nd-edition-brian-s-everitt/
Multivariate Time Series Analysis in Climate and
Environmental Research 1st Edition Zhihua Zhang (Auth.)
https://textbookfull.com/product/multivariate-time-series-
analysis-in-climate-and-environmental-research-1st-edition-
zhihua-zhang-auth/
Practical Statistics for Educators 6th Edition Ruth

Ravid
https://textbookfull.com/product/practical-statistics-for-
educators-6th-edition-ruth-ravid/
Genetics : analysis & principles 6th Edition Robert J.

Brooker
https://textbookfull.com/product/genetics-analysis-
principles-6th-edition-robert-j-brooker/
Longitudinal Multivariate Psychology Emilio Ferrer
https://textbookfull.com/product/longitudinal-multivariate-
psychology-emilio-ferrer/
Recent Developments in Multivariate and Random Matrix

Analysis Festschrift in Honour of Dietrich von Rosen
Thomas Holgersson
https://textbookfull.com/product/recent-developments-in-
multivariate-and-random-matrix-analysis-festschrift-in-honour-of-
dietrich-von-rosen-thomas-holgersson/
Practical Multivariate
Analysis
Sixth Edition
Practical Multivariate
Analysis
Sixth Edition
Abdelmonem Afifi
Susanne May
Robin A. Donatello
Virginia A. Clark
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
c 2020 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-1-138-70222-6 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made
to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all
materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all
material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been
obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future
reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized
in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, mi-
crofilming, and recording, or in any information storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com
(http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA
01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For
organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identi-
fication and explanation without intent to infringe.
Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com
and the CRC Press Web site at

http://www.crcpress.com
Contents
Preface xi
Authors xv
I Preparation for Analysis 1

1 What is multivariate analysis? 3
1.1 Defining multivariate analysis 3
1.2 Examples of multivariate analyses 3
1.3 Exploratory versus confirmatory analyses 6
1.4 Multivariate analyses discussed in this book 6
1.5 Organization and content of the book 9
2 Characterizing data for analysis 11

2.1 Variables: Their definition, classification, and use 11
2.2 Defining statistical variables 11
2.3 Stevens’s classification of variables 12
2.4 How variables are used in data analysis 14
2.5 Examples of classifying variables 15
2.6 Other characteristics of data 15
2.7 Summary 15
2.8 Problems 15
3 Preparing for data analysis 17

3.1 Processing data so they can be analyzed 17
3.2 Choice of a statistical package 18
3.3 Techniques for data entry 19
3.4 Organizing the data 23
3.5 Reproducible research and literate programming 29
3.6 Example: depression study 31
3.7 Summary 33
3.8 Problems 33
4 Data visualization 37
4.1 Introduction 37
4.2 Univariate data 38
4.3 Bivariate data 45
4.4 Multivariate data 50
4.5 Discussion of computer programs 52
4.6 What to watch out for 54
4.7 Summary 56
4.8 Problems 56
v
vi CONTENTS
5 Data screening and transformations 59
5.1 Transformations, assessing normality and independence 59
5.2 Common transformations 59
5.3 Selecting appropriate transformations 62
5.4 Assessing independence 69
5.6 Summary 71
5.7 Problems 72
6 Selecting appropriate analyses 75

6.1 Which analyses to perform? 75
6.2 Why selection is often difficult 75
6.3 Appropriate statistical measures 76
6.4 Selecting appropriate multivariate analyses 79
6.5 Summary 80
6.6 Problems 80
II Regression Analysis 85
7 Simple regression and correlation 87
7.1 Chapter outline 87
7.2 When are regression and correlation used? 87
7.3 Data example 88
7.4 Regression methods: fixed-X case 89
7.5 Regression and correlation: variable-X case 93
7.6 Interpretation: fixed-X case 93
7.7 Interpretation: variable-X case 94
7.8 Other available computer output 98
7.9 Robustness and transformations for regression 103
7.10 Other types of regression 105
7.11 Special applications of regression 107
7.14 Summary 112
7.15 Problems 112
8 Multiple regression and correlation 115

8.2 When are regression and correlation used? 115
8.3 Data example 116
8.4 Regression methods: fixed-X case 117
8.5 Regression and correlation: variable-X case 119
8.6 Interpretation: fixed-X case 124
8.7 Interpretation: variable-X case 126
8.8 Regression diagnostics and transformations 128
8.9 Other options in computer programs 132
8.12 Summary 140
8.13 Problems 141
CONTENTS vii
9 Variable selection in regression 145
9.2 When are variable selection methods used? 145
9.4 Criteria for variable selection 149
9.5 A general F test 152
9.6 Stepwise regression 153
9.7 Lasso regression 159
9.9 Discussion of strategies 164
9.11 Summary 167
9.12 Problems 168
10 Special regression topics 171

10.2 Missing values in regression analysis 171
10.3 Dummy variables 177
10.4 Constraints on parameters 184
10.5 Regression analysis with multicollinearity 186
10.6 Ridge regression 187
10.7 Summary 190
10.8 Problems 191
11 Discriminant analysis 195

11.2 When is discriminant analysis used? 195
11.4 Basic concepts of classification 197
11.5 Theoretical background 202
11.6 Interpretation 204
11.7 Adjusting the dividing point 207
11.8 How good is the discrimination? 209
11.9 Testing variable contributions 210
11.10 Variable selection 211
11.13 Summary 214
11.14 Problems 214
12 Logistic regression 217

12.2 When is logistic regression used? 217
12.4 Basic concepts of logistic regression 219
12.5 Interpretation: categorical variables 220
12.6 Interpretation: continuous variables 222
12.7 Interpretation: interactions 223
12.8 Refining and evaluating logistic regression 229
12.9 Nominal and ordinal logistic regression 238
12.10 Applications of logistic regression 243
12.11 Poisson regression 246
viii CONTENTS
12.14 Summary 251
12.15 Problems 252
13 Regression analysis with survival data 255

13.2 When is survival analysis used? 255
13.3 Data examples 256
13.4 Survival functions 256
13.5 Common survival distributions 262
13.6 Comparing survival among groups 262
13.7 The log-linear regression model 264
13.8 The Cox regression model 266
13.9 Comparing regression models 274
13.12 Summary 278
13.13 Problems 278
14 Principal components analysis 281

14.2 When is principal components analysis used? 281
14.4 Basic concepts 282
14.5 Interpretation 285
14.6 Other uses 292
14.9 Summary 295
14.10 Problems 296
15 Factor analysis 297

15.2 When is factor analysis used? 297
15.5 Initial extraction: principal components 300
15.6 Initial extraction: iterated components 303
15.7 Factor rotations 305
15.8 Assigning factor scores 309
15.9 Application of factor analysis 310
15.12 Summary 313
15.13 Problems 314
16 Cluster analysis 317

16.2 When is cluster analysis used? 317
16.4 Basic concepts: initial analysis 318
16.5 Analytical clustering techniques 324
CONTENTS ix
16.6 Cluster analysis for financial data set 328
16.9 Summary 336
16.10 Problems 336
17 Log-linear analysis 339

17.2 When is log-linear analysis used? 339
17.4 Notation and sample considerations 341
17.5 Tests and models for two-way tables 343
17.6 Example of a two-way table 345
17.7 Models for multiway tables 347
17.8 Exploratory model building 350
17.9 Assessing specific models 354
17.10 Sample size issues 355
17.11 The logit model 356
17.14 Summary 359
17.15Problems 360
18 Correlated outcomes regression 361

18.2 When is correlated outcomes regression used? 361
18.3 Data examples 362
18.5 Regression of clustered data with a continuous outcome 369
18.6 Regression of clustered data with a binary outcome 373
18.7 Regression of longitudinal data 375
18.8 Generalized estimating equations analysis of correlated data 379
18.11 Summary 386
18.12 Problems 386
Appendix A 389
A.1 Data sets and how to obtain them 389
A.2 Chemical companies’ financial data 389
A.3 Depression study data 389
A.4 Financial performance cluster–analysis data 389
A.5 Lung cancer survival data 390
A.6 Lung function data 390
A.7 Parental HIV data 390
A.8 Northridge earthquake data 391
A.9 School data 391
A.10 Mice data 391
Bibliography 393
Index 411
Preface
The first edition of this book appeared in 1984 under the title “Computer Aided Multivariate Anal-
ysis.” The title was chosen in order to distinguish it from other books that were more theoretically
oriented. By the time we published the fifth edition in 2012, it was impossible to think of a book on
multivariate analysis for scientists and applied researchers that is not computer oriented. We there-
fore decided at that time to change the title to “Practical Multivariate Analysis” to better characterize
the nature of the book. Today, we are pleased to present the sixth edition.
We wrote this book for investigators, specifically behavioral scientists, biomedical scientists,
and industrial or academic researchers, who wish to perform multivariate statistical analyses and
understand the results. We expect the readers to be able to perform and understand the results,
but also expect them to know when to ask for help from an expert on the subject. The book can
either be used as a self-guided textbook or as a text in an applied course in multivariate analysis.
In addition, we believe that the book can be helpful to many statisticians who have been trained
in conventional mathematical statistics who are now working as statistical consultants and need to
explain multivariate statistical concepts to clients with a limited background in mathematics.
We do not present mathematical derivations of the techniques; rather we rely on geometric and
graphical arguments and on examples to illustrate them. The mathematical level has been deliber-
ately kept low. While the derivations of the techniques are referenced, we concentrate on applica-
tions to real-life problems, which we feel are the ‘fun’ part of multivariate analysis. To this end, we
assume that the reader will use a packaged software program to perform the analysis. We discuss
specifically how each of four popular and comprehensive software packages can be used for this
purpose. These packages are R, SAS, SPSS, and STATA. The book can be used, however, in con-
junction with all other software packages since our presentation explains the output of most standard
statistical programs.
We assume that the reader has taken a basic course in statistics that includes tests of hypotheses
and covers one-way analysis of variance.
Approach of this book

We wrote the book in a modular fashion. Part One, consisting of six chapters, provides examples
of studies requiring multivariate analysis techniques, and discusses characterizing data for analysis,
computer programs, data entry, data management, data clean-up, missing values, and transforma-
tions. It also includes a new chapter on graphics and data visualization and presents a rough guide
to assist in the choice of an appropriate multivariate analysis. We included these topics since many
investigators have more difficulty with these preliminary steps than with running the multivariate
analyses themselves. Also, if these steps are not done with care, the results of the statistical analysis
can be faulty.
In the rest of the chapters, we follow a standard format. The first four sections of each chapter
include a discussion of when the technique is used, a data example, and the basic assumptions and
concepts of the technique. In subsequent sections, we present more detailed aspects of the analysis.
At the end of each chapter, we give a summary table showing which features are available in the
four software packages. We also include a section entitled ‘What to watch out for’ to warn the reader
about common problems related to data analysis. In those sections, we rely on our own experiences
in consulting and those detailed in the literature to supplement the formal treatment of the subject.
xi
xii PREFACE
Part Two covers regression analysis. Chapter 7 deals with simple linear regression and is in-
cluded for review purposes to introduce our notation and to provide a more complete discussion
of outliers and diagnostics than is found in some elementary texts. Chapters 8-10 are concerned
with multiple linear regression. Multiple linear regression is used very heavily in practice and pro-
vides the foundation for understanding many concepts relating to residual analysis, transformations,
choice of variables, missing values, dummy variables, and multicollinearity. Since these concepts
are essential to a good grasp of multivariate analysis, we thought it useful to include these chapters
in the book.
Chapters 11-18 might be considered the heart of multivariate analysis. They include chapters on
discriminant analysis, logistic regression analysis, survival analysis, principal components analysis,
factor analysis, cluster analysis, log-linear analysis, and correlated outcomes regression. The mul-
tivariate analyses have been discussed more as separate techniques than as special cases of some
general framework. The advantage of this approach is that it allows us to concentrate on explaining
how to analyze a certain type of data from readily available computer programs to answer realistic
questions. It also enables the reader to approach each chapter independently. We did include inter-
spersed discussions of how the different analyses relate to each other in an effort to describe the ‘big
picture’ of multivariate analysis.
How to use the book

We have received many helpful suggestions from instructors and reviewers on how to order these
chapters for reading or teaching purposes. For example, one instructor uses the following order in
teaching: principal components, factor analysis, and then cluster analysis. Another prefers present-
ing a detailed treatment of multiple regression followed by logistic regression and survival analysis.
Instructors and self-learning readers have a wide choice of other orderings of the material because
the chapters are largely self contained.
What’s new in the Sixth Edition

During the nearly thirty-six years since we wrote the first edition of this book, tremendous advances
have taken place in the field of computing and software development. These advances have made
it possible to quickly perform any of the multivariate analyses that were available only in theory at
that time. They also spurred the invention of new multivariate analyses as well as new options for
many of the standard methods. In this edition, we have taken advantage of these developments and
made many changes as described below.
For each of the techniques discussed, we used the most recent software versions available and
discussed the most modern ways of performing the analysis. In each chapter, we updated the ref-
erences to today’s literature (while still including the fundamental original references). In terms
of statistical software, we discontinued description of S-Plus because of the more wide-spread use
of the similar package R. Also, we no longer include Statistica since it is largely not used by our
intended readers.
In addition to the above-described modifications, we included comments to distinguish between
exploratory and confirmatory analyses in Chapter 1 and throughout the book. We also expanded
the discussion of missing values in Chapter 3 and added a discussion of literate programming and
reproducible research.
As mentioned above, we added a new chapter (Chapter 4) on graphics and data visualization.
In Chapter 9, we updated our discussion of variable selection and added a description of Lasso,
a more recent method than the ones already included. In Chapter 10, we added a description of
MICE, a multiple imputation approach for dealing with missing values. In Chapter 18, we added a
description of the generalized estimating equations (GEE) method for handling correlated data and
compared it to the mixed model approach. Finally, in each chapter we updated and/or expanded the
summary table of the options available in the four statistical packages to make it consistent with the
most recent software versions.
PREFACE xiii
Data sets used for examples and problems are described throughout the book as needed and
summarized in Appendix A. Two web sites are also available. The first one is the CRC web site:
http://www.crcpress.com/product/isbn/9781138702226. From this site, you can down-
load all the data sets used in the book by clicking on the Downloads/Updates tab. The other web
site that is available to all readers is: https://stats.idre.ucla.edu/other/examples/pma6.
This site, developed by the UCLA Institute for Digital Research and Education (IDRE), includes
the data sets in the formats of various statistical software packages available in the links included in
the Appendix A part of the table of contents in that web page. It also includes illustrations of exam-
ples in most chapters, complete with code for three of the four software packages used in the book.
Please note that the current site is done for the 5th edition and it is hoped that it will be updated for
the sixth edition. We encourage readers to obtain data from either web site and frequently refer to
the solutions given in the UCLA web site for practice.
Acknowledgements
We would like to express our appreciation to our colleagues and former students and staff that
helped us over the years, both in the planning and preparation of the various editions. These include
our colleagues Drs. Carol Aneshensel, Roger Detels, Robert Elashoff, Ralph Frerichs, Mary Ann
Hill, and Roberta Madison. Our former students include Drs. Stella Grosser, Luohua Jiang, Jack
Lee, Steven Lewis, Tim Morgan, Leanne Streja, and David Zhang. Our former staff includes Ms.
Dorothy Breininger, Jackie Champion, and Anne Eiseman. In addition, we would like to thank Ms.
Meike Jantzen and Mr. Jack Fogliasso for their help with the references and typesetting.
We also thank Rob Calver and Lara Spieker from CRC Press for their very capable assistance
in the preparation of the sixth edition.
We especially appreciate the efforts of the staff of the UCLA Institute for Digital Research and
Education in putting together the UCLA web site of examples from the book (referenced above).
Our deep gratitude goes to our spouses, Marianne Afifi, Bruce Jacobson, Ian Donatello, and
Welden Clark, for their patience and encouragement throughout the stages of conception, writing,
and production of the book. Special thanks go to Welden Clark for his expert assistance and trou-
bleshooting of earlier electronic versions of the manuscript.
Abdelmonem Afifi
Susanne May
Robin A. Donatello
Virginia A. Clark
Authors
Abdelmonem Afifi, Ph.D., has been Professor of Biostatistics in the School of Public Health, Uni-
versity of California, Los Angeles (UCLA) since 1965, and served as the Dean of the School from
1985 until 2000. His research includes multivariate and multilevel data analysis, handling miss-
ing observations in regression and discriminant analyses, meta-analysis, and model selection. Over
the years, he taught well-attended courses in biostatistics for public health students and clinical re-
search physicians, and doctoral-level courses in multivariate statistics and multilevel modeling. He
has authored many publications in statistics and health related fields, including two widely used
books (with multiple editions) on multivariate analysis. He received several prestigious awards for
excellence in teaching and research.
Susanne May, Ph.D., is a Professor in the Department of Biostatistics at the University of Wash-
ington in Seattle. Her areas of expertise and interest include clinical trials, survival analysis, and
longitudinal data analysis. She has more than 20 years of experience as a statistical collaborator
and consultant on health related research projects. In addition to a number of methodological and
applied publications, she is a coauthor (with Drs. Hosmer and Lemeshow) of Applied Survival
Analysis: Regression Modeling of Time-to-Event Data. Dr. May has taught courses on introductory
statistics, clinical trials, and survival analysis.
Robin A. Donatello, Dr.P.H., is an Associate Professor in the Department of Mathematics and

Statistics and the developer of the Data Science Initiative at California State University, Chico. Her
areas of interest include applied research in the public health and natural science fields. She has
expertise in data visualization, techniques to address missing and erroneous data, implementing
reproducible research workflows, computational statistics, and data science. Dr. Donatello teaches
undergraduate and graduate level courses in statistical programming, applied statistics, and data
science.
Virginia A. Clark, Ph.D., was professor emerita of Biostatistics and Biomathematics at UCLA.
For 27 years, she taught courses in multivariate analysis and survival analysis, among others. In
addition to this book, she is coauthor of four books on survival analysis, linear models and analysis
of variance, and survey research, as well as an introductory book on biostatistics. She published
extensively in statistical and health science journals.
xv
Part I
Preparation for Analysis
1
Chapter 1
What is multivariate analysis?
1.1 Defining multivariate analysis

The expression multivariate analysis is used to describe analyses of data that are multivariate in
the sense that numerous observations or variables are obtained for each individual or unit studied. In
a typical survey 30 to 100 questions are asked of each respondent. In describing the financial status
of a company, an investor may wish to examine five to ten measures of the company’s performance.
Commonly, the answers to some of these measures are interrelated. The challenge of disentangling
complicated interrelationships among various measures on the same individual or unit and of inter-
preting these results is what makes multivariate analysis a rewarding activity for the investigator.
Often results are obtained that could not be attained without multivariate analysis.
In the next section of this chapter several studies are described in which the use of multivariate
analysis is essential to understanding the underlying problem. Section 1.3 provides a rationale for
making a distinction between confirmatory and exploratory analyses. Section 1.4 gives a listing and
a very brief description of the multivariate analysis techniques discussed in this book. Section 1.5
then outlines the organization of the book.
1.2 Examples of multivariate analyses

The studies described in the following subsections illustrate various multivariate analysis tech-
niques. These are used later in the book as examples.
Depression study example

The data for the depression study have been obtained from a complex, random, multiethnic sample
of 1000 adult residents of Los Angeles County. The study was a panel or longitudinal design where
the same respondents were interviewed four times between May 1979 and July 1980. About three-
fourths of the respondents were re-interviewed for all four interviews. The field work for the survey
was conducted by professional interviewers from the Institute for Social Science Research at the
University of California in Los Angeles.
This research is an epidemiological study of depression and help-seeking behavior among free-
living (noninstitutionalized) adults. The major objectives are to provide estimates of the prevalence
and incidence of depression and to identify causal factors and outcomes associated with this con-
dition. The factors examined include demographic variables, life events stressors, physical health
status, health care use, medication use, lifestyle, and social support networks. The major instrument
used for classifying depression is the Depression Index (CESD) of the National Institute of Mental
Health, Center of Epidemiological Studies. A discussion of this index and the resulting prevalence
of depression in this sample is given in Frerichs et al. (1981).
The longitudinal design of the study offers advantages for assessing causal priorities since the
time sequence allows us to rule out certain potential causal links. Nonexperimental data of this type
cannot directly be used to establish causal relationships, but models based on an explicit theoretical
3
4 CHAPTER 1. WHAT IS MULTIVARIATE ANALYSIS?
framework can be tested to determine if they are consistent with the data. An example of such model
testing is given in Aneshensel and Frerichs (1982).
Data from the first time period of the depression study are described in Chapter 3. Only a subset
of the factors measured on a subsample of the respondents is included in this book’s web site in
order to keep the data set easily comprehensible. These data are used several times in subsequent
chapters to illustrate some of the multivariate techniques presented in this book.
Parental HIV study

The data from the parental HIV study have been obtained from a clinical trial to evaluate an inter-
vention given to increase coping skills (Rotheram-Borus et al., 2001). The purpose of the interven-
tion was to improve behavioral, social, and health outcomes for parents with HIV/AIDS and their
children. Parents and their adolescent children were recruited from the New York City Division of
Aids Services (DAS). Adolescents were eligible for the study if they were between the ages of 11
and 18 and if the parents and adolescents had given informed consent. Individual interviews were
conducted every three months for the first two years and every six months thereafter. Information
obtained in the interviews included background characteristics, sexual behavior, alcohol and drug
use, medical and reproductive history, and a number of psychological scales.
A subset of the data from the study is available on this book’s web site. To protect the identity of
the participating adolescents we used the following procedures. We randomly chose one adolescent
per family. In addition, we reduced the sample further by choosing a random subset of the original
sample. Adolescent case numbers were assigned randomly without regard to the original order or
any other numbers in the original data set.
Data from the baseline assessment will be used for problems as well as to illustrate various
multivariate analysis techniques.
Northridge earthquake study

On the morning of January 17, 1994 a magnitude 6.7 earthquake centered in Northridge, CA awoke
Los Angeles and Ventura County residents. Between August 1994 and May 1996, 1830 residents
were interviewed about what happened to them in the earthquake. The study uses a telephone sur-
vey lasting approximately 48 minutes to assess the residents’ experiences in and responses to the
Northridge earthquake. Data from 506 residents are included in the data set posted on the book web
site, and described in Appendix A.
Subjects were asked where they were, how they reacted, where they obtained information,
whether their property was damaged or whether they experienced injury, and what agencies they
were in contact with. The questionnaire included the Brief Symptom Inventory (BSI), a measure
of psychological functioning used in community studies, and questions on emotional distress. Sub-
jects were also asked about the impact of the damage to the transportation system as a result of the
earthquake. Investigators not only wanted to learn about the experiences of the Southern California
residents in the Northridge earthquake, but also wished to compare their findings to similar studies
of the Los Angeles residents surveyed after the Whittier Narrows earthquake on October 1, 1987,
and Bay Area residents interviewed after the Loma Prieta earthquake on October 17, 1989.
The Northridge earthquake data set is used in problems at the end of several chapters of the book
to illustrate a number of multivariate techniques. Multivariate analyses of these data include, for
example, exploring pre- and post-earthquake preparedness activities as well as taking into account
several factors relating to the subject and the property (Nguyen et al., 2006).
Bank loan study

The managers of a bank need some way to improve their prediction of which borrowers will suc-
cessfully pay back a type of bank loan. They have data from the past on the characteristics of persons
1.2. EXAMPLES OF MULTIVARIATE ANALYSES 5
to whom the bank has lent money and the subsequent record of how well the person has repaid the
loan. Loan payers can be classified into several types: those who met all of the terms of the loan,
those who eventually repaid the loan but often did not meet deadlines, and those who simply de-
faulted. They also have information on age, sex, income, other indebtedness, length of residence,
type of residence, family size, occupation, and the reason for the loan. The question is, can a simple
rating system be devised that will help the bank personnel improve their prediction rate and lessen
the time it takes to approve loans? The methods described in Chapter 12 and Chapter 13 can be used
to answer this question.
Lung function study

The purpose of this lung function study of chronic respiratory disease is to determine the effects
of various types of smog on lung function of children and adults in the Los Angeles area. Because
they could not randomly assign people to live in areas that had different levels of pollutants, the
investigators were very concerned about the interaction that might exist between the locations where
persons chose to live and their values on various lung function tests. The investigators picked four
areas of quite different types of air pollution and measured various demographic and other responses
on all persons over seven years old who live there. These areas were chosen so that they are close to
an air-monitoring station.
The researchers took measurements at two points in time and used the change in lung function
over time as well as the levels at the two periods as outcome measures to assess the effects of air
pollution. The investigators had to do the lung function tests by using a mobile unit in the field, and
much effort went into problems of validating the accuracy of the field observations. A discussion
of the particular lung function measurements used for one of the four areas can be found in Detels
et al. (1975). In the analysis of the data, adjustments must be made for sex, age, height, and smoking
status of each person.
Over 15,000 respondents have been examined and interviewed in this study. The data set is being
used to answer numerous questions concerning effects of air pollution, smoking, occupation, etc. on
different lung function measurements. For example, since the investigators obtained measurements
on all family members seven years old and older, it is possible to assess the effects of having parents
who smoke on the lung function of their children (Tashkin et al., 1984). Studies of this type require
multivariate analyses so that investigators can arrive at plausible scientific conclusions that could
explain the resulting lung function levels.
This data set is described in Appendix A. Lung function and associated data for nonsmoking
families for the father, mother, and up to three children ages 7–17 are available from the book’s web
site.
School data set

The school data set is a publicly available data set that is provided by the National Center
for Educational Statistics. The data come from the National Education Longitudinal Study of
1988 (called NELS:88). The study collected data beginning with 8th graders and conducted ini-
tial interviews and four follow-up interviews which were performed every other year. The data
used here contain only initial interview data. They represent a random subsample of 23 schools
with 519 students out of more than a thousand schools with almost twenty five thousand stu-
dents. Extensive documentation of all aspects of the study is available at the following web site:
http://nces.ed.gov/surveys/NELS88/. The longitudinal component of NELS:88 has been
used to investigate change in students’ lives and school-related and other outcomes. The focus
on the initial interview data provides the opportunity to examine associations between school and
student-related factors and students’ academic performance in a cross-sectional manner. This type
of analysis will be illustrated in Chapter 18.
1.3 Exploratory versus confirmatory analyses
A crucial component for most research studies and analyses is the testing of hypotheses. For some
types of studies, hypotheses are specified in detail prior to study start (a priori) and then remain
unchanged. This is typically the case, e.g., for clinical trials and other designed experiments. For
other types of studies, some hypotheses might be specified in advance while others are generated
only after study start and potentially after reviewing some or all of the study data. This is often
the case for observational studies. In this section, we make a distinction between two conceptually
different approaches to analysis and reporting based on whether the primary goal of a study is to
confirm prespecified hypotheses or to explore hypotheses that have not been prespecified.
The following is a motivating example provided by Fleming (2010). He describes an experience
where he walked into a maternity ward (when they still had such) while visiting a friend who had
just given birth. He noticed that there were 22 babies, but only 2 of one gender while the other 20
were of the other gender. As a statistician, he dutifully calculated the p-value for the likelihood of
seeing such (or worse) imbalance if in truth there are 50% of each. The two-sided p-value turns out
to be 0.0001, indicating a very small likelihood (1 in 10,000) of such or more extreme imbalance
being observed if in truth there are 50% of each. This is an example of where the hypothesis was
generated after seeing the data. We will call such hypotheses exploratory.
Following Fleming, researchers might want to go out and test an exploratory hypothesis in an-
other setting or with new data. In the example above, one might want to go to another maternity
ward to collect further evidence of a strong imbalance in gender distribution at birth. Imagine that
in a second (confirmatory) maternity ward there might be exactly equal numbers for each gender
(e.g. 11 boys and 11 girls). Testing the same hypothesis in this setting will not yield any statistically
significant difference from the presumed 50%. Nevertheless, one might be tempted to simply com-
bine the two studies. A corresponding two-sided p-value remains statistically significant (p-value
< 0.01).
The above example might appear silly, because few researchers will believe that the distribution
of gender at birth (without human interference) is very different from 50%. Nevertheless, there
are many published research articles which test and present the results for hypotheses that were
generated by looking at data and noticing ‘unusual’ results. Without a clear distinction between
whether hypotheses were specified a priori or not, it is difficult to interpret the p-values provided.
Results from confirmatory analyses provide much stronger evidence than results from ex-
ploratory analyses. Accordingly, interpretation of results from confirmatory analyses can be stated
using much stronger language than interpretation of results from exploratory analyses. Furthermore,
results from exploratory analyses should not be combined with results from confirmatory analyses
(e.g. in meta analyses), because the random high bias (Fleming, 2010) will remain (albeit atten-
uated). To avoid random high bias when combining data or estimates from multiple studies only
data/estimates from confirmatory analyses should be combined. However, this requires clear iden-
tification of whether confirmatory or exploratory analysis was performed for each individual study
and/or analysis.
Many authors have pointed out that the medical literature is replete with studies that cannot be
reproduced (Breslow, 1999; Munafò et al., 2017). As argued by Breslow (1999), reproducibility of
studies, and in particular epidemiologic studies, can be improved if hypotheses are specified a priori
and the nature of the study (exploratory versus confirmatory) is clearly specified.
Throughout this book, we distinguish between the two approaches to multivariate analyses and
presentations of results and provide examples for each.
1.4 Multivariate analyses discussed in this book

In this section a brief description of the major multivariate techniques covered in this book is pre-
sented. To keep the statistical vocabulary to a minimum, we illustrate the descriptions by examples.
1.4. MULTIVARIATE ANALYSES DISCUSSED IN THIS BOOK 7
Simple linear regression
A nutritionist wishes to study the effects of early calcium intake on the bone density of post-
menopausal women. She can measure the bone density of the arm (radial bone), in grams per square
centimeter, by using a noninvasive device. Women who are at risk of hip fractures because of too
low a bone density will tend to show low arm bone density also. The nutritionist intends to sample a
group of elderly churchgoing women. For women over 65 years of age, she will plot calcium intake
as a teenager (obtained by asking the women about their consumption of high-calcium foods during
their teens) on the horizontal axis and arm bone density (measured) on the vertical axis. She expects
the radial bone density to be lower in women who had a lower calcium intake. The nutritionist plans
to fit a simple linear regression equation and test whether the slope of the regression line is zero. In
this example a single outcome factor is being predicted by a single predictor factor.
Simple linear regression as used in this case would not be considered multivariate by some
statisticians, but it is included in this book to introduce the topic of multiple regression.
Multiple linear regression

A manager is interested in determining which factors predict the dollar value of sales of the firm’s
personal computers. Aggregate data on population size, income, educational level, proportion of
population living in metropolitan areas, etc. have been collected for 30 areas. As a first step, a
multiple linear regression equation is computed, where dollar sales is the outcome variable and
the other factors are considered as candidates for predictor variables. A linear combination of the
predictors is used to predict the outcome or response variable.
Discriminant function analysis

A large sample of initially disease-free men over 50 years of age from a community has been
followed to see who subsequently has a diagnosed heart attack. At the initial visit, blood was drawn
from each man, and numerous other determinations were made, including body mass index, serum
cholesterol, phospholipids, and blood glucose. The investigator would like to determine a linear
function of these and possibly other measurements that would be useful in predicting who would
and who would not get a heart attack within ten years. That is, the investigator wishes to derive a
classification (discriminant) function that would help determine whether or not a middle-aged man
is likely to have a heart attack.
Logistic regression
An online movie streaming service has classified movies into two distinct groups according to
whether they have a high or low proportion of the viewing audience when shown. The company
also records data on features such as the length of the movie, the genre, and the characteristics
of the actors. An analyst would use logistic regression because some of the data do not meet the
assumptions for statistical inference used in discriminant function analysis, but they do meet the
assumptions for logistic regression. From logistic regression we derive an equation to estimate the
probability of capturing a high proportion of the target audience.
Poisson regression
In a health survey, middle school students were asked how many visits they made to the dentist in
the last year. The investigators are concerned that many students in this community are not receiving
adequate dental care. They want to determine what characterizes how frequently students go to the
dentist so that they can design a program to improve utilization of dental care. Visits per year are
count data and Poisson regression analysis provides a good tool for analyzing this type of data.
Poisson regression is covered in the logistic regression chapter.
Survival analysis
An administrator of a large health maintenance organization (HMO) has collected data for a number
of years on length of employment in years for their physicians who are either family practitioners or
internists. Some of the physicians are still employed, but many have left. For those still employed,
the administrator can only know that their ultimate length of employment will be greater than their
current length of employment. The administrator wishes to describe the distribution of length of
employment for each type of physician, determine the possible effects of factors such as gender and
location of work, and test whether or not the length of employment is the same for two specialties.
Survival analysis, or event history analysis (as it is often called by behavioral scientists), can be used
to analyze the distribution of time to an event such as quitting work, having a relapse of a disease,
or dying of cancer.
Principal components analysis

An investigator has made a number of measurements of lung function on a sample of adult males
who do not smoke. In these tests each man is told to inhale deeply and then blow out as fast and
as much as possible into a spirometer, which makes a trace of the volume of air expired over time.
The maximum or forced vital capacity (FVC) is measured as the difference between maximum
inspiration and maximum expiration. Also, the amount of air expired in the first second (FEV1), the
forced mid-expiratory flow rate (FEF 25–75), the maximal expiratory flow rate at 50% of forced vital
capacity (V50), and other measures of lung function are calculated from this trace. Since all these
measures are made from the same flow–volume curve for each man, they are highly interrelated.
From past experience it is known that some of these measures are more interrelated than others and
that they measure airway resistance in different sections of the airway.
The investigator performs a principal components analysis to determine whether a new set of
measurements called principal components can be obtained. These principal components will be
linear functions of the original lung function measurements and will be uncorrelated with each other.
It is hoped that the first two or three principal components will explain most of the variation in the
original lung function measurements among the men. Also, it is anticipated that some operational
meaning can be attached to these linear functions that will aid in their interpretation. The investigator
may decide to do future analyses on these uncorrelated principal components rather than on the
original data. One advantage of this method is that often fewer principal components are needed
than original variables. Also, since the principal components are uncorrelated, future computations
and explanations can be simplified.
Factor analysis
An investigator has asked each respondent in a survey whether he or she strongly agrees, agrees, is
undecided, disagrees, or strongly disagrees with 15 statements concerning attitudes toward inflation.
As a first step, the investigator will do a factor analysis on the resulting data to determine which
statements belong together in sets that are uncorrelated with other sets. The particular statements
that form a single set will be examined to obtain a better understanding of attitudes toward inflation.
Scores derived from each set or factor will be used in subsequent analyses to predict consumer
spending.
Cluster analysis
Investigators have made numerous measurements on a sample of patients who have been classified
as being depressed. They wish to determine, on the basis of their measurements, whether these
patients can be classified by type of depression. That is, is it possible to determine distinct types of
depressed patients by performing a cluster analysis on patient scores on various tests?
Another random document with
no related content on Scribd:
[760] G., siguira.
[761] G., deçimo terçio.
ARGUMENTO
DEL DEÇIMO
QUARTO CANTO
DEL GALLO[762]
En el deçimo quarto canto que se

sigue el auctor concluye con la
subida del çielo y propone
tratar la bajada del infierno[763]
declarando muchas cosas que
açerca dél tuuieron los
gentiles historiadores y poetas
antiguos.
Miçilo.—Ya estoy esperando, ¡o

graçioso gallo y celestial Menipo!
que con tu dulçe y eloquente
canto satisfagas mi spirito tan
deseoso de saber las cosas del
çielo como de estar allá. Por lo
qual te ruego no te sea
pesadumbre auer de satisfazer mi
alma que tanto cuelga de lo que
la has oy de dezir.
Gallo.—No puedo, Miçilo, negar
oy tu petiçion, y ansi digo que si
bien me acuerdo me pediste ayer
te dixesse el asiento y orden que
los angeles y bienauenturados
tienen en el cielo se conoçe allá
entre ellos la ventaja de su
bienauenturança. Para lo qual
deues entender que todo aquel
lugar en que angeles y santos
estan ante Dios está relumbrando
de oro muy marauilloso que
excede sin comparaçion al de
acá, juntamente con el resplandor
inestimable de que su cogeta da
el çielo en que está, como te dixe
en el canto passado; y este lugar
está todo adornado de muy
preciosas margaritas
conuenientes a semejante
estancia. Estan pues todos
aquellos moradores ocupados en
ver a Dios, del qual como de vna
fuente perenal proçede y emana
sumo goço y alegria la qual nunca
los da hastio; pero mientra mas
della gozan mas la desean. En
esto está su bienauenturança y la
ventaja conoçela en sí cada qual
en la más, o menos comunicaçion
en que se les da Dios. Cada vno
está contento con ver a Dios, y
ninguno tiene cuenta con la
ventaja que otro le pueda[764]
tener, porque alli ni ay delantera,
ni lugar en que la preheminençia
se pueda conoçer. No ay asientos
ni sillas, porque el spiritu no
reçibe cansançio sentado ni en
pie, ni ocupa lugar, y do quiera
que el bienauenturado está tiene
delante y a su lado y junto a si a
Dios, y ninguno está tan çerca de
si mesmo como está Dios dél. De
manera que sillas y lugares y
orden y preheminençia del çielo
no está en otra cosa sino en el
pecho de Dios, quanto a su mayor
o menor comunicaçion; y todo lo
demas que vosotros en este caso
por acá dezis es por via de
metaphora, o manera de dezir,
porque lo podais mejor entender
en vuestra manera de hablar. En
esta presençia vniuersal de Dios
que te he dado a entender están
en coros los santos ante su
magestad, a los quales todos mi
angel me guió por los ver. Estaua
en lo mas çercano (a lo que me
pareçió) al trono y acatamiento de
Dios la madre benditissima del
Saluador rodeada de aquella
compañia de los viejos padres de
la religion cristiana, doze
apostoles y discipulos de Cristo y
euangelistas, rodeados de
angeles que con gran musica y
melodia de diuersos instrumentos
y admirables bozes continuan sin
nunca çesar gloria a Dios. Siguen
a estos grandes compañas de
martires con palmas en las manos
y vnas guirnaldas de roble
çelestial en las cabezas, que
denotaua su fortaleza con que
sufrieron los martirios por Cristo.
Por el semejante estos estauan
acompañados de la mesma
abundançia de musica, y
enbelesados y arrebatados en la
vision diuina. Estaua luego vna
inumerable multitud de
confessores, pontifiçes, perlados,
saçerdotes y religiosos que en
vidas honestas y recogidas
acabaron y se fueron a gozar de
Dios. En vn muy florido y ameno
prado de flores muy graçiosas y
de toda hermosura y deleyte
estaua vna gran compaña de
damas, de las quales demas de
su veldad echauan de si vn tan
admirable resplandor que pribara
todo juizio humano si de beatitud
no comunicara. Estas, sentadas
en torno en aquella çelestial
verdura, hazian gran cuenta de
vna prniçipal guia que las
entonaua y ponia en una musica
que con altissimo orden loaua á
Dios. Tenian todas muy graçiosas
guirnaldas en sus cabeças,
entretexidas rosas, violetas,
jazmines, halhelies y de otro
infinito genero de flores naçidas
allá que no se podian marchitar ni
corromper. Dellas tañian organos,
dellas clauicordios, monacordios,
clauiçimbanos y otras diuersas
sonaxas acompañados[765] con
vozes de gran suauidad. Estas,
me dixo mi angel que era la
bianauenturada Santa Ursula con
su compañia de virgenes; porque
demas de sus honze mil auia alli
otro inumerable cuento dellas.
Aqui conoçi las almas de mis
padres y parientes y de otras
muchas personas señaladas que
yo acá conoçi, que dexo yo agora
de nombrar por no te ser
importuno. A las quales conoçi
por vna çierta manera de
alumbramiento que por su bondad
Dios me comunicó, la cual es vna
manera de conoçerse los
bienauenturados entre sí para su
mayor gozo y gloriosa
comunicaçion. En esta alta y
soberana conuersaçion que tengo
contado estuue ocho dias por
preuillegio y don soberano de
Dios.
Miçilo.—Por çierto, gallo, mucho
me has dicho; y tanto que
humano pensamiento nunca tal
conçibió; bien pareçe que has
estado allá; por lo qual bien te
podemos[766] llamar çelestial.
Dime agora que deseo mucho
saber; allá en el cielo ay noches y
dias differentes entre sí?
Gallo.—No, pero despues
venido acá me saludauan mis
amigos como ausente de tanto
tiempo, y por la cuenta que hallé
que contauan en el mes. Que allá
todo es luz, claridad, alegria y
plazer. No ay tinieblas, obscuridad
ni noche donde está Dios que es
luz y lumbre eterna a los que
viben allá. En estos ocho dias vi,
hablé y comuniqué con todos mis
parientes, amigos y conocidos, y
a todos los abracé con mucho
plazer y alegria, y me preguntaron
por los parientes y amigos que
tenian acá, y yo los[767] dezia
todo el bien dellos con que más
los podia complazer y deleytar, y
no era en mi mano dezirles cosas
que los pudiesse entristecer,
avnque de ninguna cosa
reçibieran ellos turbaçion ya que
se la dixera: porque allá estan tan
conformes con la voluntad de
Dios que ninguna cosa que acá
suçeda los puede turbar, porque
tienen entendido que proçede
todo de Dios, porque en Dios y
ellos sola ay vna voluntad y
querer.
Miçilo.—Dime agora, gallo, ¿qué
manera de habla y lenguaje vsan
allá?
Gallo.—Mira, Miçilo, que los
bienauenturados que no tienen
sus cuerpos allá no hablan
lenguaje ni por boz esterior:
porque esta solo se puede hazer
y formar por miembros que como
instrumentos dio naturaleza al
cuerpo para se dar a entender
como lengua, dientes y paladar.
Pero las almas que no tienen
cuerpo, cada qual queriendo
puede comunicar y manifestar sus
coçibimientos sin lengua a quien
le plaze, tan claros como cada
vno se puede asimesmo
entender, y ansi Cristo y la virgen
Maria y San Juan euangelista que
tienen sus cuerpos allá hablan
con bozes como nosotros
hablamos aqui, y ansi será
despues del juizio vniuersal de
todos los buenos que tiene
consigo Dios, que hablarán como
agora nosotros quando despues
del juizio tuuieren sus cuerpos
allá. Pero en el entretanto con
sola su alma se pueden entender.
Miçilo.—Dime más que deseo
saber: ¿si esas almas desos
bienauenturados, si algun tiempo
vienen acá?
Gallo.—Quando yo subi allá
muchas almas de buenos
subieron a gozar, en cuya
compañia entramos en el çielo:
pero al boluer ninguna vi que
boluiese aca: porque creo que no
seria cordura que siendo el alma
del defunto libertada de tan cruel
carçel y mazmorra como es la del
mundo, poseyendo tanto deleyte
y libertad allá desee ni quiera
boluer acá. Bien es de presumir
que el demonio muchas vezes
viene al mundo haziendo[768]
ylusiones y apariciones diziendo
que es algun defunto por
infamarle, o por engañar a sus
parientes.
Miçilo.—Pues dime, gallo: ¿qué
dezian allá en el çielo de las bulas
y indulgençias? Que casi quieren
dezir los theologos deste tiempo
que el Papa puede robar el
purgatorio absolutamente.
Gallo.—Dexemos esas cosas,
Miçilo, que no conuiene que se
diga todo a ti; y sabe que otro
lenguaje es el que se trata acá
differente del que passa allá. Que
muchas cosas tiene en el çielo
Dios y haze, cuya verdad y fin
reserua para sí, porque quiere él,
y porque deue ansi de conuenir
para el suçeso, orden y
dispusiçion del mundo y a la
grandeza de su magestad, y
nuestra saluaçion. Por lo qual no
deuen los hombres escudriñar en
las cosas la causa, fin y voluntad
de Dios, pero deuense en todo
remitir a su infinito y eterno saber,
y prinçipalmente en las cosas que
determina y tiene la iglesia y ley
que profesas; no inquieras más
porque es ocasion de herrar; y
boluiendo al proçeso de mi
peregrinaçion sabras que como
huuimos andado todas las
estançias y choros de angeles y
sanctos me tomó el ángel de mi
guia por la mano y me dixo: vn
gran don te otorga Dios como a
señalado amigo suyo, el qual
deues estimar con las gracias que
te ha hecho hasta aqui; y es que
te quiere comunicar vna vision de
grandes y admirables cosas que
estan por venir; y diziendo esto
llegamos á vn templo de
admirable magestad, el qual
sobre la puerta prinçipal tenia vna
letra que a quantos la leyan
mostraua dezir. Este es el templo
de propheçia y diuinaçion. Era por
defuera adornado de toda
hermosura, edificado de jaspes
muy claros, de ambar y veril
transparente más que vidrio muy
preçioso. Era tan admirable su
resplandor que turbaua la vista; y
como entramos dentro y vi tanta
magestad no me pude contener
sin me derrocar a los pies de mi
angel queriendole adorar, y él me
leuantó diziendome: no hagas tal
cosa, que soy criatura como tú.
Leuantate y adora al criador y
hazedor de todo esto, que tan
gran merçed te conçedio. Era
fundado y adornado por dentro
este diuino templo de muchas
piedras preçiosas: de zafires,
calçedonias, esmeraldas, jaçintos,
rubies, carbuncos, topacios,
perlas, crisotoles, diamantes,
sardo y veril; y luego se me
representó en diuina vision todo el
poder de la tierra quanto del
oriente al poniente, medio dia y
septentrion se puede imaginar, y
estando ansi atento por ver lo que
se me mostraua vi deçendir de lo
alto de los montes Ripheos a las
llanuras de Traçia vna grande y
disforme vestia llena de cuernos y
cabeças, con cuyo siluo y veneno
tenia corrompida y contaminada
la mayor parte del mundo: arabes,
egiçios, syros y persas: hasta
Trasiluania y Bohemia: teutonicos,
anglos y galicos pueblos. Esta
trae cabalgando sobre sí vn
monstruoso serpiente que la guia
y ampara, adornado de mil
colores y nombres de gran
soberuia, y estos juntos son
criados para examen, prueba y
toque de los verdaderos fieles y
secaçes de Dios, y será el estado
y señorio desta fiera más
estendido por causa de las
cobdiçias y disensiones y
intereses de los principes de la
tierra, porque ocupados en ellos
tiene mas lugar sin auer quien le
aya de resistir. Lleuaua este
serpiente en su cabeça vna gran
corona adornada de muchas
piedras preçiosas, y vestido de
purpura y de muy ricos jaezes, y
en la mano un çeptro imperial con
el qual amenaça subjetar todo el
uniuerso. Lleuaua en vna divisa y
estandarte vna letra de gran
soberuia que dize. Ego regno a
Gange et Indo vsque in omnes
fines terre. Que quiere dezir. Yo
reino desde[769] los rios Ganges y
Indus hasta los fines de la tierra.
Lleuaua las manos y ropas
teñidas de sangre de fieles, y
dauale a beuer en vasos de oro y
de plata a sus gentes por más las
encrueleçer. Entonçes sonaron
truenos, grandes terremotos y
relampagos que ponian gran
temor y espanto, que pareçia
desolarse el trono y templo y venir
todo al suelo, y tan grande que
nunca los hombres vieron cosas
de tan grande admiraçion, y fue
tanta que yo cay atonito y
espantado a los pies de mi angel.
El qual leuantandome por la mano
me dixo. ¿De qué te espantas y te
marauillas? Pues mira con gran
atencion, que aunque este
monstruo y vestia tiene agora
gran soberuia muy presto caerá; y
no lo acabó de dezir quando
mirando vi salir de las montañas
hespericas vn gran leon coronado
y de gran magestad que con su
bramido juntó gran muchedumbre
de fieras generosas y brauas que
estan sobre la tierra, las cuales
juntas vinieron contra el fiero
serpiente resistiendo su furia; y a
otro bramido que el fuerte leon dio
juntó en los valles teutonicos
todos los viejos fieles que auia en
la tierra; por cuya sentençia
(aunque con alguna dilaçion) fue
condenada la vestia y sus
secaçes á muerte cruel, y ansi vi
que a deshora dio vn terrible
trueno que toda la tierra tenbló, y
deçendiendo de la gran montaña
vn espantoso y admirable fuego
los abrasa todos conuertiendolos
en zeniza y pauesa. En tanta
manera que en breue tiempo ni
pareçió vestia ni secaz, ni avn
rastro de auer sido alli; y ansi todo
cumplido vi deçendir de la alta
montaña gran compaña de
angeles que cantando con gran
melodia subieron a los çielos al
leon, donde le coronó Dios y le
asentó para sienpre jamas junto á
sí; y acabada la vision me mandó
Dios llamar ante su tribunal y que
propussiese la causa porque auia
subido allá, porque cualquiera
cosa que yo pidiesse se me haria
la razonable satisfazion.
Miçilo.—Querria que antes que
pasasses adelante me
declarasses esa tu vision o
propheçia. ¿Quién se entiende
por la vestia que deçendio de
aquellas montañas, monstruo y
leon?
Gallo.—La interpretaçion deste
enigma no es para ti: a los que
toca se les dará. Vamos adelante
que me queda mucho por dezir.
Como ante Dios fue puesto me
humillé de rodillas ante su tribunal
y luego propuse ansi. Sacra y
diuina magestad, omnipotente
Dios. Porque no ay quien no
enmudezca viendo vuestra
incomparable çelsitud, querria,
señor, demandaros de merçed,
que de alguno de vuestros
cortesanos más acostunbrados a
hablar ante vuestra grandeza
mandassedes leer esta petiçion;
la qual estendiendo la mano
mostré; y luego salio alli delante
el euangelista San Juan, que creo
que lo tenia por offiçio, y ansi en
alta voz començó.
Sacra y diuina magestad,
omnipotente Dios. Vuestro
Icaromenipo, griego de naçion, la
más humilde criatura que en el
mundo teneis, besso vuestro
sacro tribunal y suplico a vuestra
divina magestad tenga por bien
de saber, en como el vuestro
mundo está en necesidad que le
remedieis mientra no tuuieredes
por bien de le destruir llegado el
juizio vniuersal; el tiempo del qual
esta segun nuestra fe reseruado a
vuestro diuino saber. Soy venido
de parte de todos aquellos que en
el mundo tenemos deseos de
alcanzar la vuestra alta sabiduria
y especular con nuestro miserable
injenio los secretos incumbrados
de nuestra naturaleza. Para lo
qual sabra vuestra magestad, que
avnque de noche y de dia por
grandes cuentos de años no
hagamos sino trabajar
estudiando, no se puede por
ningun injenio quanto quiera que
sea perpicaçissimo alcançar
alguna parte por pequeña que
sea en estas buenas letras, artes
y sçiençias. Porque han salido
agora en el mundo vn genero de
hombres somnoliento, dormilon
imaginatiuo, rixoso, vanaglorioso,
lleno de ambiçion y soberuia, y
estos con gran presunçion de sí
mesmos hanse dotado de
grandes títulos de maestros
philosophos y theologos, diziendo
que ellos solos saben y entienden
en todas las sçiençias y artes la
suma verdad; riendose a la
contina de todo quanto hablan,
dizen, comunican, tratan, visten la
otra gente del comun. Diziendo
que todos deuanean y estan
locos, sino ellos solos que tienen
y alcançan la regla y verdad del
vivir; y venidos al enseñar de sus
sçiençias, muestran segun
pareçe, querernos confundir[770].
Porque han inuentado vnos no sé
qué géneros de setas y opiniones
que nos lançan en toda confusion.
Unos se llaman reales y otros
nominales. Que dexado aparte las
niñerias y arguçias de
sophistas[771], actos
sinchategorematicos, y reglas de
instar del Maestro Enzinas y los
sophismas de Gaspar Lax y las
sumulas de Zelaya y Coroneles
que absolutamente, señor, deueis
mandar destruir, y que ellos y sus
auctores no salgan mas a luz. En
la philosophia es verguença de
dezir la diuersidad de prinçipios
naturales que ponen; insecables
atomos, inumerables formas,
diuersidad de materias, ydeas.
Tantas questiones de vacuo y
infinito que no estan debajo de
numero conque se puedan contar.
En la theologia ya no ay sino
relaçiones, segundas intinçiones,
entia rationis; cosas que
solamente tienen ser en el
entendimiento y imaginaçion[772];
en fin cosas que no tienen ser. Es
venido el negoçio a tal estado que
ya diuididas estas gentes en
quadrillas, glosan y declaran
segun sus dos opiniones real y
nominal, vuestra sagrada
Escriptura y Ley; y segun tengo
visto, Señor, en esta xornada que
he hecho acá, que en todo
devanean y sueñan, sin nunca
despertar; y esto, sagrada
magestad, suçede en gran
confusion de los que nos damos
al estudio de las sçiençias[773]. En
lo qual creo que entiende
Sathanas por la perdiçion y daño
del comun. En esto pues
suplicamos a vuestra sagrada
magestad proueais que Luçifer
mande a Sathanas que sobresea
y no se entremeta en causar tan
gran mal, y los auctores se
prendan destas setas, y se les
mande tener perpetuo silençio, y
que sus libros y scripturas en que
estan sus barbaras opiniones las
mandeis quemar y destruir, que
no parezcan más; y pedimos en
todo se nos sea hecha entera
justiçia. Para la qual imploramos
el soberano poder de vuestra
diuina magestad.
Luego como la petiçion fue leyda
proueyo Dios que yo y el mi angel
fuessemos por el infierno y
notificassemos a Luzifer lo
hiziesse ansi como se pedia por
mí, y mandó que se lleuasse
luego de alli al mundo al consejo
de la Inquisiçion y que lo
cumpliessen y hiziessen cunplir
conforme a la petiçion[774]. El qual
aucto luego escriuio San Juan en
las espaldas de la peticion, y la
refrendó y rubricó de su mano
como por Dios omnipotonte fue
proueydo; y luego abraçando a
todos nuestros amigos y parientes
y conoçidos, despidiendonos[775]
de todos ellos nos salimos del
çielo para nos bajar, y quando nos
fueron abiertas las puertas de los
çielos para salir hallamos junto a
ellas infinita multitud de almas
que con grandes fuerças y
inportunidad nos estorbauan, que
ellas por entrar no nos dexauan
salir; hasta que un angel con gran
poder, furia y magestad las apartó
de alli, y yo pregunté a mi angel
qué gente era aquella que estaua
aqui, que con tanto deseo y
inportunidad hazian por entrar y
no las abrian; y el me respondio
que eran las almas de los que en
el mundo tienen toda la vida
buenos deseos de hazer bien,
hazer obras de virtud, hazer
penitençia y recogerse en lugares
santos y buenos con deseo de se
saluar y en toda su vida no
passan de alli ni hazen más que
prometer y mostrar que desean
hazer mucho bien sin nunca
començar, ni avn se aparejar a
padeçer. A estos tales danles la
gloria en la mesma forma, porque
los ponen a la puerta del parayso
con el mesmo deseo de entrar, y
aqui tienen la mayor pena que se
puede imaginar: porque tanto
quanto mucho desearon hazer
bien sin nunca lo començar tanto
mucho más en infinito sin
comparaçion les atormenta el
deseo de entrar sin nunca los
querer abrir; y en el tormento
deste deseo prouee Dios de su
gran justiçia y poder, porque en
esta manera los quiere castigar
para siempre jamas abrasandoles
con el fuego de la justiçia diuina.
Pues como del çielo salimos
lleuóme mi angel y guia por un
camino sin huella ni sendero y
avn sin señal de auer pisado ni
caminado por él alguno, de que
me marauillé, y preguntele qual
fuesse la causa de aquella
esterilidad y respondiome que no
se continuaua mucho despues
que Cristo passó por alli quando
resuçitó, y la compaña de los
santos padres que entonçes sacó
del limbo. Aunque tanbien le
passan los angeles que se
bueluen al çielo dexando despues
de la muerte sus clientulos y
encomendados allá. Repliquele
yo: ¿dime angel, el purgatorio no
está a esta parte? Respondiome:
si está: pero avn los que de ay
passan son tan pocos que no le
bastan trillar ni asenderar. Por
çierto mucho deseo he tenido,
Miçilo, de llegar hasta aqui.
Miçilo.—En verdad yo lo
deseaua mucho más, porque
espero que con tu injeniosa
eloquençia me has de hazer
presente a cosas espantosas y de
grande admiraçion que deseamos
acá los honbres saber. Espero de
ti que harás verdadera narraçion
como de çierta esperiençia, y no
de cosas fabulosas y mentirosas
que los poetas y hombres
prestigiosos acostumbran fingir
por nos lo más encareçer.
Gallo.—Mucho me obligas ¡o
Miçilo! a te complazer quando veo
en ti la confianza que tienes
dezirte yo verdad; y ansi protesto
por la deydad angélica que en
esta xornada me acompañó de no
te contar cosa que salga de lo
que realmente vi y mi guia me
mostró, porque no me atreuere a
hazer tan alto spiritu testigo de
falsedad y fiçion. Contarte he el
sitio y dispusiçion del lugar:
penas, tormentos, furias,
carçeles, mazmorras, fuego y
atormentadores que a la contina
atormentan alli. En conclusion
descriuirte he la suma y puesto
del estado infernal, con aquellas
mesmas sombras, espantos,
miedos, tristezas, gritos, lloros,
llantos y miseria[776] que los
condenados padeçen allí, y
trabajaré por te lo pintar y
proponer con tanta esaxeraçion y
orden de palabras que te haré las
cosas tan presentes aqui como
las tube yo estando allá. Pero
primero quiero que sepas que no
ay allá aquel Pluton, Proserpina,
Æaco y Cançerbero, ni Minos, ni
Rhadamanto[777], juezes
infernales. Ni las lagunas ni rios

Multivariate Analysis 6th Edition Afifi

Uploaded by

Copyright:

Available Formats

Multivariate Analysis 6th Edition Afifi

Uploaded by

Document Information

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Multivariate Analysis 6th Edition Afifi

Uploaded by

Copyright:

Available Formats

Download and Read online, DOWNLOAD EBOOK, [PDF EBOOK EPUB ], Ebooks

download, Read Ebook EPUB/KINDE, Download Book Format PDF

Practical multivariate analysis 6th Edition Afifi

Practical multivariate analysis 6th Edition Afifi

Multivariate Data Analysis Joseph F. Hair

Multivariate Analysis for the Behavioral Sciences Kimmo

Multivariate analysis for the behavioral sciences 2nd

Practical Statistics for Educators 6th Edition Ruth

Genetics : analysis & principles 6th Edition Robert J.

Longitudinal Multivariate Psychology Emilio Ferrer

Recent Developments in Multivariate and Random Matrix

c 2020 by Taylor & Francis Group, LLC

No claim to original U.S. Government works

Printed on acid-free paper

International Standard Book Number-13: 978-1-138-70222-6 (Hardback)

Visit the Taylor & Francis Web site at

and the CRC Press Web site at

I Preparation for Analysis 1

2 Characterizing data for analysis 11

3 Preparing for data analysis 17

6 Selecting appropriate analyses 75

8 Multiple regression and correlation 115

10 Special regression topics 171

11 Discriminant analysis 195

12 Logistic regression 217

13 Regression analysis with survival data 255

14 Principal components analysis 281

15 Factor analysis 297

16 Cluster analysis 317

17 Log-linear analysis 339

18 Correlated outcomes regression 361

Approach of this book

How to use the book

What’s new in the Sixth Edition

Robin A. Donatello, Dr.P.H., is an Associate Professor in the Department of Mathematics and

Preparation for Analysis

What is multivariate analysis?

1.1 Defining multivariate analysis

1.2 Examples of multivariate analyses

Depression study example

Parental HIV study

Northridge earthquake study

Bank loan study

Lung function study

School data set

1.4 Multivariate analyses discussed in this book

Multiple linear regression

Discriminant function analysis

Principal components analysis

En el deçimo quarto canto que se

Miçilo.—Ya estoy esperando, ¡o

You might also like