Spss CATEGORIES 13.0 User Manual

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SPSS Categories® 13.0

Jacqueline J. Meulman

Jacqueline J. Meulman Willem J. Heiser

Willem J. Heiser SPSS Inc

SPSS Inc

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Preface

SPSS 13.0 is a comprehensive system for analyzing data. The Categories optional add-on mod The Categories add-on module must be used with the SPSS 13.0 Base system and is completely integrated into that system.

Installation

To install the Categories add-on module, run the License Authorization Wizard using the authorization code that you received from SPSS Inc. For more information, see the insta

Compatibility

SPSS is designed to run on many computer systems. See the installation instructions that cam requirements.

ule provides the additional analytic techniques described in this manual.

llation instructions supplied with the SPSS Base system.

e with your system for specific information on minimum and recommended

Serial N

Your ser this serial number when you contact SPSS Inc. for information regarding support, payment, or an upgraded system. The serial number was provided with your Base system

Customer Service

If you have any questions concerning your shipment or account, contact your local office your serial number ready for identification.

umbers

ial number is your identification number with SPSS Inc. You will need

, listed on the SPSS Web site at http://www.spss.com/worldwide. Please have

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Training Seminars

SPSS Inc. provides both public and onsite training seminars. All seminars feature hands-on workshops. Seminars will be offered in major cities on a regular basis. For more informa

tion on these seminars, contact your local office, listed on the SPSS

Web site at http://www.spss.com/worldwide.

Technical S

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s of SPSS Technical Support are available to registered customers. Customers may contact Technical Support for assistance in using SPSS or for installation help for one of the supported hardware environments. To reach Technical Support, s

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SS Web site.

The SPSS Statistical Procedures Companion, by Marija Norušis, has been published by Prentice Hall. A new version of this book, updated for SPSS 13.0, is planned

.TheSPSS Advanced Statistical Procedures Companion,alsobasedon SPSS 13.0, is forthcoming. The SPSS Guide to Data Analysis for SPSS 13.0 is also in development. Announcements of publications available exclusively through Prentice Hall wi your home country, and then click

ll be available on the SPSS Web site at http://www.spss.com/estore (select

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About This Manual

This manual documents the graphical user interface for the procedures included in the Categories add-on module. Illustrations of dialog boxes are taken from SPSS for Windows. Di

alog boxes in other operating systems are similar. Detailed information about the command syntax for features in this module is provided in the SPSS Command Syntax Reference, available from the Help menu.

Contacting SPSS

If you would like to be on our mailing list, contact one of our offices, listed on our Web site at http://www.spss.com/worldwide.

Acknowledgments

The optimal scaling procedures and their SPSS implementation were developed by the Data Theory Scaling System Group (DTSS), consisting of members of the departme

nts of Education and Psychology of the Faculty of Social and Behavioral

Sciences at Leiden University.

Willem Heiser, Jacqueline Meulman, Gerda van den Berg, and Patrick Groenen

were invo

lved with the original 1990 procedures. Jacqueline Meulman and Peter Neufeglise participated in the development of procedures for categorical regression, correspondence analysis, categorical principal components analysis, and multidi

mensional scaling. In addition, Anita van der Kooij contributed especially to CATREG, CORRESPONDENCE, and CATPCA,andFrankBusingandWillem Heiser, to the PROXSCAL procedure. The development of PROXSCAL has profited from tec

hnical comments and suggestions from Jacques Commandeur and Patrick

Groenen.

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Contents

Part I: User

's Guide

1 I ntroduction to SPSS O ptimal Scaling

Procedures for Categorical Data 1

WhatIsOptimalScaling?....................................... 1

WhyUseOptimalScaling? ...................................... 2

OptimalScalingLevelandMeasurementLevel ...................... 2

SelectingtheOptimalScalingLevel............................ 3

Tran sfor

CategoryCodes .......................................... 5

Which Proc

Categoric

CategoricalPrincipalComponentsAnalysis .................... 10

NonlinearCanonicalCorrelationAnalysis ...................... 11

Correspon

MultipleCorrespondenceAnalysis ........................... 14

MultidimensionalScaling .................................. 15

AspectRatioinOptimalScalingCharts ........................... 16

RecommendedReadings ...................................... 16

mationPlots . ... ... ... ... ... ... ... ... ... ... ... ... .. 4

edureIsBestforYourApplication? ...................... 8

alRegression..................................... 9

denceAnalysis.................................. 12

2 Categoric

CategoricalRegression ....................................... 19

DefineScaleinCategoricalRegression ........................... 21

CategoricalRegressionDiscretization ............................ 23

al Regression (CATREG) 19

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CategoricalRegressionMissingValues ........................... 24

CategoricalRegressionOptions ................................. 25

CategoricalRegressionOutput.................................. 26

CategoricalRegressionSave................................... 28

CategoricalRegressionTransformationPlots....................... 29

CATREGCommandAdditionalFeatures............................ 29

3 Categorica

l Principal Components Analysis

(CATPCA) 31

DefineScaleandWeightinCATPCA.............................. 34

Categorical Princ ipal Components Analysis Discretization . . . . . . . . . . . . . 36

Categorical Princ ipal Components Analysis Missing Values . . . . . . . . . . . . 38

Categorical Princ ipal Components Analysis Options . . . . . . . . . . . . . . . . . . 39

Categorical Princ ipal Components Analysis Output . . . . . . . . . . . . . . . . . . . 42

Categorical Princ ipal Components Analysis Save . . . . . . . . . . . . . . . . . . . . 44

Categorical Princ ipal Components Analysis Object Plots. . . . . . . . . . . . . . . 45

Categorical Princ ipal Components Analysis Category Plots. . . . . . . . . . . . . 46

Categorical Princ ipal Components Analysis Loadin g Plots . . . . . . . . . . . . . 47

CATPCACommandAdditionalFeatures............................ 48

4 Nonlinear Canonical Correlation Analysis

(OVERALS) 4

Nonlinear

Define Ran

CanonicalCorrelationAnalysis.......................... 49

geandScale....................................... 53

ge............................................... 54

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NonlinearCanonicalCorrelationAnalysisOptions ................... 54

OVERALSCommandAdditionalFeatures........................... 56

5 Correspond

DefineRowRangeinCorrespondenceAnalysis..................... 61

DefineColumnRangeinCorrespondenceAnalysis................... 62

CorrespondenceAnalysisModel ................................ 63

CorrespondenceAnalysisStatistics.............................. 66

CorrespondenceAnalysisPlots ................................. 67

CORRESPONDENCECommandAdditionalFeatures................... 68

6 Multiple C

Define Variable Weight in Multiple Correspondence Analysis . . . . . . . . . . . 73

MultipleCorrespondenceAnalysisDiscretization.................... 74

MultipleCorrespondenceAnalysisMissingValues................... 75

MultipleCorrespondenceAnalysisOptions......................... 77

MultipleCorrespondenceAnalysisOutput ......................... 80

MultipleCorrespondenceAnalysisSave........................... 82

MultipleCorrespondenceAnalysisObjectPlots..................... 82

MultipleCorrespondenceAnalysisVariablePlots.................... 83

MULTIPLE CORRESPONDENCE Command Additional Features . . . . . . . . . . 85

ence Analysis 59

orrespondence Analysis 71

7 Multidime

ProximitiesinMatricesacrossColumns........................... 90

nsional Scaling (PROXSCAL) 87

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ProximitiesinColumns........................................ 91

ProximitiesinOneColumn ..................................... 92

CreateProximitiesfromData ................................... 93

MeasuresDialogBox......................................... 94

DefineaMultidimensionalScalingModel.......................... 95

MultidimensionalScalingRestrictions ............................ 97

MultidimensionalScalingOptions................................ 98

MultidimensionalScalingPlots,Version1.......................... 99

MultidimensionalScalingPlots,Version2......................... 101

MultidimensionalScalingOutput ............................... 101

PROXSCALCommandAdditionalFeatures ........................ 103

Part II: Exam

ples

8 Categorical Regression 107

Example: C

Example: Oz

RecommendedReadings ..................................... 157

arpetCleanerData ................................. 107

A Sta ndard L

ACategoricalRegressionAnalysis .......................... 117

Discretizi

SelectionofTransformationType............................ 132

OptimalityoftheQuantifications ............................ 145

Effects of T

inearRegressionAnalysis....................... 108

oneData........................................ 130

ngVariables .................................... 131

ransformations................................. 148

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9 Categorical

Example: Examining Interrelations of Social Systems . . . . . . . . . . . . . . . . 159

RunningtheAnalysis..................................... 160

NumberofDimensions ................................... 165

Quantifica

ObjectScores.......................................... 169

ComponentLoadings..................................... 170

Additional D

Example:SymptomatologyofEatingDisorders..................... 175

RunningtheAnalysis..................................... 177

Transformat

ModelSummary ........................................ 194

ComponentLoadings..................................... 194

Object Score s

ExaminingtheStructureoftheCourseofIllness ................ 198

Recommended R

Principal Components Analysis 159

tions ......................................... 167

imensions.................................... 172

ionPlots..................................... 190

.......................................... 196

eadings ..................................... 213

10 Nonlinear Canonical Correlation Analysis 217

Example: An

Examining t

AccountingforSimilaritybetweenSets....................... 226

ComponentLoadings..................................... 230

Transformat

SingleversusMultipleCategoryCoordinates................... 235

CentroidsandProjectedCentroids........................... 236

An Alternati

GeneralSuggestions..................................... 247

Recommended R

AnalysisofSurveyResults.......................... 217

heData...................................... 218

ionPlots..................................... 231

veAnalysis................................... 241

eadings ..................................... 248

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11 Corresponde

Normalization.............................................. 250

Example:SmokingBehaviorbyJobCategory...................... 251

RunningtheAnalysis..................................... 252

CorrespondenceTable.................................... 255

Dimensional

Biplot................................................. 257

ProfilesandDistances.................................... 258

Row and Colum

PermutationsoftheCorrespondenceTable.................... 262

ConfidenceStatistics..................................... 263

Supplementa

Example:PerceptionsofCoffeeBrands .......................... 270

RunningtheAnalysis..................................... 271

Dimensionali

Contributions........................................... 277

Plots................................................. 279

Symmetrical N

Example:FlyingMileagebetweenCities.......................... 283

CorrespondenceTable.................................... 292

Row and Column S

RecommendedReadings ..................................... 294

nce Analysis 249

ity.......................................... 256

nScores.................................. 260

ryProfiles................................... 264

ty.......................................... 276

ormalization ................................ 281

cores.................................. 293

12 Multiple Correspondence Analysis 297

Example:CharacteristicsofHardware........................... 297

RunningtheAnalysis..................................... 298

Model Summary

ObjectScores.......................................... 302

DiscriminationMeasures.................................. 304

........................................ 301

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CategoryQuantifications.................................. 305

AMoreDetailedLookatObjectScores....................... 307

Omission of O

RecommendedReadings ..................................... 315

utliers...................................... 310

13 Multidimensional Scaling 317

Example:AnExaminationofKinshipTerms........................ 317

ChoosingtheNumberofDimensions......................... 318

AThree-Di

A Three-Dimensional Solution with Nondefault Transformations . . . . 335

Discussion............................................. 340

RecommendedReadings ..................................... 341

mensionalSolution.............................. 328

Bibliography 343

Index 351

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Part 1: User's Guide

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Chapter

Introductio

ntoSPSS

Optimal Scaling P rocedures for Categorical Data

SPSS Categories procedures use optimal scaling to analyze data that are difficult or impossible for standard statistical procedures to analyze. This chapter describes what each proc relationships between the procedures, and the relationships of these procedures to their standard statistical counterparts.

Note:The Theory Scaling System Group (DTSS), consisting of members of the departments of Education and Psychology, Faculty of Social and Behavioral Sciences, Leiden Univers

What Is Optimal Scaling?

edure does, the situations in which each procedure is most appropriate, the

se procedures and their SPSS implementation were developed by the Data

ity.

The idea behind optimal scaling is to assign numerical quantifications to the

ries of each variable, thus allowing standard procedures to be used to obtain

catego a solution on the quantified variables.

The optimal scale values are assigned to categories of each variable based on the

zing criterion of the procedure in use. Unlike the original labels of the nominal

optimi or ordinal variables in the analysis, these scale values have metric properties.

In most Categories procedures, the optimal quantification for each scaled variable

ined through an iterative method called alternating least squares in which,

is obta after the current quantifications are used to find a solution, the quantifications are updated using that solution. The updated quantifications are then used to find a new

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Chapter 1

solution, which is used to update the quantifications, and so on, until some criterion is reached that signals the process to stop.

Why Use Optimal Scaling?

Categorical data are often found in marketing research, survey research, and research in the social and behavioral sciences. In fact, many researchers deal almost exclusive

data, they often do not perform well for data sets that feature:

 Too few observations  Too many variables  Toomanyvaluespervariable

ly with categorical data.

While adaptations of most standard models exist specifically to analyze categorical

By quantify situations. Moreover, they are useful even when specialized techniques are appropriate.

output is often based on graphical displays. Optimal scaling techniques offer excellent exploratory analyses, which complement other SPSS models well. By narrowing the focus of yo the basis of an analysis that centers on interpretation of model parameters.

Optimal S

This can b When specifying the level, you specify not the level at which variables are measured but the level at which they are scaled. The idea is that the variables to be quantified may have

For Cate

 The nom

ing categories, optimal scaling techniques avoid problems in these

Rather tha

n interpreting parameter estimates, the interpretation of optimal scaling

ur investigation, visualizing your data through optimal scaling can form

caling Level and Measurement Level

e a very confusing concept when you first use Categories procedures.

nonlinear relations regardless of how they are measured.

gories purposes, there are three basic levels of measurement:

inal level implies that a variable’s values represent unordered categories.

Examples of variables that might be nominal are region, zip code area, religious affiliation, and multiple choice categories.

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Introduction t



The ordinal level implies that a variable’s values represent ordered categories.

Examples in

clude attitude scales representing degree of satisfaction or confidence

o SPSS Optimal Scaling Procedures for Categorical Data

and preference rating scores.

 The numerical level implies that a variable’s values represent ordered categories

with a meaningful metric so that distance comparisons between categories are appropriat

For example

e. Examples include age in years and income in thousands of dollars.

, suppose that the variables region, job,andage are coded as shown

in the following table.

Tab l e 1-1

Coding scheme for region, job , and age

Region Job Age

1 North 1 intern 20 twenty years old 2 South 2 sales rep 22 twenty-two years old 3East 3 4 West 27 twenty-seven years old

manager

twenty-five years old

The values shown represent the categories of each variable. Region would be a nominal variable. There are four categories of region, with no intrinsic ordering. Values 1

through 4 simply represent the four categories; the coding scheme is completely arbitrary. Job,ontheotherhand,couldbeassumedtobeanordinal variable. The original categories form a progression from intern to manager. Larger codes r

epresent a job higher on the corporate ladder. However, only the order information is known—nothing can be said about the distance between adjacent categories. In contrast, age could be assumed to be a numerical variable. In the case

, the distances between the values are intrinsically meaningful. The distance

of age between 20 and 22 is the same as the distance between 25 and 27, while the distance between 22 and 25 is greater than either of these.

Selecting the Optimal Scaling Level

It is important to understand that there are no intrinsic properties of a variable that automatically predefine what optimal scalinglevelyoushouldspecifyforit.Youcan

ore your data in any way that makes sense and makes interpretation easier. By

expl

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Chapter 1

analyzing a numerical-level variable at the ordinal level, for example, the use of a nonlinear transformation may allow a solution in fewer dimensions.

The followin

g two examples illustrate how the “obvious” level of measurement might not be the best optimal scaling level. Suppose that a variable sorts objects into age groups. Although age can be scaled as a numerical variable, it may be true that for peo

ple younger than 25 safety has a positive relation with age, whereas for people older than 60 safety has a negative relation with age. In this case, it might be better to treat age as a nominal variable.

As another

example, a variable that sorts persons by political preference appears to be essentially nominal. However, if you order the parties from political left to political right, you might want the quantification of parties to respect this o rder by using an

ordinal level of analysis.

Even though there are no predefined properties of a variable that make it exclusively one level or another, there are some general guidelines to help the novice user. Wit

h single-nominal quantification, you don’t usually know the order of the categories but you want the analysis to impose one. If the order of the categories is known, you should try ordinal quantification. If the categories are unorderable, you might try

multiple-nominal quantification.

Transformation Plots

The different levels at which each variable can be scaled impose different restrictions on the qu quantifications and the original categories resulting from the selected optimal scaling level. For example, a linear transformation plot results when a variable is treated as numeri Transformationplots for variables treated nominally that are U-shaped (or the inverse) display a quadratic relationship. Nominal variables could also yield transformation plots w following figure displays a sample transformation plot.

optim collapsing these categories into one category may be warranted. Alternatively, if a variable treated as nominal receives quantifications that display an increasing trend, an ord

antifications. Transformation plots illustrate the relationship between the

cal. Variables treated as ordinal result in a nondecreasing transformation plot.

ithout apparent trends by changing the order of the categories c ompletely. The

Transformation plots a re particularly suited to determining how well the selected

al scaling level performs. If several categories receive similar quantifications,

inal transformation may result in a similar fit. If that trend is linear, numerical

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Introduction t

o SPSS Optimal Scaling Procedures for Categorical Data

treatment may be appropriate. However, if collapsing categories or changing scaling levels is warranted, the analysis will not change significantly.

Figure 1-1

Transformation plot of price (numerical)

Category Codes

Somecareshouldbetakenwhencodingcategorical variables because some coding schemes may yield unwanted output or incomplete analyses. Possible coding schemes for job

Tab l e 1

ative coding schemes for job

Altern

Category A B C D

intern 1 1 sales rep 2 2 6

manager

are displayed in the following table.

-2

Scheme

3773

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Chapter 1

Some Categories procedures require that the range of every variable used be defined. Any value outside this range is treated as a missing value. The minimum category value is alw

ays 1. The maximum category value is supplied by the user. This value is not the number of categories for a variable—it is the largest category value. For example, in the table, scheme A has a maximum category value of 3 and scheme B has a maxim

um category value of 7, yet both schemes code the same three categories.

The variable range determines which categories will be omitted from the analysis. Any categories with codes outside the defined range are omitted from the analysis. This is a si

mple method for omitting categories but can result in unwanted analyses. An incorrectly defined maximum category can omit valid categories from the analysis. For example, for scheme B, defining the maximum category value to be 3 indicates that job h

as categories coded from 1 to 3; the manager category is treated as missing. Because no category has actually been coded 3, the third category in the analysis contains no cases. If you wanted to omit all manager categories, this analysis would be appro

priate. However, if managers are to be included, the maximum category must

be defined as 7, and missing values must be coded with values above 7 or below 1.

For variables treated as nominal or ordinal, the range of the categories does not

he results. For nominal variables, only the label and not the value associated

affect t with that label is important. For ordinal variables, the order of the categories is preserved in the quantifications; the category values themselves are not important. All coding

schemes resulting in the same category ordering will have identical results. For example, the first three schemes in the table are functionally equivalent if job is analyzed at an ordinal level. The order of the categories is identical in these schemes. Scheme

D, on the other hand, inverts the second and third categories and will yield

different results than the other schemes.

Although many coding schemes for a variable are functionally equivalent, schemes

mall differences between codes are preferred because the codes have an

with s impact on the amount of output produced by a procedure. All categories coded with values between 1 and the user-defined maximum are valid. If any of these

ories are empty, the corresponding quantifications will be either system missing

categ or zero, depending on the procedure. Although neither of these assignments affect the analyses, output is produced for these categories. Thus, for scheme B, job has

categories that receive system-missing values. For scheme C, there are also four

four categories receiving system-missing indicators. In contrast, for scheme A there are no system-missing quantifications. Using consecutive integers as codes for variables

ated as nominal or ordinal results in much less output without affecting the results.

tre

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Introduction t

o SPSS Optimal Scaling Procedures for Categorical Data

Coding schemes for variables treated as numerical are more restricted than the ordinal case. For these variables, the differences between consecutive categories are important.

Tab l e 1-3

Alternativ

Category A B C

Any recod

The following table displays three coding schemes for age.

e coding schemes for age

Scheme

20 20 1 1 22 22 3 2 25 25 27 27 8 4

ing of numerical variables must preserve the differences between the categories. Using the original values is one method for ensuring preservation of differences. However, this can result in many categories having system-missing

ors. For example, scheme A employs the original observed values. For all

indicat Categories procedures except for Correspondence Analysis, the maximum category value is 27 and the minimum category value is set to 1. The first 19 categories are empty a

nd receive system-missing indicators. The output can quickly become rather cumbersome if the maximum category is much greater than 1 and there are many empty categories between 1 and the maximum.

To redu

ce the amount of output, recoding can be done. However, in the numerical case, the Automatic Recode facility should not be used. Coding to consecutive integers results in differences of 1 between all consecutive categories, and, as a

t, all quantifications will be equally spaced. The metric characteristics deemed

resul important when treating a variable as numerical are destroyed by recoding to consecutive integers. For example, scheme C in the table corresponds to automatically

ding age. The difference between categories 22 and 25 has changed from three to

reco one, and the quantifications will reflect the latter difference.

An alternative recoding scheme that preserves the differences between categories

subtract the smallest category value from every category and add 1 to each

is to difference. Scheme B results from this transformation. The smallest category value, 20, has been subtracted from each category, and 1 was added to each result. The

sformed codes have a minimum of 1, and all differences are identical to the

tran original data. The maximum category value is now 8, and the zero quantifications before the first nonzero quantification a re all eliminated. Yet, the nonzero

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Chapter 1

quantifications corresponding to each category resulting from scheme B are identical to the quantifications from scheme A.

Which Procedure Is Best for Your Application?

The techniques embodied in four of these procedures (Correspondence Analysis, Multiple Correspondence Analysis, Categorical Principal Components Analysis, and Nonlinear data analysis known as dimension reduction. That is, relationships between variables are represented in a few dimensions—say two or three—as often as possible. would be too difficult to fathom in their original richness and complexity. In market research applications, these techniques can be a form of perceptual mapping.A major adv levels of optimal scaling.

Categorical Regression describes the relationship between a categorical response variable each predictor variable on the response variable is described by the corresponding regression weight. As in the other procedures, data can be analyzed with different levels o

Multidimensional Scaling describes relationshipsbetweenobjectsinasfew dimensions as possible, starting either with a matrix of proximities between the objects

Canonical Correlation Analysis) fall into the general area of multivariate

This enables you to describe structures or patterns in the relationships that

antage of these procedures is that they accommodate data with different

and a combination of categorical predictor variables. The influence of

f optimal scaling.

or with the original data from which the proximities are computed.

ing are brief guidelines for each of the procedures:

 Use Cat

egorical Regression to predict the values of a categorical dependent

variable from a combination of categorical independent variables.

 Use Categorical Principal Components Analysis to account for patterns of

variation in a single set of variables of mixed optimal scaling levels.

 Use Nonlinear Canonical Correlation Analysis to assess the extent to which two

or more

 Use Co

sets of variables of mixed optimal scaling levels are correlated.

rrespondence Analysis to analyze two-way contingency tables or data that can be expressed as a two-way table, such as brand preference or sociometric choice data.

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

 Use Multidimensional Scaling to analyze proximity data to find a least-squares

Categorica

The use of Ca is to predict a dependent (response) variable from a set of independent (predictor) variables. As with all optimal scaling procedures, scale values are assigned to each categ to the regression. The solution of a categorical regression maximizes the squared correlation between the transformed response and the weighted combination of transfor

Relation to other Categories procedures. Categorical regression with optimal scaling is

comparable to optimal scaling canonical correlation analysis with two sets, one of which con is derived by comparing each set to an unknown variable that lies somewhere between all of the sets. In categorical regression, similarity of the transformed response and the line

Introduction t

o SPSS Optimal Scaling Procedures for Categorical Data

Use Multiple Correspondence Analysis to analyze a categorical multivariate data

matrix when

you are willing to make no stronger assumption that all variables are

analyzed at the nominal level.

representation of the objects in a low-dimensional space.

l Regression

tegorical Regression is most appropriate when the goal of your analysis

ory of every variable such that these values are optimal with respect

med predictors.

tains only the dependent variable. In the latter technique, similarity of sets

ar combination of transformed predictors is assessed directly.

Relation to standard techniques. In standard linear regression, categorical variables can

either be recoded as indicator variables or be treated in the same fashion as interval

ariables. In the first approach, the model contains a separate intercept and slope

level v for each combination of the levels of the categorical variables. This results in a large number of parameters to interpret. In the second approach, only one parameter is

ted for each variable. However, the arbitrary nature of the category codings

estima makes generalizations impossible.

If some of the variables are not continuous, alternative analyses are available. If

sponse is continuous and the predictors are categorical, analysis of variance

the re is often employed. If the response is categorical and the predictors are continuous, logistic regression or discriminant analysis may be appropriate. I f the response and

edictors are both categorical, loglinear models are often used.

the pr

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Chapter 1

Regression with optimal scaling offers three scaling levels for each variable. Combinations of these levels can account for a wide range of nonlinear relationships for which an offers greater flexibility than the standard approaches with minimal added complexity.

In addition, nonlinear transformations of the predictors usually reduce the dependenci matrix for the predictors with the eigenvalues of the correlation matrix for the optimally scaled predictors, the latter set will usually be less variable than the former. In other wo of the predictor correlation matrix smaller and the smaller eigenvalues larger.

y single “standard” method is ill-suited. Consequently, optimal scaling

es among the predictors. If you compare the eigenvaluesof the correlation

rds, in categorical regression, optimal scaling makes the larger eigenvalues

Categoric

al Principal Components Analysis

The use of you want to account for patterns of variation in a single set of variables of mixed optimal scaling levels. This technique attempts to reduce the dimensionality of asetofva values are assigned to each category of every variable so that these values are optimal with respect to the principal components solution. Objects in the analysis receive compone patterns among the objects in the analysis and can reveal unusual objects in the data. The solution of a categorical principal components analysis maximizes the correl of components (dimensions) specified.

An important application of categorical principal components is to examine prefer preference. In the usual SPSS data configuration, rows are individuals, columns are measurements for the items, and the scores across rows are preference scores (on a 0 to 10 s may want to treat the individuals as variables. Using the Transpose procedure, you can transpose the data. The raters become the variables, and all variables are declared

nal. There is no objection to using more variables than objects in CATPCA.

ordi

Relation to other Categories procedures. If all variables are declared multiple nominal,

categorical principal components analysis produces an analysis equivalent to a mult

Categorical Principal Components Analysis is most appropriate when

riables while accounting for as much of the variation as possible. Scale

nt scores based on the quantified data. Plots of the component scores reveal

ations of the object scores with each of the quantified variables for the number

ence data, in which respondents rank or rate a number of items with respect to

cale, for example), making the data row-conditional. For preference data, you

iple correspondence analysis run on the same variables. Thus, categorical

Page 27

Nonlinear

Introduction t

o SPSS Optimal Scaling Procedures for Categorical Data

principal components analysis can be seen as a type of multiple correspondence analysis in which some of the variables are declared ordinal or numerical.

Relation to

standard techn iques.

If all variables are scaled on the numerical level, categorical principal components analysis is equivalent to standard principal components analysis.

More genera

lly, categorical principal components analysis is an alternative to computing the correlations between non-numerical scales and analyzing them using a standard principal components or factor-analysis approach. Naive use of the usual Pearson co

rrelation coefficient as a measure of association for ordinal data can lead to

nontrivial bias in estimation of the correlations.

Canonical Correlation Analysis

Nonlinear different applications. The goal of nonlinear canonical correlation analysis is to analyze the relationships between two or more sets of variables instead of between the varia may have two sets of variables, where one set of variables might be demographic background items on a set of respondents and a second set might be responses to a set of at ordinal, and numerical. Optimal scaling canonical correlation analysis determines the similarity among the sets by simultaneously comparing the canonical variables from each se

Canonical Correlation Analysis is a very general procedure with many

bles themselves, as in principal components analysis. For example, you

titude items. The scaling levels in the analysis can be any mix of nominal,

t to a compromise set of scores assigned to the objects.

Relation to other Categories procedures. If there are two or more sets of variables

with only one variable per set, optimal scaling canonical correlation analysis is

lent to optimal scaling principal components analysis. If all variables in

equiva a one-variable-per-set analysis are multiple nominal, optimal scaling canonical correlation analysis is equivalent to multiple correspondence analysis. If there are two

f variables, one of which contains only one variable, optimal scaling canonical

sets o correlation analysis is equivalent to categorical regression with optimal scaling.

Relation to standard techniques. Standard canonical correlation analysis is a statistical

ique that finds a linear combination of one set of variables and a linear

techn combination of a second set of variables that are maximally correlated. Given this set of linear combinations, canonical correlation analysis can find subsequent

Page 28

Chapter 1

independent sets of linear combinations, referred to as canonical variables, up to a maximum number equal to the number of variables in the smaller set.

If there are t

wo sets of variables in the analysis and all variables are defined to be numerical, optimal scaling canonical correlation analysis is equivalent to a standard canonical correlation analysis. Although SPSS does not have a canonical correlation analysis pr

ocedure, many of the relevant statistics can be obtained from multivariate

analysis of variance.

Optimal scaling canonical correlation analysis has various other applications.

If you have

two sets of variables and one of the sets contains a nominal variable declared as single nominal, optimal scaling canonical correlation analysis results can be interpreted in a similar fashion to regression analysis. If you consider the variable to be multi

ple nominal, the optimal scaling analysis is an alternative to discriminant analysis. Grouping the variables in more than two sets provides a variety of ways to analyze your data.

Correspondence Analysis

The goal of correspondence analysis is to make biplots for correspondence tables. In a correspondence table, the row and column variables are assumed to represent unorder Both variables are inspected for their nominal information only. That is, the only consideration is the fact that some objects are in the same category while others are not. same variable.

tables in the correspondence analysis solution is the minimum of r minus 1 or c minus 1, whichever is less. In other words, you could perfectly represent the row categories or the col speaking, however, you would like to represent the row and column categories of a two-way table in a low-dimensional space, say two dimensions, for the reason that t spatial representations.

stati are represented in the low-dimensional representation. Provided that the quality of

ed categories; therefore, the nominal optimal scaling level is always used.

Nothing is assumed about the distance or order between categories of the

One specific use of correspondence analysis is the analysis of two-way contingency

.Ifatablehasr active rows and c active columns, the number of dimensions

umn categories of a contingency table in a space of dimensions. Practically

wo-dimensional plots are more easily comprehensible than multidimensional

When fewer than the maximum number of possible dimensions is used, the

stics produced in the analysis describe how well the row and column categories

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Introduction t

o SPSS Optimal Scaling Procedures for Categorical Data

representation of the two-dimensional solution is good, you can examine plots of the row points and the column points to learn which categories of the row variable are similar, wh

ich categories of the column variable are similar, and which row and

column categories are similar to each other.

Relation to other Categories procedures. Simple correspondence analysis is limited to

two-way tab

les. If there are more than two variables of interest, you can combine variables to create interaction variables. For example, for the variables region, job, and age, you can combine region and job to create a new variable rejob with the 12 categorie

s shown in the following table. This new variable forms a two-way table

with age (12 rows, 4 columns), which can be analyzed in correspondence analysis.

Tab l e 1-4

Combinations of region and job

Category Code Category Definition Category Code Category Definition

1 North, intern 7 East, intern 2 North, sales rep 8 East, sales rep 3 North, manager 9 East, manager 4 South, intern 10 West, intern 5 6 South, manager 12 West, manager

South, sales rep 11 West, sales rep

One shortcoming of this approach is that any pair of variables can be combined. We can combine job and age, yielding another 12-category variable. Or we can combine

and age, which results in a new 16-category variable. Each of these interaction

region

variables forms a two-way table with the remaining variable. Correspondence analyses of these three tables will not yield identical results, yet each is a valid

ach. Furthermore, if there are four or more variables, two-way tables comparing

appro an interaction variable with another interaction variable can be constructed. The number of possible tables to analyze can get quite large, even for a few variables. You

elect one of these tables to analyze, or you can analyze all of them. Alternatively,

can s the Multiple Correspondence Analysis procedure can be used to examine all of the variables simultaneously without the need to construct interaction variables.

tion to standard techniques.

Rela

The SPSS Crosstabs procedure can also be used to analyze contingency tables, with independence as a common focus in the analyses. However, even in small tables, detecting the cause of departures from independence

be difficult. The utility of correspondence analysis lies in displaying such

may

Page 30

Chapter 1

patterns for two-way tables of any size. If there is an association between the row and column variables—that is, if the chi-square value is significant—correspondence analysis ma

y help reveal the nature of the relationship.

Multiple Correspondence Analysis

Multiple Correspondence Analysis tries to produce a solution in which o bjects within thesameca are plotted far apart. Each object is as close as possible to the category points of categories that apply to the object. In this way, the categories divide the objects into homogeneo objects in the same categories into the same subgroups.

For a one-dimensional solution, multiple correspondence analysis assigns optimal scale val that overall, on average, the categories have maximum spread. For a two-dimensional solution, multiple correspondence analysis finds a second set of quantifications of the categori spread, and so on. Because categories of a variable receive as many scorings as there are dimensions, the variables in the analysis are assumed to be multiple nominal in optimal

Multiple correspondence analysis also assigns scores to the objects in the analysis in such a way that the category quantifications are the averages, or centroids, of the object s

tegory are plotted close together and objects in different categories

us subgroups. Variables are considered homogeneous when they classify

ues (category quantifications) to each category of each variable in such a way

es of each variable unrelated to the first set, attempting again to maximize

scaling level.

cores of objects in that category.

Relation to other Categories procedures. Multiple correspondence analysis is also

known as homogeneity analysis or dual scaling. It gives comparable, but not

cal, results to correspondence analysis when there are only two variables.

identi Correspondence analysis produces unique output summarizing the fit and quality of representation of the solution, including stability information. Thus, correspondence

sis is usually preferable to multiple correspondence analysis in the two-variable

analy case. Another difference between the two procedures is that the input to multiple correspondence analysis is a data matrix, where the rows are objects and the columns

riables, while the input to correspondence analysis can be the same data matrix,

are va a general proximity matrix, or a joint contingency table, which is an aggregated matrix in which both the rows and columns represent categories of variables. Multiple

espondence analysis can also be thought of as principal components analysis of

corr data scaled at the multiple nominal level.

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Multidi

Introduction t

Relation to standard techniques. Multiple correspondence analysis can be thought of

o SPSS Optimal Scaling Procedures for Categorical Data

as the analysis of a multiway contingency table. Multiway contingency tables can also be anal

yzed with the SPSS Crosstabs procedure, but Crosstabs gives separate summary statistics for each category of each control variable. With multiple correspondence analysis, it is often possible to summarize the relationship between all of the v

ariables with a single two-dimensional plot. An advanced use of multiple correspondence analysis is to replace the original category values with the optimal scale values from the first dimension and perform a secondary multivariate analysis. Since mult

iple correspondence analysis replaces category labels with numerical scale values, many different procedures that require numerical data can be applied after the multiple correspondence analysis. For example, the Factor Analysis procedure produces

a first principal component that is equivalent to the first dimension of multiple correspondence analysis. The component scores in the first dimension are equal to the object scores, and the squared component loadings are equal to the discrim

ination measures. The second multiple correspondence analysis dimension,

however, is not equal to the second dimension of factor analysis.

mensional Scaling

The use analysis is to find the structure in a set of distance measures between objects or cases. This is accomplished by assigning observations to specific locations in a concep match the given (dis)similarities as closely as possible. The result is a least-squares representation of the objects in that low-dimensional space, which, in many cases, will h

of Multidimensional Scaling is most appropriate when the goal of your

tual low-dimensional space so that the distances between points in the space

elp you further understand your data.

Relation to other Categories procedures. When you have multivariate data from which

you create distances and then analyze with multidimensional scaling, the results

milar to analyzing the data using categorical principal components analysis

are si with object principal normalization. This kind of PCA is also known as principal coordinates analysis.

tion to standard techniques.

Rela

The Categories Multidimensional Scaling procedure (PROXSCAL) offers several improvements upon the scaling procedure available in the Base system (ALSCAL). PROXSCAL offers an accelerated algorithm for

ain models and allows you to put restrictions on the common space. Moreover,

cert

Page 32

Chapter 1

PROXSCAL attempts to minimize normalized raw stress rather than S-stress (also referred to as strain). The normalized raw stress is generally preferred because it is a measure bas

ed on the distances, while the S-stress is based on the squared distances.

Aspect Ratio in Optimal Scaling Charts

Aspect ratio in optimal scaling plots is isotropic. In a two-dimensional plot, the distance r unit in dimension 2. If you change the range of a dimension in a two-dimensional plot, the system changes the size of the other dimension to keep the physical distances equal. Iso

epresenting one unit in dimension 1 is equal to the distance representing one

tropic aspect ratio cannot be overridden for the optimal scaling procedures.

Categorical Regression

Categorical regression quantifies categorical data by assigning numerical values to the catego variables. Categorical regression is also known by the acronym CATREG, for categorical regression.

Standard differences between a response (dependent) variable and a weighted combination of predictor (independent) variables. Variables are typically quantitative, with (nominal) categori variables serve to separate groups of cases, and the technique estimates separate sets of parameters for each group. The estimated coefficients reflect how changes in the pr combination of predictor values.

An alternative approach involves regressing the response on the categorical predict variable. However, for categorical variables, the category values are arbitrary. Coding the categories in different ways yield different coefficients, making comparisons across

CATREG extends the standard approach by simultaneously scaling nominal, ordinal, and numerical variables. The procedure quantifies categorical variables so that th treats quantified categorical variables in the same way as numerical variables. Using nonlinear transformations allow variables to be analyzed at a variety of levels to find the be

ries, resulting in an optimal linear regression equation for the transformed

linear regression analysis involves minimizing the sum of squared

cal data recoded to binary or contrast variables. As a result, categorical

edictors affect the response. Prediction of the response is possible for any

or values themselves. Consequently, one coefficient is estimated for each

analyses of the same variables difficult.

e quantifications reflect characteristics of the original categories. The procedure

st-fitting model.

Example. Categorical regression could be used to describe how job satisfaction

depends on job category, geographic region, and amount of travel. You might find

high levels of satisfaction correspond to managers and low travel. The resulting

that

Page 36

Chapter 2

regression equation could be used to predict job satisfaction for any combination of the three independent variables.

Statistics

and plots.

Frequencies, regression coefficients, ANOVA table, iteration history, category quantifications, correlations between untransformed p redictors, correlations between transformed predictors, residual plots, and transformation plots.

Data. CATRE

G operates on category indicator variables. The category indicators should be positive integers. You can use the Discretization dialog box to convert fractional-value variables and string variables into positive integers.

ns.

Assumptio

Only one response variable is allowed, but the maximum number of predictor variables is 200. The data must contain at least three valid cases, and the number of valid cases must exceed the number of predictor variables plus one.

Related pr

ocedures.

CATREG is equivalent to categorical canonical correlation analysis with optimal scaling (OVERALS) with two sets, one of which contains only one variable. Scaling all variables at the numerical level corresponds to standard multiple

To Obtain a Categorical Regression

E From the menus choose:

Analyze

Regressi

regression analysis.

Optimal S

caling...

Page 37

Categorical Re

Figure 2-1

Categorical Regression dialog box

E Select the dependent variable and independent variable(s). E Click OK.

gression (CATREG)

Optionally, change the scaling level for each variable.

Define Scale in Categorical Regression

You can set the optimal scaling level for the dependent and independent variables. By default, they are scaled as second-degree monotonic splines (ordinal) with two interior knots. Additionally, you can set the weight for analysis variables.

Page 38

Chapter 2

Figure 2-2

Define Scale dialog box

Optimal Scaling Level. You can also select the scaling level for quantifying each

variable

 Spline Or

dinal.

The order of the categories of the observed variable is preserved in the optimally scaled variable. Category points will be on a straight line (vector) through the origin. The resulting transformation is a smooth monotonic piecewise polynomi

al of the chosen degree. The pieces are specified by the user-specified

number and procedure-determined placement of the interior knots.

 Spline Nominal. The only information in the observed variable that is preserved in

the optimally scaled variable is the grouping of objects in categories. The order of the ca

tegories of the observed variable is not preserved. Category points will be on a straight line (vector) through the origin. The resulting transformation is a smooth, possibly nonmonotonic, piecewise polynomial of the chosen degree. The piec

es are specified by the user-specified number and procedure-determined placement of the interior knots.

 Ordinal. The order of the categories of the observed variable is preserved in the

optimally scaled variable. Category points will be on a straight line (vector)

h the origin. The resulting transformation fits better than the spline ordinal

throug transformation but is less smooth.

 Nominal. The only information in the observed variable that is preserved in the

optimally scaled variable is the grouping of objects in categories. The order of

tegories of the observed variable is not preserved. Category points will be

the ca

Page 39

 Numeric. Ca

Categoric

The Discre variables. Fractional-value variables are grouped into seven categories (or into the number of distinct values of the variable if this number is less than seven) with an approxim always converted into positive integers by assigning category indicators according to ascending alphanumeric order. Discretization for string variables applies to these in arethenusedintheanalysis.

Categorical Re

on a straight line (vector) through the origin. The resulting transformation fits better than

The order of the categories and the equal distances between category numbers of the observed variable are preserved in the optimally scaled variable. Category points will at the numeric level, the analysis is analogous to standard principal components analysis.

the spline nominal transformation but is less smooth.

tegories are treated as ordered and equally spaced (interval level).

be on a straight line (vector) through the origin. When all variables are

gression (CATREG)

al Regression Discretization

tizationdialogboxallowsyoutoselectamethodofrecodingyour

ately normal distribution unless otherwise specified. String variables are

tegers. Other variables are left alone by default. The discretized variables

Page 40

Chapter 2

Figure 2-3

Discretization dialog box

Method

 Groupi

 Rankin

 Multip

Choose between grouping, ranking, and multiplying.

ng.

Recode into a specified number of categories or recode by interval.

The variable is discretized by ranking the cases.

lying.

The current values of the variable are standardized, multiplied by 10,

rounded, and have a constant added so that the lowest discretized value is 1.

Groupin

 Number

The following options are availablewhen discretizing variables by grouping:

of categories.

Specify a number of categories and whether the values of the variable should follow an approximately normal or uniform distribution across those categories.

 Equal intervals. Variables are recoded into categories defined by these equally

ntervals. You must specify the length of the intervals.

sized i

Categorical Regression Missing Values

The Missing Values dialog box allows you to choose the strategy for handling missing

in analysis variables and supplementary variables.

values

Page 41

Categorical Re

Figure 2-4

Missing Values dialog box

Strategy. Choose to exclude objects with missing values (listwise deletion) or impute

gression (CATREG)

missing values (active treatment).

 Exclude objects with missing values on this variable. Objects with missing values

on the selected variable are excluded from the analysis. This strategy is not available for supplementary variables.

 Impute missing values. Objects with missing values on the selected variable have

those values imputed. You can choose the method of imputation. Select

Mode to

replace missing values with the most frequent category. When there are multiple modes, the one with the smallest category indicator is used. Select

Extra category

to replace missing values with the same quantification of an extra category. This implies that objects with a missing value on this variable are considered to belong to the same (extra) category.

Categorical Regression Options

The Options dialog box allows you to select the initial configuration style, specify iteration and convergence criteria, select supplementary objects, and set the labeling of plots.

Page 42

Chapter 2

Figure 2-5

Options dialog box

Supplementary Objects. This allows you to specify the objects that you want to treat as

supplem

entary. Simply type the number of a supplementary object and click

Add.

You can not weight supplementary objects (specified weights are ignored).

Initial Configuration. If no variables are treated as nominal, select the Numerical

configu

ration. If at least one variable is treated as nominal, select the

configuration.

Criteria. You can specify the maximum number of iterations that the regression may

ugh in its computations. You can also select a convergence criterion value. The

go thro regression stops iterating if the difference in total fit between the last two iterations is less than the convergence value or if the maximum number of iterations is reached.

Label P

lots By.

Allows you to specify whether variables and value labels or variable names and values will be used in the plots. You can also specify a maximum length for labels.

Categorical Regression Output

The Output dialog box allows you to select the statistics to display in the output.

Random

Page 43

Categorical Re

Figure 2-6

Output dialog box

gression (CATREG)

Tables. Produces tables for:

 Multiple R. Includes R

,adjustedR2, and adjusted R2taking the optimal scaling

into account.

 Coefficients. This option givesthree tables: a Coefficientstable that includes betas,

standard error of the betas, t values, and significance; a Coefficients-Optimal Scaling table with the standard error of the betas taking the optimal scaling degrees of freedom into account; and a table with the zero-order, part, and partial correlation, Pratt’s relative importance measure for the transformed predictors, and the tolerance before and after transformation.

 Iteration history. For each iteration, including the starting values for the algorithm,

the multiple R and regression error are shown.TheincreaseinmultipleR is listed starting from the first iteration.

 Correlations of the original variables. A matrix showing the correlations between

the untransformed variables is displayed.

Page 44

Chapter 2



Correlations of the transformed variables. A matrix showing the correlations

between the

 ANOVA. This

transformed variables is displayed.

option includes regression and residual sums of squares, mean squares, and F. T wo ANOVA tables are displayed: one with degrees of freedom for the regression equal to the number of predictor variables and one with degrees of freedom f

Category Quantifications. Tables showing the transformed values of the selected

or the regression taking the optimal scaling into account.

variables are displayed.

Descriptiv

of the selected variables are displayed.

Categoric

The Save di

Figure 2-7

Save dialog box

Save to Working File. You can save the transformed values of the variables,

model-predicted values, and residuals to the working file.

e Statistics.

Tables showing the frequencies, missing values, and modes

al Regression Save

alog box allows you to save results to the working file or an external file.

Save to

External File.

to external files.

You can save the discretized data and transformed variables

Page 45

Categorical

The Plots di transformation and residual plots.

Figure 2-8

Plots dialo

Transformation Plots. For each of these variables, the category quantifications are

plotted horizontal axis but do not affect the computations. These categories are identified by breaks in the line connecting the quantifications.

Categorical Re

gression (CATREG)

Regression Transformation Plots

alog box allows you to specify the variables that will produce

gbox

against the original category values. Empty categories appear on the

al Plots.

Residu

variable predicted from all predictor variables except the predictor variable in question) are plotted against category indicators and the optimal category

fications multiplied with beta against category indicators.

quanti

For each of these variables, residuals (computed for the dependent

CATREG Command Additional Features

You can customize your categorical regression if you paste your selections into

ax window and edit the resulting

asynt language also allows you to:

 Specify rootnames for the transformed variables when saving them to the working

data f

ile (with the

SAVE subcommand).

CATREG command syntax. SPSS command

Page 46

Page 47

Chapter

Categorical Components Analysis (CATPCA

This proce dimensionality of the data. Categorical principal components analysis is also known by the acronym CATPCA, for categorical principal components analysis.

The goal of into a smaller set of uncorrelated components that represent most of the information found in the original variables. The technique is most useful when a large number of variab (subjects and units). By reducing the dimensionality, you interpret a few components rather than a large number of variables.

Standard numeric variables. On the other hand, the optimal-scaling approach allows variables to be scaled at different levels. Categorical variables are optimally quantified in the specifi can be modeled.

dure simultaneously quantifies categorical variables while reducing the

principal components analysis is to reduce an original set of variables

les prohibits effective interpretation of the relationships between objects

principal components analysis assumes linear relationships between

ed dimensionality. As a result, nonlinear relationships between variables

Principal

)

Example. Categorical principal components analysis could be used to graphically

display (high, medium, and low), and job satisfaction. You might find that two dimensions account for a large amount of variance. The first dimension might separate job catego from amount of travel. You also might find that high job satisfaction is related to a medium amount of travel.

Statis

variance accounted for by centroid coordinates, vector coordinates, total per variable and per dimension, component loadings for vector-quantified variables, category quant

the relationship between job category, job division, region, amount of travel

ry from region, whereas the second dimension might separate job division

tics and plots.

ifications and coordinates, iteration history, correlations of the transformed

Frequencies, missing values, optimal scaling level, mode,

Page 48

Chapter 3

variables and eigenvalues of the correlation matrix, correlations of the original variables and eigenvalues of the correlation matrix, object scores, category plots, joint category pl

ots, transformation plots, residual plots, projected centroid plots, object

plots, biplots, triplots, and component loadings plots.

Data. String variable values are always converted into positive integers by ascending

alphanumer

ic order. User-defined missing values, system-missing values, and values less than 1 are considered missing; you can recode or add a constant to variables with values less than 1 to make them nonmissing.

ns.

Assumptio

The data must contain at least t hree valid cases. The analysis is based on positive integer data. The discretization option will automatically categorize a fractional-valued variable by grouping its values into categories with a close to “normal”

distribution and will automatically convert values of string variables into

positive integers. You can specify other discretization schemes.

Related procedures. Scaling all variables at the numeric level corresponds to standard

principa

l components analysis. Alternate plotting features are available by using the transformed variables in a standard linear principal components analysis. If all variables have multiple nominal scaling levels, categorical principal components

s is identical to multiple correspondence analysis. If sets of variables are of

analysi interest, categorical (nonlinear) canonical correlation analysis should be used.

To Obtai

E From th

Analyze

Data Reduction

Optimal Scaling...

n a Categorica l Principal Components Analysis

e menus choose:

Page 49

Categorical Pr

Figure 3-1

Optimal Scaling dialog box

E Select Some variable(s) not multiple nominal. E Select One set. E Click Define.

incipal Components Analysis (CATPCA)

Page 50

Chapter 3

Figure 3-2

Categorical Principal Components dialog box

E Select at least two analysis variables and specify the number of dimensions in the

solution.

E Click OK.

You may optionally specify supplementary variables, which are fitted into the solution found, or labeling variables for the plots.

Define Scale and Weight in CATPCA

You can set the optimal scaling level for analysis variables and supplementary variables. By default, they are scaled as second-degree monotonic splines (ordinal) with two interior knots. Additionally, you can set the weight for analysis variables.

Page 51

Categorical Pr

Figure 3-3

Define Scale a nd Weight dialog box

Variable weight. You can choose to define a weight for each variable. The value

incipal Components Analysis (CATPCA)

specified must be a positive integer. The default value is 1.

Optimal S

caling Level.

You can also select the scaling level to be used to quantify

each variable.

 Spline ordinal. The order of the categories of the observed variable is preserved in

the optim

ally scaled variable. Category points will be on a straight line (vector) through the origin. The resulting transformation is a smooth monotonic piecewise polynomial of the chosen degree. The pieces are specified by the user-specified number a

 Spline

nd procedure-determined placement of the interior knots.

nominal.

The only information in the observed variable that is preserved in the optimally scaled variable is the grouping of objects in categories. The order of the categories of the observed variable is not preserved. Category points will be on a s

traight line (vector) through the origin. The resulting transformation is a smooth, possibly nonmonotonic, piecewise polynomial of the chosen degree. The pieces are specified by the user-specified number and procedure-determined placem

 Multi

ent of the interior knots.

ple nominal.

he centroid of the objects in the particular categories. Multiple indicates

be in t that different sets of quantifications are obtained for each dimension.

Page 52

Chapter 3



Ordinal. The order of the categories of the observed variable is preserved in the

optimally s through the origin. The resulting transformation fits better than the spline ordinal transformation but is less smooth.

 Nominal. The only information in the observed variable that is preserved in the

optimally s the categories of the observed variable is not preserved. Category points will be on a straight line (vector) through the origin. The resulting transformation fits better tha

 Numeric.

The order of the categories and the equal distances between category numbers of the observed variable are preserved in the optimally scaled variable. Category points wi at the numeric level, the analysis is analogous to standard principal components analysis.

caled variable. Category points will be on a straight line (vector)

caled variable is the grouping of objects in categories. The order of

n the spline nominal transformation but is less smooth.

Categories are treated as ordered and equally spaced (interval level).

ll be on a straight line (vector) through the origin. When all variables are

Categori

cal Principal Components Analysis Discretization

The Disc variables. Fractional-valued variables are grouped into seven categories (or into the number of distinct values of the variable if this number is less than seven) with an approxi always converted into positive integers by assigning category indicators according to ascending alphanumeric order. Discretization for string variables applies to these i arethenusedintheanalysis.

retizationdialogboxallowsyoutoselectamethodofrecodingyour

mately normal distribution, unless specified otherwise. String variables are

ntegers. Other variables are left alone by default. The discretized variables

Page 53

incipal Components Analysis (CATPCA)

Figure 3-4

Discretization dialog box

Method

 Groupi

 Rankin

 Multip

Choose between grouping, ranking, and multiplying.

ng.

Recode into a specified number of categories or recode by interval.

The variable is discretized by ranking the cases.

lying.

The current values of the variable are standardized, multiplied by 10,

Categorical Pr

rounded, and have a constant added such that the lowest discretized value is 1.

Groupin

The following options are available when you are discretizing variables

by grouping:

 Number of categories. Specify a number of categories and whether the values of

iable should follow an approximately normal or uniform distribution

the var across those categories.

 Equal intervals. Variables are recoded into categories defined by these equally

sized intervals. You must specify the length of the intervals.

Page 54

Chapter 3

Categorical

The Missing values in analysis variables and supplementary variables.

Figure 3-5

Missing Val

Principal Components Analysis Missing Values

Values dialog box allows you to choose the strategy for handling missing

ues dialog box

g Value Strategy.

Missin

Choose to exclude missing values (passive treatment), impute missing values (active treatment), or exclude objects with missing values (listwise deletion).

 Exclude missing values; for correlations impute after quantification. Objects with

missing values on the selected variable do not contribute to the analysis for this

ble. If all variables are given passive treatment, then objects with missing

varia values on all variables are treated as supplementary. If correlations are specified in the Output dialog box, then (after analysis) missing values are imputed with the

frequent category,or mode, of the variable for the correlations of the original

most variables. For the correlations of the optimally scaled variables, you can choose the method of imputation. Select

ptimally scaled variable. Select

the o

Mode to replace missing values with the mode of

Extra category to replace missing values with

Page 55

Categori

Categorical Pr

incipal Components Analysis (CATPCA)

the quantification of an extra category. This implies that objects with a missing value on thi

 Impute miss

those values imputed. You can choose the method of imputation. Select

s variable are considered to belong to the same (extra) category.

ing values.

Objects with missing values on the selected variable have

Mode to

replace missing values with the most frequent category. When there are multiple modes, the o

ne with the smallest category indicator is used. Select

Extra category

to replace missing values with the same quantification of an extra category. This implies that objects with a missing value on this variable are considered to belong to t

 Exclude o

he same (extra) category.

bjects with missing values on this variable.

Objects with missing values on the selected variable are excluded from the analysis. This strategy is not available for supplementary variables.

cal Principal Components Analysis Options

The Optio and convergence criteria, select a normalization method, choose the method for labeling plots, and specify supplementary objects.

ns dialog box allows you to select the initial configuration, specify iteration

Page 56

Chapter 3

Figure 3-6

Options dialog box

Supplementary Objects. Specify the case number of the object, or the first and last case

numbers of a range of objects, that you want to make supplementary and then click

Add. Continue until you have specified all of your supplementary objects. If an object

is specified as supplementary, then case weights are ignored for that object.

Normalization Method. You can specify one of five options for normalizing the

object scores and the variables. Only one normalization method can be used in a given analysis.

 Variable Principal. This option optimizes the association between variables. The

coordinates of the variables in the object space are the component loadings (correlations with principal components, such as dimensions and object scores). This is useful when you are primarily interested in the correlation between the variables.

Page 57

Categorical Pr



Object Principal. This option optimizes distances between objects. This is useful

when you are

primarily interested in differences or similarities between the

incipal Components Analysis (CATPCA)

objects.

 Symmetrical. Use this normalization option if you are primarily interested in the

relation between objects and variables.

 Independent. Use this normalization option if you want to examine distances

between obj

 Custom. Yo

ects and correlations between variables separately.

u can specify any real value in the closed interval [–1, 1]. A value of 1 is equal to the Object Principal method, a value of 0 is equal to the Symmetrical method, and a value of –1 is equal to the Variable Principal method. By specifyin

g a value greater than –1 and less than 1, you can spread the eigenvalue over both objects and variables. This method is useful for making a tailor-made biplot or triplot.

Criteria. You can specify the maximum number of iterations the procedure can go

through i

n its computations. You can also select a convergence criterion value. The algorithm stops iterating if the difference in total fit between the last two iterations is less than the convergence value or if the maximum number of iterations is reached.

ts By.

Label Plo

Allows you to specify whether variables and value labels or variable names and values will be used in the plots. You can also specify a maximum length for labels.

Plot Dime

 Display

nsions.

Allows you to control the dimensions displayed in the output.

all dimensions in the solution.

All dimensions in the solution are displayed

in a scatterplot matrix.

 Restrict the number of dimensions. The displayed dimensions are restricted to

plotted pairs. If you restrict the dimensions, you must select the lowest and highest

dimensions to be plotted. The lowest dimension can range from 1 to the number of dimensions in the solution minus 1 and is plotted against higher dimensions. The highest dimension value can range from 2 to the number of dimens

ions in the solution and indicates the highest dimension to be used in plotting the dimension pairs. This specification applies to all requested multidimensional plots.

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Chapter 3

Configuration. You can read data from a file containing the coordinates of a

configuration. The first variable in the file should contain the coordinates for the first dimen

sion, the second variable should contain the coordinates for the second

dimension, and so on.

 Initial. The configuration in the file specified will be used as the starting point

of the analy

 Fixed. The

sis.

configuration in the file specified will be used to fit in the variables. The variables that are fitted in must be selected as analysis variables, but because the configuration is fixed, they are treated as supplementary variables (so they do not need to

be selected as supplementary variables).

Categorical Principal Components Analysis Output

The Output dialog box allows you to produce tables for object scores, component loadings variance accounted for per variable and per dimension, category quantifications for selected variables, and descriptive statistics for selected variables.

, iteration history, correlations of original and transformed variables, the

Page 59

Categorical Pr

Figure 3-7

Output dialog box

incipal Components Analysis (CATPCA)

Object scores. Displays the object scores and has the following options:

 Include Categories Of. Displays the category indicators of the analysis variables

selected.

 Label Object Scores By. From the list of variables specified as labeling variables,

you can select one to label the objects.

Component loadings. Displays the component loadings for all variables that were not

given multiple nominal scaling levels.

Iteration history. For each iteration, the variance accounted for, loss, and increase

in variance accounted for are shown.

Correlations of original variables. Shows the correlation matrix of the original variables

and the eigenvalues of that matrix.

Correlations of transformed variables. Shows the correlation matrix of the transformed

(optimally scaled) variables and the eigenvalues of that matrix.

Page 60

Chapter 3

Variance accounted for. Displays the amount of variance accounted for by centroid

coordinates, vector coordinates, and total (centroid and vector coordinates combined) per variabl

e and per dimension.

Category Qu

antifications.

Gives the category quantifications and coordinates for each

dimension of the variable(s) selected.

Descriptive Statistics. Displays frequencies, number of missing values, and mode of

the variable(s) selected.

Categorical Principal Components Analysis Save

The Save dialog box allows you to add the transformed variables, object scores, and approximations to the working data file or as new variables in external files and save the discre

Figure 3-8

Save dialog box

tized data as new variables in an external data file.

Save. Save selections to the working data file. If any variable has been given the

multiple nominal scaling level, the number of dimensions to be saved must be specified.

Save to External File. Save selections to a new external file. Specify a filename for

each selected option by clicking

File. Each file specified must have a different name.

Page 61

Categorical

The Object a desired and the variables for which plots will be produced.

Figure 3-9

Object and V

Categorical Pr

incipal Components Analysis (CATPCA)

Principal Components Analysis Object Plots

nd Variable Plots dialog box allows you to specify the types of plots

ariable Plots dialog box

Object points. A plot of the object points is displayed.

s and variables (biplot).

Object

The object points are plotted with your choice of the

variable coordinates—component loadings or variable centroids.

Objects, loadings, and centroids (triplot). The object points are plotted with the

ids of multiple nominal-scaling-level variables and the component loadings

centro of other variables.

Biplot and Triplot Variables. You can choose to use all variables for the biplots and

ots or select a subset.

tripl

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Chapter 3

Label Objects. You can choose to have objects labeled with the categories of selected

variables ( you may choose category indicator values or value labels in the Options dialog box)

or with their case numbers. One plot is produced per variable if

Variable is

selected.

Categorica

The Catego variables for which plots will be produced.

Figure 3-1

Category P

l Principal Components Analysis Category Plots

ry Plots dialog box allows you to specify the types of plots desired and the

lots dialog box

Category Plots. For each variable selected, a plot of the centroid and vector coordinates

is plotted. For variables with multiple nominal scaling levels, categories are in the

oids of the objects in the particular categories. For all other scaling levels,

centr categories are on a vector through the origin.

Joint Category Plots. This is a single plot of the centroid and vector coordinates of

selected variable.

each

Page 63

Categorical Pr

Transformation Plots. Displays a plot of the optimal category quantifications versus the

incipal Components Analysis (CATPCA)

category indicators. You can specify the numberofdimensionsdesiredforvariables with multip

le nominal scaling levels; one plot will be generated for each dimension.

You can also choose to display residual plots for each variable selected.

Project Centroids Of. You may choose a variable and project its centroids onto selected

variables.

Variables with multiple nominal scaling levels cannot be selected to project on. When this plot is requested, a table with the coordinates of the projected centroids is also displayed.

Categorical Principal Components Analysis Loading Plots

The Loading Plots dialog box allows you to specify the variables which will be included in the plot, and whether or not to include centroids in the plot.

Figure 3-11

Loading Plots dialog box

Display component loadings. If selected, a plot of the component loadings is displayed. Loading Variables. You can choose to use all variables for the component loadings

r select a subset.

plot o

Page 64

Chapter 3

Include centroids. Variables with multiple nominal scaling levels do not have

component loadings, but you may choose to include the centroids of those variables in the plot.

You can choose to use all multiple nominal variables or select a subset.

CATPCA Command Additional Features

You can customize your categorical principal components analysis if you paste your selection SPSS command language also allows you to:

 Specify rootnames for the transformed variables, object scores, and

 Specify a maximum length for labels for each plot separately (with the PLOT

 Specify a separate variable list for residual plots (with the PLOT subcommand).

s into a syntax window and edit the resulting

approxima

tionswhensavingthemtotheworkingdatafile(withthe

subcommand).

CATPCA command syntax.

SAVE

Page 65

Chapter

Nonlinear Ca

nonical

Correlation Analysis (OVERALS)

Nonlinear

Canonical Correlation Analysis

Nonlinea correlation analysis with optimal scaling. The purpose of this procedure is to determine how similar sets of categorical variables are to one another. Nonlinear canonica

where the second set does not contain a single response variable, but multiple ones. The goal among two sets of numerical variables in a low dimensional space. Initially, the variables in each set are linearly combined such that the linear combinations have amaxima are determined that are uncorrelated with the previous combinations and that have the largest correlation possible.

First, OVERALS allows more than two sets of variables. Second, variables can be scaled as either nominal, ordinal, or numerical. As a result, nonlinear relationships betwee between the variable sets, the sets are compared to an unknown compromise set defined by the object scores.

r canonical correlation analysis corresponds to categorical canonical

l correlation analysis is also known by the acronym OVERALS.

Standard canonical correlation analysis is an extension of multiple regression,

is to explain as much as possible of the variance in the relationships

l correlation. Given these combinations, subsequent linear combinations

The opt

imal scaling approach expands the standard analysis in three crucial ways.

n variables can be analyzed. Finally, instead of maximizing correlations

le.

Examp

used to graphically display the relationship between one set of variables containing job category and years of education and another set of variables containing region of resi

Categorical canonical correlation analysis with optimal scaling could be

dence and gender. You might find that years of education and region of residence

Page 66

Chapter 4

discriminate better than the remaining variables. You might also find that years of education discriminates best on the first dimension.

Statistics

and plots.

Frequencies, centroids, iteration history, object scores, category quantifications, weights, component loadings, single and multiple fit, object scores plots, category coordinates plots, component loadings plots, category centroids plots, transforma

Data. Use integers to code categorical variables (nominal or ordinal scaling level). To

tion plots.

minimize output, use consecutive integers beginning with 1 to code each variable. Variables

scaled at the numerical level should not be recoded to consecutive integers. To minimize output, for each variable scaled at the numerical level, subtract the smallest observed value from every value and add 1. Fractional values are truncated after the

Assumptions. Variables can be classified into two or more sets. Variables in the

decimal.

analysis are scaled as multiple nominal, single nominal, ordinal, or numerical. The maximum n

umber of dimensions used in the procedure depends on the optimal scaling level of the variables. If all variables are specified as ordinal, single nominal, or numerical, the maximum number of dimensions is the minimum of the number of observa

tions minus 1 and the total number of variables. However, if only two sets of variables are defined, the maximum number of dimensions is the number of variables in the smaller set. If some variables are multiple nominal, the maximum number of dimens

ions is the total number of multiple nominal categories plus the number of nonmultiple nominal variables minus the number of multiple nominal variables. For example, if the analysis involves five variables, one of which is multiple nominal with

tegories, the maximum number of dimensions is (4 + 4 – 1), or 7. If you

four ca specify a number greater than the maximum, the maximum value is used.

Related procedures. If each set contains one variable, nonlinear canonical correlation

sis is equivalent to principal components analysis with optimal scaling. If

analy each of these variables is multiple nominal, the analysis corresponds to multiple correspondence analysis. If two sets of variables are involved and one of the sets

ins only one variable, the analysis is identical to categorical regression with

conta optimal scaling.

Page 67

Nonlinear Cano

To Obtain a Nonlinea r Canonical Correlation Analysis

E From the menus choose:

Analyze

Data Reduction

Optimal Scaling...

Figure 4-1

Optimal Scaling dialog box

nical Correlation Analysis (OVERALS)

E Select either All variables multiple nominal or Some variable(s) not multiple nominal. E Select Multiple sets. E Click Define.

Page 68

Chapter 4

Figure 4-2

Nonlinear Canonical Correlation Analysis (OVERALS) dialog box

E Define at least two sets of variables. Select the variable(s) that you want to include in

the first set. To move to the next set, click Next, and select the variables that you want to include in the second set. You can add additional sets as desired. Click

Previous to

return to the previously defined variable set.

E Define the value range and measurement scale (optimal scaling level) for each

selected variable.

E Click OK.

Optionally, you can:

 Select one or more variables to provide point labels for object scores plots. Each

variable produces a separate plot, with the points labeled by the values of that variable. You must define a range for each of these plot label variables. Using the dialog box, a single variable cannot be used both in the analysis and as a labeling variable. If labeling the object scores plot with a variable used in the analysis is desired, use the Compute facility on the Transform menu to create a copy of

Page 69

that variable. Use the new variable to label the plot. Alternatively, command syntax can b

 Specify the

eused.

number of dimensions you want in the solution. In general, choose as few dimensions as needed to explainmost of the variation. If the analysis involves more than two dimensions, SPSS produces three-dimensional plots of the first three dimen

sions. Other dimensions can be displayed by editing the chart.

Define Range and Scale

Figure 4-3

Define Range and Scale dialog box

You must define a range for each variable. The maximum value specified must be an integer. Fractional data values are truncated in the analysis. A category value that is outside

Automatic Recode facility on the Transform menu to create consecutive categories

beginning with 1 for variables treated as nominal or ordinal. Recoding to consecutive intege output for variables treated as numerical, for each variable, subtract the minimum value from every value and add 1.

of the specified range is ignored in the analysis. To minimize output, use the

rs is not recommended for variables scaled at the numerical level. To minimize

Nonlinear Cano

nical Correlation Analysis (OVERALS)

You mus

 Ordin

t also select the scaling to be used to quantify each variable.

al.

The order of the categories of the observed variable is preserved in

the quantified variable.

 Single nominal. In the quantified variable, objects in the same category receive

thesamescore.

 Multiple nominal. The quantifications can be different for each dimension.  Discrete numeric. Categories are treated as ordered and equally spaced. The

ences between category numbers and the order of the categories of the

differ observed variable are preserved in the quantified variable.

Page 70

Chapter 4

Define Range

Figure 4-4

Define Range dialog box

You must define a range for each variable. The maximum value specified must be an integer. Fractional data values are truncated in the analysis. A category value that is outsid use the Automatic Recode facility on the Transform menu to create consecutive categories beginning with 1.

You must a plots. However, labels for categories with data values outside of the defined range for the variable do appear on the plots.

e of the specified range is ignored in the analysis. To minimize output,

lso define a range for each variable used to label the object scores

Nonlinear Canonical Correlation Analysis Options

The Options dialog box allows you to select optional statistics and plots, save object scores as new variables in the working data file, specify iteration and convergence

a, and specify an initial configuration for the analysis.

criteri

Page 71

Nonlinear Cano

Figure 4-5

Options dialog box

nical Correlation Analysis (OVERALS)

Display. Available statistics include marginal frequencies (counts), centroids, iteration

history, weights and component loadings, category quantifications, object scores, and single and multiple fit statistics.

 Centroids. Category quantifications, and the projected and the actual averages of

the object scores for the objects (cases) included in each set for those belonging to the same category of the variable.

 Weights and Component Loadings. The regression coefficients in each dimension

for every quantified variable in a set, where the object scores are regressed on the quantified variables, and the projection of the quantified variable in the o bject space. Provides an indication of the contribution each variable makes to the dimension within each set.

 Single and Mul tiple Fit. Measures of goodness of fit of the single- and

multiple-category coordinates/category quantificationswith respect to the objects.

 Category Quantifications. Optimal scale values assigned to the categories of

avariable.

 Object Scores. Optimal score assigned to an object (case) in a particular

dimension.

Plot. Yo u can produce plots of category coordinates, object scores, component

loadings, category centroids, and transformations.

Page 72

Chapter 4

Save object scores. You can save the object scores as new variables in the working

data file. Object scores are saved for the number of dimensions specified in the main dialog

Use random initial configuration. A random initial configuration should be used if all

box.

or some of the variables are single nominal. If this option is not selected, a nested initial con

Criteria. You can specify the maximum number of iterations the nonlinear canonical

figuration is used.

correlation analysis can go through in its computations. You can also select a convergen

ce criterion value. The analysis stops iterating if the difference in total fit between the last two iterations is less than the convergence value or if the maximum number of iterations is reached.

OVERALS Command Additional Features

You can customize your nonlinear canonical correlation analysis if you paste your selections into a syntax window and edit the resulting OVERALS command syntax. SPSS comm

 Specify

dimensions (using the

 Specify the number of value label characters used to label points on the plots

(with the

 Designate more than five variables as labeling variables for object scores plots

(with th

 Select

plots (with the

 Select variables to provide point labels for the quantification score plot (with the

PLOT subcommand).

 Specify the number of cases to be included in the analysis, if you do not want to

use all c

 Specif

subcommand).

 Specify the number of dimensions to be saved, rather than saving all extracted

dimensions (with the

and language also allows you to:

the dimension pairs to be plotted, rather than plotting all extracted

NDIM keywordonthePLOT subcommand).

PLOT subcommand).

variables used in the analysis as labeling variables for the object scores

PLOT subcommand).

ases in the working data file (with the

y rootnames for variables created by saving object scores (with the

SAVE subcommand).

NOBSERVATIONS subcommand).

SAVE

Page 73

Nonlinear Cano



Write category quantifications to a matrix file (using the MATRIX subcommand).

 Produce low-resolution plots that may be easier to read than the usual

high-resolu

 Produce cen

tion plots (using the

troid and transformation plots for specified variables only (with the

SET command).

nical Correlation Analysis (OVERALS)

PLOT subcommand).

Page 74

Page 75

Chapter

Corresponde

One of the goals of correspondence analysis is to describe the relationships between two nominal variables in a correspondence table in a low-dimensional space, while simultaneously describing the relationships between the categories for each variable. For each variable, the distances between category points in a plot reflect the relationships between the categories with similar categories plotted close to each other. Projecting points for one variable on the vector from the origin to a category point for the other variable describe the relationship between the variables.

An analysis of contingency tables often includes examining row and column profiles and testing for independencevia the chi-square statistic. However, the number of profiles can be quite large, and the chi-square test does not reveal the dependence structure. The Crosstabs procedure offers several measures of association and tests of association but cannot graphically represent any relationships between the variables.

Factor analysis is a standard technique for describing relationships between variables in a low-dimensional space. However, factor analysis requires interval data, and the number of observations should be five times the number of variables. Correspondence analysis, on the other hand, assumes nominal variables and can describe the relationships between categories of each variable, as well as the relationship between the variables. In addition, correspondence analysis can be used to analyze any table of positive correspondence measures.

nce Analysis

Example. Correspondence analysis could be used to graphically display the

relationship between staff category and smoking habits. You might find that with regard to smoking, junior managers differ from secretaries, but secretaries do not differ from senior managers. You might also find that heavy smoking is associated with junior managers, whereas light smoking is associated with secretaries.

Page 76

Chapter 5

Statistics and plots . Correspondence measures, row and column profiles, singular

values, row and column scores, inertia, mass, row and column score confidence statistics

, singular value confidence statistics, transformation plots, row point plots,

column point plots, and biplots.

Data. Categorical variables to be analyzed are scaled nominally. For aggregated data

or for a corr

espondence measure other than frequencies, use a weighting variable with

positive similarity values. Alternatively, for table data, use syntax to read the t able.

Assumptions. The maximum number of dimensions used in the procedure depends

on the numb

er of active rows and column categories and the number of equality constraints. If no equality constraints are used and all categories are active, the maximum dimensionality is one fewer than the number of categories for the variable with the f

ewest categories. For example, if one variable has five categories and the other has four, the maximum number of dimensions is three. Supplementary categories are not active. For example, if one variable has five categories, two of which are

supplementary, and the other variable has four categories, the maximum number of dimensions is two. Treat all sets of categories that are constrained to be equal as one category. For example, if a variable has fivecategories, three of which are constra

ined to be equal, that variableshould be treated as having three categories when determining the maximum dimensionality. Two of the categories are unconstrained, and the third category corresponds to the three constrained categories. If you specify anumbe

r of dimensions greater than the maximum, the maximum value is used.

Related procedures. If more than two variables are involved, use multiple

correspondence analysis. If the variables should be scaled ordinally, use categorical

pal components analysis.

princi

To Obtain a Correspondence Analysis

E From the menus choose:

Analyz

Data Re

duction... pondence Analysis...

Corres

Page 77

Figure 5-1

Correspondence Analysis dialo g box

E Select a row variable. E Select a column variable. E Define the ranges for the variables. E Click OK.

Correspondenc

eAnalysis

Define Row Range in Correspondence Analysis

You must define a range for the row variable. The minimum and maximum values specified must be integers. Fractional data values are truncated in the analysis. A categor

y value that is outside of the specified range is ignored in the analysis.

Page 78

Chapter 5

Figure 5-2

Define Row Range dialog box

All categories are initially unconstrained and active. You can constrain row categories to equal other row categories, or you can define a row category as supplementary.

 Categories must be equal. Categories must have equal scores. Use equality

constraints if the obtained order for the categories is undesirable or counter

intuitive. The maximum number of row categories that can be constrained to be equal is the total number of active row categories minus 1. To impose different equality constraints on sets of categories, use syntax. For example, use syntax

toconstraincategories1and2tobeequalandcategories3and4tobe

equal.

 Category is supplemental. Supplementary categories do not influence the analysis

but are represented in the space defined by the active categories. Supplementary

ries play no role in defining the dimensions. The maximum number of

catego supplementary row categories is the total number of row categories minus 2.

Define Column Range in Correspondence Analysis

You must define a range for the column variable. The minimum and maximum values specified must be integers. Fractional data values are truncated in the analysis. A

ory value that is outside of the specified range is ignored in the analysis.

categ

Page 79

Correspondenc

Figure 5-3

Define Column Range dialog box

eAnalysis

All categories are initially unconstrained and active. You can constrain column categories to equal other column categories, or you can define a column category as suppl

 Catego

ementary.

ries must be equal.

Categories must have equal scores. Use equality constraints if the obtained order for the categories is undesirable or counterintuitive. The maximum number of column categories that can be

ained to be equal is the total number of active column categories minus 1.

constr To impose different equality constraints on sets of categories, use syntax. For example, use syntax to constrain categories 1 and 2 to be equal and categories 3and4t

 Categ

obeequal.

ory is supplemental.

Supplementary categories do not influence the analysis but are represented in the space defined by the active categories. Supplementary categories play no role in defining the dimensions. The maximum number of

ementary column categories is the total number of column categories

suppl minus 2.

Correspondence Analysis Model

The Model dialog box allows you to specify the number of dimensions, the distance measure, the standardization method, and the normalization method.

Page 80

Chapter 5

Figure 5-4

Model dialog box

Dimensions in solution. Specify the number of dimensions. In general, choose as few

dimensions as needed to explain most of the variation. The maximum number of dimensions depends on the number of active categories used in the analysis and on the equality constraints. The maximum number of dimensions is the smaller of:

 The number of active row categories minus the number of row categories

constrained to be equal, plus the number of constrained row category sets.

 The number of active column categories minus the number of column categories

constrained to be equal, plus the number of constrained column category sets.

Distance Measure. You can select the measure of distance among the rows and

columns of the correspondence table. Choose one of the following alternatives:

 Chi square. Use a weighted profile distance, where the weight is the mass of the

rows or columns. This measure is required for standard correspondence analysis.

 Euclidean. Use the square root of the sum of squared differences between pairs of

rows and pairs of columns.

Standardization Method. Choose one of the following alternatives:

 Row and column means are removed. Both the rows and columns are centered.

This method is required for standard correspondence analysis.

Page 81

Correspondenc



Row m eans are removed. Only the rows are centered.

 Column means are removed. Only the columns are centered.  Row totals are equalized and means are removed. Before centering the rows, the

row margins a

 Column tota

re equalized.

ls are equalized and means are removed.

Before centering the columns,

eAnalysis

the column margins are equalized.

Normalizat

 Symmetrica

ion M ethod.

Choose one of the following alternatives:

For each dimension, the row scores are the weighted average of the column scores divided by the matching singular value, and the column scores are the weighted average of row scores divided by the matching singular value. Use this met

hod if you want to examine the differences or similarities between

the categories of the two variables.

 Principal. The distances between row points and column points are

approximations of the distances in the correspondence table according to the selected d

istance measure. Use this method if you want to examine differences between categories of either or both variables instead of differences between the two variables.

 Row principal. The distances between row points are approximations of the

distance

s in the correspondence table according to the selected distance measure. The row scores are the weighted average of the column scores. Use this method if you want to examine differences or similarities between categories of the row varia

 Column p

ble.

rincipal.

The distances between column points are approximations of the distances in the correspondence table according to the selected distance measure. The column scores are the weighted average of the row scores. Use this method i

f you want to examine differences or similarities between categories of

the column variable.

 Custom. You must specify a value between –1 and 1. A value of –1 corresponds

to column principal. A value of 1 corresponds to row principal. A value of 0

ponds to symmetrical. All other values spread the inertia over both the

corres row and column scores to varying degrees. This method is useful for making tailor-made biplots.

Page 82

Chapter 5

Corresponde

The Statistics dialog box allows you to specify the numerical output produced.

Figure 5-5

Statistics d

Correspondence table. A crosstabulation of the input variables with row and column

marginal

Overview of row points. For each row category, the scores, mass, inertia, contribution

to the inertia of the dimension, and the contribution of the dimension to the inertia of the poi

Overview of column points. For each column category, the scores, mass, inertia,

contribution to the inertia of the dimension, and the contribution of the dimension to the iner

nce Analysis Statistics

ialog box

totals.

nt.

tia of the point.

Row profiles. For each row category, the distribution across the categories of the

column variable.

Column p

rofiles.

For each column category, the distribution across the categories of

the row variable.

Permutations of the correspondence table. The correspondence table reorganized such

that the

rows and columns are in increasing order according to the scores on the first dimension. Optionally, you can specify the maximum dimension number for which permuted tables will be produced. A permuted table for each dimension from 1tothe

Confidence Statistics for Row points. Includes standard deviation and correlations

number specified is produced.

for all nonsupplementary row points.

Page 83

Confidence Statistics for Column points. Includes standard deviation and correlations

for all nonsupplementary column points.

Correspondence Analysis Plots

The Plots dialog box allows you to specify which plots are produced.

Figure 5-6

Plots dialog box

Correspondenc

eAnalysis

Scatterplots. Produces a matrix of all pairwise plots of the dimensions. Available

scatterplots include:

 Biplot. Produces a matrix of joint plots of the row and column points. If principal

normalization is selected, the biplot is not available.

 Row points. Produces a matrix of plots of the row points.  Column points. Produces a matrix of plots of the column points.

Optionally, you can specify how many value label characters to use when labeling the

s. This value must be a non-negative integer less than or equal to 20.

point

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Chapter 5

Line plots. Produces a plot for every dimension of the selected variable. Available

line plots include:

 Transformed row categories. Produces a plot of the original row category values

against their corresponding row scores.

 Transformed column categories. Produces a plot of the original column category

values agai

nst their corresponding column scores.

Optionally, you can specify how many value label characters to use when labeling the category axis. This value must be a non-negative integer less than or equal to 20.

ions.

Plot Dimens

 Display al

Allows you to control the dimensions displayed in the output.

l dimensions in the solution.

All dimensions in the solution are displayed

in a scatterplot matrix.

 Restrict the number of dimensions. The displayed dimensions are restricted to

plotted pairs. If you restrict the dimensions, you must select the lowest and highest di

mensions to be plotted. The lowest dimension can range from 1 to the number of dimensions in the solution minus 1, and is plotted against higher dimensions. The highest dimension value can range from 2 to the number of dimensi

ons in the solution, and indicates the highest dimension to be used in plotting the dimension pairs. This specification applies to all requested multidimensional plots.

CORRESPO

You can customize your correspondence analysis if you paste your selections into a syntax window and edit the resulting CORRESPONDENCE command syntax. SPSS command

 Specif

 Specify the number of value-label characters used to label points for each type of

 Write a

NDENCE Command Additional Features

language also allows you to:

y table data as input instead of using casewise data (using the

=ALL

scatterplot matrix or biplot matrix (with the

line plo

OUTFILE subcommand).

subcommand).

PLOT subcommand).

t (with the

PLOT subcommand).

matrix of row and column scores to an SPSS matrix data file (with the

TABLE

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Correspondenc



Write a matrix of c onfidence statistics (variances and covariances) for the

singular va

lues and the scores to an SPSS matrix data file (with the

eAnalysis

OUTFILE

subcommand).

 Specify multiple sets of categories to be equal (with the EQUAL subcommand).

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Chapter

Multiple Cor

Multiple Correspondence Analysis quantifies nominal (categorical) data by assigning numerical values to the cases (objects) and categories so that objects within the same category are close together and objects in different categories are far apart. Each object is as close as possible to the category points of categories that apply to the object. In this way, the categories divide the objects into homogeneous subgroups. Variables are considered homogeneous when they classify objects in the same categories into the same subgroups.

Example. Multiple Correspondence Analysis could be used to graphically display the

relationship between job category, minority classification, and gender. You might find that minority classification and gender discriminate between people but that job category does not. You might also find that the Latino and African-American categories are similar to each other.

Statistics and plots. Object scores, discrimination measures, iteration history,

correlations of original and transformed variables, category quantifications, descriptive statistics, object points plots, biplots, category plots, joint category plots, transformation plots, and discrimination measures plots.

respondence Analysis

Data. String variable values are always converted into positive integers by ascending

alphanumeric order. User-defined missing values, system-missing values, and values less than 1 are considered missing; you can recode or add a constant to variables with values less than 1 to make them nonmissing.

Assumptions. All variables have the multiple nominal scaling level. The data must

contain at least three valid cases. The analysis is based on positive integer data. The discretization option will automatically categorize a fractional-valued variable by grouping its values into categories with a close to “normal” distribution and will automatically convert values of string variables into positive integers. You can specify other discretization schemes.

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Chapter 6

Related procedures. For two variables, Multiple Correspondence Analysis is analogous

to Correspondence Analysis. If you believethat variables possess ordinal or numerical properties

, Categorical Principal Components Analysis should be used. If sets of

variables are of interest, Nonlinear Canonical Correlation Analysis should be used.

To Obtain a M

E From the me

Analyze

Data Reduction

Optimal Scaling...

Figure 6-1

Optimal Sca

E Select All variables multiple nominal. E Select One set.

ultiple Correspondence Analysis

nus choose:

ling dialog box

E Click Define.

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Multiple Corre

Figure 6-2

Multiple Correspondence Analysis dialog box

E Select at least two analysis variables and specify the number of dimensions in the

spondence Analysis

solution.

E Click OK.

You may optionally specify supplementary variables, which are fitted into the solution found, or labeling variables for the plots.

Define Variable Weight in M ultiple Correspondence Analysis

You can set the weight for analysis variables.

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Chapter 6

Figure 6-3

Define Variable Weight dialog box

Variable weight. You can choose to define a weight for each variable. The value

specified must be a positive integer. The default value is 1.

Multiple Correspondence Analysis Discretization

The Discretization dialog box allows youtoselectamethodofrecodingyour variables. Fractional-valued variables are grouped into seven categories (or into the number of approximately normal distribution unless otherwise specified. String variables are always converted into positive integers by assigning category indicators according to ascen these integers. Other variables are left alone by default. The discretized variables arethenusedintheanalysis.

distinct values of the variable if this number is less than seven) with an

ding alphanumeric order. Discretization for string variables applies to

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Multiple Corre

Figure 6-4

Discretization dialog box

Method. Choose between grouping, ranking, and multiplying.

 Grouping. Recode into a specified number of categories or recode by interval.  Ranking. The variable is discretized by ranking the cases.  Multiplying. The current values of the variable are standardized, multiplied by 10,

spondence Analysis

rounded, and have a constant added so that the lowest discretized value is 1.

Grouping. The following options are available when discretizing variables by grouping:

 Number of categories. Specify a number of categories and whether the values of

the variable should follow an approximately normal or uniform distribution

those categories.

across

 Equal i

ntervals.

Variables are recoded into categories defined by these equally

sized intervals. You must specify the length of the intervals.

Multiple Correspondence Analysis Missing Values

The Missing Values dialog box allows you to choose the strategy for handling missing values in analysis variables and supplementary variables.

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Chapter 6

Figure 6-5

Missing Values dialog box

Missing Value Strategy. Choose to exclude missing values (passive treatment), impute

missing values (active treatment), or exclude objects with missing values (listwise

on).

deleti

 Exclu

de missing values; for correlations impute after quantification.

Objects with missing values on the selected variable do not contribute to the analysis for this variable. If all variables are given passive treatment, then objects with missing

s on all variables are treated as supplementary. If correlations are specified

value in the Output dialog box, then (after analysis) missing values are imputed with the most frequent category, or mode, of the variable for the correlations of the original

bles. For the correlations of the optimally scaled variables, you can choose

varia the method of imputation. Select the optimally scaled variable. Select

uantification of an extra category. This implies that objects with a missing

the q

Mode to replace missing values with the mode of

Extra category to replace missing values with

value on this variable are considered to belong to the same (extra) category.

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

Impute missing values. Objects with missing values on the selected variable have

those value

s imputed. You can choose the method of imputation. Select replace missing values with the most frequent category. When there are multiple modes, the one with the smallest category indicator is used. Select to replace m

issing values with the same quantification of an extra category. This implies that objects with a missing value on this variable are considered to belong to the same (extra) category.

 Exclude objects with missing values on this variable. Objects with missing values

on the sele

cted variable are excluded from the analysis. This strategy is not

available for supplementary variables.

Multiple Correspondence Analysis Options

The Options dialog box allows you to select the initial configuration, specify iteration and convergence criteria, select a normalization method, choose the method for labeling

plots, and specify supplementary objects.

Multiple Corre

spondence Analysis

Mode to

Extra category

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Chapter 6

Figure 6-6

Options dialog box

Supplementary Objects. Specify the case number of the object, or the first and last case

numbers of a range of objects, that you want to make supplementary, and then click

Add. Continue until you have specified all of your supplementary objects. If an object

is specified as supplementary, then case weights are ignored for that object.

Normalization Method. You can specify one of five options for normalizing the

object scores and the variables. Only one normalization method can be used in a given analysis.

 Variable Principal. This option optimizes the association between variables. The

coordinates of the variables in the object space are the component loadings (correlations with principal components, such as dimensions and object scores). This is useful when you are interested primarily in the correlation between the variables.

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Multiple Corre



Object Principal. This option optimizes distances between objects. This is useful

when you are

interested primarily in differences or similarities between the

spondence Analysis

objects.

 Symmetrical. Use this normalization option if you are interested primarily in the

relation between objects and variables.

 Independent. Use this normalization option if you want to examine distances

between obj

 Custom. Yo

ects and correlations between variables separately.

g a value greater than –1 and less than 1, you can spread the eigenvalue over both objects and variables. This method is useful for making a tailor-made biplot or triplot.

Criteria. You can specify the maximum number of iterations the procedure can go

through i

ts By.

Label Plo

Allows you to specify whether variables and value labels or variable names and values will be used in the plots. You can also specify a maximum length for labels.

Plot Dime

 Display

nsions.

Allows you to control the dimensions displayed in the output.

all dimensions in the solution.

All dimensions in the solution are displayed

in a scatterplot matrix.

 Restrict the number of dimensions. The displayed dimensions are restricted to

plotted pairs. If you restrict the dimensions, you must select the lowest and highest

ions in the solution and indicates the highest dimension to be used in plotting the dimension pairs. This specification applies to all requested multidimensional plots.

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Chapter 6

Configuration. You can read data from a file containing the coordinates of a

configuration. The first variable in the file should contain the coordinates for the first dimen

sion, the second variable should contain the coordinates for the second

dimension, and so on.

 Initial. The configuration in the file specified will be used as the starting point

of the analy

 Fixed. The

sis.

configuration in the file specified will be used to fit in the variables. The variables that are fitted in must be selected as analysis variables, but, because the configuration is fixed, they are treated as supplementary variables (so they do not need to

be selected as supplementary variables).

Multiple Correspondence Analysis Output

The Output dialog box allows you to produce tables for object scores, discrimination measures category quantifications for selected variables, and descriptive statistics for selected variables.

, iteration history, correlations of original and transformed variables,

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Figure 6-7

Output dialog box

Multiple Corre

spondence Analysis

Object scores. Displays the object scores, including mass, inertia, and contributions,

and has

 Inclu

the following options:

de Cate gories Of.

Displays the category indicators of the analysis variables

selected.

 Label Object Scores By. From the list of variables specified as labeling variables,

you can select one to label the objects.

Discrimination measures. Displays the discrimination measures per variable and

per dim

Iteration history. For each iteration, the variance accounted for, loss, and increase

ension.

in variance accounted for are shown.

lations of original variables.

Corre

Shows the correlation matrix of the original variables

and the eigenvalues of that matrix.

Correlations of transformed variables. Shows the correlation matrix of the transformed

mally scaled) variables and the eigenvalues of that matrix.

(opti

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Chapter 6

Category Quantifications and Contributions. Gives the category quantifications

(coordinates), including mass, inertia, and contributions, for each dimension of the variabl

Descriptive Statistics. Displays frequencies, number of missing values, and mode of

the variable(s) selected.

e(s) selected.

Multiple Correspondence Analysis Save

The Save dialog box allows you to add the transformed variables and object scores to the working data file or as new variables in external files and to save the discretized data as new

Figure 6-8

Save dialog box

variables in an external data file.

Save. S

saved must be specified.

Save to External File. Save selections to a new external file. Specify a filename for

each se

ave selections to the working data file. The number of dimensions to be

lected option by clicking

File. Each file specified must have a different name.

Multiple Correspondence Analysis Object Plots

The Object Plots dialog box allows you to specify the types of plots desired and the

bles for which plots will be produced.

varia

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Figure 6-9

Object Plots dialog box

Object points. A plot of the object points is displayed.

Multiple Corre

spondence Analysis

Objects and centroids (biplot). The object points are plotted with the variable centroids. Biplot Variables. You can choose to use all variables for the biplots or select a subset. Label Objects. You can choose to have objects labeled with the categories of selected

variables ( you may choose category indicator values or value labels in the Options dialog box) or with their case numbers. One plot is produced per variable if selected.

Multiple Correspondence Analysis Variable Plots

The Variable Plots dialog box allows you to specify the types of plots desired and the variables for which plots will be produced.

Variable is

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Chapter 6

Figure 6-10

Variable Plots dialog box

Category Plots. For each variable selected, a plot of the centroid coordinates is plotted.

Categories are in the centroids of the objects in the particular categories.

Joint Category Plots. This is a single plot of the centroid coordinates of each selected

variable.

Transformation Plots. Displays a plot of the optimal category quantifications versus

the category indicators. You can specify the number of dimensions desired; one plot will be generated for each dimension. You can also choose to display residual plots for each variable selected.

Discrimination M easures. Produces a single plot of the discrimination measures for

the selected variables.

Spss CATEGORIES 13.0 User Manual

What Is Optimal Scaling?

Why Use Optimal Scaling?

Selecting the Optimal Scaling Level

Transformation Plots

Category Codes

Which Procedure Is Best for Your Application?

Correspondence Analysis

Multiple Correspondence Analysis

Aspect Ratio in Optimal Scaling Charts

Recommended Readings

Categorical Regression

Define Scale in Categorical Regression

Categorical Regression Missing Values

Categorical Regression Options

Categorical Regression Output

CATREG Command Additional Features

Define Scale and Weight in CATPCA

Categorical Principal Components Analysis Output

Categorical Principal Components Analysis Save

Categorical Principal Components Analysis Loading Plots

CATPCA Command Additional Features

Define Range and Scale

Define Range

Nonlinear Canonical Correlation Analysis Options

OVERALS Command Additional Features

Define Row Range in Correspondence Analysis

Define Column Range in Correspondence Analysis

Correspondence Analysis Model

Correspondence Analysis Plots

Define Variable Weight in M ultiple Correspondence Analysis

Multiple Correspondence Analysis Discretization

Multiple Correspondence Analysis Missing Values

Multiple Correspondence Analysis Options

Multiple Correspondence Analysis Output

Multiple Correspondence Analysis Save

Multiple Correspondence Analysis Object Plots

Multiple Correspondence Analysis Variable Plots