Spss CATEGORIES 11.0 User Manual

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SPSS Categories® 11.0

Jacqueline J. Meulman Willem J. Heiser SPSS Inc.

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SPSS Categories

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ISBN 1-56827-276-6

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Preface

SPSS 11.0 is a powerful software package for microcomputer data management and analysis. The Categories option is an add-on enhancement that provides a comprehensive set of procedures for optimal scaling. The procedures in Categories must be used with the SPSS 11.0 Base and are completely integrated into that system.

The Categories option includes procedures for:

• Categorical regression

• Categorical principal components analysis

• Nonlinear canonical correlation analysis

• Correspondence analysis

• Homogeneity analysis

• Multidimensional scaling

Installation

Compatibility

Serial Numbers

To install Categories, follow the instructions for adding and removing features in the installation instructions supplied with the SPSS Base. (To start, double-click on the SPSS Setup icon.)

The SPSS system is designed to operate on many computer systems. See the materials that came with your system for specific information on minimum and recommended requirements.

Your serial number is your identification number with SPSS Inc. You will need this serial number when you call SPSS Inc. for information regarding support, payment, or an upgraded system. The serial number was provided with your Base system. Before using the system, please copy this number to the registration card.

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Registration Card

Don’t put it off: fill out and send us your registration card. Until we receive your registration card, you have an unregistered system. Even if you have previously sent a card to us, please fill out and return the card enclosed in your Categories package. Registering your system entitles you to:

• Technical support services

• New product announcements and upgrade announcements

Customer Service

If you have any questions concerning your shipment or account, contact your local office, listed on page vi. Please have your serial number ready for identification when calling.

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SPSS Inc. provides both public and onsite training seminars for SPSS. All seminars feature hands-on workshops. SPSS seminars will be offered in major U.S. and European cities on a regular basis. For more information on these seminars, call your local office, listed on page vi.

Technical Support

The services of SPSS Technical Support are available to registered customers. Customers may call Technical Support for assistance in using SPSS products or for installation help for one of the supported hardware environments. To reach Technical Support, see the SPSS Web site at http://www.spss.com, or call your local office, listed on page vi. Be prepared to identify yourself, your organization, and the serial number of your system.

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Tell Us Your Thoughts

Your comments are important. Please let us know about your experiences with SPSS products. We especially like to hear about new and interesting applications using the SPSS system. Please send e-mail to [email protected], or write to SPSS Inc., Attn: Director of Product Planning, 233 South Wacker Drive, 11th Floor, Chicago, IL 60606-6412.

About This Manual

This manual is divided into two sections. The first section documents the graphical user interface and provides examples of the statistical techniques available. In addition, this section offers advice on interpreting the output. The second part of the manual is a Syntax Reference section that provides complete command syntax for all of the commands included in the Categories option. Most features of the system can be accessed through the dialog box interface, but some functionality can be accessed only through command syntax.

This manual contains two indexes: a subject index and a syntax index. The subject index covers both sections of the manual. The syntax index applies only to the Syntax Reference section.

Acknowledgments

Contacting SPSS

The optimal scaling procedures and their SPSS implementation were developed by the Data Theory Scaling System Group (DTSS), consisting of members of the departments 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 involved 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 multidimensional scaling. In addition, Anita van der Kooij contributed especially to CATREG, CORRESPONDENCE, and CATPCA, and Frank Busing and Willem Heiser, to the PROXSCAL procedure. The development of PROXSCAL has profited from technical comments and suggestions from Jacques Commandeur and Patrick Groenen.

If you would like to be on our mailing list, contact one of our offices, listed on page vi, or visit our Web site at http://www.spss.com. We will send you a copy of our newsletter and let you know about SPSS Inc. activities in your area.

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Contents

1 Introduction to SPSS Optimal Scaling Procedures for

Categorical Data 1

What Is Optimal Scaling? 1

Why Use Optimal Scaling? 1

Optimal Scaling Level and Measurement Level 2

Selecting the Optimal Scaling Level 3 Transformation Plots 4 Category Codes 5

Which Procedure Is Best for Your Application? 7

Categorical Regression 8 Categorical Principal Components Analysis 9 Nonlinear Canonical Correlation Analysis 9 Correspondence Analysis 10 Homogeneity Analysis 12 Multidimensional Scaling 13

Displays with More than Two Dimensions 14

Three-Dimensional Scatterplots 14 Scatterplot Matrices 15

Aspect Ratio in Optimal Scaling Charts 16

2 Categorical Regression (CATREG) 17

To Obtain a Categorical Regression 18

Define Scale in Categorical Regression 19 To Define the Scale in CATREG 20 Categorical Regression Discretization 20 Categorical Regression Missing Values 22 Categorical Regression Options 23 Categorical Regression Output 24 Categorical Regression Save 25 Categorical Regression Plots 26 CATREG Command Additional Features 26

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3 Categorical Principal Components Analysis (CATPCA) 27

To Obtain a Categorical Principal Components Analysis 28

Define Scale and Weight in CATPCA 29 To Define the Scale and Weight in CATPCA 31 Categorical Principal Components Discretization 32 Categorical Principal Components Missing Values 33 Categorical Principal Components Category Plots 34 Categorical Principal Components Object and Variable Plots 35 Categorical Principal Components Loading Plots 36 Categorical Principal Components Output 37 Categorical Principal Components Save 38 Categorical Principal Components Options 39 CATPCA Command Additional Features 40

4 Nonlinear Canonical Correlation Analysis (OVERALS) 41

To Obtain a Nonlinear Canonical Correlation Analysis 42

Define Range and Scale in OVERALS 44 To Define an Optimal Scaling Range and Scale in OVERALS 45 Define Range in OVERALS 45 To Define an Optimal Scaling Range in OVERALS 46 Nonlinear Canonical Correlation Analysis Options 46 OVERALS Command Additional Features 48

5 Correspondence Analysis 49

To Obtain a Correspondence Analysis 50

Define Row Range in Correspondence Analysis 51 To Define a Row Range in Correspondence Analysis 51 Define Column Range in Correspondence Analysis 52 To Define a Column Range in Correspondence Analysis 53 Correspondence Analysis Model 54 Correspondence Analysis Statistics 56 Correspondence Analysis Plots 57 CORRESPONDENCE Command Additional Features 58

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6 Homogeneity Analysis (HOMALS) 59

To Obtain a Homogeneity Analysis 60

Define Range in Homogeneity Analysis 62 To Define an Optimal Scaling Range in Homogeneity Analysis 62 Homogeneity Analysis Options 63 HOMALS Command Additional Features 64

7 Multidimensional Scaling (PROXSCAL) 65

To Obtain a Multidimensional Scaling 66

Proximities in Matrices across Columns 67

Proximities in Columns 68

Proximities in One Column 69

Create Proximities from Data 70

Measures Dialog Box 71 Define a Multidimensional Scaling Model 72 Multidimensional Scaling Restrictions 74 Multidimensional Scaling Options 75 Multidimensional Scaling Plots, Version 1 76 Multidimensional Scaling Plots, Version 2 78 Multidimensional Scaling Output 78 PROXSCAL Command Additional Features 80

8 Categorical Regression Examples 81

Example 1: Carpet Cleaner Data 81

A Standard Linear Regression Analysis 82 A Categorical Regression Analysis 84

Example 2: Ozone Data 93

Categorizing Variables 94 Selection of Transformation Type 95 Optimality of the Quantifications 101 Effects of Transformations 102

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9 Categorical Principal Components Analysis Examples 107

Example 1: Interrelations of Social Systems 108

Number of Dimensions 110 Quantifications 111 Object Scores 112 Component Loadings 114 Additional Dimensions 115

Example 2: Symptomatology of Eating Disorders 118

Transformation Plots 121 Model Summary 122 Component Loadings 123 Object Scores 124 Examining the Structure of the Course of Illness 126

10 Nonlinear Canonical Correlation Analysis Examples 131

Example: An Analysis of Survey Results 131

Examining the Data 132 Accounting for Similarity between Sets 134 Component Loadings 138 Transformation Plots 138 Single versus Multiple Category Coordinates 140 Centroids and Projected Centroids 142

An Alternative Analysis 145

General Suggestions 149

11 Correspondence Analysis Examples 151

Example 1: Smoking Behavior by Job Category 152

Profiles and Distances 156 Inertia 157 Row and Column Scores 158 Dimensionality 159 Supplementary Profiles 160 Contributions 162 Permutations of the Correspondence Table 165 Confidence Statistics 166 Normalization 168

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Example 2: Perceptions of Coffee Brands 171

Principal Normalization 172 Dimensionality 172 Contributions 173 Plots 175 Symmetrical Normalization 177

Example 3: Flying Mileage between Cities 178

Row and Column Scores 181

12 Homogeneity Analysi s Examples 183

Example: Characteristics of Hardware 184

Multiple Dimensions 185 Object Scores 186 Discrimination Measures 188 Category Quantifications 189 A More Detailed Look at Object Scores 191 Omission of Outliers 194

13 Multidimensional Scaling Examples 197

Example: An Examination of Kinship Terms 197

Choosing the Number of Dimensions 198 A Three-Dimensional Solution 199 A Three-Dimensional Solution with Nondefault Transformations 205 Discussion 208

Syntax Reference 209

Introduction 211 ANACOR 215 CATPCA 225 CATREG 243 CORRESPONDENCE 255 HOMALS 267 OVERALS 275

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PRINCALS 285 PROXSCAL 295

Bibliography 313

Subject Index 317

Syntax Index 323

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Introduction to SPSS Optimal Scaling Procedures for Categorical Data

SPSS Categories procedures use optimal scaling to analyze data that are difficult or impossible for “standard” statistical procedures to analyze. each procedure does, the situations in which each procedure is most appropriate, the relationships between the procedures, and the relationships of these procedures to their “standard” statistical counterparts.

What Is Optimal Scaling?

The idea behind optimal scaling is to assign numerical quantifications to the categories of each variable, thus allowing “standard” procedures to be used to obtain a solution on the quantified variables.

The optimal scale values are assigned to categories of each variable based on the optimizing criterion of the procedure in use. Unlike the original labels of the nominal or ordinal variables in the analysis, these scale values have metric properties.

In most Categories procedures, the optimal quantification for each scaled variable is obtained through an iterative method called alternating least squares in which, 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 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?

This chapter describes what

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

1. These procedures and their SPSS implementation were developed by the Data Theory Scaling System Group (DTSS), consisting of members of the departments of Education and Psychology, Faculty of Social and Behavioral Sciences, Leiden University.

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While adaptations of most standard models exist specifically to analyze categorical

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

• Too few observations

• Too many variables

• Too many values per variable

By quantifying categories, optimal scaling techniques avoid problems in these situations. Moreover, they are useful even when specialized techniques are appropriate.

Rather than interpreting parameter estimates, the interpretation of optimal scaling 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 your investigation, visualizing your data through optimal scaling can form the basis of an analysis that centers on interpretation of model parameters.

Optimal Scaling Level and Measurement Level

This can be a very confusing concept when you first use Categories procedures. 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 nonlinear relations regardless of how they are measured.

For Categories purposes, there are three basic levels of measurement:

•The nominal level implies that a variable’s values represent unordered categories.

Examples of variables that might be nominal are region, zip code area, religious af-

filiation, and multiple choice categories.

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

amples include attitude scales representing degree of satisfaction or confidence 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 appropriate. Examples include age in years and income in thousands of dollars.

For example, suppose the variables

Table 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 3 East 3 manager 25 twenty-five years old 4 West 27 twenty-seven years old

region, job, and age are coded as shown in Table 1.1.

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Introduction to SPSS Optimal Scaling Procedures for Categorical Data 3

The values shown represent the categories of each variable. Region would be a nominal variable. There are four categories of through 4 simply represent the four categories; the coding scheme is completely arbitrary.

Job, on the other hand, could be assumed to be an ordinal variable. The original

categories form a progession from intern to manager. Larger codes represent 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, to be a numerical variable. In the case of trinsically meaningful. The distance 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 scaling level you should specify for it. You can explore your data in any way that makes sense and makes interpretation easier. By 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 following 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 people 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 order 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. With 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.

region, with no intrinsic ordering. Values 1

age could be assumed

age, the distances between the values are in-

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Transformation Plots

The different levels at which each variable can be scaled impose different restrictions on the quantifications. Transformation plots illustrate the relationship between the 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 numerical. Variables treated as ordinal result in a nondecreasing transformation plot. Transformation plots for variables treated nominally that are U-shaped (or the reverse) display a quadratic relationship. Nominal variables could also yield transformation plots without apparent trends by changing the order of the categories completely. Figure 1.1 displays a sample transformation plot.

Transformation plots are particularly suited to determining how well the selected optimal scaling level performs. If several categories receive similar quantifications, collapsing these categories into one category may be warranted. Alternatively, if a variable treated as nominal receives quantifications that display an increasing trend, an ordinal transformation may result in a similar fit. If that trend is linear, numerical 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)

Although HOMALS does not currently offer the transformation plot in Figure 1.1 as an option, creating this plot is a straightforward procedure. For each variable, create a new variable containing the values (and labels) of the categories. Create a new variable containing the quantifications for that variable from the Categories analysis. Use the line facility with the data in the chart representing the values of individual cases. You can use the quantifications for the line in the chart and use the categories to label the chart.

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Category Codes

Some care should be taken when coding categorical variables, because some coding schemes may yield unwanted output or incomplete analyses. Possible coding schemes for

Table 1.2 Alternative coding schemes for job

Category A B C D

intern 1 1 5 1 sales rep 2 2 6 5 manager 3 7 7 3

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 always 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 Table 1.2, scheme A has a maximum category of 3 and scheme B has a maximum category value of 7, yet both schemes code the same three categories.

Any categories with codes outside the defined range are omitted from the analysis. This is a simple 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 has categories coded from 1 to 3; the manager category is treated as missing. Because no category has actually been coded cases. If you wanted to omit all manager categories, this analysis would be appropriate. 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.

the results. For nominal variables, only the label and not the value associated 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 Table 1.2 are functionally equivalent if 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.

Introduction to SPSS Optimal Scaling Procedures for Categorical Data 5

job are displayed in Table 1.2.

Scheme

The variable range determines which categories will be omitted from the analysis.

job

3, the third category in the analysis contains no

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

job is analyzed at an ordinal

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Although many coding schemes for a variable are functionally equivalent, schemes with small differences between codes are preferred because the codes have an 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 categories are empty, the corresponding quantifications will be either system missing 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 four categories that receive

system-missing values. For scheme C, there are also four categories receiving systemmissing indicators. In contrast, for scheme A there are no system-missing quantifications. Using consecutive integers as codes for variables treated as nominal or ordinal results in much less output without affecting the results.

Coding schemes for variables treated as numerical are more restricted than the ordinal case. For these variables, the differences between consecutive categories are important. Table 1.3 displays three coding schemes for

Table 1.3 Alternative coding schemes for age

Scheme

Category A B C

20 20 1 1 22 22 3 2 25 25 6 3 27 27 8 4

age.

Any recoding 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 indicators. For example, scheme A in Table 1.3 employs the original observed values. For all 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 and 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 reduce 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 result, all quantifications will be equally spaced. The metric characteristics deemed important when treating a variable as numerical are destroyed by recoding to consecutive integers. For example, scheme C in Table 1.3 corresponds to automatically recoding

age. The differ-

ence between categories 22 and 25 has changed from three to one, and the quantifications will reflect the latter difference.

An alternative recoding scheme that preserves the differences between categories is to subtract the smallest category value from every category and add one to each difference.

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Introduction to SPSS Optimal Scaling Procedures for Categorical Data 7

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 transformed codes have a minimum of 1, and all differences are identical to the original data. The maximum category value is now eight, and the zero quantifications before the first nonzero quantification are all eliminated. Yet, the nonzero quantifications corresponding to each category resulting from scheme B are identical to the quantifications from scheme A.

Which Procedure Is Best f or Your Application?

The techniques embodied in four of these procedures (Correspondence Analysis, Homogeneity Analysis, Categorical Principal Components Analysis, and Nonlinear Canonical Correlation Analysis) fall into the general area of multivariate 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. This enables you to describe structures or patterns in the relationships that 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 advantage of these procedures is that they accommo- date data with different levels of optimal scaling.

Categorical Regression describes the relationship between a categorical response variable and a combination of categorical predictor variables. The influence of 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 of optimal scaling.

Multidimensional Scaling describes relationships between objects in as few dimensions as possible, starting either with a matrix of proximities between the objects or with the original data from which the proximities are computed.

Following are brief guidelines for each of the procedures:

• Use Categorical 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 sets of variables of mixed optimal scaling levels are correlated.

• Use Correspondence 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.

• Use Homogeneity 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.

• Use Multidimensional Scaling to analyze proximity data to find a least-squares representation of the objects in a low-dimensional space.

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Categorical Regression

The use of Categorical Regression is most appropriate when the goal of your analysis 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 category of every variable such that these values are optimal with respect to the regression. The solution of a categorical regression maximizes the squared correlation between the transformed response and the weighted combination of transformed predictors.

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

comparable to optimal scaling canonical correlation analysis with two sets, one of which contains only the dependent variable. In the latter technique, similarity of sets 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 linear 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 can be treated in the same fashion as interval level variables. In the first approach, the model contains a separate intercept and slope 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 estimated for each variable. However, the arbitrary nature of the category codings makes generalizations impossible.

If some of the variables are not continuous, alternative analyses are available. If the response is continuous and the predictors are categorical, analysis of variance is often employed. If the response is categorical and the predictors are continuous, logistic regression or discriminant analysis may be appropriate. If the response and the predictors are both categorical, loglinear models are often used.

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 any single “standard” method is ill-suited. Consequently, optimal scaling offers greater flexibility than the standard approaches with minimal added complexity.

In addition, nonlinear transformations of the predictors usually reduce the dependencies among the predictors. If you compare the eigenvalues of the correlation 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 words, in categorical regression, optimal scaling makes the larger eigenvalues of the predictor correlation matrix smaller and the smaller eigenvalues larger.

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Introduction to SPSS Optimal Scaling Procedures for Categorical Data 9

Categorical Prin cipal Components Analysis

The use of Categorical Principal Components Analysis is most appropriate when 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 a set of variables while accounting for as much of the variation as possible. Scale values are assigned to each category of every variable such that these values are optimal with respect to the principal components solution. Objects in the analysis receive component scores based on the quantified data. Plots of the component scores reveal 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 correlations of the object scores with each of the quantified variables, for the number of components (dimensions) specified.

An important application of categorical principal components is to examine preference data, in which respondents rank or rate a number of items with respect to 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 scale, for example), making the data row-conditional. For preference data, you may want to treat the individuals as variables. Using the transpose the data. The raters become the variables, and all variables are declared ordinal. There is no objection to using more variables than objects in CATPCA.

TRANSPOSE procedure, you can

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

categorical principal components analysis produces an analysis equivalent to a homogeneity analysis run on the same variables. Thus, categorical principal components analysis can be seen as a type of homogeneity analysis in which some of the variables are declared ordinal or numerical.

Relation to standard techniques. If all variables are scaled on the numerical level, cate-

gorical principal components analysis is equivalent to standard principal components analysis.

More generally, 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 correlation coefficient as a measure of association for ordinal data can lead to nontrivial bias in estimation of the correlations.

Nonlinear Canonical Correlation Analysis

Nonlinear canonical correlation analysis is a very general procedure with many 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 variables themselves, as in principal components analysis. For example, you may have two sets of variables, where

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one set of variables might be demographic background items on a set of respondents, while a second set of variables might be responses to a set of attitude items. The scaling levels in the analysis can be any mix of nominal, ordinal, and numerical. Optimal scaling canonical correlation analysis determines the similarity among the sets by simultaneously comparing the canonical variables from each set 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 equivalent to optimal scaling principal components analysis. If all variables in a one-variable-per-set analysis are multiple nominal, optimal scaling canonical correlation analysis is equivalent to homogeneity analysis. If there are two sets of variables, one of which contains only one variable, optimal scaling canonical correlation analysis is equivalent to categorical regression with optimal scaling.

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

technique that finds a linear combination of one set of variables and a linear combination of a second set of variables that are maximally correlated. Given this set of linear combinations, canonical correlation analysis can find subsequent 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 two 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 procedure, 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 multiple 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 unordered categories; therefore, the nominal optimal scaling level is always used. 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. Nothing is assumed about the distance or order between categories of the same variable.

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Introduction to SPSS Optimal Scaling Procedures for Categorical Data 11

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

tables. If a table has r active rows and c active columns, the number of dimensions 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 column categories of a contingency table in a space of dimensions. Practically

min r c,()1–

speaking, however, you would like to represent the row and column categories of a twoway table in a low-dimensional space, say two dimensions, for the reason that twodimensional plots are more easily comprehensible than multidimensional spatial representations.

When fewer than the maximum number of possible dimensions is used, the statistics produced in the analysis describe how well the row and column categories are represented in the low-dimensional representation. Provided that the quality of 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, which 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 tables. If there are more than two variables of interest, you can combine variables to create interaction variables. For example, for the variables in Table 1.1, you can combine

region and job to create a new variable rejob with the 12 categories in Table

1.4. This new variable forms a two-way table with age (12 rows, 4 columns), which can be analyzed in correspondence analysis.

Table 1.4 Combination s of region and job

Category Code

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 South, sales rep 11 West, sales rep

6 South, manager 12 West, manager

Category Definition

Category Code

Category Definition

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 region and age, which results in a new 16 category variable. Each of these interaction 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 approach. Furthermore, if there are four or more variables, two-way tables comparing 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 can select one of these tables to

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analyze, or you can analyze all of them. Alternatively, the Homogeneity Analysis procedure can be used to examine all of the variables simultaneously without the need to construct interaction variables.

Relation to standard techniques. The SPSS Crosstabs procedure can also be used to an-

alyze contingency tables, with independence as a common focus in the analyses. However, even in small tables, detecting the cause of departures from independence may be difficult. The utility of correspondence analysis lies in displaying such patterns for twoway 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 may help reveal the nature of the relationship.

Homogene ity Analysis

Homogeneity analysis tries to produce a solution in which objects within the same category are plotted close together and objects in different categories 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 homogeneous subgroups. Variables are considered homogeneous when they classify objects in the same categories into the same subgroups.

For a one-dimensional solution, homogeneity analysis assigns optimal scale values (category quantifications) to each category of each variable in such a way that overall, on average, the categories have maximum spread. For a two-dimensional solution, homogeneity analysis finds a second set of quantifications of the categories of each variable unrelated to the first set, attempting again to maximize 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 scaling level.

Homogeneity 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 scores of objects in that category.

Relation to other Categories procedures. Homogeneity analysis is also known as multi-

ple correspondence analysis or dual scaling. It gives comparable, but not identical, results to correspondence analysis when there are only two variables. Correspondence analysis produces unique output summarizing the fit and quality of representation of the solution, including stability information. Thus, correspondence analysis is usually preferable to homogeneity analysis in the two-variable case. Another difference between the two procedures is that the input to homogeneity analysis is a data matrix, where the rows are objects and the columns are variables, while the input to correspondence analysis can be the same data matrix, a general proximity matrix, or a joint contingency table, which is an aggregated matrix where both the rows and columns represent categories of variables.

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

data scaled at the multiple nominal level.

Relation to standard techniques. Homogeneity analysis can be thought of as the analysis

of a multiway contingency table. Multiway contingency tables can also be analyzed with the SPSS Crosstabs procedure, but Crosstabs gives separate summary statistics for each category of each control variable. With homogeneity analysis, it is often possible to summarize the relationship between all the variables with a single two-dimensional plot.

An advanced use of homogeneity analysis is to replace the original category values with the optimal scale values from the first dimension and perform a secondary multivariate analysis. Since homogeneity analysis replaces category labels with numerical scale values, many different procedures that require numerical data can be applied after the homogeneity analysis. For example, the Factor Analysis procedure produces a first principal component that is equivalent to the first dimension of homogeneity analysis. The component scores in the first dimension are equal to the object scores, and the squared component loadings are equal to the discrimination measures. The second homogeneity analysis dimension, however, is not equal to the second dimension of factor analysis.

Multidimensional Scaling

Introduction to SPSS Optimal Scaling Procedures for Categorical Data 13

The use of multidimensional scaling is most appropriate when the goal of your 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 conceptual low-dimensional space such that the distances between points in the space 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 help 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 are similar to analyzing the data using categorical principal components analysis with object principal normalization. This kind of PCA is also known as principal coordinates analysis.

Relation to standard techniques. 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 certain models and allows you to put restrictions on the common space. Moreover, 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 based on the distances, while the S-stress is based on the squared distances.

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Displays with More than Two Dimensions

All of the optimal scaling procedures involving dimension reduction allow you to select the number of dimensions included in the analysis. For analyses with three or more dimensions, these procedures produce three-dimensional scatterplots or matrices of scatterplots.

Three-Dimensional Scatterplots

Figure 1.2 shows a three-dimensional plot of object scores produced by a homogeneity analysis with four dimensions.

Figure 1.2 Three-dimensional plot of object scores

Although only the first three dimensions are displayed on the scatterplot, information about all dimensions is included when the chart is created. You can choose to display different combinations of dimensions on the scatterplot by selecting Series menu in the Chart Editor. Figure 1.3 shows the 3-D Scatterplot Displayed Data dialog box, with dimension 4 selected to be displayed in place of dimension 3. The plot displaying these selections is shown in Figure 1.4.

Displayed from the

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Introduction to SPSS Optimal Scaling Procedures for Categorical Data 15

Figure 1.3 3-D Scatterplot Displayed Data dialog box

Figure 1.4 Three-dimensional plot of object scores, displaying dimensions 1, 2, and 4

Scatterplot Matrices

To view more than three dimensions on a single plot, it is useful to graphically display two-dimensional scatterplots for all dimensions in a matrix scatterplot. To convert a chart into a scatterplot matrix, use the scatterplot gallery in the Chart Editor. This option offers great flexibility in converting one chart type to another. A scatterplot matrix displaying four dimensions is shown in Figure 1.5.

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Figure 1.5 Scatterplot matrix displaying object scores for four dimensions

In contrast to the other dimension reduction techniques, correspondence analysis produces a matrix of scatterplots similar to Figure 1.5 for all analyses. If you desire individual two- or three-dimensional scatterplots, use the Scatter option on the Gallery menu in the Chart Editor. Alternatively, to omit or add dimensions to an existing scatterplot matrix, use the for information on editing charts and using the chart gallery).

Displayed command on the Series menu (see the SPSS Base User’s Guide

Aspect Ratio in Optimal Scaling Charts

Aspect ratio in optimal scaling plots is isotropic. In a two-dimensional plot, the distance representing one unit in dimension 1 is equal to the distance representing one 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. Isotropic aspect ratio cannot be overridden for the optimal scaling procedures.

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Categorical Regression (CATREG)

Categorical regression quantifies categorical data by assigning numerical values to the categories, resulting in an optimal linear regression equation for the transformed variables. Categorical regression is also known by the acronym CATREG, for categorical regression.

Standard linear regression analysis involves minimizing the sum of squared differences between a response (dependent) variable and a weighted combination of predictor (independent) variables. Variables are typically quantitative, with (nominal) categorical data recoded to binary or contrast variables. As a result, categorical 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 predictors affect the response. Prediction of the response is possible for any combination of predictor values.

An alternative approach involves regressing the response on the categorical predictor values themselves. Consequently, one coefficient is estimated for each variable. However, for categorical variables, the category values are arbitrary. Coding the categories in different ways yield different coefficients, making comparisons across analyses of the same variables difficult.

CATREG extends the standard approach by simultaneously scaling nominal, ordinal, and numerical variables. The procedure quantifies categorical variables such that the quantifications reflect characteristics of the original categories. The procedure 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 best-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 that high levels of satisfaction correspond to managers and low travel. The resulting 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 predictors, correlations between transformed predictors, residual plots, and transformation plots.

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Data. CATREG 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.

Assumptions. 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 procedures. 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 regression analysis.

To Obtain a Categorical Regression

 From the menus choose:

Analyze

Regression

Optimal Scaling…

Figure 2.1 Categorical Regression dialog box

Select the dependent variable and independent variable(s).



 Click OK.

Optionally, change the scaling level for each variable.

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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.

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

• Spline 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) 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 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 categories 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 pieces 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 opti-

mally scaled variable. Category points will be on a straight line (vector) 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 opti-

mally 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 straight line (vector) through the origin. The resulting transformation fits better than the spline nominal transformation but is less smooth.

• Numeric. Categories are treated as ordered and equally spaced (interval level). 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 be on a straight line (vector) through the origin. When all variables are at the numeric level, the analysis is analogous to standard principal components analysis.

Categorical Regression (CATREG) 19

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To Define the Scale in CATREG

 Select one or more variables on the variables list in the Categorical Regression dialog box.

 Click Define Scale.

Figure 2.2 Categorical Regression Define Scale dialog box

Select the optimal scaling level to be used in the analysis.



 Click Continue.

Categorical Regression Discretization

The Discretization dialog box allows you to select a method of recoding your 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 approximately normal distribution, unless specified otherwise. String variables are always converted into positive integers by assigning category indicators according to ascending alphanumeric order. Discretization for string variables applies to these integers. Other variables are left alone by default. The discretized variables are then used in the analysis.

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Categorical Regression (CATREG) 21

Figure 2.3 Categorical Regression Discretization dialog box

Method. Choose between grouping, ranking, or 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,

rounded, and have a constant added such 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 across those categories.

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

intervals. You must specify the length of the intervals.

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Categorical Regression Missing Values

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

Figure 2.4 Categorical Regression Missing Values dialog box

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

• 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 re-

place 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.

• 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.

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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.

Figure 2.5 Categorical Regression Options dialog box

Categorical Regression (CATREG) 23

Supplementary Objects. This allows you to specify the objects that you want to treat as supplementary. Simply type the number of a supplementary object and click

Add. You

cannot weight supplementary objects (specified weights are ignored).

Initial Configuration. If no variables are treated as nominal, select the uration. If at least one variable is treated as nominal, select the

Numerical config-

Random configuration.

Criteria. You can specify the maximum number of iterations the regression may go through in its computations. You can also select a convergence criterion value. The 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 Plots 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.

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Categorical Regression Ou tput

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

Figure 2.6 Categorical Regression Output dialog box

Tab le s. Produces tables for:

• Multiple R. Includes R

, adjusted R2, and adjusted R2 taking the optimal scaling into

account.

• Coefficients. This option gives three tables: a Coefficients table 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. The increase in multiple R is listed starting from the first iteration.

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

untransformed variables is displayed.

• Correlations of the transformed variables. A matrix showing the correlations

between the transformed variables is displayed.

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• ANOVA. This option includes regression and residual sums of squares, mean squares,

and F. Two 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 for the regression taking the optimal scaling into account.

Category Quantifications. Tables showing the transformed values of the selected variables are displayed.

Descriptive Statistics. Tables showing the frequencies, missing values, and modes of the selected variables are displayed.

Categorical Regression Save

The Save dialog box allows you to save results to the working file or an external file.

Figure 2.7 Categorical Regression Save dialog box

Categorical Regression (CATREG) 25

Save to Working File. You can save the transformed values of the variables, modelpredicted values, and residuals to the working file.

Save to External File. You can save the discretized data and transformed variables to external files.

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Categorical Regression Plots

The Plot dialog box allows you to specify the variables that will produce transformation and residual plots.

Figure 2.8 Categorical Regression Plot dialog box

Transformation Plots. For each of these variables, the category quantifications are plotted against the original category values. Empty categories appear on the horizontal axis but do not affect the computations. These categories are identified by breaks in the line connecting the quantifications.

Residual Plots. For each of these variables, residuals (computed for the dependent variable predicted from all predictor variables except the predictor variable in question) are plotted against category indicators and the optimal category quantifications multiplied with beta against category indicators.

CATREG Command Additional Features

You can customize your categorical regression if you paste your selections into a syntax window and edit the resulting also allows you to:

• Specify rootnames for the transformed variables when saving them to the working data file (with the

SAVE subcommand).

CATREG command syntax. SPSS command language

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Categorical Principal Components Analysis (CATPCA)

This procedure simultaneously quantifies categorical variables while reducing the dimensionality of the data. Categorical principal components analysis is also known by the acronym CATPCA, for categorical principal components analysis.

The goal of principal components analysis is to reduce an original set of variables 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 variables prohibits effective interpretation of the relationships between objects (subjects and units). By reducing the dimensionality, you interpret a few components rather than a large number of variables.

Standard principal components analysis assumes linear relationships between 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 specified dimensionality. As a result, nonlinear relationships between variables can be modeled.

Example. Categorical principal components analysis could be used to graphically display the relationship between job category, job division, region, amount of travel (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 category from region, whereas the second dimension might separate job division from amount of travel. You also might find that high job satisfaction is related to a medium amount of travel.

Statistics and plots. Frequencies, missing values, optimal scaling level, mode, variance accounted for by centroid coordinates, vector coordinates, total per variable and per dimension, component loadings for vector-quantified variables, category quantifications and coordinates, iteration history, correlations of the transformed variables and eigenvalues of the correlation matrix, correlations of the original variables and eigenvalues of the correlation matrix, object scores, category plots, joint category plots, 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 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.

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Assumptions. The data must contain at least three valid cases. The analysis is based on positive integer data. The discretization option will automatically categorize a fractionalvalue 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 principal 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 analysis is identical to homogeneity analysis. If sets of variables are of interest, categorical (nonlinear) canonical correlation analysis should be used.

To Obtain a Categorical Principal Components Analysis

 From the menus choose:

Analyze

Data Reduction

Optimal Scaling...

Figure 3.1 Optimal Scaling dialog box

Select Some variable(s) not multiple nominal.



 Select One set.

 Click Define.

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Categorical Principal Components Analysis (CATPCA) 29

Figure 3.2 Categorical Principal Components dialog box

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



solution.

 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.

Variable weight.

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

Optimal Scaling Level. You can also select the scaling level to be used to quantify each variable.

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

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

• 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 categories 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 pieces are specified by the user-specified number and procedure-determined placement of the interior knots.

• Multiple 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 in the centroid of the objects in the particular categories. Multiple indicates that different sets of quantifications are obtained for each dimension.

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

mally scaled variable. Category points will be on a straight line (vector) 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 opti-

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

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Categorical Principal Components Analysis (CATPCA) 31

To Define the Scale and Weight in CATPCA

 Select a variable in the Analysis Variables list in the Categorical Principal Components

dialog box.

 Click Define Scale and Weight.

Figure 3.3 Categorical Principal Components Define Scale and Weight dialog box

 Enter the weight value for the variable.

 Select the optimal scaling level to be used in the analysis. If you choose a spline trans-

formation, you must also specify the degree of the polynomial and the number of interior knots.

 Click Continue.

You can alternatively define the scaling level for supplementary variables by selecting them from the list and clicking

Define Scale.

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Categorical Principal Components Discretization

The Discretization dialog box allows you to select a method of recoding your 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 approximately normal distribution, unless specified otherwise. String variables are always converted into positive integers by assigning category indicators according to ascending alphanumeric order. Discretization for string variables applies to these integers. Other variables are left alone by default. The discretized variables are then used in the analysis.

Figure 3.4 Categorical Principal Components Discretization dialog box

Method. Choose between grouping, ranking, and multiplying.