# Chapter 2014 Multivariate Statistical Analysis II

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363

Scott M. Smith and Gerald S. Albaum, An Introduction to Marketing Research, 2010

Chapter 14MULTIVARIATE ANALYSIS IIFACTOR ANALYSIS, CLUSTERING METHODS,

MULTIDIMENSIONAL SCALING AND CONJOINT ANALYSISIn this chapter we discuss two techniques that do not require data to be partitioned into

criterion and predictor variables. Rather, it is the entire set of interdependent relationships thatare of interest. We discuss factor analysis as a methodology that identifies the commonalityexisting in sets of variables. This methodology is useful to identify consumer lifestyle andpersonality types.

Continuing with analyses that do not partition the data, a second set of methods iseffective in clustering respondents to identify market segments. The fundamentals of clusteranalysis are described using examples of respondents and objects grouped because they havesimilar variable scores.

Third, we discuss two sets of multivariate techniques, multidimensional scaling andconjoint analysis, that are particularly well suited (and were originally developed) for measuringhuman perceptions and preferences. Multidimensional scaling methodology is closely related tofactor analysis, while conjoint analysis uses a variety of techniques (including analysis ofvariance designs and regression analysis) to estimate parameters, and both techniques are relatedto psychological scaling (discussed in Chapter 9). The use of both multidimensional scaling andconjoint analysis in marketing is widespread.

AN INTRODUCTION TO THE BASIC CONCEPTS OF FACTOR ANALYSISFactor analysis is a generic name given to a class of techniques whose purpose often

consists of data reduction and summarization. Used in this way, the objective is to represent a setof observed variables, persons, or occasions in the form of a smaller number of hypothetical,underlying, and unknown dimensions calledfactors.

Factor analysis operates on the data matrix. The form of the data matrix can be flipped(transposed) or sliced to produce different types, or modes, of factor analysis. The most widelyused mode of factor analysis is theR-technique (relationships among items or variables areexamined), followed distantly by theQ-technique (persons or observations are examined). These,together with other modes, are identified in Exhibit 14.1. Creative marketing researchers mayfindS- andT-techniques helpful when analyzing purchasing behavior or advertising recall data.TheP- andO-techniques might be appropriate for looking at the life cycle of a product class, orperhaps even changes in demographic characteristics of identified market segments.

EXHIBIT 14.1 Modes of Factor Analysis

Six distinct modes of factor analysis have been identified (Stewart, 1981, p. 53). The alternative modes of

factor analysis can be portrayed graphically. The original data set is viewed as a variables/persons/occasions

matrix (a). R-type and Q-type techniques deal with the variables/persons dichotomy (b). In contrast, P-type

and O-type analyses are used for the occasions/ variables situation and S-type and T-type are used when the

occasions/persons relationship is of interest (c).

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Factor analysis doesnotentail making predictions using criterion and predictor variables;rather, interest is centered on summarizing the relationships involving awholeset of variables.Factor analysis has three main qualities:

1. The analyst is interested in examining the strength of the overall association among variables,in the sense that a smaller set of factors (linear composites of the original variable) may beable to preserve most of the information in the full data set. Often ones interest will stressdescription of the data rather than statistical inference.

2. No attempt is made to divide the variables into criterion versus predictor sets.3. The models typically assume that the data are interval-scaled.

The major substantive purpose of factor analysis is to search for (and sometimes test)structure in the form of constructs, ordimensions, assumed to underlie the measured variables.This search for structure is accomplished by literally partitioning the total variance associatedwith each variable into two components: (a) common factors and (b) unique factors.Commonfactorsare the underlying structure that contributes to explaining two or more variables. Inaddition, each variable is usually influenced by unique individual characteristics not shared withother variables, and by external forces that are systematic (non-random) and not measured(possibly business environment variables). This non-common factor variance is called auniquefactor and is specific to an individual variable. Graphically, this may appear as diagrammed inFigure 14.1, in which four variables are reduced to two factors that summarize the majority ofthe underlying structure, and four unique factors containing information unique to each variablealone.

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The structure of the factors identified by the analysis will, of course, differ for each dataset analyzed. In some applications the researcher may find that the variables are so highlycorrelated that a single factor results. In other applications the variables may exhibit lowcorrelations and result in weak or ill-defined factors. In response to these eventualities, theresearcher may revise the variable list and add additional variables to the factor analysis. The

process of adding and eliminating variables is common in factor analysis when the objective ofthe analysis is to identify those variables most central to the construct and to produce results thatare both valid and reliable. Behavioral and consumer researchers have employed these methodsto develop measurement instruments such as personality profiles, lifestyle indexes, or measuresof consumer shopping involvement. Thus, in addition to serving as a data reduction tool, factoranalysis may be used for to develop behavioral measurement scales.

We use a numerical example to illustrate the basic ideas of factor analysis. A grocerychain was interested in the attitudes (in the form of images) that customers and potentialcustomers had of their stores. A survey of 169 customers was conducted to assess images. Theinformation obtained included 14 items that were rated using a seven-category semanticdifferential scale. These items are shown in Table 14.1. The resulting data set would then be a

matrix of 169 rows (respondents) by 14 columns (semantic differential scales). These data willbe analyzed asR-type factor analysis.

Figure 14.1 The Concept of Factor Analysis

Table 14.1 Bipolar Dimensions Used in Semantic Differential Scales for Grocery Chain Study

Inconvenient locationConvenient locationLow-quality productsHigh-quality products

ModernOld-fashionedUnfriendly clerksFriendly clerks

Sophisticated customersUnsophisticated customersClutteredSpacious

Fast check-outSlow check-outUnorganized layoutOrganized layout

Enjoyable shopping experienceUnenjoyable shopping experienceBad reputationGood reputation

Good selection of productsBad selection of productsDullExciting

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IDENTIFYING THE FACTORSIf we now input the raw data into a factor analysis program, correlations between the

variables are computed, as is the analysis. Some relevant concepts and definitions for this type ofanalysis are presented in Exhibit 14.2.

A factor analysis of the 14 grocery-chain observed variables produces a smaller number

of underlying dimensions (factors) that account for most of the variance. It may be helpful tocharacterize each of the 14 original variables as having an equal single unit of variance that isredistributed to 14 underlying dimensions or factors. In every factor analysis solution, thenumber of input variables equals the number of common factors plus the number of uniquefactors to which the variance is redistributed. In factor analysis, the analysis first determines howmany of the 14 underlying dimensions or factors are common, and then the common factors areinterpreted.

EXHIBIT 14.2 Some Concepts and Definitions ofR-Type Factor Analysis

Factor Analysis: A set of techniques for finding the underlying relationships between manyvariables and condensing the variables into a smaller number of dimensions

called factors.

Factor: A variable or construct that is not directly observable, but is developed as alinear combination of observed variables.

Factor Loading: The correlation between a measured variable and a factor. It is computed bycorrelating factor scores with observed manifest variable scores.

Factor Score: A value for each factor that is assigned to each individual person or object fromwhich data was collected. It is derived from a summation of the derived weightsapplied to the original data variables.

Communality (h2): The common variance of each variable summarized by the factors, or the

amount (percent) of each variable that is explained by the factors. Theuniqueness component of a variables variance is 1-h

2.

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