# A Facet Theoretical Approach For Testing Measurement and Structural Theories: an Application of Confirmatory Mds

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Lutz Hildebrandt (1986) ,"A Facet Theoretical Approach For Testing Measurement and Structural Theories: an Application of Confirmatory Mds", in NA - Advances in Consumer Research Volume 13, eds. Richard J. Lutz, Provo, UT : Association for Consumer Research, Pages: 523-528.

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http://acrwebsite.org/volumes/6545/volumes/v13/NA-13

[Parts of the article were written while the author was a visiting professor at Auburn University in Alabama.]

The analysis of covariance structures has become a widely accepted tool to test measurement and structural theories in a holistic way. Due to-underlying restrictive assumptions of the methodology, however, its usage is seldom justified in the analysis of behavioral data. This article proposes the use of facet theory and constrained multidimensional scaling as an alternative in confirmatory data analysis.

INTRODUCTION

During the last decade confirmatory methods in data analysis, especially methods which allow an integrative test of measurement and structural relations have attracted many social scientists. Marketing researchers have focused their attention particularly on one approach which is based upon the analysis of covariance structure (Joereskog 1981). The widely available software LISREL V/VI (Joereskog/Soerbom 1981, 1984) allows a simultaneous test of a measurement model and a model representing the causal hypothesis. In addition, Bagozzi (1980, 1984) provided philosophical arguments for the structural equation methodology and developed on this ground a general prospectus for theory construction in marketing. However, especially the LISREL approach in its maximum likelihood version has been widely criticized, due to its restrictive assumptions. Requirements like multivariate normal distributed variables or linearity etc. are seldom met by the data. The available test-statistic is affected by the sample size and violations of these assumptions (Fornell 1982, 1983). Later developed programs like EQS (Bentler 1984) or LVPLS (Lohmoeller 1985) do not have the restrictive distributional assumptions, but also assume linearity. The methods in general require metric data, although nonmetric extensions exist (Muthen 1983; Wold and Bartholet 1981).

The discussion about the application of the linear models is quite intensive. The facet theoretical approach, however, which is also based on an integration of measurement and theory testing is seldom discussed in consumer research. The methodology majorly is developed out of Guttman's (1954, 1957) work on scaling.Beginning with an investigation of the content of a theory, the statistical structure of the corresponding empirical observations can be predicted and tested by a multidimensional scaling procedure (see for instance Borg 1977, 1985).

Guttman's approach for theory testing can be distinguished from the methodology based on covariance structure analysis in two major instances. First, whereas the letter methodology begins with the integration of measurement and theory on the statistical and model specification level, Guttman's integrative approach starts on the research design level. Facet theory serves as a tool to explicitly define the measurement space on the basis of content. Metatheoretical rules additionally provide a set of structural hypotheses Second, the proposed multivariate procedure to test the structural hypothesis is a nonmetric multidimensional scaling method based on soft assumptions. Correlations and other types of measures of contingencies and/or (dis)similarities may serve to test the hypothesis. The method constrained monotone distance analysis (CMDA) to perform confirmatory multidimensional scaling (Borg and Lingoes 1980) is nonmetric and invariant to changes in the magnitude of the measures, as long as the order relation of the input data is maintained.

It should be mentioned that the methodology allows only derived causal interpretations of the results (see Denison and Fornell 1985). As in covariance structure analysis the available test-statistics should rather be interpreted as measures of fit. Instead of that a quasi-statistical model based on an efficacy coefficient can be used to evaluate a model (Lingoes and Borg 1983). In this article we will first introduce this methodology more closely. Then a worked-out example is performed, selecting items from an attitudinal study.

THE BASICS OF FACET THEORY

One element of Gutman's metatheoretical approach (1959) is the systematic research design based on facets as a prerequisite to test hypotheses. To clarify the concept of a facet or of 'facet theory', a definition outlined by Borg (1977, ?- 65) can be used: "Facet theory is a general methodology for investigation in the social sciences: it provides a general framework for the precise definition of an universe of observations, which is directly related to both the specification of the various elements of empirical studies (stimuli, subjects, responses) ar.d to theories about the structure of those observations".

The basic idea is that a theoretical problem can be decomposed into the number of underlying conceptual facets. The term 'facet' refers to a 'set consisting of a finite number of elements' (Foa 1965). The cartesian product of a finite number of facets for item construction corresponds to the factorial design for experimentation. Examples of facets include the set of components of an attitude, a set of attitudinal objects or a set of different methods to measure one attitude. As such the well-known design of an MTMM-matrix represents the combination of two facets: a method facet and a trait facet. The cartesian product of the two facets provides the basis for data collection (see also Mellenbergh et al 1979).

Based on facet theory Guttman also develops his concept of a theory as 'hypothesis of correspondence between a definitional system for a universe of observations and an aspect of the empirical structure of those observations, together with a rationale for such a hypothesis' (Gratch 1973, . 35).

The definitional system has to explain the domain of observation and provides a meaningful basis for the replication of research and validation of results. The definitional basis for the measurement of the theoretical construct is provided by the use of a facet design related to the basis-paradigms of research in social sciences. This consists of three basic facets: a population (P), a set of stimuli or variables (S), and a set of responses (R). The population facet (P) describes the characteristics of the population. The second facet (S) defines the content of the variables under investigation. The third facet (R) specifies the categories of the answers.

As a framework to define the area under investigation serves the so-called mapping sentence (Guttman 1959; Shye 1978). A mapping sentence consists of a number of domain facets and a facet which represents the range of answers. The relation of the facets in the mapping sentence can be formalized as follows:

Let A_{i} be a set of elements and S = A_{1} x A_{2} x ... x A_{n} be the cartesian set which can be built by A_{k} (k=1,...,n), then s e S holds, if and only if s is a structuple of the kind that

s= a_{i1}, a_{i2}, ..., a_{in}

having

a_{ik} e A_{k} (k=1,2,...,n)

every A_{k} then is a facet of S. The whole mapping sentence can be formulated as:

p x s |-> r

(domain) (range)

where P is the population facet, S = A_{1} x A_{2} x ... x A_{n} the cartesian set of the facets of content, which serves to define the stimuli, and R the range of answering. The elements of a facet are named structs, ,he elements of the cartesian product of the facets structuples.

The mapping sentence does not constitute a theory. The hypotheses are to be formulated according to the lawfulness of the empirical structure of the variables (Guttman 1959). An example of a mapping sentence from satisfaction research is given by figure 1. The construct under investigation is the consumer's satisfaction with different aspects of retailing in the consumer's area.

A MAPPING SENTENCE OF CONSUMER SUPPLY SATISFACTION

The cartesian product of the defined content facets AXBxC provides all possible combinations of the facet elements which may be relevant to the domain under investigation. However, not all structuples (facet combinations) in general make sense (e.g. Jordan and Horn 1975). The system may serve as a basis for constructing items, but it can as well be used to select items from former empirical research. Using this approach,the researcher assures that also different types of studies are comparable on the general level, even if the focus of each single study may be different. The source of the construction of facets may be either theory or also ad hoc thinking (Borg 1985). Although the approach seems rather simple, it provides the ground to develop some metahypotheses.

Metahypothesis

The systematic definition of variables according to underlying substantive knowledge enables us to test their structural relations. In addition, it is assumed that some of the concepts of facet theory are so closely related to psychological processes that they provide meta-theoretical rules concerning the statistical relationships of the variables involved. Foa (1965) suggests two principles for predicting results from a facet structure of the variables: the principle of contiguity, and the concept of semantic principal components.

The principle of contiguity states that variables which are more similar in their facet structure (e.g. on the level of content) will also be more related empirically. An example is given by a combination of three facets, where each facet consists of two elements. The cartesian product of the facets - the space of content - results in eight structuples (a_{1}, b_{1}, c_{1} ; i,j,k = 1,2) to define the variables. If one uses correlation coefficients to capture the relations of three variables, the contiguity principle (under the assumption of actual weights of the facets) has to lead to the following inequality, if the data are valid:

r(a_{1}b_{1}c_{1} ; a_{1}b_{2}c_{1}) > r(a_{1}b_{1}c_{1} ; a_{1}b_{2}c_{2})

Using the principle it is possible to derive an ordering from the variables which in the simplest case is linear. in a study of Guttman (1959) four variables (items) are measured, because of the underlying structuples it was possible to arrange the items according to their contiguity. Each structuple is different to its neighbor by one element

var 1 a_{1} b_{1} c_{1}

var 2 a_{1} b_{2} c_{1}

var 3 a_{1} b_{2} c_{2}

var 4 a_{2} b_{2} c_{2}

On the semantical (facet-)level we observe the following similarities: var 1 and var 2; var 2 and var 3; var 3 and var 4. Less similar are var 1 and var 3 or var 2 and var 4. Dissimilar are var 1 and var 4.

The principle of contiguity is confirmed empirically, if the correlation matrix of the variables shows a simplex structure. This structure is given if we have a decreasing ordering of the correlation coefficients in the off diagonal columns and rows toward the end points. The geometrical mapping of a simplex is a line or some linear ordering. This is exactly the same which is required if one builds a Guttman-scale. Therefore, the above example results in scale, which is named 'semantic' Guttman scale by Foa (1965)

Contiguity, however, is only a necessary condition to predict a simplex structure. In a definitional system having three facets we are only able to define partial orderings. If we have dichotomous items, then the system of contiguity has the following structure (see also Guttman/Schlesinger 1967):

THE ORDERING IN A DEFINITIONAL SYSTEM OF THREE BINARY ITEMS

If all eight structuples are ordered so that each structuple is contiguit with its neighbor, and if there is only a difference in one element, then the first structuple becomes similar again with the last structuple. Each of the variables will correlate higher with its neighbors than with any other variable. The result is a circumplex structure in the data (Borg 1977 or Foa 1965). The geometrical representation of a circumplex is a circle.

In some cases, however, the principle of contiguity does not produce a unique order. If we assume equal weight of the facets, the facets are interchangeable. The order of the variables are dependent of the choice of the first facet and the first element. To solve this problem, Foa (1965) introduced the concept of the semantic principle component into facet theory, in analogy to Guttman's (1954) principle components of perfect scales. It states ghat in a sequence of several structuples the facet which changes its elements the least (in general the first facet) serves as the first semantic principle component. If we have more than two facets, the frequence of the variation in the elements of a facet determines its relative weight. However, the concept of semantic principle components is defined very insufficiently. No order hypothesis can be derived without additional assumptions (Lantermann 19809.

Structural Hypothesis

Simplex and its extension, the circumplex, are basic structures and should be mapped only by using the relevant items (Borg 1981, p. 143). They have their importance primarily in scale construction.

In consumer research it should be of more importance to test the more general hypothesis, if the discriminations created by the facets in S, P, and/or R correspond to empirical regularities. Less restrictive versions of contiguity serve as the basis to relate total configurations to the definitorial system of a facet design. In an ongoing study of Hildebrandt (1985) the mapping sentence of figure l served to define the domain of the items concerning supply satisfaction of convenience goods (b1). The facet A can be regarded as ordered. Facet B is only represented by one element, and facet C is unordered. Using a weak conception of contiguity for a structural hypothesis it is required that a) items which have the same element in facet C should be represented in the same area or region of a two-dimensional MDS configuration, and b) facet A should discriminate items of type a and type a2 in this region. Both should lie in different partial regions. In addition a valid representation should discriminate facet A and facet C in a way that the areas or point-clusters are non-overlapping and the representation is simple.

This spatial representation of the data is only possible in a two-dimensional space and a structure, which is called a radex (Guttman and Schlesinger 1967). The facet A serves as a modulating facet, facet C as a polarizing facet. The radex hypothesis which should be confirmed by the data using (N)MDS is represented in figure 3. The order expectation of the C facet is specified according to results from former studies.

A RADEX HYPOTHESIS OF SUPPLY SATISFACTION

Higher order facet structures and the development of complex spatial hypothesis as for instance the duplex, multiplex, cylindrex etc. primarily depend on Guttman's work and are discussed in a recent Paper by Gorg (1985).

TESTING STRUCTURAL HYPOTHESIS

Applying facet theory and developing a set of structural hypotheses presents the first step to leave the stage of data reduction and exploration in multidimensional scaling. The hypothesized spatial structure based on a meta-hypothesis may be supplemented by some substantive assumptions about the relation of the constructs under investigation (see e.g. Fornell and Denison 1982). However, if we apply a classical (N)MDS method like MINISSA (Lingoes and Roskam 1975) to investigate if the empirical data confirm our structural hypothesis, we still remain in the theory formation process. Applying (N)MDS to our data, we leave the theoretical level and optimize a 'blind' stress which only accounts for the similarity structure. The resulting spatial representation then may satisfy the minimization criteria very well, but may not confirm our set of hypotheses. On the other hand we can not be sure whether there does not exist another configuration which satisfies the optimation criteria equally well and in addition confirms the structural hypothesis.

These interpretational difficulties in MDS solutions caused by local optima and indeterminacy are overcome by confirmatory MDS. Approaches for confirmatory MDS allow to put external constraints on a distance structure. The constraints might be derived from a facet design, from measurement requirements, and/or substantive theory (Fornell and Denison 1982). Constrained MDS solutions may be produced e.g. by programs like KYST (Carroll et al. 1980) or CMDA (Lingoes and Borg 1980). The last is applied in this article. Technically CMDA requires an acceptable (K >.15) solution of an unconstrained MDS, for instance by MINISSA. This unconstrained configuration is imposed by a set of constraints to produce a spatial solution according to our theory. In addition to the spatial representation of our data CMDA provides measures to decide, if the constrained and unconstrained distance configurations are equivalent representations of the original proximity data.

In their first approach Lingoes and Borg (1980) developed a statistical test based on Hotelling's t-test. This test is criticized because of its poor robustness against assumptional violations. In a more recent article instead of that Lingoes and Borg (1983) propose a set of decision criteria for the acceptance/rejection decision. However, it should be noted that the criteria are rather rules of thumb than strong statistical criteria. The following coefficients are relevant:

As criteria for acceptance are proposed:

In cases where the ratio is between 1 and 3 additional criteria should be used to build a weighting factor A for K. For the calculation of this factor the reader is referred to Lingoes and Borg (1983, p. 47). The reason for the introduction of the multiplier A is that the condition of a larger sample size, larger matrix and a relative large number of constraints etc. should improve the chance for acceptance of a given model. It is proposed: If

we accept the equivalence hypothesis.

APPLYING THE FACET THEORETICAL APPROACH

The clarify the holistic way of data analysis using the facet theoretical approach an item selection procedure is carried out to test hypotheses about the information behavior of young consumers. The research bases on the general hypothesis that consumers who are better informed in general are also better informed regarding product and consumer issues. A concept which tries to explain this relationship is 'the propensity to use information ' (Wiswede 1975). In this study the concept is measured on the basis of attitudinal items toward different kinds of information. It is assumed that the better the attitude toward an information type , the better the information use behavior (resulting in a more rational buying behavior). The study then states a causal hierarchy for the 'propensity to use information' regarding three types of information :

general information --> product related information --> consumer information

Based on these conceptual considerations the following mapping sentence is used to define the area of investigation :

The attitude is defined as a three-partite concept. The cartesian product of the facets of content defines all possible combinations of facet elements. It serves as the frame of reference to select items out of an existing consumer study (Tschammer-Osten and Warnecke 1978).

THE CARTESIAN PRODUCT OF FACETS

The items were selected by using a comprehensive system of categories with regard to the semantic meaning of cognitive, affective, and behavioral items. For the criterion set to distinguish different sources of information served the closeness of an information type to buying behavior. The selection procedure resulted in a set of nine items, which in part represent the same structuples, and on the other hand some of the structuples of the facet design were not available in the data set. The structuples covered bs the available items are underlined.

We can now try to derive some structural hypotheses based on the facets represented by the items. The contiguity principle states that items with the same facet structure should be closest together. The derivation of an ordering depends on the definition of the first principle component and the specification of the first element in the ordering of the facets. Going back to our basic hypothesis we can assume that the facet B discriminating the information sources should be the dominant facet. Therefore, we specify that the attitude toward general information has to be the first structuple of the ordering, and we derive the following scheme :

Variables with the same facet structures are x_{3} and x_{4}, x_{6} and x_{7}, x_{8} and x_{9}. They should be closest together. The largest dissimilarities are to be expected between x_{2} and x_{3}, and x_{5} and x_{6}. The facet structure in the items representing the same constructs are more similar than between the constructs. The information facet should generate clusters of items. If we apply the logic of an MTMM-analysis, we will get to the same assumption.

In addition we can formulate a simplex expectation between the items x_{1}, x_{2}, x_{9}. To test the structural hypothesis we first generated a MINISSA solution, which produced a two-dimensional configuration based on an alienation K = 0.9, which is a good fit.

MINISSA-UNCONSTRAINED SOLUTION

Analyzed were the Pearson correlations of the nine items of a 248 respondents' sample. The MINISSA solution does confirm the simplex expectation. Some of the expected similarities are also represented by the distances. However, the solution does not confirm a strong clustering expectation generated by the elements of the B facet. Especially for the item x_{6} we can not decide, if it represents a 'product' related item or a 'consumer' related item.

For constraining the solution in a CMDA analysis according to our general research hypothesis and the structural expectation derived from the facet theory we use a strong concept of contiguity. The spatial solution should be constrained so that each point within an item cluster (representing the B facet) should be nearer to all other points within that cluster than to any point outside of that cluster. The relations between the clusters should represent the assumed causal ordering (see upper triangular in the correlation matrix).

CMDA STRONG CLUSTER - STRONG TEEORY SOLUTION

The CMDA solution shown in figure 7 represents the clear clustering structure and also the stated hierarchical relation behind our three concepts. However, the solution does not satisfy the criteria for acceptance (see figure 8)

An additional analysis without relational constraints according to the assumed hierarchical structure generates a slightly better solution. However, the source of invalidity remains item x_{6}. A possible strategy might be to accept a weaker type of contiguity. If we find an acceptable solution we may calculate the centroids of the clusters as representations of the underlying constructs, and we can use euclidian) distances to have a measure of their (causal) relations (see: Fornell and Denison 1982; Denison and Fornell 1985).

DISCUSSION

The facet theoretical approach of Guttman integrates the basic steps of empirical research : design - measurement hypothesis testing. It is more comprehensive than the 'holistic' methodology for the analysis of covariance structure. Shortcomings may be that there is only an indirect way for a causal interpretation of the results and the lack of a sufficient statistical test. But with regard to the last problem we should take into account that in social sciences the assumptions to apply a statistical test like the LISREL X_{2} are seldom met. If the assumptions however, are met, the theory testing part of the structural equation methodology is more powerful. The facet-theoretical approach using CMDA has its advantage if we can not assume linearity or if we do not have metric or quasi metric data. It also has advantages, if the researcher is involved in large variable sets.

THE MATRIX OF THE CORRELATIONS / ORDER RESTRICTIONS

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