# The Index of Consumer Satisfaction: Methodology

##### Citation:

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James C. Lingoes and Martin Pfaff (1972) ,"The Index of Consumer Satisfaction: Methodology", in SV - Proceedings of the Third Annual Conference of the Association for Consumer Research, eds. M. Venkatesan, Chicago, IL : Association for Consumer Research, Pages: 689-712.
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[An earlier version of this paper was presented as "Measurement of Subjective Welfare and Satisfaction" at the 84th Annual Meeting of the American Economic Association jointly with the Association for the Study of the Grants Economy, New Orleans, Dec. 27, 1971. The research was financed by Contract No. 12-17-05-1-610 from the U.S. Department of Agriculture.]

I. INTRODUCTION

1. Subjective Market Failure and Consumer Dissatisfaction

The intellectual landscape of the past decade has been marked by dramatically contrasting signposts: On the one hand, economic measures of the growth of national product and income have pointed to a society of affluence. On the other hand, the storm signals about urban unrest, social disintegration, and widespread poverty in urban ghettos and rural enclaves has suggested that there are certain basic features of American society which cry for remedy.

In the area of economic ant market performance, we notice a growing dissatisfaction of consumers with goods and services offered in the market place. This may be due more to a rising level of awareness fostered by news media than by any deterioration in the objective conditions under which these goods and services are being provided. Nonetheless, if consumer satisfaction is viewed as the objective of market actions in an open society, it has to be taken into account when formulating public and corporate policy.

The economy theory of "market failures" or "market inefficiencies" recognizes that deviations from an economic ideal of welfare maximization can be brought about by (a) monopoly forces which distort prices and output levels; (b) indivisibilities or "lumpiness" in production or consumption which make impossible the fine marginal adjustments postulated by welfare theory; (c) increasing returns to scale which obviate competition since a single firm can supply a market at a cost lower than several competitors; and (d) externalities in production or consumption which lead to a divergence between private and social benefits and costs. To these we may add market imperfections due to limited information, cyclical instabilities, and so on. These theories, however, assume implicitly that diminution of social welfare occurs because of forces largely outside the individual consumer. In fact, the consumer sector is treated largely as a more or less homogeneous group.

Such a view neglects, however, the wide variance in the degree of (dis) satisfaction among different consumers. What is needed therefore is a theory of "subjective" welfare: Dissatisfaction of the individual consumer results from the discrepancy between ideal and actual attribute combinations. Such a discrepancy, moreover, can result from both outside and within the consumer: If, for example, the perceived ideal changes, dissatisfaction can go up even if the so-called objective circumstances do not change at all! This view of subjective market failure treats the consumer group as highly heterogeneous.

The Index of Consumer Satisfaction is a measure of this subjective welfare of consumers as influenced by the attributes and circumstances under which goods and services are offered in the market. In this paper we describe the methodological foundations of (a) representing market goods and services in terms meaningful to the subjective experience of the consumer (the representation problem); (b) predicting overall satisfactions (the composition problem); and (c) aggregating individual satisfaction scores into group satisfaction and potentially into an overall satisfaction measure for the nation (the aggregation problem).

2. Measurement of Economic and Market Performance

It is perhaps the conflict between overall progress signified in economic accounting systems, and social disharmony and misery as expressed by other measures on the social plane, which has motivated the growing concern and the re-examination of our existing measures of economic and social performance.

Institutes, conferences and professional meetings have been addressed to the question of whether the national economic accounts are deficient in any way and how they could possibly be improved to reflect some of these economic and social problems. The Conference on Research in Income and Wealth, (sponsored by the National Bureau of Economic Research at Princeton, New Jersey, November 4-6, 1971) is one of the more recent examples of this search for new ways of measuring economic and social performance. One of the major themes that came out of this conference was the need to view the performance of the economy and of society in terms of the positions of specific sub-groups and social sub-strata of total society rather than in an overall fashion. Furthermore, the need to recognize social costs in the framework of national accounts that result, for example, from the deterioration of the environment, is widely felt. No doubt, the presently used framework of national accounts has taken note of the government in the economy; however, it is largely a market-based system and it neglects the wide role of nonmarket transactions. [This is one of the major areas of concern for the Association for the Study of the Grants Economy which has studied not only monetary transfers between different groups but also the nature of non-market production. See for example, Pfaff & Pfaff, and Boulding.] Others again have pointed to the need to recognize the various imputations, such as imputed rent on owner-occupied dwellings, the services of housewives, consumer durables, etc. in the national income accounts explicitly (Ruggles & Ruggles, 1970).

Even if all of these proposals for the extension of economic measures of the economic performance of our society are adopted, they fall short of describing some significant aspects of the quality of life and of social existence. The reason is simply that such measures rely on dollar measurements of transaction flows; they reflect social problems such as discrimination, inequality, crime, etc., through social gains or losses which would be added on to the economic gains or losses reflected in the national income accounts.

This recognition, together with an increasing awareness of the social problems of our society, have increased the level of interest in a system of social indicators which would be a more direct reflection of the areas of social concern. No doubt there are conflicting views as to what social indicators should do and how they should relate to other measures of economic and social performance in our society. Most would agree, however, that social indicators should be relatively comprehensive; ant that they should relate to a system of overall social accounts which offers perhaps some parallels to the presently used system of economic accounts. [For a bibliography on social indicators, see Knezo.] This point of view, however, is bound to be somewhat controversial at a time when a general theory or model of how social processes relate to economic process has not yet gained wide acceptance. [For a model of this type see Pfaff.]

Several measures have been employed or proposed to assess the performance of the market system or of its components. The type of measure employed depends on the theory or model of the market that the investigator holds either implicitly or explicitly (Pfaff, 1968).

Within any of these approaches, the process of measurement is influenced by the concept of market performance that the investigator adopts: He can evaluate the system by imposing his own "external" values; or else, he can observe the value patterns prevalent in the system itself and judge its performance by these "internal values" (Douglas, 1968; Cook, 1968). In any case, he is forced to embrace some kind of norm or yardstick, without which measurement is impossible. Even if he falls back on the traditional economic norm of "efficiency" or productivity, he is no less normative: In the face of conflicting social goals, the selection of any one norm represents a normative act par excellence.

Similar considerations apply to the measurement of the performance of the non-market sector, including the public economy.

Among the measures of market performance we find the 1. "value-added method", 2. physical productivity analysis, 3. flow analysis, 4. market structure analysis, and 5. the application of welfare economics (Buzzell, 1959; Pfaff, 1969).

3. The Measurement of Utility and Welfare

The second major approach to the problem of assessing the performance of the economy and of society tends to get at the very end or objective of the while economic and social process which, in an open--society, is generally held to be the satisfaction of the needs of the population. No doubt, this focus has a long tradition in economic theory where the utility or the @'want-satisfying power" of a commodity or service was the central focus of the theory of value. Indeed, if the utility of various economic and social acts could be assessed empirically, a system for the evaluation of performance par excellence would be at hand. Unfortunately, while the concept of utility has been the main foundation for the theory of value, it has not really been extended to practical social policy.

Ever since Robbins' An Essay on the Nature and Significance of Economic Science (Robbins, 1932), it was held that value judgments should not be part of "objective" or scientific analysis. This position is reflected in the work of Kaldor, Hicks, and others who sought to formulate a value-free New Welfare Economics (Kaldor, 1939; Hicks, 1939, Reder, 1947). However, their approach contained ethical assumptions like consumer sovereignty, as was pointed out later by Bergson (1938) and Baumol (1946-47). Bergson proceeded to explicate the role of value judgments in the form of his Economic Welfare Function. Assuming that all non-economic, i.e., social or institutional variables (like government and social structure) remain constant, economic welfare is determined simply by economic variables; or, it is assumed that changes in the values of economic variables are not large enough to bring about changes in the institutional environment. Thus, "The Economic Welfare Function simply traces the effects on the individual's welfare of changes occurring in one part of the environment, the other portions remaining unchanged" (Rothenberg, 1961, p. 10).

Social welfare functions have been applied in areas where alternative policies were to be evaluated in the face of conflicting ends. Many criticize however the basis on which these judgments are formulated. In their view the de-facto policies of government cannot necessarily be taken as an expression of a social welfare judgment.

Others again would reject the concept of welfare function on theoretical grounds, based on Arrow's celebrated Impossibility Theorem (Arrow, 1951). A reinterpretation by Rothenberg appears to offer a way out of this theoretical impasse (Rothenberg, 1961). In the face of the problems associated with the measurement of utility, several scholars have turned to an attempt to measure welfare rather than utilities. No doubt, the concept of "measurable welfare" is somewhat different from utility, but it may provide the very alternative which is required for public policy making. One of the most interesting of these attempts has been the work of the United Nations Research Institute of Social Development which has consistently attempted to formulate indices of the level of living, of welfare, and so on (UNRISD). In one of these reports, Jan Drewnowski defines measurable welfare as "a quantitative expression for the satisfaction of needs of a population," (Drewnowski, 1968). He intends to formulate such a measure that should present a macro-concept which is fully measurable in cardinal terms, summable, and comparable interpersonally, internationally and through time Analogous to the concept of utility, he interprets measurable welfare as the dependent variable of a preference function:

The Level of Welfare is the sum total of characteristics of the person or the population expressed in quantitative form and referring to the state of satisfaction of the needs of that person (Drewnowski, 1968).

Drewnowski proposes to measure the level of welfare indirectly by relating different component measures which are expressions of their somatic status (physical development level), to the educational status, and the social status. He touches on the problem of weighting: He points out that in the absence of information to the contrary, the various subcomponents would be given equal weights within an overall component, and various components will be again equally weighted for a total "level of welfare" index measure. The kinds of indicators he would use are essentially cardinal measures of the health status expressed, for example, through life expectancy, or percent distribution of healthy versus sick people. Similarly, he would measure educational status by literacy rate, educational attainment rate, and so on. In the area of social status he attempts to find numerical expressions for activities and social trends which would have profound ramifications for the level of welfare of a given society; among these he lists the level of integration within the nation, social groups, and family, as well as the level of participation of various social actors.

4. "Subjective" Measures of Economic and Market Performance

As an alternative to these indirect and "objective" measures of welfare we proceeded to measure the level of satisfaction of the needs of individuals and persons not by some abstract indicator but on the basis of an individual's own Perception of his own satisfaction. This approach then amounts to a direct and "subjective" approach to the measurement of welfare, based on the respondents' own evaluation of their satisfaction with particular aspects of goods or services provided by the market or the public economy (Pfaff & Pfaff 1969). We attempted to measure the performance of the market economy by what we have termed an Index of Consumer Satisfaction which measures a cross section of the population's evaluation of the various services provided by the market-distribution system. Furthermore, we measured the performance of the public economy, that is, the services provided by the public sector, by the citizens' evaluation of the attributes of such services and the relationship of these attributes to their own needs and expectations. This Index of Citizen Satisfaction has only been explored initially, and a pilot test of this Index is contemplated for the near future. These indices of consumer and citizen's satisfaction are, therefore, social indicators which attempt to measure the human satisfaction which results from the operations and performance of the market and public economy. They aim at judging the quality of life produced by various products and services available to consumers through market offerings on the one hand, and by the various programs of government which have economy and noneconomic consequences for different groups in society, on the other. The ultimate aim of these indices is to provide measures of the satisfaction with the different aspects of life experienced by consumers and citizens. A major interest in this study is the specific and general aspects of consumer dissatisfaction or citizen dissatisfaction with the quality of life, and how such satisfactions or dissatisfactions are distributed across specific racial, ethnic, age, income and other groups, defined on the basis of psychological, social, economic and cultural criteria.

Subjective measurement of satisfaction has a long tradition at the hands of industrial psychologists who were concerned with the more specific area of job satisfaction. Their aim was to measure the relationship between satisfaction and productivity. However, "no really substantial, reliable or general correlation between satisfaction and productivity has been established" (Smith, Kendall, & Hulin, 1969, p. 3; Brayfield & Crockett, 1955; Smith & Cranny, 1968).

These measures of work or job satisfaction are concerned with the input side of the economic process. At the other extreme, we have what may be construed as measures of the output of the social system, viz. the happiness with life in general, and with specific facets of life in particular. Among these, Hadley Cantril (1965) studied the happiness, and the hopes, and fears of persons in fourteen countries, Bradburn, (1969), and Caplovitz, (1965) and Wilson (1967) discuss the distribution of avowed happiness across different groups. Inkeles (1960) concluded that individuals of higher socio-economic status tend to be more satisfied with their lives than those of lower status. More recently, Easterlin (MIMEO) synthesized the findings based on cross-section analyses of individuals' self-assessment of happiness. He inferred a strong positive relationship between happiness and income within a society, but noted the absence of such a relationship across countries at a given point in time. Gurin, Veroff, and Field (1960) study happiness in the context of the overall mental health of Americans.

There has been no study, however, of the satisfaction of consumers with the performance of specific aspects of the market system on the one hand, and there are only a few approaches to the study of citizens' satisfaction with public goods provided by the government, on the other (Stagner, 1970).

The methodological foundation for the computation of these Indices of Consumer and Citizen Satisfaction was provided by nonmetric scaling techniques, as developed by Guttman, Lingoes, and others.

The computation of these indices was based on a set of procedures appropriate to the assumptions which we felt were justified by the data collected. In so doing we aimed at achieving reliable, valid, and univocal scales. We shall now turn to a description of the methods employed and their underlying rationale.

II. RESEARCH STRATEGY

We impose the logical requirement that individuals or groups having higher values than others on avowed overall satisfaction are also more satisfied (or, at least, see or report themselves as being more satisfied) in respect to all the comPonentS that determine the overall satisfaction index. Formally, we are imposing a homogeneity criterion on our indices, such that any pair of individuals/groups which are nearer one another than either are to any other individual/group (in terms of their indices of satisfaction), will have profiles over the components of the index which are closer together. The index is, therefore, predictive of its constituents.

To the extent that a set of variables is factorially complex (requiring more than one dimension to explain the inter-relations) we have a partial order defined on the individuals' scores. This implies that a given person may be higher than another on one dimension but the reverse may be true on another dimension. Take,for example, the case of one baseball player being a better hitter than runner and the reverse for another player; how can we compare these hypothetical players as to baseball talent? Can we arrive at some measure to make the following statement sensible: "Lou has more baseball talent than Joe."? In essence, can we integrate or map a complex domain onto a line so that the order of scores is meaningful? Our work-a-day world in which decisions must be made (e.g., salaries to be paid and honors to be bestowed in the case of our baseball players) requires that we derive some "rule" which will permit us to compare the incomparable with a minimum 1088 and maximum sense. Many of the techniques to be described below had their primary justification for use in their ability to fulfill this goal.

In contrast with such measures as Drewnowski (1968) used, such as life expectancy and percent distributions of healthy versus sick people, our "measures" of subjectively perceived (or reported) satisfaction with various product attributes, products, and product classes cannot be assumed to have the same statistical properties or measurement status. As an example, take the following five-point scale:

Very Satisfied

Somewhat Satisfied

Neither Satisfied nor Dissatisfied

Somewhat Dissatisfied

Very Dissatisfied

What numbers shall we assign to the above various verbal descriptions? Some would say that the integers 1 to 5 would be appropriate, while others with equal justification might wish to assign the following number sequence: 1, 3, 4, 5, and 7. Each of these assignments entails assumptions about the relationship between language and a number system. Each refers to some hypothesis regarding the regression system which relates these values among themselves and to other variables (which takes place in the context of establishing functional or statistical relationships). The 1-5 assignment assumes equal distances between adjacent points, while the second correspondence assumes equal distances for only part of the scale (the middle) and insists that the end points are twice as far from their immediate neighbors as are points in the middle of the range. Both, in this instance, assume a symmetric scale. Other assignments are possible and need not be restricted to linear scales, e.g., 1, 2, 5, 2.5, and .5, for the case of a nonsymmetric and curvilinear scale. In any event, every such assignment carries implications regarding regression estimates (Guttman, 1971) whether we are aware of them or not. If we do not presuppose specified properties of the scale (e.g., the exact spacing between points), then various techniques become applicable for deriving this spacing. We are then involved in the process of scaling, which is the subject matter to be discussed under the topic of scoring methods.

1. Dimensional Simplification

A number of techniques are available for studying a set of interrelationships and their patterning. These may be broadly classified under the headings of metric and nonmetric methods, and each can be further subdivided into dimensional and adimensional or typological (clustering) procedures. The chief difference between the class of metric and the class of nonmetric methods lies in the level or quality of the data collected, or, better yet, it rests on the level which we assume to exist, since the numbers themselves are "dumb" when it comes to communicating their origin or use. If we use metric procedures we assume that the difference between, say, a correlation of .4 and .5 is equal to that between .8 and .9, and we judge the goodness of fit for the solution obtained in respect to the degree to which such equalities are preserved. A nonmetric approach, on the other hand, is based on the assumption that only the order of these relationships is meaningful. It would consider any solution which would be order preserving as being satisfactory. For example, .2, .25, 1.3, and 2, corresponding to the above four correlations seriatim.

The preceding distinction can be characterized in terms of the concept of a loss function, which is a measure of the errors we are willing to tolerate for whatever purposes we have in mind in analyzing the data in the first instance. Implicit in many metric techniques is the minimization of the 1088 function:

whereas, the comparable nonmetric loss function would be:

where R* is a monotone transformation of R (the input correlations), e.g., R permuted into the order of R, where R is the matrix of reproduced coefficients based on m dimensions.

It might be obvious to many that loss functions play a critical role in constructing measures of central tendency in statistics, or that they are implicit in everyday situations where some loss might be incurred as a result of making the wrong guess (e.g., in gambling). But most are unaware of their implications in the field of multivariate analysis. This may be true partly because the errors involved do not have clearcut costs associated with them. A gambler knows that if he guesses more often wrongly than correctly, he will be out-of pocket. The researcher, by insisting on an inappropriate loss functions, generally pays in the coin of complexity. Our preference for nonmetric over metric procedures is based on the fact that the former greatly simplifies our interpretation of patterns. It results in fewer dimensions, in general, and the assumptions for its use are more consonant with the types of data collected by social scientists.

Both dimensional and adimensional approaches are useful under given conditions. If the pattern of coefficients is such as to result in disjoint subsets, then the adimensional or typological approach is appropriate, otherwise, the dimensional approach is in order. Without analyzing the data, however, one cannot usually say in advance whether one's approach should be dimensional or adimensional. Clustering procedures used in tandem with dimensional analyses are often most productive, as long as some decision rule is present for rejecting one or the other hypotheses. The typical clustering procedure, like item analysis, always permits one to admit the hypothesis. There are clusters or items for constructing a scale. This state of affairs is, of course, undesirable, and we offer one possible solution to correct it. Dimensional analysis presupposes a continuous space in which points or vectors may lie and, as such, can be given a geometric representation. Adimensional analysis, on the other hand, is based on the assumption of discontinuity; it has its natural representation in terms of unordered classes. A third possibility, lying between these extremes, are metric spaces, which in fact (but not in principle), have dense regions of points scattered throughout it (like the heavens) and, as such, may be fruitfully viewed as consisting of ordered clusters.

The following three nonmetric techniques have been found useful in studying data structures derived from satisfaction ratings, from the point of view of both interpretation and as a desirable preliminary to constructing homogeneous indices: a) monotone distance analysis (Guttman, 1968; Roskam & Lingoes, 1970; Lingoes & Roskam, 1971); b) monotone vector analysis (Lingoes, 1966; Lingoes & Guttman, 1967); and, c) probability evaluated partition analysis - a clustering procedure (Lingoes & Cooper, 1971). We shall briefly discuss each of these methods in turn.

a) Monotone Distance Analysis

Let us start with a given arbitrary symmetric matrix of coefficients, all of which are assumed to be comparable; this would be the case, for example, with correlation coefficients. The goal of monotone distance analysis is to determine the smallest space in which the points lie, such that the distances among the points are isotonic (order isomorphic) with the order existing among the coefficients. Short of a perfect order-preserving, monotonic transformation of the input values, we seek the best possible fit for a stated number of dimensions, by minimizing the loss function:

where D is the generalized Euclidean distance in m dimensions and d* is a monotone transformation of those distances in terms of the order present in the input matrix R, for:

where X is a n (points) x m (dimensions) rectangular coordinate set of real numbers. If we are dealing with similarity data, such as correlations, then small distances correspond to large correlations, e.g., dij<dkl whenever rij>rkl (for all i, j. k, and 1; i = j and k=1). On the other hand, if our coefficients measure dissimilarity, then dij<dkl whenever rij<rkl. Our geometric representation reflects the order information in R and we interpret the space in terms of proximity. Points which lie close together in the space have something in common. On the basis of what we know about the variables and our hypotheses, we attempt to determine the law of formation giving rise to the particular observed configuration (Green and Carmone, 1970). A short example will illustrate some of the principles.

Suppose we had the following correlation matrix for five variables:

X

.02 X

.00 .02 X

-.02 .00 .02 X

-.04-.02 .00 .02 X

Can we find a nonmetric solution in terms of a set of distances which will strictly preserve the above order? According to the Guttman-Lingoes theorem (Lingoes, 1971), n-2 dimensions will always satisfy our nonmetric requirement and such solutions are neither informative nor unique. Our question should rather be, can we find an order-preserving solution in fewer than n-2 dimensions? The answer, of course, is "yes", since the matrix was constructed to yield a one-dimensional solution. This implies that the coordinates: (0,1,2,3,4) result in the distance matrix

X

1 X

2 1 X

3 2 1 X

4 3 2 1 X

which is inversely monotonic with the above correlation matrix.

Serving as a contrast is the metric solution for the above correlations, i.e., (-.2, -.1, .0, .1, .2), for dimension I and (.0, .1, .2, .1, .0), for dimension II, reproducing the exact values in R. In general, a nonmetric model will require fewer (usually much fewer) dimensions to reflect the pattern in R than will a metric model; thus, the process of interpretation is facilitated. It will be noted in this artificial example that the nonmetric solution also preserves equality for intervals. But such is not the general rule, since any monotone transformation (and not merely linear as in the present example) is equally good from the nonmetric point of view. For less than perfect fits, however, the particular monotone transformation which reduces dimensionality tends to be unique. As the number of points increase relative to the number of dimensions in which a solution is sought, the nonmetric order constraints act as if they were metric in nature.

b) Monotone Vector Analysis

In contrast to the distance model described, we can define a set of scalar products:

which will minimize logs function (2) above. Rather than determining distances which are monotone with R, we are finding vectors in the space so positioned that the product of their lengths and the cosine of the angle subtending them will yield a get of values monotonic with the input values (which are generally correlations or covariances). If we had employed (5) with (1), of course, we would be seeking a metric solution, i.e., we would be doing a factor analysis. Since most researchers have some familiarity with factoring procedures, monotone vector analysis serves as a relatively painless introduction to the area of nonmetric techniques.

Didactic motivations apart, however, there are some matrices for which a vector solution is more parsimonious than one based on the distance model. Spearman's unit rank hierarchy, for example, yields one dimension when decomposed by either the metric or nonmetric vector models, whereas n-2 dimensions are required with the distance model. There are other matrices for which the opposite holds and some yield equally parsimonious solutions for the vector and distance models. In no instance, however, would a metric solution be more Parsimonious than an appropriately applied nonmetric procedure. The differences in dimensionality obtained by different models is in a large part due to communality issues. This is a topic which we can but mention rather than discourse upon here (Guttman, 1967). We can, however, put forth some guidelines for selection of models.

In general, if one has some particular composition model in mind then the method of analysis should be in agreement with that model. For example, similarity/dissimilarity ratings are consonant with the concept of closeness or proximity. Therefore, a monotone distance analysis is appropriate to such indices. Correlational data, on the other hand, quite naturally evoke the concept of covariation; as a consequence, a monotone vector analysis is the method of choice (Coombs, 1964; Roskam, 1968). In the absence of a composition model or theory about the data collected, one is free to select that method of analysis which most parsimoniously and clearly reflects those features of the data (e.g., patterning) which are to be interpreted. The foregoing would appear to some as a cavalierly stated dictum; it might require one to perform several analyses before attaining a satisfactory representation. If the method leads to some new insights, this may be justification enough for one's pragmatism.

There are some more specific considerations, however, in model selection. Whenever there are more than three dimensions necessary to obtain a reasonable fit when using the distance model, the vector model may offer more purchase or interpretative leverage on the solution; it also provides a means for arriving at scores for individuals. Other factors may affect dimensionality. For example, choice of coefficient or non-monotone transformations of the coefficient, such as using r rather than r, or even reflecting negative r's to make them positive when applying the distance model, fall among these. But we shall not dwell on these issues here. Suffice it to say that it is important not only to match method and mode; it is equally important to understand, prior to their analysis, what the coefficients mean either in terms of proximity or covariation. If one does not take such considerations into account, one's comprehension of the configuration or geometric representation will be limited (Lingoes, 1970).

As an aid in comprehending spaces of large dimensionality or as a method for analyzing data assumed to be basically discontinuous or "clumpy", we shall now briefly outline another nonmetric technique.

c) Probability Evaluated Partitions

Let us start with a matrix of either proximity or concomitance (covarying) data, all of whose values are deemed to be comparable. The aim of probability evaluated partition analysis (or PEP) is to determine the minimum number of disjoint sets which are consistent with the hypothesis that their emergence or existence would not be reasonably expected if the space in which they could be embedded were indeed continuous. That is to say, we are proposing the null hypothesis that the space is continuous or that the points are non-clusterable, against the alternative hypothesis that the space is discontinuous or that the points are clusterable. Implicit in this formulation is the concept of statistical inference, where it is assumed that the probability is 1/2 that any given coefficient is above or below some specified cut-off value. The essential flavor of the procedure can be rather quickly captured by providing an example for analysis. Suppose we had the following matrix of coefficients and wished to test the hypothesis of continuity:

Our first step in decomposing the above matrix is to assign rank-order values to the above coefficients (from 1 to (n2), where 1 is assigned to the largest value if we are dealing with dissimilarity data or to the smallest value for similarities). Considering the relationships among these nine points to be similarities results in the following matrix of rank-order values:

, where fractional values denote tied ranks.

Our next step involves assigning zeroes to our rank matrix in terms of the above rank order until that point is reached where the matrix becomes disjoint. This implies that there is a partitioning into two sets, such that all members in one set are related to at least one other member in that set and no members from one set are related to members of the other set. In the above example, we find that 25 zeroes must be assigned before we can get the desired separation, resulting in the partition consisting of points (1,2,3,4,5) and another having members (6,7,8,9), which can be displayed graphically in terms of the original coefficients as:

It will be noted that the largest coefficient between any member of set A and set B is 18 (corresponding to the rank-order value of 25), the element in row 7 and column 2 of the above matrices, respectively. All links in A and B are greater than 18, (those points having links < 18 are not graphed). Partition B can be differentiated from A in terms of the number of connections or links relative to the maximum number of connections for an order n graph, i.e., the coefficient of tightness for A is 5/10 or .5, while for B this coefficient is 6/6 or 1.0. No attempt was made in the graphs shown above to make distance proportional to the input coefficients, although this might be Possible in this instance.

The only novel feature in the algorithm to this point is the use of the ordinal rather than metric information for creating the division. The final, but crucial, step involves attaching some probability value to the partition, which would allow us to either terminate or continue the process of partitioning. Based on the Cooper-Lingoes decomposition theorem for computing probabilities of labeled disconnected graphs, we find p - .322 for 25 or fewer deletions (zeroes) resulting in a disconnected graph. If we had chosen an alpha level of .05 for testing the null hypothesis, we would have to accept the hypothesis of continuity, i.e., this matrix is non-clusterable. If p were <a , we would have a basis for subpartitioning each of the partitions obtained until no further partitions resulted having acceptable p's. The clusters obtained from PEP are invariant up to a monotonic transformation of the input values.

The preceding three methods were used for understanding the observed relationships among satisfaction scores and as a preliminary for deciding which variables to use in constructing our scales. We employed them since the scaling procedures are sensitive to lack of homogeneity among the items scaled and the process of aggregation over variables introduces ambiguity of interpretation whenever heterogeneity exists. [All methods in the Guttman-Lingoes series have been programmed for computers. For a description of these methods see Lingoes.]

We shall now turn our attention to constructing scales and indices based upon relatively homogeneous subsets of items.

2. Scaling

We shall be speaking of two different ways of arriving at scores for individuals: a) raw scores (RS) on satisfaction ratings, which assumes that the 7 positions on the scale are equally spaced and are orderable in terms of most to least; and, b) a second system for obtaining scores which are optimal, subject to the explicit constraint that they also be monotonic with RS, was used which we shall call optimal monotonic scores (OMS). Such a scaling stretches the intervals between scores, but maintains their order, so that either the average correlation among the set of variables analyzed will be maximal under the monotonicity restriction (Lingoes, 1972) or a given variable is best predicted by a set of independent variables (Lingoes, 1972, In Press). By scaling the data in this manner we are in effect saying that we will not assume, out of hand, that the difference between a 1 and 2 rating (one unit) is the same as the difference between a 5 and 6 in rated satisfaction.

Rescaling the RS into OMS results in a metric which will be optimal from the point of view that no other scoring system will yield larger product moment coefficients of correlation among the variables so scaled or will yield a higher multiple correlation (subject to the monotonicity constraint). The main implication of this scaling is that if we wish to aggregate across variables or individuals, then we should attempt to maximize homogeneity or predictability so that the resulting scores are as unambiguous as Possible.

The degree of homogeneity/predictability achieved in the OMS scales is a function of sample, sample size, the number of categories, the number of individuals falling in each of the categories (the frequency distribution), and the set of variables included in any particular OMS scaling. Some of these parameters can be controlled and have an effect on homogeneity or predictability.

These two scoring systems (RS & OMS) were studied in terms of both correlations and predictions to see to what extent which of them might be better for achieving a reliable and valid Index of Consumer Satisfaction. We shall discuss these analyses in the sections to follow in the context of the aggregation problem. First, we shall address the procedures followed in constructing optimal monotonic scores.

a) Optimal Monotonic Scores (Maximizing Homogeneity)

The present method (CM-III) for obtaining OMS is best used when the items to be scaled are dimensionally simplified, i.e., when they approach unidimensionality. This again underlines the importance of our preliminary study of data structures.

The essence of the OMS CM-III Procedure is to perform differential stretchings of the RS scale intervals, such that the product moment correlation coefficient will adequately reflect the interdependencies in one's data. The algorithm consists of two basic steps: 1) determining the least-squares predictors over a set of variables for all individuals, and, 2) permuting these predictors into the order required for each of the n variables, i.e., performing a monotone transformation. Thus, we secure a single scoring system differing only in the order in which these scores appear for each variable (a necessary condition for maximizing a set of intercorrelations). When our loss function is minimized we will have determined the best monotone transformation of our scores. Our new scales will be invariant up to a linear transformation. We obtain equal interval scales for each of the items in the set. If there are mild nonlinearities in the n variable regression system, CM-III will "iron them out", as it were. The method will tend to yield linear regressions for every pair of variables (a necessary and sufficient condition for maximizing intercorrelations). CM-III is a direct solution to the relational simplex problem posed by Cattell (1962). If one can attain a perfect fit for one's data, then all variables will be perfectly correlated. Short of a perfect fit, however, the average intercorrelation will be maximized.

The following simple two variable problem carried through only one iteration will help the reader grasp the essentials of the CM-III OMS procedure. Consider the ten paired RS for variables x and y:

If we wished to best predict a given individual's score over a set of variables, the mean will be the best predictor (it minimizes the sum of the squared deviations). To obtain the monotone transformations of x and y, i.e., x* and y*, respectively, we merely permute the mean vector into the order of x and y, thus:

and, then, to ensure strict monotonicity, we tie the above transformed values. wherever ties exist in x and y, e.g., Ss 2 & 3 have tied x values, so:

As a result of just one iteration we have increased the correlation between these two variables from .845 to .915, representing about a 12.5% increase in predictability (x from y or y from x). Further iterations would, of course, improve upon this increment. The following graph shows how the above procedure performed differential stretchings on the x and y intervals:

It should be apparent from the above graph that variable x underwent more distortion than did y in achieving the increment in linear correlation.

The above technique for homogenizing regression systems can be carried out for two or more variables, which are essentially unidimensional or which cover a unified domain of behavior or attitude. We shall now demonstrate the CM-III scaling procedure on some real data, i.e., on satisfaction ratings obtained from 342 women in an initial survey we carried out under sponsorship of the U. S. Department of Agriculture. Our variables in this example are five satisfaction scales for preselected attributes, i.e., packaging, taste, nutritional value, availability, and price for the product of luncheon meats.

In table 1 below we give the CM-III OMS values corresponding to the original RS assignments 1 to 7 (after we have added in the respective means for the five attributes based on RS, since the CM-III Procedure yields values whose mean is zero and whose variance is unity). We also list the delta (increment) for the six intervals. so that the differential stretchings will be apparent.

CM-III OMS VALUES FOR LUNCHEON MEATS ATTRIBUTES

Apart from the substantive findings (e.g., these women were most dissatisfied with the price of luncheon meats and most satisfied with their availability), we can see that each attribute has been rescaled by CM-III such that the intervals appear quite different from those present in the RS 1-7 scale. As a result of the above monotonic transformation we increased the average correlation among these five attributes from .427 to .442 (an increment of 1.2% in variance), which suggests, at least in this instance, that for all practical purposes we had a linear regression system and could have just as well have used the raw scores, themselves. In other instances, of course, we might achieve greater gains in homogeneity if nonlinearities were present.

In Table 2 below we display the product moment correlations for both before (RS) and after (OMS) scaling to afford the reader some insight into which variable/s was/were affected most by altering scale intervals.

CORRELATIONS AMONG RS (ABOVE) AND OMS (BELOW)

The average correlation coefficient was increased most for item 4 (Availability) and least for item 5 (Price), which suggests a mild departure in linearity of regression for the attribute of availability. The increase in homogeneity would not be deemed sufficient, however, to warrant the application of CM-III for these five attributes, although it does no harm to use the OMS in place of the RS despite the insignificant increment.

b) Optimal MonotOnic Scores (Maximizing Predictability)

In contrast with our previous procedure for maximizing the interrelations among a set of variables (the homogeneity issue, whose primary application is in the area of aggregation of indices), our present concern deals with the problem of prediction. These two problems do not necessarily have the same solution, a matter which we shall address after presenting the CM-V method for obtaining OMS.

Given a set of n fixed independent variables, what is the best linear combination of those variables which will predict the responses to a dependent variable? This is, of course, the standard problem of linear regression analysis, i.e., we have the following linear model (when operating on standardized variables):

where the x's are our standardized set of independent variables, the 6's are the normalized be-ta weights, and the y's are the predicted scores (i=1,2,... ,N observations, j=1,2,...,n independent variables).

Suppose, however, that the dependent variable, y, had undergone some unknown monotonic transformation. Under this condition we would obtain something less than monotonic transformation. Under this condition we would obtain something less than the best in terms of predictability. That is, if we could somehow recover that monotonic transformation, apply its inverse to make the regression system linear, then we would increase predictability. This, in essence, is the goal of CM-V OMS. We achieve this aim by iteratively computing: 1) the multiple correlation between the fixed independent variables and our current estimate of y*, a monotone transformation of the dependent variable, and, 2) permuting the predicted scores into the order of y. When the correlation between predicted and monotone scores is a maximum, the process comes to a halt and we conclude by giving the prediction equation as well as the scale values for y*.

In applying CM-V to data it is not necessary (and, indeed, may be contraindicated) that the set of x's be homogeneous or unidimensional (although they may well be from a unified behavior domain relevant to the predictor). All that is required is that independent portions of the criteria be sampled by each of the x's, for otherwise all of the predictability in the system will be carried by a subset of the x's (as in a simplex structure, for example). A computationally simple example does not readily come to mind, so we shall illustrate CM-V by using our previous data on the five attribute satisfaction scales (OMS from CM-III) to predict rated satisfaction for luncheon meats. To the extent we are successful in predicting luncheon meat satisfaction, then one's satisfaction with luncheon meats can be considered to be a linear combination of one's satisfaction with the attributes of luncheon meats. If the multiple correlation is sufficiently high, then we have evidence that the appropriate attributes had been selected. On the other hand, if R (the multiple correlation) is too low, then either further attributes are needed to increase predictability or we have the wrong model.

We list the zero order correlations between the five OMS attributes and the RS ratings on luncheon meats satisfaction seriatim: .39 (packaging), .77- (taste), .64 (nutritional value), .36 (availability), and .47 (price). The corresponding beta weights are: .03, .60, .13, -.00, and .13, yielding R - .785. After a few CM-V iterationS we achieved an R of .789, representing a modest increase of less than one per cent (.6%) in predictive variance. The zero order correlations between the five OMS attributes and the monotone transformation of the luncheon meats satisfaction scale (with their beta weights) are: .41 (.06), .77 (.60), .65 (.13), .38 (.02), and .47 (.11). Here again we find that the system is essentially linear and no substantial benefit accrues from scaling. Indeed, practically all of the predictive variance is covered by but the single attribute of taste (r - .77), which strongly suggests that we may well have a simplex structure (Guttman, 1954) in this instance.

One implication of the simplex model in the present case is that if we wish to improve our predictability of luncheon meats satisfaction, we should seek an attribute like taste, which is correlated with taste but has a higher correlation with luncheon meats satisfaction. We will not dwell further on this issue, other than to point out that our exercise was not completely without merit, since it served to illustrate our methods and was suggestive of a more fruitful explanatory model. For other products, however, a more complex model has been found necessary. We shall conclude this section by giving the OMS for luncheon meats derived from CM-V (after adding in the mean of 2.98) and the scale interval increments, i.e.:

III. CONCLUSION

1. Two Models for constructing Indices of Consumer Satisfaction

We shall now formalize our two approaches for constructing an index of consumer satisfaction (ICS) based on our discussion of the two methods (CM-III and CM-V) used for obtaining OMS. Although we shall confine our discussion to products and their attributes, the rationale is simply generalized to product classes and global satisfaction ratings.

The following notation will be helpful in discussing our two models:

Let

(7) A = (aijk) = the matrix of raw satisfaction ratings (RS) for the attributes of products (i=1,2,...,N observations; j=1,2,...,n attributes; k=1,2,...,m products);

(8) A* = f(A) = a monotone transformation of the raw satisfaction scores for attributes, i.e., the OMS (via CM-III for each product separately);

(9) p = (pik) = the matrix of RS for products (i=1,2,...,N; k=1,2,.. .,m); and,

(10) P* = f(P) = a monotone transformation of product RS (via CM-V for each product, the dependent variable,and its associated scaled attributes, the independent variables, separately).

Now if, as a result of our analysis, we found that a particular product's satisfaction rating was predictable from one or at most two attributes (as we did find in respect to luncheon meats), we would include the product and its attributes in a CM-III analysis to yield the following measure of satisfaction for person i on product k:

where P* now represents the OMS(via CM-III). The reader will easily recognize Eq. (11) as the simple aggregation model, whose rationale stems from a simplex structure.

If, on the other hand, we found that the structure was not that of a simplex, i.e., the network of interrelationships among the attributes was complex, then the appropriate model for obtaining subject i's satisfaction score on product k would be:

or the linear-monotone regression model of CM-V, where s*ik represents the OMS and eik is the residual or error score.

In the first or aggregation model we will be in error to the extent that the set of attributes and product departs from a simplex structure. In the second or linear-monotone regression model we will incur error to the degree that the residual score comPonent is large (i.e., R is low). In either event, we shall have achieved the best possible scores based on the data we have from the point of view of either homogeneity or predictability, respectively.

We defer a discussion of the details for constructing an ICS, since it simply involves an expansion of the above treatment for product classes and global ratings, summing over individuals (or subsets of individuals for group comparisons), and, possibly, weighting the normed index for a base period to take into account the economic importance of the various products.

In this paper we surveyed in general terms the contributions that traditional measures of economic performance can make to the measurement of economic and market performance. We noted, however, that these measures should be extended to take into account many phenomena which can be expressed in monetary and physical terms. Beyond that, social indicators are called for which monitor the quality of life and of social existence, including subjective welfare and satisfaction.

We discussed the methodological problems and opportunities arising from nonmetric scaling techniques, with special reference to our work on Indices of Consumer Satisfaction and Indices of Citizen Satisfaction. Dimensional simplification via monotone distance analysis, monotone vector analysis, and probability evaluated partitions techniques were described in general terms. Therefrom we went on to chart our approaches to scaling and aggregation of responses, including the uses of optimal monotone scores.

The methodology outlined in the latter part of this paper suggests a useful approach to supplementing objective indices. It offers a handle on the difficult problem of representing goods and services in terms perceived by the consumer and citizen. It takes these as inputs into a procedure for aggregating subjective satisfaction scores. These, in turn,, lead to profiles of satisfaction or to an overall Index of Satisfaction.

Up to this point we have only investigated subjective welfare and satisfaction at one instant of time in respect to a very select set of items and for a small sample of individuals only. It would be important, both theoretically and practically, to extend our results for generality over time, items, and samples.

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----------------------------------------

##### Authors

James C. Lingoes, University of Michigan [Associate Professor of Psychology, University of Michigan.]

Martin Pfaff, Wayne State University ant University of Augsburg [Professor of Economics and Operations Research, Wayne State University and Professor of Economics. University of Augsburg.]

##### Volume

SV - Proceedings of the Third Annual Conference of the Association for Consumer Research | 1972

##### Share Proceeding

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