The Relationship Between Cognitive Models of Choice and Non-Metric Multidimensional Scaling


Flemming Hansen and Thomas Bolland (1971) ,"The Relationship Between Cognitive Models of Choice and Non-Metric Multidimensional Scaling", in SV - Proceedings of the Second Annual Conference of the Association for Consumer Research, eds. David M. Gardner, College Park, MD : Association for Consumer Research, Pages: 376-388.

Proceedings of the Second Annual Conference of the Association for Consumer Research, 1971     Pages 376-388


Flemming Hansen, University of New Hampshire [Now Marketing Director, T. Bak-Jensen A/S, Copenhagen, Denmark]

Thomas Bolland, University of New Hampshire

[The research to be reported here has been supported by a CURF Research. Grant from the University of New Hampshire, Central University Research Funds.]


Belief-value-type models of consumer behavior have been proposed by several authors (Bither and Miller 1969, Bass and Talarzyk 1969, Sheth 1969, and Hansen 1967 and 1969). Basically these models assume that the consumers choice among a number of alternatives can be predicted from the attractiveness of the alternatives where the attractiveness of alternative "i" is defined as


and where "Vj" is the "jth" of "r" values (or choice criteria) salient in the choice process and where "Bji" is the strength of the belief that alternative "i" is instrumental to the "jth" value. A critical problem with these models relates to the identification of the belief-value dimensions based upon which consumer choices should be predicted. (Vj's)

Partly, it is difficult to define variables which are not interrelated; partly, the researcher can never be certain, not even when a large number of belief-value dimensions are included, that the most important ones are among those he has picked.

Another problem relates to the number of variables which are needed in order to make satisfactory predictions. Often it has been reported that even though a large number of dimensions have been included in the study, only a few of these account for the majority of the predictive power of the model (Sheth 1969). This suggests that the identification of the few most important variables is paramount to the functioning of the model.

To summarize the present state of the development of these Fishbein (1967) - Rosenberg (1957) oriented consumer choice models, one may say that a highly valuable conceptual framework is emerging, but important estimation problems are still unsolved.

In the area of non-metric multidimensional scaling the situation is almost opposite. Simultaneously with the arrival of the belief-value choice model on the marketing scene, other researchers (for a review see Green and Carmone 1970) have directed their attention to this family of relatively new "descriptive" procedures.

These models have the advantage of requiring relatively weak input data (similarity and/or preference orderings) from which perceptual dimensions are developed.

Regardless of the type of input data, the aim of this kind of scaling analysis is the construction of an n-dimensional mapping of the alternatives for which data have been obtained. The nature of this representation is such that the rank order of distances between the alternatives in the n-dimensional space reproduce as closely as possible the rank order of the similarities data originally measured. Of course, the larger the number of dimensions which one allows in the solution, the better the computed distance rankings correspond with the original similarities rankings.

The goal of these procedures then, in brief, can be said to be:

1. to tell how many dimensions are required in order to obtain a representation of a predetermined accuracy.

2. to provide a plot of the alternatives in the revealed space.

Commonly, it has been found that consumers' perceptions of brands, products and the like can be described in terms of relatively few perceptual dimensions. Moreover, it is argued that the perception along these dimensions is critical to the choices which the consumer makes and so commonly the dimensions are interpreted as value - like variables.

So far the two lines of research referred to above have developed relatively independently of each other. It seems natural to suggest, however, that the perceptual dimensions revealed in multidimensional scaling somehow correspond to the belief-value dimensions critical to the cognitive choice model. It has been the purpose of the research reported here to examine to what an extent such a relationship exists. This is done by testing the following fundamental hypothesis:

"The perceptual dimensions revealed by non-metric multidimensional scaling are the same as those belief-value dimensions which account for a majority of the correct choice predictions which can be made with the cognitive choice model".


The procedure followed is very straightforward. Data in a form suggested by each of the two models and/or procedures were collected from the same respondents on the same topics. This data could then be analyzed by means of both multidimensional non-metric scaling techniques and via the procedure used in the traditional Rosenberg (1957) type of attitude and choice prediction. ( See,for example, Hansen 1969.)

Subjects for the study were students at the University of New Hampshire and residents of Durham, New Hampshire. Two sets of test objects or stimuli were used. One set consisted of alternative car-wash facilities, the other of several rathskellers (student hangouts selling beer).

In what we shall call the "Beer-Hall" study the questionnaire was centered around three types of information. First the respondents were faced with 11 value dimensions supposedly relevant to choices among such restaurants. The dimensions used were, for example, such things as "service", "quality of food", "prices", etc. Each respondent was asked to rate, on ten point scales, the importance of these factors in the choice of a restaurant. Following this the three restaurants included in the study were evaluated along the same eleven dimensions. These evaluations constituted the measures of the independent variables in the belief-value model.

Secondly, direct similarity data were obtained on the three alternatives. Since only three alternatives were studied similarity data could easily be obtained by comparisons of all pairs of pairs of stimuli. That is, questions of the following type were asked: "Do you think restaurants A and B are more alike that restaurants B and C?"

And, thirdly, information was obtained concerning respondents' actual choice behavior and their preferences. Here it was asked what restaurant was visited most frequently, which one was most liked, which one was visited last, etc.

To obtain as homogenous a sample as possible respondents in this "Beer-Hall" study were restricted to male junior and senior students, above 21 years of age and living in dormitories. Originally, a sample of 100 students had been selected, but the interviewing which had begun just prior to the students strike in the spring of 1970 had to be stopped before all interviews had been completed. At the time when the students left campus 68 students had been approached from whom 43 complete interviews were obtained. Of these, 3 had to be discarded because of missing information, leaving 40 respondents.

In the "Car Wash" study the alternative stimuli were "coin-operated self-service", "coin-operated automatic", "at home", '|garage", and "charity". The sample was composed of 100 randomly selected car owners registered with the service department of the university. Again here, the student strike made it impossible to complete the interviewing and so in the final analyses only 37 complete interviews were included. In this study value importance and instrumentality of each alternative were measured as in the "Beer-Hall" study, this time along 9 selected dimensions such as convenience", "quality of job", price", etc.

For the measurement of similarity data the anchor point method was used (Taylor 1969). The data obtained were then converted to individual rank order similarity matrices using the TRICON I program (Carmone, 1968). And, finally, behavioral and preference data were obtained from each respondent.


Prediction of Choice

Partly as a check on the applicability of the cognitive choice model to the type of products studies here, and partly to identify those belief-value dimensions which are critical to the respondents, predictions of actual behavior were carried out based upon the data which were collected.

Such predictions can be based upon the attractiveness scores computed in accordance with (I), or they can be based upon the belief (instrumentality) data alone:


Both computations were carried out, and the results are shown in Table 1. In all cases an overall score is computed for all alternatives and a correct prediction is considered to be one where the alternative with the highest score is the one most frequently used.



In the Beer-Hall study (with 3 alternatives) 80-85% of the predictions were correct. In the Car Wash study (with 5 alternatives) approximately 60-70% of the predictions were correct.

To identify which of the 11 dimensions in the Beer-Hall study were of major importance for the prediction of the choices, a regression analysis was carried out. The overall preference ratings for the alternatives (which correspond closely with the behavioral choices) were correlated with the belief scores. Based upon the B-coefficients obtained from this analysis, it can be concluded that two closely related socially-oriented variables, "social" and "kind of people", accounted for a majority of the correct predictions.



In the Car Wash study the overall preference ratings measured on a 15-point scale for the alternatives were also correlated with the evaluations of the alternatives along the nine dimensions used. The results of these analyses are shown in Table 2.

In the same table results are also presented from a regression analysis where the dependent variable was the difference in overall preference ratings for the two alternatives and the independent variables were the corresponding differences in belief scores. For both regressions the estimated, B-coefficients, t-ratios and simple correlations of dependent and independent variables are presented. In each column the three highest ranking values (in absolute values) are underlined. It appears that the only two variables consistently showing up as important are "quality of appearance" and "effort required".

It should be mentioned, however, that other variables in the analysis may have some importance also, and that the two variables mentioned are slightly inter-correlated. In the two analyses their correlation coefficients are -.213 and -.325.

It is the purpose in the remaining part of this section to examine to what an extent these critical variables, for the choice predictions, resemble the perceptual dimensions revealed by non-metric multidimensional scaling.

The Relationship Between Perceptual and Belief-value Dimensions

The choice predictions were based upon respondents' own rating of the importance of 11 (Beer-Hall) and 9 (Car Wash) choice values together with their rating of the instrumentality of the alternatives along the same dimensions. As mentioned previously direct similarity data were obtained for both the 3 Beer-Hall alternatives and for the 5 Car Wash alternatives. To examine how the perceptual dimensions revealed by non-metric scaling compare with the more important conceptual belief dimensions the following analyses were carried out.

In the Beer-Hall study the direct similarity data were fed into the TORSCA-9 program for non-metric multidimensional scaling. (Young and Torgerson, 1967; Young 1968). This analysis of this data revealed an unidimensional solution with a stress measure of 0.00. Considering the number of alternatives this is not surprising. More interesting, however is the fact that the ordering of the alternatives agrees closely with their ordering along the more important social image dimensions which were critical in the choice prediction.

In the Car Wash study the converted anchor-point similarity data were used as input for the Torsca program. The two dimensional solutions (stress = 0.0 to 0.01) for different subgroups of the sample are very similar and all of them look very much like the solution based upon all respondents which is shown in Figure 1.

In this representation, the location of the five alternatives can be read directly from the map, which yields easily to an interpretation along the lines suggested by the analysis of beliefs. Looking upon the location of the alternatives one sees that the "Effort Required Dimension" makes good sense ranking as it does from "washing the car at home" to "Coin operated automatic". Along the "quality of appearance" dimension the ordering is surprising to the authors. But it is in good agreement with the rank order based upon the beliefs of the respondents as shown in Table 3.

From Table 3 it appears, however, that the agreement is not perfect, even though some improvement can be obtained when the observation is taken into consideration that the two belief dimensions are slightly correlated, so that one should not expect orthogonal axes. However, even with this in mind the results are such that further analysis of the relationship between the two sets of data is warranted.






Comparisons at the Individual Level

Green (1970) makes a distinction between indirect and direct similarity data, the latter based upon respondents own judgments of similarities among alternatives. Indirect similarity data can be constructed in several ways, some of which utilize belief/value-importance data.

First it is possible to derive similarity measures from the instrumentality data. It is natural to assign a very high similarity to two alternatives which are rated almost identically. On the other hand, if the instrumentalities of two alternatives are very different the alternatives should be said to be very dissimilar and consequently they should be assigned a low similarity rating. One way of representing this is by calculating the differences in rating for two alternatives on each dimension and then summing these over all dimensions. When this is done a similarity matrix for each individual can be constructed. Using the notations of equation (I) this would mean that the distance between two alternatives is computed as: r


It is possible also to apply the same procedure to the weighted belief measures. That is, to the differences in beliefs ratings weighted with the respondents rating of the importance of the dimensions. This leads to the following expression:


Comparisons between the direct and the indirect similarity data can be carried out in different ways. The individual rank order similarity data based upon evaluations (beliefs), can be compared individually with the direct similarity data, they can be compared as aggregated matrices, and they can be used in a complete scaling analysis, the results of which are compared with the results of the same analysis carried out on the direct similarity data.

The problem with these comparisons is that they are very difficult to evaluate when carried out on the raw data, whereas one cannot be certain, when the comparison is made based upon some manipulated version of the raw data, whether the differences which are revealed result from differences in the raw data, or whether they to some extent are ascribable to the manipulations themselves. Here comparisons are tried first at the raw data level.

For each respondent in the Car Wash study anchor point matrices are constructed based upon the belief data. Then these matrices are compared with those obtained directly from the respondents. The nature of this comparison is illustrated in Table 4.



The overall difference between matrices of the type shown in Table 4 can be computed as:


(the figure "12" in the example is a measure of this difference). This difference is critical for the evaluation of the quality of anchor point matrices constructed from the belief value data. When this figure is low there is good agreement between the direct and the indirect data; when it is high the opposite is true. Actually for all subjects the average difference is 20.16 compared with an expected difference of 22.5 which would have resulted had the data in the reconstructed matrix been random numbers. This difference is small and only approaching significance (a X2 test gives p < 0.06) When the same analysis is carried out on the Beer-Hall data it results in x2 value of 15.74 with three degrees of freedom (p < 0.005). For the above analysis the belief image data were used alone. Green and Carmone (1969) report that the weights individuals assign to dimensions may be of critical importance when comparisons are made between analyses based upon preference data and upon direct similarity data. It may be that the belief data obtained in the present study in some way reflect individual preferences also.

If this is the case, and if the Green and Carmone (1969) conclusion is valid, weighting the belief data with the importance values should improve the indirect anchor point similarity data. This can be done by using formula (IV) for the computation of the distances between alternative car wash facilities. When that is done the agreement between direct and indirect similarity data improves slightly, but not significantly. This applies to the Car Wash data as well as to the Beer-Hall data.

Seemingly, in the Beer-Hall study the agreement between direct and indirect similarity data is good, whereas in the Car Wash study the results are more questionable. To examine the nature of these data further the following analyses were carried out.

Aggregated Comparisons of Car Wash Data

Even though the individual data do not compare very well, the aggregated data may still do so. Also a number of modified transformation rules might improve the results. To this end the indirect measures were used as above to construct similarity matrices. This was done using formula III as well as formula IV. A squared distance concept was also used. Specifically, corresponding to formula III, this concept calculates the distance between alternatives i and I as:


and corresponding to formula IV, this distance is calculated as:


Corresponding to each distance matrix for each individual a simple rank order matrix of these distances was constructed. This gives a total of eight different measures for each subject. To analyze these different indirect similarity data, correlations were computed between the derived distance matrices obtained by using these 8 different matrices, aggregated over individuals, as input to the TORSCA-9 program, and the similarly derived distance matrix based on the direct similarity (anchor point) data. The analyses were carried out for 3,2, and 1 dimensional solutions, and for each of these it was done on all subjects and on subgroups of subjects composed of individuals with different references as to most frequently used type of car wash facility. All these analyses turn out to give very similar results with regard to the relative quality of the different indirect similarity concepts. As an example the 2 dimensional case with all respondents included is presented here. In Table s the computed distances are reproduced and in Table 6 the correlations are presented. From the tables it can be seen that the 8 different methods of generating indirect similarity matrices all provide very similar results. The agreement with the direct similarity data is less perfect, but still quite good: All correlations are larger than .60, and the differences among them are not large. It is reasonable to conclude that the different modifications of the simple indirect distance concept (III) do not improve the results significantly. With regard to the analyses carried out on the different respondents, it turns out that considerably better agreement between direct and indirect similarity data is found among those respondents using other than the "home wash" alternative most frequently (r=.719), than among those washing their car at home most frequently (r=.628).





Finally comparisons can be made by visual inspection of the representations based upon direct similarity data and indirect similarity data. This is done in Fig. 2 for representations derived from data for all respondents using formula III. From the figure it appears that the two representations do agree somewhat (to the extent that with both procedures identical stimuli are placed in the same quadrants) but some discrepancies are also obvious.




Comparisons were made between the dimensions revealed by multidimensional nonmetric scaling, and those that were of of major importance in a belief-valuebased choice-prediction. In the two studies reported, the dimensions revealed by nonmetric multidimensional scaling do resemble those that are of major importance in the prediction of choice.

To explore the similarity between the two approaches further comparisons were made between the direct similarity data used in the nonmetric-scaling approach and the indirect similarity data which can be derived from the belief value-ratings. When this is done two conclusions emerge. First, the representations derived from non-metric multi-dimensional scaling procedure applied to indirect similarity data is not highly sensitive to the approach used for constructing the input data (at least not among 8 alternative approaches tried). Secondly, whereas the agreement between direct and indirect data is poor at an individual level, somewhat better agreement is found at an aggregated level.

This agreement, however, is far from perfect.


One major problem, in the comparisons between belief-value, image, or preference data on the one hand and direct similarity data on the other, relates to the meaning one attaches to the concept "important dimensions". It is perfectly possible that a certain dimension is extremely important to the respondents but, since the alternatives do not differ along this dimension some techniques may not reveal these dimensions at all. It is plausible that this applies to representations based upon direct similarity data; and the common observation that even very complex stimuli sets often result in satisfactory two or three dimensional solutions may find its rationale in this very fact. In the present approach the use made of the belief-value data is such that major emphasis is placed upon those dimensions along which major differences between the alternatives exist, but still the possibility cannot be ruled out that those dimensions that appear to be of major importance in the explanation of choice are not the same as those upon which respondents place major emphasis when comparing the alternatives

Another possible explanation for the observed discrepancy between the two sets of data may rest with the data collection procedures. Apart from the problems the student strike presented for the study reported here, the possibility exists that even though, conceptually the two approaches do deal with the same phenomena, the measurement techniques used in one of the two, or in both, areas are Imperfect. to such an extent that this inevitably will result in discrepancies.

In general three types of research would improve our understanding of the relationship between the kind of results presented by non-metric multidimensional scaling and the kind of insights into consumer behavior gained by direct studies of belief-value types of variables:

1. Exploration of the relationship between the two approaches in a wider range of product areas and among different segments of consumers

2. Further comparative studies of reliability, validity, and stability of the measurement techniques used in the two areas, and

3. Further insight into the extent to which differential stretching of the perceptual axes (Green and Carmone 1969) is capable of eliminating the discrepancies between representations derived from different types of data.

Hopefully future research will throw further light on these issues.


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Flemming Hansen, University of New Hampshire [Now Marketing Director, T. Bak-Jensen A/S, Copenhagen, Denmark]
Thomas Bolland, University of New Hampshire


SV - Proceedings of the Second Annual Conference of the Association for Consumer Research | 1971

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