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Faculty Working Papers 
 
 MULTIDIMENSIONAL SCALING OF BINARY 
 DATA FOR HOMOGENEOUS GROUPS 
 
 Robert Redlnger and Jagdish N. Sheth 
 
 # 32 3 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana-Champaign 
 
FACULTY WORKING PAPERS 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 
 August 16, 1976 
 
 MULTIDIMENSIONAL SCALING OF BINARY 
 DATA FOR HOMOGENEOUS GROUPS 
 
 Robert Redinger and Jagdish N. Sheth 
 
 9 325 
 
MULTIDIMENSIONAL SCALING OF BINARY DATA 
 FOR HOMOGENEOUS GROUPS 
 
 Robert Redinger and Jagdish N. Sheth 
 INTRODUCTION 
 
 Inspired by measurement in the hard sciences, the first developed 
 techniques in multidimensional scaling (c.f., 20) required the input 
 data to be metric. However, the necessity of using metric data as input 
 required strong assumptions about the underlying psychological processes 
 (9, 11). One method of scaling psychological data while relaxing the 
 assumptions of the input data and the concomitant cognitive processes 
 is to collect lower order data (ordinal), find a function to transform 
 this data into a metric representation, and then input this transformed 
 data into existing metric multidimensional scaling techniques. Shepard 
 (13, 14) discusses the problems attendant with this approach and as an 
 alternative presents a method of multidimensional scaling (refined by 
 Kruskal (7, 8) ) that requires only ordinal data as input, yet produces 
 scales with metric properties. 
 
 The major advantage of nonmetric versus metric multidimensional 
 scaling is a relaxation in the assumptions of the underlying psychological 
 processes an individual uses in making judgements. As Shepard (11) noted, 
 qualitative judgements can be made with greater ease, assurance, validity, 
 and reliability than can quantitative judgements. However, several problems 
 can be identified with these nonmetric multidimensional scaling techniques. 
 
 First, an assumption of metric techniques is that the respondent be 
 consistent throughout the task with respect to the criteria used and the 
 quantification of that criteria. Nonmetric techniques, while they do not 
 require quantification, retain the assumption of consistancy of criteria. 
 Shepard (12) found that similarity judgements are likely to be influenced 
 by attention fluctuations, and Torgerson (18) reported that the judgements 
 
- 2 - 
 
 nay be affected by contextual effects. 
 
 Second, although the nonmetric methods require only ordinal properties 
 in the data, t.ie assumptions of ordint.lity must be met. If the basic 
 ordinal properties (properties that are empirically testable) are exhibited 
 by the data, the researcher is justified in using geometric models for 
 scaling. Thus, the use of nonmetric techniques depends on the validity 
 of the underlying ordinal assumptions (1). The more difficult the task, 
 the more likely it is the underlying assumptions of the psychological 
 process and of consistancy will not be met. 
 
 Task difficulty can be resolved primarily as a function of the number 
 of stimuli and the requirements of the task. As the number of stimuli 
 increases, the difficulty of the task increases. The rank ordering of 
 similarities of all possible pairs (990) of forty-five stimuli is a more 
 difficult task than the rank ordering of all possible pairs (45) of ten 
 stimuli. Rao and Katz (10) state that methods of collecting similarities 
 data (magnitude estimation, ranking of all possible pairs, n-dimensional 
 rank ordering) for large stimulus sets are cumbersome and may render judgements 
 meaningless. 
 
 Further, different techniques require different types of data. The 
 less invariant the data is to be (metric vs. ordinal), the more restrictive 
 the assumptions of the underlying process, and hence, the task will be more 
 difficult. For example, the question "How much greater is A than B?", which 
 would yield interval data, is a more difficult task than that represented by 
 the question "Which is greater, A or B?", which would yield ordinal data. 
 
 The third problem associated with nonmetric techniques is that these 
 methods require assumptions on the part of the researcher as to the dimension- 
 ality of the underlying process and the metric to be used for calculating 
 distances and scaling stimuli. The calculations in these techniques are 
 
- 3 - 
 
 based on the minimization of some criterion of error. Hence, if the under- 
 lying model (i.e., dimensionality and metric) is inappropriate, the procedures 
 will calculate results capitalizing on the noise in the data, making interpre- 
 tation difficult and statistical inferences to populations cr across similar 
 experiments unlikely (1). 
 
 What is needed then are simpler data collection procedures to handle 
 the first two problems and simpler analytic procedures (at least in terms of 
 fewest assumptions) to handle the third problem. Due to the large number of 
 stimuli necessary for many marketing studies, attention has focused en providing 
 alternative methods of collecting ordinal (similarities) data, methods which 
 basically involve a reduction in the number of judgements the individual must 
 make (10, ). However, en alternative solution is to reduce the difficulty 
 of the task by further relaxing the assumptions underlying the psychological 
 process implicit in the data collection technique. Rather than collecting 
 ordinal data, the researcher can obtain nominal (classifactory) data or, 
 in the simplest case of two classes, binary data. Green, Wind, and Jain (5) 
 analysed associative data by assuming the association frequency represented a 
 proximity measure of the stimuli and utilized existing geometric scaling 
 models to arrive at configurations. They found the technique resulted in 
 high dimensionality which was difficult to interpret. They met the first 
 condition of simpler data but not the second condition of simpler analytic 
 strategy which suggests that an alternative method of analysis for associative 
 data may also be appropriate. 
 
 The remainder of the paper describes a method of scaling associative 
 (specifically binary) data which (1) requires as input only binary similarities 
 data thereby increasing the consistancy of the data while relaxing the assumptions 
 of the underlying cognitive process, and (2) does not require prior specifi- 
 cation of a geometric model (dimensionality and metric). After a discussion 
 
- 4 - 
 
 of the technique, the method is applied to the scaling of soft drinks and the 
 results compered with the results from a standard multidimensional scaling 
 method. Finally, the unresolved problems associeated with this technique 
 and the implications of the technique for marketing research are discussed. 
 
 DESCRIPTION OP THE MODEL 
 
 Binary data may be collected in a variety of ways, ultimately represented 
 as the assignment of the stimuli to one of two groups. Judgements can be made 
 regarding an object's possession of an attribute, or an object belonging to 
 a group. To collect binary similarities data respondents would judge whether 
 a pair of stimuli were similar or not similar. Accumulating judgements over 
 individuals, a frequency distribution of similarity of stimulus-pairs is 
 obtained. Guttman (6) noted that a multivariate frequency distribution is 
 scalable if one can derive from the distribution a quantitative variable 
 with which to characterize the objects in the population so that each attribute 
 is a simple function of that quantitative variable. Justified by the 
 arguement that factor analysis can be legitimately applied to any symmetric 
 table, Burt (3! describes a technique by which qualitative data can be 
 factor analyzed. Sheth (15) has adapted this technique for the analysis of 
 brand loyalty. 
 
 Suppose we wish to estimate the attribute space of n products and 
 
 then scale the products within that space relying on binary similarities 
 
 data for input. The similarity judgements are obtained by asking M individuals 
 
 whether a product-pair is similar (coded 1) or not similar (coded 0) for each 
 
 of the N - n(n-l)/2 product-pairs. The data can be represented in an M x N 
 
 matrix Y, where each cell, y. . , represents the judgement of similarity of 
 — i»k 
 
 product-pair k by individual i. 
 
- 5 - 
 
 product-pair (k) 
 1 2 . . . N 
 
 1 ... 1 
 
 1 1 ... 
 
 ... 1 
 
 1 
 
 2 
 
 3 
 
 M 
 
 individual 
 (i) 
 
 In estimating the relevant attribute space, a necessary assumption is 
 that all the individuals use the same space in making judgements. To test 
 this assumption, a points of view analysis (22) using Eckart and Young's 
 theorem of matrix approximation (4) is performed. An individual by individual 
 matrix, C, is calculated 
 
 - M X K 
 
 ifrlxN ■=• N x M 
 
 where each cell, c. ., represents the number of times individuals i and j 
 
 both rated a product-pair as similar. C turns out to be nothing more than a 
 
 square symmetric contingency table. These absolute joint frequencies are a 
 
 function of the number of product pairs rated. To eliminate this sample 
 
 size bias, the frequencies are standat Jized by computing the relative joint 
 
 frequencies, p . ■ c. ./N . Dividing these relative joint frequencies by 
 
 the standard deviation (p.p.) results in a set of proportionate values 
 
 1 J 
 
 : i,j - p ifj ' ( <vy 
 
 c . / (c.c.r 
 
 3-.J i J 
 
 This is equivalent to pre- and post- multiplying C, the contingency table, 
 by a diagonal matrix £ 2 with elements l/(c.) . Thus, we obtain a square 
 
 symmetric matrix R, which is positive, semi -definite; 
 
 1. 
 
 '2 
 
 -Js 
 
 ,-h 
 
 ,-h 
 
 R • D C D~* = D"' Y Y' D~' ■ MM' where M « D " ' Y 
 and being symmetric, R has gramnian properties (2, 17). This standardization 
 yields l's in the diagonal, hence R may be directly applied to principal 
 

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6 - 
 
 components analysis, resulting in each individual R being expressed as a 
 linear comination of factor scores, F. 
 
 R. ■ a. „F„ + a. _ F_ + . . . + a. F 
 j j,l 1 j, 2 2 j,m m 
 
 Using the factor scores, groups of individuals with assumed similar psychological 
 attribute spaces can be formed. The subsequent scaling of products within 
 an attribute space should be applied separately to each homogeneous group. 
 
 Scaling by Factor Analysis 
 
 Summing over individuals, a product by product square symmetric 
 contingency table X is created for the group. Again, to eliminate sample size 
 bias, X is standardized by calculating relative frequencies and dividing by 
 the standard deviations. 
 
 x i!j - x i tj ' (x i v 1 * 
 
 This standardized matrix, X is positive, semi-definite, and being symmetrical 
 has grammian properties. Since the standardization yields l's in the main 
 diagonal, the matrix may be used directly in principal components analysis. 
 X may be directly factored into the product of principal components U and 
 
 A 2 • 
 
 a matrix of characteristic roots A_ in the following manner. Since X is 
 
 grammian, a matrix M can be found such that X «= MM'. Defining U and 
 
 W as transformation matricies such that U = U~ and W » W~ , let M ■ U A W, 
 
 Thar * 2 
 
 ' 21 = ii £' * ( ^ d E ) ( Si' A u • ) » u A_ u • 
 
 Each variable, X. can then be expressed as a linear combination of scores on 
 the principal components r and the product-moment correlations between the 
 factors and the variables A. 
 
 X* - A F ; where A ■ U A , and F = _A" U« X* . 
 
 The resulting principal component vectors, which are orthogonal, represent 
 
- 7 - 
 
 the underlying dimensions in the psychological process. Because the results are 
 unique only up to affine transformations, the principal component vectors may 
 be rotated to aid in identification, "hat is, a square symmetric matrix T 
 with the restriction that IT' * I can be found such that 
 
 • 2 
 
 X = U T A T'U' « A F , where A « U T A 
 
 Factor scores for each product on the underlying dimension can be calculated 
 using either the rotated solution 
 
 F « (T» A' A T )" T» A' X* 
 
 1 
 or the unrotated solution 
 
 t - ( a» a r 1 ^ 1 X* . 
 
 The factor scores represent the desired scale values of each of the products 
 on the underlying dimensions, and a geometrical representation can be 
 obtained from a plot of these scores. If the factor scores are computed 
 after rotation, the rotation must be non-distance destroying or the resultant 
 scale values will be meaningless. 
 
 There ~re several advantages o^ this method of scaling over the more 
 traditional nonmetric multidimensional algorithms. First, this technique is no 
 based on a criterion of error. Whereas geometric models attempt to best fit 
 
 the data, that is to find a solution with interpoint distances whose rank order 
 
 1 -1 • 
 
 This calculation is derived from the relationships A = U A_ and F = _A U* X 
 
 as follows: 
 
 LIS 1 ■ H. 
 
 A' A _A -1 = A' U 
 
 -1 -1 
 
 -A " (A* b) L % LL ' s i nce ( A' A ) is invertible 
 
 -1 • -1 • 
 
 thus, F = (A* A) A' U U' X = (A 1 A) A' X because UU' « I. 
 
 Similarly for the rotated factors. 
 
- 8 - 
 
 most closely approximates the rank order of the original data, this factor 
 analytic technique attempts to explain the maximum amount of variation in the 
 data. Second, in multidimensional scaling algorithms, the resultant scales 
 on any dimension are dependent on the number of dimensions specified. However, 
 the scale values of an item on a factor is independent of the number of factors 
 specified because the factors are extracted sequentially in order of the amount 
 of variation explained. Finally, traditional methods can obtain a local minimum. 
 That is, the techniques are dependent on the initial configuration specified by 
 the researcher, even if it is only a random placement. Factor analysis requires 
 no such initial starting point. 
 
 AN APPLICATION 
 
 The products used for this experiment were fifteen soft drinks: Coke, 
 Pepsi, Royal Crown, Dr Pepper, Tab, Diet Pepsi, Seven-up, Sprite, Squirt, 
 Diet Seven-up, Root Beer, Grape, Cherry, orange, and Lemon-Lime. Soft 
 drinks were chosen because of subject experience with the product class, 
 recognition by brand name only was possible, and the set of all possible 
 soft drinks with which subjects were familiar was large. 
 
 A total of seventy subjects were divided into two equal groups. The 
 first group was presented a list of all possible pairs (105) of the fifteen 
 soft drinks and asked to indicate whether or not they considered the pair 
 to be similar cr not. The responses (yes for similar, no for not similar) 
 constituted the binary data. One month later, each member of the group was 
 presented a deck of cards, each card containing a pair of soft drinks. The 
 subjects were asked to rank order the cards so that the top card was the pair 
 judged most similar, the second card the next most similar, and so forth. It 
 was further suggested that the subjects use a stepwise procedure to complete 
 
- 9 - 
 
 the task, first sorting the cards into two piles of similar and disimilar 
 pairs, each p .le then sorted into two piles, and so forth. After eight 
 piles or so had been created, they were to rank the cards in each pile, 
 combine piles one at a time, check the ordering of the new complete pile, 
 and when completed, go through the deck one last time to be certain they 
 were satisfied with the ordering. At the end of the task, the subjects were 
 asked to describe the process arid the criteria they used in completing the 
 task, the perceived difficulty of the task, and their confidence in being able 
 to replicate the process consistently. Several subjects were then given a second 
 deck and asked to perform the task again as a measure of reliability. A similar 
 procedure was used with the second group, except the order of the tasks was 
 reversed (i.e., ordinal data were collected first). Thus, for each individual, 
 two sets of data were collected, namely a product by product matrix of binary 
 similarities data and a product by product matrix of rank order similarities data. 
 
 Although both techniques required judgements on 105 pairs of products 
 the binary data task took less than one-fourth the time to complete than the 
 rank order task. Further, the rank-roder task was perceived as more difficult 
 than the binary task. Alternative methods of collecting rank order data are 
 available, however, this stepwise method was chosen so that the results would 
 be as "accurate" as possible. Also, the respondents indicated they felt that 
 they were consistant in their use of criteria for judging similarities 
 throughout both the binary and the rank order tasks, however, the indepth 
 questioning concerning the rank order task indicated that they were not 
 consistent. 
 
 RESULTS AND DISCUSSION 
 
 Points of view analysis was performed on both sets of data, and in both 
 instances, only one group appeared with no outliers . If more than one group 
 
- 10 - 
 
 had appeared, then separate scaling would have been performed for each subgroup. 
 In this instance, all individuals were included in each analysis. Further, the 
 data were analyzed separately for each group to determine if the order of the tasks 
 had any effect on the results. There appeared to be no order effect, based on 
 visual comparison of the resulting maps, so the two groups were combined and 
 an analysis using the total sample was performed. Because of the similarity, 
 only the results from the analysis of the total sample is presented. 
 
 The Rank Order Data. A group similarities matrix was calculated with 
 cell entries consisting of the average rank order for that product-pairj 
 this matrix was used as input for TORSCA with the three dimensional results 
 presented in Figure 1. As previously mentioned, this technique requires the 
 prior specification of a model (metric and dimensionality) and of an initial 
 configuration. For this study the Euclidean distance function was chosen 
 and 2-, 3-, 4-, and 5- dimensional solutions calculated, each starting from 
 a random initial configuration. The scale values of a solution are dependent 
 on the number of dimensions, hence, a necessary task for the researcher in applying 
 these techniques is to choose the number of dimensions. A possible approach is 
 to choose the dimensionality based on interpretability and the information 
 provided. Stress values, measuring the goodness of fit of the data, can also 
 be used. Stress values for the 2-, 3-, and 4-dimensional configurations v/ere 
 .240, .160, and .107 respectively. Frimarily for the purpose of comparison with 
 the binary data solutions, the three dimensional solution is presented. 
 
 As is apparant from an examination of Figure 1, there is no easy 2nd 
 obvious interpretation of the results. This further demonstrates a problem with 
 geometric models , namely interpretation of the results. Several possible methods 
 to aid in the identification process include factor analyzing the data and usi: g 
 the factor loadings, ot to collect evaluations of each product on various 
 prespecified criteria and then fitting regression lines using this data to the 
 
- 11 - 
 
 obtained perceptual space. Also of interest in this example is that although 
 the stress value decreased for the 4- and 5 -dimensional solutions, interpretation 
 was not enhanced by the addition of the extra dimensions. This leads us to 
 conclude the underlying model implicit in the technique may not be appropriate. 
 The recourse for the researcher is to continue to try additional models in 
 the hope of obtaining a meaningful solution. 
 
 The Binary Data . The method of analysis described in this paper was 
 applied to the group contingency table. The factor analysis procedure yielded 
 three factors which explained slightly more than 80% of the variance. The plots 
 of the rotated factor scores appear in Figure 2. 
 
 As opposed to the rank-order solutions, interpretation of these dim- 
 ensions seems relatively apparant. The first dimension appears to be a 
 cola (alternatively a dark-colored) dimension with seven products— Coke, 
 Pepsi, Royal Crown, Tab, Diet Pepsi, Dr Pepper , and Root Beer-*loading 
 heavily. (Note, interpretation is aided in this technique by the use of the 
 factor loadings). The second dimension appears to be an *un-cola" dimension 
 (a lemon-lime, citrus flavored dimension) with five products- -Seven-up, Sprite, 
 Squirt, Diet Seven-up, and Lemon-lime, loading heavily. The third dimension 
 appears to be a fruit-flavored (other than lemon-lime) dimension with three 
 products- -cherry, grape, and orange— loading heavily and two products— Root 
 Beer and Lemon- lime— loading slightly. Although not instructed to do so, the 
 subjects seem to have used flavor as a major criteria in judging similarities 
 resulting in three underlying flavor dimensions. Examination of the four 
 dimensional solution (which did not significantly increase the percent variance 
 explained) yielded the same three dimensions plus a diet dimension with 
 three products— Tab, Diet Pepsi, and Diet Seven-up— loading heavily on the 
 fourth factor. 
 
- 12 - 
 
 Discussion 
 
 The purpose of both scaling techniques is to obtain a geometrical repre- 
 sentation of the psychological space of soft drinks. In this study three 
 dimensions were chosen for both techniques (1) for the purpose of comparison, 
 
 (2) because three dimensions suited the criteria used in each technique, and 
 
 (3) because a priori, three dimensions seemed appropriate (although not the 
 resulting dimensions). As is evident from a quick, examination of Figures 
 
 1 and 2, the two methods did not yield similar results. Consequently, it is 
 desirable to explain why these differences exist and to determine which 
 mapping, if either k more nearly represents the true psychological space. 
 V;"e believe that the map resulting from the binary data provides a close 
 representation of the psychological space while the map from the rank order 
 data is relatively meaningless. This belief is substantiated through the 
 examination of several comparative criteria of validity: cross validity, 
 face validity, external validity, and predicting validity. 
 
 (1) The criterion of cross validity implies consistancy of results 
 across replications or across subgroups of the same population. In this 
 instance two separate sets of data applicable to each technique were 
 originally collected. When analysed separately the binary data yielded almost 
 identical three dimensional perceptual maps. However, the maps derived from 
 the two sets of rank order, while similar in the amount of dispersion exhibited, 
 were completely different with respect to the relationships between the 
 products. Thus, cross validation would support the binary technique since it 
 yielded consistent results, but not the rank order technique. 
 
 (2) Results of a study have face validity if on inspection they are 
 similar to what one might expect them to be. A priori, we hypothesized that 
 
 the psychological space would be represented by three dimensions: a cola (color) 
 
- 13 
 
 dimension with coke and seven-up at the opposite ends, a diet dimension, 
 and a fruit flavor dimension. As noted, the map from the rank-order data 
 was not interpretable , thus having no face validity. On the other hand, the 
 binary data resulted in a map very nearly representing our intuitive picture 
 of the space. If we would have considered the cola-~lemon dimension as being 
 actually two orthogonal dimensions, then the four-dimensional binary solution 
 would have almost exactly duplicated our a priori notions. 
 
 (3) As a measure of external validity, each subject was asked, after 
 completing the rank order task, to state the criteria used during that task 
 in the judgements of similarity. The most frequently mentioned criteria 
 were cola, lemon flavor (seven-up like), diet, and fruit flavored. The 
 technique using the binary data clearly extracted these dimensions, thus 
 reproducing the stated criteria of the subjects. However, the rank order 
 maps failed to even come close to these stated dimensions, eventhough these 
 external elicitations occurred immediately after the subjects performed the 
 rank order task. 
 
 (4) Finally, predictive validity of the model can be obtained by having 
 subjects produce geometric product spaces. Subjects were asked to physically 
 place the products in three dimensional product spaces. Again, most subjects' 
 traps were very nearly the same as those obtained by the binary scaling method. 
 The only exceptions being a few subjects whose maps w^re more nearly similar to 
 cur a priori dimensions of cola, fruit flavor, and diet. 
 
 The question then arises as to why a method utilizing weaker data (binary) 
 produced results which across a variety of criteria were judged superior to 
 those resulting from a technique utilizing stronger (ordinal) data. The 
 first reason could be due to the different analytic procedures of the two 
 methods. The traditional multidimensional scaling technique required prior 
 
14 - 
 
 specification of the model, and in this instance, our specification may have 
 been incorrect thereby yielding meaningless results. Further, the obtained 
 results may have been one of several local minimums, dependent on the 
 prespecified initial configuration. The factor analytic technique has 
 niether of these problems since it requires no prior specification of a 
 model or a starting configuration. 
 
 A second reason for the superiority of the binary data may be due to the 
 differences in the data collection techniques. Both methods require consistency 
 of criteria by the individual throughout the task and both must be applied to 
 homogeneous groups of individuals. Thus, both methods must have data that 
 is highly consistant both within and between individuals. In the collection of 
 the binary data, the task was rather simple. Subjects were able to complete 
 the task in about ten minutes and afterwards indicated that they were able to 
 use the same criteria in making judgements throughout the task. Further, 
 all subjects generally used the same criteria or at least judged the same 
 pairs to be similar most of the time. In contrast, the rank order task was 
 very difficult. On the average, the task required forty-five minutes to 
 complete and all the subjects stated that they might have changed criteria 
 during the course of the task. Subjects further indicated that they did not 
 believe they would be consistent over trials, a fact verified by repeat testing. 
 Thus, the rank order task resulted in data highly inconsistent within subjects. 
 To demonstrate the problem of between individual consistancy, the average 
 rank order (input to the TORSCA program;) and the range of the rank orderings 
 for each of the 105 product-pairs are presented in Table I. This represents 
 a major problem; however, even if the between individual differences could be 
 reduced, it is doubtful that meaningful results could be obtained from the rank 
 order data because of the within individual inconsistency. The consistancy 
 problem results directly from the number of stimuli and the difficulty of the task. 
 
- 15 - 
 
 CONCLUSION 
 
 Limitations of the Model 
 
 First, the technique presented is only applicable for group data. 
 While some traditional multidimensional scaling techniques can be applied 
 to similarities data for an individual as well as a group, this method requires 
 relative frequencies as input, hence analysis can only be performed on 
 group data. However, we are currently investigating a statistical pro- 
 cedure for mapping individual binary responses. 
 
 Second, this technique is only applicable in those instances where 
 binary responses are appropriate. Other forms of associative data are not 
 directly usable because of the need to form a frequency distribution of 
 responses. Further, preference type data, often used in marketing applications 
 of multidimensional scaling could not be scaled with this technique because 
 the responses would not be binary. 
 
 Finally, this method also suffers from the problem of lack of invariance 
 common in the nonmetric multidimensional scaling techniques. Since the results 
 of this technique are unique only up l 5 affine trans formations, the axes 
 chosen are somewhat arbitrary. Further, there is no exact criteria for chosing 
 the number of dimensions. However, the criteria that do exist for this method 
 are perhaps better substantiated than in other methods. 
 
 Summary and Implications 
 
 The implications for marketing research are many. The costs associated 
 with binary data collection would be less. Less time is required per individual 
 and compliance to cooperate in the task is higher; both should yield lower 
 costs. Thus, even if binary data and ordinal data produced identical results, 
 the use of binary techniques would be advantageous from a cost-benefit point 
 of view. 
 
- 16 - 
 
 Somewhat similar to cost effectiveness is the task effectiveness of 
 this method. It is easier to maintain concentration for shorter tasks, all 
 things else being equal. Further, considerably more information can be 
 obtained in comparable time periods. Because the task difficulty is lower, 
 within individual consistency will be higher. 
 
 Third, nonmetric methods are based on a criterion that minimizes some 
 form of error, which results in a problem of statistical inference, especially 
 if the underlying model (dimensionality and metric) is incorrect. The use 
 of the frequency distributions of the binary data represents a method 
 whereby statistical inference theory is applicable, which, through sampling, 
 could result in generalizations to populations. In addition, if through points 
 of view analysis, subgroups with different psychological spaces are found, 
 statistical tests of differences between these subgroups are possible. 
 
 Finally, since marketing research typically involves large stimulus sets, 
 if scaling. is to provide useful analysis for the researcher, methodologies 
 must be employed which have underlying assumptions that can be met. If the 
 assumptions underlying a technique are not met, the validity of the results 
 is questionable. Binary scaling represents one technique with assumptions that 
 are more likely to be .-net, thus providing greater confidence in the validity 
 of the results. 
 
TABLE I 
 
 
 
 
 
 
 MEAN AND 
 
 RANGE OP RANK 
 
 ORDER 
 
 DATA 
 
 
 
 
 
 
 Drinks 
 
 Average 
 
 Rank 
 Order 
 
 Range of 
 
 Rank 
 
 Order j 
 
 Drinks 
 
 Average 
 Rank 
 Order 
 
 Range of 
 
 Rank 
 
 Orders 
 
 Dri 
 
 nks 
 
 Average 
 
 Rank 
 Order 
 
 Range of 
 
 Rank 
 
 Orders 
 
 2 
 
 1 
 
 49.933 
 
 19 
 
 _ 
 
 103 
 
 3 
 
 2 
 
 45.400 
 
 7 
 
 m 
 
 101 
 
 3 
 
 1 
 
 58.733 
 
 15 
 
 - 103 
 
 4 
 
 3 
 
 47.233 
 
 14 
 
 «» 
 
 102 
 
 4 
 
 2 
 
 50.700 
 
 10 
 
 - 
 
 102 
 
 4 
 
 1 
 
 60.300 
 
 7 
 
 - 99 
 
 5 
 
 4 
 
 61.300 
 
 25 
 
 tm 
 
 105 
 
 5 
 
 3 
 
 62.267 
 
 14 
 
 - 
 
 103 
 
 5 
 
 2 
 
 65.667 
 
 26 
 
 - 105 
 
 5 
 
 1 
 
 67.633 
 
 23 
 
 «• 
 
 104 
 
 6 
 
 5 
 
 70.467 
 
 40 
 
 m 
 
 105 
 
 6 
 
 4 
 
 65.167 
 
 24 
 
 - 100 
 
 6 
 
 3 
 
 62.400 
 
 26 
 
 - 
 
 96 
 
 6 
 
 2 
 
 55.500 
 
 16 
 
 - 
 
 102 
 
 6 
 
 1 
 
 18.200 
 
 2 
 
 - 70 
 
 7 
 
 6 
 
 20.433 
 
 4 
 
 - 
 
 69 
 
 7 
 
 5 
 
 70.000 
 
 12 
 
 - 
 
 103 
 
 7 
 
 4 
 
 67.467 
 
 8 
 
 - 103 
 
 7 
 
 3 
 
 58.483 
 
 6 
 
 - 
 
 93 
 
 7 
 
 2 
 
 54.967 
 
 5 
 
 - 
 
 104 
 
 7 
 
 1 
 
 11.333 
 
 1 
 
 - 40 
 
 8 
 
 7 
 
 17.900 
 
 3 
 
 - 
 
 42 
 
 8 
 
 6 
 
 31.967 
 
 6 
 
 mt 
 
 87 
 
 8 
 
 5 
 
 73.100 
 
 20 
 
 - 101 
 
 9 
 
 4 
 
 64.467 
 
 27 
 
 - 
 
 105 
 
 8 
 
 3 
 
 60.633 
 
 31 
 
 - 
 
 92 
 
 3 
 
 2 
 
 56.067 
 
 16 
 
 - 105 
 
 8 
 
 1 
 
 20.133 
 
 1 
 
 - 
 
 59 
 
 9 
 
 8 
 
 17.600 
 
 1 
 
 - 
 
 49 
 
 9 
 
 7 
 
 9.400 
 
 1 
 
 - 31 
 
 9 
 
 6 
 
 10.967 
 
 1 
 
 - 
 
 69 
 
 9 
 
 5 
 
 69.067 
 
 33 
 
 - 
 
 102 
 
 9 
 
 4 
 
 66.553 
 
 21 
 
 - 103 
 
 9 
 
 3 
 
 58.333 
 
 27 
 
 ■ 
 
 94 
 
 9 
 
 2 
 
 56.000 
 
 13 
 
 «#• 
 
 102 
 
 9 
 
 1 
 
 10.733 
 
 1 
 
 - 32 
 
 10 
 
 9 
 
 66.367 
 
 16 
 
 ■ 
 
 105 
 
 10 
 
 8 
 
 75.433 
 
 29 
 
 - 
 
 105 
 
 10 
 
 7 
 
 71.167 
 
 29 
 
 - 101 
 
 10 
 
 6 
 
 55.033 
 
 6 
 
 - 
 
 105 
 
 10 
 
 5 
 
 49.100 
 
 16 
 
 - 
 
 98 
 
 10 
 
 4 
 
 64.300 
 
 29 
 
 - 104 
 
 10 
 
 3 
 
 63.033 
 
 21 
 
 - 
 
 102 
 
 10 
 
 2 
 
 75.867 
 
 35 
 
 - 
 
 103 
 
 10 
 
 1 
 
 70.867 
 
 28 
 
 - 105 
 
 11 
 
 10 
 
 18.200 
 
 1 
 
 n 
 
 89 
 
 11 
 
 9 
 
 70.167 
 
 10 
 
 «* 
 
 102 
 
 11 
 
 8 
 
 74.900 
 
 29 
 
 - 105 
 
 11 
 
 7 
 
 63.767 
 
 9 
 
 ■i 
 
 102 
 
 11 
 
 6 
 
 52.800 
 
 4 
 
 - 
 
 104 
 
 11 
 
 5 
 
 42.300 
 
 12 
 
 - 104 
 
 11 
 
 4 
 
 65.967 
 
 24 
 
 - 
 
 105 
 
 11 
 
 3 
 
 58.667 
 
 9 
 
 - 
 
 99 
 
 11 
 
 2 
 
 72.100 
 
 23 
 
 - 103 
 
 11 
 
 1 
 
 74.833 
 
 24 
 
 - 
 
 104 
 
 12 
 
 11 
 
 45.067 
 
 14 
 
 «* 
 
 95 
 
 12 
 
 10 
 
 45.267 
 
 15 
 
 - 96 
 
 12 
 
 9 
 
 66.400 
 
 9 
 
 ■ 
 
 105 
 
 12 
 
 8 
 
 72.633 
 
 11 
 
 *» 
 
 104 
 
 12 
 
 7 
 
 71.033 
 
 24 
 
 - 105 
 
 12 
 
 6 
 
 72.700 
 
 33 
 
 - 
 
 104 
 
 12 
 
 5 
 
 43.600 
 
 7 
 
 - 
 
 100 
 
 12 
 
 4 
 
 58.967 
 
 27 
 
 - 105 
 
 12 
 
 3 
 
 49.033 
 
 4 
 
 - 
 
 102 
 
 12 
 
 2 
 
 71.200 
 
 37 
 
 ■-• 
 
 103 
 
 12 
 
 1 
 
 71.633 
 
 10 
 
 - 103 
 
 13 
 
 12 
 
 36.800 
 
 6 
 
 - 
 
 94 
 
 13 
 
 11 
 
 22.767 
 
 5 
 
 *» 
 
 101 
 
 13 
 
 10 
 
 19.367 
 
 4 
 
 - 86 
 
 13 
 
 9 
 
 63.433 
 
 13 
 
 - 
 
 103 
 
 13 
 
 8 
 
 73.733 
 
 25 
 
 - 
 
 105 
 
 13 
 
 7 
 
 64.967 
 
 22 
 
 - 102 
 
 13 
 
 6 
 
 66.567 
 
 19 
 
 *• 
 
 102 
 
 13 
 
 5 
 
 41.467 
 
 9 
 
 •* 
 
 105 
 
 13 
 
 4 
 
 64.333 
 
 37 
 
 - 102 
 
 13 
 
 3 
 
 60.200 
 
 18 
 
 - 
 
 94 
 
 13 
 
 2 
 
 73.600 
 
 45 
 
 — 
 
 103 
 
 13 
 
 1 
 
 69.367 
 
 14 
 
 - 101 
 
 14 
 
 13 
 
 7.367 
 
 1 
 
 - 
 
 26 
 
 14 
 
 12 
 
 33.733 
 
 2 
 
 •». 
 
 93 
 
 14 
 
 11 
 
 18.667 
 
 9 
 
 - 80 
 
 14 
 
 10 
 
 13.033 
 
 1 
 
 - 
 
 79 
 
 14 
 
 9 
 
 64.800 
 
 7 
 
 - 
 
 105 
 
 14 
 
 6 
 
 70.333 
 
 15 
 
 - 105 
 
 14 
 
 7 
 
 63.133 
 
 9 
 
 - 
 
 104 
 
 14 
 
 6 
 
 68.567 
 
 22 
 
 j» 
 
 103 
 
 14 
 
 5 
 
 39.900 
 
 4 
 
 - 103 
 
 14 
 
 4 
 
 65.467 
 
 29 
 
 ■ 
 
 104 
 
 14 
 
 3 
 
 55.500 
 
 17 
 
 - 
 
 105 
 
 14 
 
 2 
 
 70.233 
 
 42 
 
 - 102 
 
 14 
 
 1 
 
 62.467 
 
 25 
 
 ■i 
 
 105 
 
 15 
 
 14 
 
 8.867 
 
 1 
 
 - 
 
 93 
 
 15 
 
 13 
 
 12.067 
 
 1 
 
 - 90 
 
 15 
 
 12 
 
 38.400 
 
 3 
 
 - 
 
 94 
 
 15 
 
 11 
 
 20.367 
 
 3 
 
 - 
 
 99 
 
 15 
 
 10 
 
 18.033 
 
 2 
 
 - 92 
 
 15 
 
 9 
 
 62.600 
 
 8 
 
 M 
 
 1Q5 
 
 15 
 
 8 
 
 74.433 
 
 26 
 
 - 
 
 105 
 
 15 
 
 7 
 
 68.100 
 
 21 
 
 - 104 
 
 15 
 
 6 
 
 70.400 
 
 20 
 
 - 
 
 103 
 
 15 
 
 5 
 
 38.867 
 
 4 
 
 - 
 
 102 
 
 15 
 
 4 
 
 62.833 
 
 26 
 
 - 105 
 
 15 
 
 3 
 
 53.767 
 
 3 
 
 - 
 
 97 
 
 15 
 
 2 
 
 73.033 
 
 43 
 
 _ 
 
 105 
 
 15 
 
 1 
 
 74.467 
 
 30 
 
 - 104 
 
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 Burt, Cyril, "The Factor Analysis of Qualitative Data," The 
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 Eckart, Carl, and Gale Young, "The Approximation of One Matrix 
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V'