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(12) Visualizing & Analyzing Customer's Preferences by Gower Plots student)CY Chen. Advisors)Dr.HanLin Li. Institute of Information Management National Chiao Tung University. ABSTRACT This research applies a graphical technique in developing a marketing planning model. The overall flow includes gathering data by the questionnaire with Likert scales, analyzing the data by Gower Plot method to define the market characteristics, customer preferences and customer behaviors. The outcomes of the analysis can be used as the guide of the marketing plan development to allocate the marketing resources and arrange the customer manipulation strategies.. ii.

(13) Y. Z. [ \. ]. Table 1- Experimental design for evaluation of carpet cleaner …………………….7 Table 2- Utility of factor…………………………………………………………….7 Table 3- Scales of measurement…………………………………………………….18 Table 4-Response data sheet of Cardianl Gower Plot………………………………20 Table 5-Reponse matrix for Ordinal Gower Plot…………………………………....21 Table 6 -- Response Matrix for Cardinal Gower Plot ………………………………21 Table 7- Response data sheet of Age below ………………………………………..28 Table 8- Response data sheet of Age between 21-30 ………………………………29 Table 9- Response data sheet of Age between 31-40 ………………………………30 Table 10- Response data sheet of Age between 41-50 ……………………………..31 Table 11- Response data sheet of Female …………………………………………..32 Table 12- Response data sheet of Male ……………………………………………..33 Table 13- Response data sheet of Plan to buy a car within 2 year ………………….34 Table 14- Response data sheet of No plan to buy a car within 2 year ………………35 Table15- Response Matrix for Ordinal Gower Plot of Age below 20 ………………36 Table16- Response Matrix for Ordinal Gower Plot of Age between 21-30 ………...36 Table17- Response Matrix for Ordinal Gower Plot of Age between 31-40 ………...36 Table18- Response Matrix for Ordinal Gower Plot of Age between 41-50 …………36 Table19- Response Matrix for Ordinal Gower Plot of Female ……………………...37 Table 20- Response Matrix for Ordinal Gower Plot of Male ……………………….37 Table 21- Response Matrix for Ordinal Gower Plot of Plan to buy a car within 2 year ………………………………………………………………………………….38 Table 22- Response Matrix for Ordinal Gower Plot of No plan to buy a car within 2 year …………………………………………………………………………………38 Table 23- Response Matrix for Cardinal Gower Plot of Age below 20 ……………44 Table 24- Response Matrix for Cardinal Gower Plot of Age between 21-30 ………44 Table 25- Response Matrix for Cardinal Gower Plot of Age between 31-40 ………44 Table 26- Response Matrix for Cardinal Gower Plot of Age between 41-50 ………45 Table 27- Response Matrix for Cardinal Gower Plot of Female ……………………45 Table 28- Response Matrix for Cardinal Gower Plot of Male ………………………45 Table 29- Response Matrix for Cardinal Gower Plot of Plan to buy a car within 2 year ………………………………………………………………………………….46 Table 30- Response Matrix for Cardinal Gower Plot of No plan to buy a car within 2 year ………………………………………………………………………………….46 Table 31- Response data sheet of Ordinal Gower Plot …………………………….52 iii.

(14) Table 32- Response Matrix for Ordinal Gower Plot ………………………………...52 Table 33- Response data sheet of Cardinal Gower Plot ………………………….54 Table 34- Response Matrix for Cardinal Gower Plot …………………………….54. iv.

(15) Z. [ \. ]. Figure 1- Result of computer analysis of experimental data………………………….8 Figure 2-- Ordinal Gower Plot of Overall condition ………………………………..22 Figure 3- Diagrammatic explanation to Ordinal Gower plot of overall condition ….23 Figure 4-Cardinal Gower plot of overall condition ………………………………….25 Figure 5- Diagrammatic explanation to Cardinal Gower plot of overall condition …26 Figure 6- Ordinal Gower Plot of Age below 20 ……………………………………..39 Figure 7- Ordinal Gower Plot of Age 21-30 ………………………………………...40 Figure 8- Ordinal Gower Plot of Age 31-40 ………………………………………...41 Figure 9- Ordinal Gower Plot of Age 41-50 ………………………………………...41 Figure 10- Ordinal Gower Plot of Female …………………………………………..42 Figure 11- Ordinal Gower Plot of Male …………………………………………….42 Figure 12- Ordinal Gower Plot of plan to buy a car within 2 years …………………43 Figure 13- Ordinal Gower Plot of no plan to buy a car within 2 years ……………..43 Figure 14- Cardinal Gower Plot of Age below 20 …………………………………..47 Figure 15- Cardinal Gower Plot of Age21-30 ………………………………………47 Figure 16- Cardinal Gower Plot of Age 31-40 ………………………………………48 Figure 17- Cardinal Gower Plot of Age41-50………………………………………..48 Figure 18- Cardinal Gower Plot of Female ………………………………………….49 Figure 19- Cardinal Gower Plot of Male ……………………………………………49 Figure 20- Cardinal Gower Plot of Plan to buy a car ……………………………….50 Figure 21- Cardinal Gower Plot of No plan to buy a car ……………………………50 Figure 22- Ordinal Gower Plot ………………………………………………………53 Figure 23-Cardinal Gower Plot ……………………………………………………...56 Figure 24- Figure 23 with coordinate data …………………………………………..56 Figure 25- Data points relocate after taking regressing line ………………………...57. v.

(16) Visualizing & Analyzing Customer’s Preferences by Gower Plots. Chun-Yi Chen. June 25, 2004. Institute of Information Management National Chiao Tung University Management Building 2, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, ROC [email protected] [email protected]. vi.

(17) Contents _. T. ……………………………………………………………………………….i. `. _. T. ………………………………………………………………………………ii. Y. Z. [. ………………………………………………………………………………...iii. Z. [. …………………………………………………………………………………v. Z. ^. [. …………………………………………………………………………………vii. 1. Introduction................................................................................................................1 2. Literature review........................................................................................................2 3. Gower Plot .................................................................................................................8 3.1. Analytic Hierarchy Process (AHP) .................................................................9 3.2. Gower Plots.....................................................................................................9 3.3. Applying Gower's technique to AHP ............................................................12 3.4. Cardinal / Ordinal Gower Plot Measure .......................................................13 3.5 Data collection ...............................................................................................16 4. Case Study I - Customer Preferences for Car Purchase...........................................19 4.1 Response data sheet (Overall condition) .......................................................19 4.2 Response data sheet (Breakdown in certain criteria).....................................27 5. Case Studies- i2 users satisfaction survey of Semiconductor Company .................51 6. Reference .................................................................................................................58 7. Appendix ..................................................................................................................59 Singular Value Decomposition ............................................................................59 Program - Ordinal Gower Plot.............................................................................60 Program - Cardinal Gower Plot ...........................................................................62 Program Fix .........................................................................................................64 Ordinal result of Case study I ..............................................................................65 Cardinal result of Case study I.............................................................................71 Cardinal result of Case study II ...........................................................................76. vii.

(18) This research applies a graphical technique in developing a marketing planning model. The overall flow includes gathering data by the questionnaire with Likert scales, analyzing the data by Gower Plot method to define the market characteristics, customer preferences and customer behaviors. The outcomes of the analysis can be used as the guide of the marketing plan development to allocate the marketing resources and arrange the customer manipulation strategies.. 1. Introduction Companies and their marketing managers frequently encounter difficulties when trying to determine what a consumer really wants about a specific product or service. In past decades, researchers have developed several measuring techniques in the mathematical psychology field, psychometrics and customer behavior patterns that can aid the manager in determining the relative importance of a product's multidimensional attributes. Companies were busy collecting and sifting mountains of data regarding preferences and behaviors, dividing consumers into ever-finer segments, and honing their products, services, and marketing pitches. But they could still not see the complete picture. Few companies have bothered to carefully look at the broader context in which a customer selects, purchases and uses products and services. They have been so focused on fine-tuning their own offerings that they fail to see whether those products and services actually meet their customers' needs. They have spent a great amount of time and money to advertise our products and services, but fail to appreciate the effect of those advertisements. Consumer preference research is a critical issue in the marketing planning process. Understanding what stimulates customer's intention, how customers process their buying decision and finally take actions to purchase the specific product/service is and should be the most important factor in every aspect of company's marketing plan. Traditionally, in the marketing research area, customer preference is mainly studied by the Conjoint technique, developed in 1960s. This research applies a graphical technique to develop of a marketing planning model. The outcome of the analysis can be used as a guide for marketing plan development when allocate marketing resources and arranging customer manipulation strategies. The purpose of this paper is to describe a graphical technique to develop a global vision of the complex relationship between message extracts from a marketing questionnaire and its analysis. 1.

(19) We combine graphical analysis with Consumer Behavior to show how to visualize respondent's data and to detect contradiction between preferences to provide a more efficient and easy way to monitor, control and even conduct the marketing plan. Genest and Zhang developed the graphical analysis named Gower Plots in 1970s. The graphical method provides an especially valuable diagnostic tool: Ordinal Gower Plot detects the contradiction between one preference and the other; and cardinal Gower Plot perceives the proportional relation of respondent judgments. The overall flow includes gathering data using a questionnaire through on-line questionnaire with Likert scales, computing and analyzing the data using the Gower Plot method to ranking customer preferences in order to define market characteristics and customer behaviors. In this article, for comparison between two visualizing tools: Conjoint analysis and Gower Plots, we first introduce Conjoint technique, the traditional tool for preference study in marketing research, following, describe Gower Polts and how the graphical method works. Finally, we will provide some empirical cases where we applied the graphical method to show how it is useful to the marketing plan and the management improvement.. 2. Literature review To emphasize that the benefit of Gower Plots, we will introduce the traditional tool for preference study in marketing research- Conjoint analysis. In past decade, many techniques, such as multidimensional scaling analysis, conjoint analysis and so on, were developed to handle those problems have a common structure that companies and their marketing managers frequently encounter in trying to figure our shat a consumer really wants in a product or service. The most famous technique is so called conjoint analysis. Conjoint analysis was developed by Paul E. Green (July- August 1975) and his collaborators as a method from the field of mathematical psychology and psychometrics that can aid the marketing manager in sorting out the relative importance of a product’s multidimensional attributes. The word “conjoint” has to do with the notion that the relative values of things considered jointly can be measured when they might not be measurable if taken one at 2.

(20) a time. Quite often respondents are asked to express the relative value to them of various alternatives by ordering the alternatives form most desirable to least desirable. The attempt in a conjoint analysis solution is to assign value to the levels of each of the attributes so that the resulting values or utilities are as monotonic as possible with the input rank-order judgments. Conjoint measurement starts with the consumer’s overall or global judgments about a set of complex alternative. It then performs the job of decomposing respondent’s original evaluations into separate and compatible utility scales by which the original global judgments can be reconstituted. The stimuli in conjoint analysis represent some predetermined combinations of attributes (ex: products or brands), and respondents are asked to make judgments about their preference for these various arrtibute combinations. The basic aim is to determine the features respondents most prefer. Respondents might use, for example, such attributes as package design, brand name, price, money-back guarantee, and so on in making judgments about which carpet cleaner they most prefer. If asked to so directly, many respondents might find it very difficult to state which attributes they were using and how they were combining them to form overall judgments. Conjoint analysis attempts to handle this problem by estimating how much each of the attributes is valued on the basis of the choice respondents make along product concepts that are varied in systematic ways. In essence respondents’ value systems are inferred form their behaviors as reflected in their choices rather than form self reports about how import each of the various attributes are to them. The procedure for determining the individual’s utilities for each of several product attributes followed in conjoint analysis is quite similar to that followed in multidimensional scaling analysis. The technique is quite dependent on the availability of a high-speed computer. Just as in multidimensional scaling, the computer program emphasis is on generating an initial solution and on modifying that solution through a series of iteration to improve the goodness-of-fit. More specifically, given a set of input judgments, the computer program will: 1. Assign arbitrary utilities to each level of each attribute; 2. Calculate the utilities for each alternative by somehow combining (most typically adding) the individual utility values; 3. Calculate the goodness of fit between the ranking of the alternatives using these derived utility values and the original ordering of the input judgments; 3.

(21) 4. Modify the utility values in a systematic way until the derived utility produce evaluations that, when ordered, correspond as closely as possible to the order of the input judgments. The following case will be taken as an example to see how conjoint measurement work. A company was interested in marketing in a new spot remover for carpets and upholstery. The technology staff has developed a new product that is designed to handle spot. Management interest centers on five attributes or factors that it expects will influence consumer preference: an applicator-type package design, brand name, price, a Good Housekeeping seal of endorsement, and a money-back guarantee. As below Table 1, the result comes from respondent’s evaluation. There are three brand names: K2R, Glory, and Bissell. Three alternative prices being considered are $1.19, $1.39, and $1.59. Since they are three alternatives for each of these factors, they are called three-level factors. The Good Housekeeping seal and money-back guarantee are two-level factors, since each is either present or not. A total of 3*3*3*2*2=108 alternatives would have to be tested if the research were to array all possible combinations of the five attributes. As an alternative, however, the research can take advantage of a special experimental design, called an orthogonal array, in which the test combinations are selected so that the independent contributions of all five factors are balanced. In this way each factor’s weight is kept separate and is not confused with those of the other factors. The table shows an orthogonal array that involves only 18 of 108 possible combinations that the company wishes to test in this case. The last column of the experimental table shows one respondent’s actual ranking of the 18 cards; rank number 1 denotes her highest evaluated concept. Note particularly that only ranked data need to be obtained and that only 18 (out of 108) combinations are evaluated. Various computer programs carry out computation of the utility scales of each attribute, which determine how influential each is in the consumers’ evaluations. The ranked data of a single respondent (or the composite ranks of a group of respondents) are entered in the program. The computer then searches for a set of scale values for each factor in the experimental design. The scale values for each level of each factor are chosen so that when they are added together the total utility of each combination will correspond to the original ranks as closely as possible.. 4.

(22) There are two problems need to be involved here. First, as mentioned previously, the experimental design shows only 18 of 108 combinations. Second, only rank-order data are supplied to the algorithms. This means that the data themselves do not determine how much more influential one attribute is than another in the consumers' choices. However, despite these limitations, the algorithms are able to find a numerical representation of the utilities, thus providing an indication of each factor's relative importance. As can be observed in Table 2, the technique obtains a utility function for each level of each factor. For example, the total utility of combination 18 is 3.1 (0.6+0.5+1.0+0.3+0.7), which is the respondent's highest evaluation of all 18 combinations listed. By focusing attention on only the package design, the company's marketing researches can see from Figure 1 that Design B displays highest utility. Moreover, all utility scales are expressed in a common unit (although their zero points are arbitrary). This means that we can compare utility ranges from factor to factor so as to get some idea of their relative importance. In the case of the spot remover, as shown in Figure 1, the utility ranges are: Package design (1.0-0.1=0.9) Brand name (0.5-0.2=0.3) Price (1.0-0.1=0.9) Good Housekeeping seal (0.3-0.2=0.1) Money-back guarantee (0.7-0.2=0.5) The lower portion of Figure 1 shows the relative size of the utility ranges express in histogram form. As noted, package design and price are the most important factors, and together they account for about two thirds of the total range in utility. The relative importance of a factor depends on the levels that are included in the design. As a crude indication of what factors to concentrate on, factor importance calculations provide a useful by-product of the main analysis regardless of such limitation. Limitations of the conjoint analysis are as the following: 1. The techniques suspect that in many instances the simpler (additive) model 5.

(23) represents a very good approximation of reality. That is, what is often called "interactions" in traditional ANOVA applications may be the result of the failure to measure the effects of independent variables on the correct scales. 2. The respondent may not behave unidimensionally toward the prespecified criterion. This could be reflected in the failure of the paired comparisons to generate a complete rank order (necessitating the use of nearest adjoining order techniques) or the failure of a one-dimensional scaling solution to accommodate the data. 3. In dealing with large-size problems, the ranking task becomes formidable. Moreover, some comparisons could be omitted, inasmuch the programs tolerate missing data. 4. As in nonnumeric scaling, conjoint solutions are susceptible to certain types of degeneracy. 5. Conjoint analysis is still a moot point as to whether direct numerical estimation procedures would lead to results comparable to those arrived at through ranking followed by conjoint measurement.. 6.

(24) Table 1- Experimental design for evaluation of carpet cleaner. Table 2- Utility of factor Package Design Factor Utility a bc X A c ba X B a bf X C. Brand Name Factor Utility a bd X K2R Glory a b e X Bissel a b g X. Retail Price Factor Utility c b a X a b d X a b c X. 7. Good Housekeeping seal Factor Utility No 0.2 Yes 0.3 X X X X. Money-back guarantee Factor Utility No 0.2 Yes 0.7 X X X X.

(25) Figure 1- Results of computer analysis of experimental data of Table 4. 3. Gower Plot The proposed method, based on the Analytic Hierarchy Process (Saaty, 1977) methodology and Gower plots (Gower, 1977), is close in spirit to the statistical technique known as multidimensional scaling, and is also a spatial representation of the result. AHP provides the alternative derived from a complete set of paired 8.

(26) comparisons expressed as a ratio scale. Gower plots provide an especially valuable diagnostic tool to visualize such data and to detect cardinal and ordinal inconsistencies in the respondent preferences.. 3.1. Analytic Hierarchy Process (AHP) AHP is a ratio scaled assessment of an agent's preferences between pairs of alternatives. One of the basic ingredients of this procedure is the evaluation of the strength of individual preference through pairwise comparison of alternatives at each level of the hierarchy. When asked to compare alternatives j and k from a collection of size nh2 with respect to a single criterion, the respondent must elicit the ratio. ωj ωk. measuring the relative dominance of item j over item k in terms of the underlying priority weights ω1 >0,… ω n >0, taken to sum up to one by convention. This give rise to n(n-1)/2 data points of the form. r jk =. ωj ε jk , ωk. (1). where ε jk is a multiplicative term accounting for inconsistencies in judgment that are usually observed in practice. Following Saaty (1977), it will be convenient to store these observations in an nin response matrix R=( r jk ), with the convention that. rkj =. 1 for all 1jj, kjn. r jk. 3.2. Gower Plots The interpretation of Gower plots constructed from the skew-symmetric matrix S=( s jk ) of size n with entries s jk = log(r jk ) . By definition, the singular value of a matrix A of rank n are the positive square roots of the eigenvalues of the symmetric matrix A' A , where' denotes transposition. In the special case where A is skew-symmetric, that is when A'=-A, the singular values of the matrix A are also 9.

(27) equal to the norm of its pure imaginary eigenvalues. As − λ must then be an eigenvalue of A whenever λ is, the matrix typically has n / 2 singular values of multiplicity 2 when n is even, and an additional singular equal to 0 when n is odd. Let λ1 ≥ ... ≥ λm ≥ 0 (and λm+1 = 0 if required) denote these values, with m standing for the integer part of n / 2 . For j = 1,..., m , let also U 2 j −1 and U 2 j be orthonormal. eigenvectors of A'A corresponding to λ2j . It is then possible to write A in the form m j =1. λ j (u 2 j −1u 2' j − u 2 j u 2' j −1 ) ,. (2). as a sum of m elementary rank 2, skew-symmetric matrix. This is a special case of the singular value decomposition theorem for arbitrary complex matrices, as described in Ch7 of the book by Horn and Johnson (1985). From a classical theorem of Wckart and Young (1936), the first l terms of Equation (2) give the best least-squares fit of rank 2l to A. In particular, the matrix A* = λ1 (UV ' −VU ' ) , with U = U 1 and V = U 2 provides the best approximation of rank 2. Plotting the vectors U = (u1 ,..., u n )' and V = ( v1 ,..., v n )' as n points Pj = (u j , v j ) in the plane should thus provide a reasonable two-dimensional representation of A. Such a graphical display, unique up to a rotation if λ1 > λ2 , is referred to herein as a Gower plot (Gower 1977, Constantine and Gower 1978). Although independent constructions that are essentially to his abound in the literature (cf., e.g., Escoufier and Grorud 1980, Chino 1990, Harshman and Lundy 1990). As pointed out by many of these authors, an adequate measure of the faithfulness of the graphical representation of A is then provided by the proportion of the variability in A that is captured by A*, viz.,. ν=. A* A. =. λ12 m. λ2 j =1 j. ,. (3). 3.2.1 Cardinal Gower Plot General guidelines for the interpretation of Gower plots can find in the paper of Gower (1977) or in the follow-up article by Constantine and Gower (1978). As 10.

(28) mentioned by these authors, the (j,k)th entry a *jk of A* is proportional to the signed area u j vk − v j u k of the parallelogram subtended at the origin by points Pj and Pk on the graphical display. Indeed if θ jk represents the directed angle between Pj and Pk , one has. a *jk = λ1 (u j vk − v j u k ) = λ1 Pj Pk sin(θ jk ). (4). This relationship can be exploited to show that the angle θ jk between two points Pj and Pk of a cardinal Gower plot is indicative of item j and k’s ordinal ranking, while the area between the points reveals something about the response’s degree of preference for one item over the other. The conclusions derived from the graph will be reliable, provided that the plot explains a large proportion of the variability contained in the data. The degree of faithfulness of the graphical representation will be measured by the quantity v defined in (3) with an appropriate choice of skew-symmetric matrix A. When the agent’s responses are cardinally consistent, the construction yields a set of points P1 = (u1 , v1 ),...., Pn = (u n , vn ) on a straight line that does not cross the origin. Lack of collinearity easily can be spotted and provides a means of detecting delinquent comparison, provided the data were ordinally consistent to begin with. A cardinal inconsistency is said to have occurred if a set of a set of distinct alternative 1j,k,ln can be found for which the identity r jl = r jk × rkl do not hold.. 3.2.2 Ordinal Gower Plot While angles and areas in the Gower plot of S respectively contain information about the ordinal and cardinal behavior of a respondent’s set of judgment, it is also possible to exploit Gower’s method to concentrate exclusively on the ordinal structure of a response matrix. Applying his technique to the tournament matrix T derived from R through Equation (1) can do this. To check more specifically for ordinal consistency, one can draw a Gower plot of the 11.

(29) nn tournament matrix (1 denotes a victory and 0stands for a defeat) T= (t jk ) defined by 1 if r jk > 1, t jk =. 0 if r jk = 1, -1 if r jk < 1,. Ideally, the points P1 ,...., Pn should be equidistant from the origin, arranged counterclockwise in order of preference with in a 180-defree arc. This is what happens when the tournament matrix T (or its parent R) is ordinally consistent. At the other extreme, the points P1 ,...., Pn will correspond to the vertices of a regular polygon inscribed in a circle centered at the origin in the case where T is a maximally intransitive tournament. An ordinal inconsistency occurs if, in addition, the implication r jk > 1 and rkl > 1. r jl > 1. is violated for some 1j,k,ln. 3.3. Applying Gower's technique to AHP The above methodology is not directly applicable to the AHP context, where a respondent's judgments rjk are coded on a ratio scale. However, a natural skew-symmetric to which Gower's technique can be applied is obtained when the responses are linearized via the transformation s jk = log(r jk ) . The Gower plot of S=( s jk ), hereafter referred to as a cardinal Gower plot, provides much insight into the structure of a set of responses. However, such plots are naturally subject to the strong interplay between ordinal and cardinal aspects of individual judgments in AHP that was underscored in the work of Genest etal. (1993). To focus on the ordinal structure 12.

(30) of the responses, one may thus wish to apply Gower's technique to tournament matrix associated with R. The resulting graph can be regarded as a complementary tool to the graphical display developed by Genest and his collaborators in their paper.. 3.4. Cardinal / Ordinal Gower Plot Measure 3.4.1 Cardinal Gower Plot Measure Let R = ( r jk ) be an n x n response matrix with entries r jk > 0 satisfying rkj = 1 / r jk for all 1 ≤ j , k ≤ n . Define the skew-symmetric S=( s jk ) with entries s jk = log(r jk ) , 1 ≤ j, k ≤ n , and let S * = ( s *jk ) = λ1 (u j v k − v j uk ) be its best least-squares approximation of rank2. Let also P1 = (u1 , v1 ),..., Pn = (u n , v n ) be the points appearing on the Gower S. (a) If P1 ,...., Pn lie within a 180 degree arc, then S* is ordinally consistent in the sense that. S *jk > 0 and S kl* > 0. S *jl > 0 ,. (5). for all possible choices of indices 1 ≤ j , k , l ≤ n . In that case, one has θ jk > 0 if and only if Pj is strictly preferred to Pk in the ordinal ranking implied by S*. (b) If P1 ,...., Pn are collinear, then S* is cardinally consistent, in the sense that. s *jk + skl* = s *jl. (6). for all possible choices of indices 1 ≤ j , k , l ≤ n . (c) The response matrix R is cardinally consistent if and only if P1 ,...., Pn are collinear and v = S * / S = 1. 13.

(31) In the view of part (a) of the above proposition, the fact that a point Pk lies within 180 degrees counterclockwise from another point Pj may be inferred to imply that item j is preferred to item k by the respondent, insofar as S* approximates S well. Thus, if R is ordinally transitive, and if the least-squares approximation is sufficiently good that. s jk >0 ⇔ s *jk > 0. (7). Pk would lie within 180 degrees counterclockwise of Pj on the Gower plot of R. In practice, it may be that equation (7) does not hold, or that S* be ordinally consistent while R is not. In fact, the preference relation induced by the θ jk ' s will not necessarily be transitive, because S* is not always ordinally consistent. As a result, there may sometime exist point Pj ,Pk and Pl such that θ jk < 180 , θ kl < 180 and. θ jl < 180 . Outside the conditions of part (b) of the proposition, the signed areas of the parallelograms subtended by pairs of points (Pl , Pk ) and (Pk , Pl ) would not generally add up to the area of the parallelogram corresponding to (Pj , Pl ). This lack of transitivity, imply by a lack of collinearity in the display, may thus be taken as a rough indication of cardinal inconsistency in the respondent's judgments. It should be emphasized that the ordinal or cardinal consistency of S* does not necessarily extend to R, unless v = 1 . 3.4.2 Ordinal Gower Plot Measure 3.4.2.1 Tournament Matrix Let B = (b jk ) be an n × n skew-symmetric matrix with entries b jk = 1 for all j < k . For n odd, let also C = ( c jk ) be another skew-symmetric matrix with entries. c jk − ( −1) j +k +1 for arbitrary j < k . (a) A tournament matrix T is said to be ordinally consistent if there exists a permutation matrix P such that T = PBP ' . (b) A tournament matrix T is said to be maximally intransitive if there exists a permutation matrix P such that T = PCP ' 3.4.2.2 Basic Features of Ordinal Gower Plots 14.

(32) Let R = ( r jk ) be an n x n response matrix with entries r jk > 0 satisfying rkj = 1 / r jk for all 1 ≤ j , k ≤ n . Let also P1 = (u1 , v1 ),..., Pn = (u n , v n ) be the points appearing on the Gower plot of the skew-symmetric matrix T defined as per equation (1). (a) If R is ordinally consistent, the points P1 ,...., Pn are then located on a circle centered at the origin and arranged counterclockwise in order of preference; the angle between two consecutive points is then equal to 180 / n degrees, so that all the Pj' s lie in a half-plane determined by a line going through the origin. (b) If R is maximally intransitive, the points P1 ,...., Pn then correspond to the vertices of a regular polygon inscribed in a circle of radius origin.. 2 / n centered at the. (c) The index v = T * / T = 1 is bounded above by cot 2 (π / 2n / T , which. reduces to 2 cot 2 (π / 2n ) / n ( n − 1) ; when there are no ties, i.e., when t jk ≠ 0 for all l ≤ j ≠ k ≤ n ; the latter bound is achieved in the two extreme cases where R is either ordinally consistent or maximally intransitive. (d) If t jk ≠ 0 for all j ≠ k , the index v is bounded below by 2 / n ; this bound is achieved whenever (T + I n )' (T + I n ) = nI n , where I n stand for the identity matrix of size n . In view of the above proposition, the angles between the points on the Gower plot of a tournament matrix are again indicative of the items' preference ranking, provided that the proportion v = T * / T. of the variability in T explained by the graphical. display is sufficiently large. In particular, it follows from part (a) of the proposition that the relation. θ jk + θ kl = θ jl ,. (8). holds for all indices j, k , l when the tournament matrix is ordinally transitive, while part (b) of the result implies that the same relation is violated to an extreme degree when the tournament matrix is maximally intransitive. However, the distance from a point to the origin can also be regarded as a measure of the influence of that item on the ordering of the alternatives and, to a large extent, on the appearance of the plot. Matrices H = T + I n that meet the condition stated in part (d) are called skew 15.

(33) Hadamard. Such matrices are known not to exist unless n equals 1,2, or a multiple of 4. Explicit examples of skew Hadamard matrices exist for small multiple of 4 (e.g., n ≤ 4 × 40 ), but their existence for large multiples of 4 remains in doubt at present. The lower bound for v is reached when T is skew Hadamard of size 4n , in which case. λ1 = ⋅ ⋅ ⋅ = λm = 4n − 1 . However, this bound is clearly not the best possible under the assumption λ1 > λ2 required for Gower plots to be uniquely defined. It should be noted that condition. v>. 1 is sufficient but not necessary to insure that the definition of the display is 2. unambiguous. The determination of the best lower bound for v under the assumption λ1 > λ2 would seem to be a difficult problem.. 3.5 Data collection The questionnaire is designed based on the Likert method. Attitude is one of the most pervasive notions in all of marketing. It plays a pivotal role in the major models describing consumer behavior, as well as in many, if not most, investigations of consumer behavior that do not rely on a formal integrated model. This research we selected the factor category base on EKB model. 3.5.1. The Likert method The Likert method of summated ratings overcomes the previous criticisms about scoring and allowing an express of intensity of feeling. The method is both constructed and used in a slightly different way than equal-appearing intervals. The basic format of the scale for the summated ratings method is the same in both construction and use. Subjects are asked to indicate their degree of agreement or disagreement with each and every statement in a series by checking the appropriate cell. The researcher attempts to develop great many statements that reflect qualities of things about the object that possible influence a person’s attitude toward it. The method is quiet different, though, in terms of the judgment sample and what is asked of the subjects. A total attitude score can be calculated for each subject using the same scoring procedure. The procedure, known as item analysis, rests on the proposition that there should be consistency in the response pattern of any individual. If the individual has a very favorable attitude toward the object, the individual should basically agree with the favorable statements and disagree with the unfavorable ones 16.

(34) and vice versa.. 3.5.2. Attitude Measurement - Scales of Measurement To properly address the subject of attitude measure, it is necessary to define measurement and to briefly review the types of scales that can be used in measure. We summarize some of the more important features of these scales, which are briefing in the following. 1. Nominal Scale One of the simplest properties of the scale of number is identity. These numbers simply identify the individual assigned the number. With a nominal scale, the only permissible operation is counting. Thus, the “Mode” is the only legitimate measure of central tendency. 2. Ordinal Scale A second property of the scale of numbers is that of order. Note that the ordinal scale implies identity, since the same number would be used for all objects that are the same. For example, this assignment would still indicate the class level of each person and the relative standing of two persons when compared in terms of who is further along in the academic program. The difference in rank says nothing about the difference in academic achievement between two ranks. We can transform an ordinal scale in any way that we wish as long as we maintain the basic ordering of the object. The ordinal scale is thus said to allow any monotonic positive transformation of the assigned numerals, because the differences in numerals are void of meaning other than order. With ordinal scales, both the “Median” and “Mode” are permissible or meaningful measures of average. 3. Interval Scale A third property of the scale of numbers is that the intervals between the numbers are meaning full in the sense that the numbers tell us how far apart the objects are with respect to the attribute. This means that the differences can be compared. One classic example of an interval scale is the temperature scale. Suppose that the low temperature for the day was 400F and the high was 800F. We cannot say that the high temperature was twice as hot as the low temperature. Thus, we cannot compare the absolute magnitude of numbers when measurement is made on the basis of an interval scale. The comparison of intervals is legitimate with an interval scale because the relationships among the differences hold regardless of the particular constants 17.

(35) chosen for a and b when transforming an interval set of numbers. With an interval scale, the “Mean”, “Median”, and “Mode” are all meaningful measures of average. 4. Ratio Scale The ratio scale differs from an interval scale in that it possesses a natural or absolute zero, one for which there is universal agreement about its location. Height and weight are examples. With a ratio scale, the comparison of the absolute magnitude of the number is legitimate. This means that with a ratio scale we can compare intervals, rank objects according to magnitude, or use the numbers to identify the objects. Ratio scales only allow the proportionate transformation of the scale value and not the addition of an arbitrary constant as do interval scales. A proportionate transformation is of the form y=bx, where x again represents the original values and y the transformed values and b is some positive constant. The “Geometric mean” as well as the more usual arithmetic mean, median, and mode are meaningful measures of average when attributes are measured on a ratio scale. Table 3- Scales of Measurement. Scale. Basic Comparison. Typical Example. Measures of Average. Nominal. Indentity. Male-Female. Mode. Ordinal. Order. Preference for brands. Median. Interval. Comparison of absolute magnitude. Units sold number of purchasers. Geometric mean. 18.

(36) 4. Case Study I - Customer Preferences for Car Purchase For experiment on and explanation of the graphical technique, we design a survey of the customer preference for car through an online questionnaire to obtain the sample data (questionnaire refers to appendix). With those response data, we can carry out the data sheet and then result of the response matrix to apply ordinal and cardinal plot. Seven attributes being considered are Price, Color, Shape, Function, Disposition, Brand and Purpose. We perform the attributes as comparison between one attribute to the other in order to minimize the size of combinations. That is, we won't ask respondents to weight each attribute respectively. Therefore, the n(n-1)/2 data points will become a ratio of the form rjk =. ωj . As we mentioned previous, the ωk. questionnaire was designed with the Likert method. We predefine the weight of each response category under the consideration of attitude score ranking calculation. Since the data points are performed as a ratio, we use the “Geometric mean” as a meaningful measures of average when attributes are measured on a ratio scale. Result can be observed as the following data sheet, response matrix and Gower plots.. 4.1 Response data sheet (Overall condition) The upper portion of Table 4 shows the response category, predefined weight, data point and comparison attribute. The left portion of Table 4 shows the result of the calculation. The last column of Table 4 shows the sample size. Originally, we asked 60 respondents for answer 21 questions through the web. Then, we can only get 40 effective copies, because there is no supervisor to monitor those respondents to exactly complete the online questionnaire. Although sample size is only 40, we still 19.

(37) can get enough data points to process the experiment. And, you will find the technique works well to carry out the results. Notice that Table 4 is under overall condition, that is, we did not put any criterion to classify the data points such as age, sex and so on. Table 4-- Response data sheet of Cardinal Gower plot Disagree: Neither agree Response Category Strongly nor disagree: disagree:. Given Weight. Comparison Data Point Attribute Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Function Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. . 0. 1. 4. 0. 3. 4. 2. 15. 8. 14. 9. 6. 2. 5. 8. 0. 0. 0. 2. 4. 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(38) . . . . Geomean Tournament. 4.00. 2.00. 1.00. 0.50. 0.25. Cardinal. Agree: Strongly Agree:. . . . .

(39) . . .

(40) . . .

(41) .

(42) . .

(43) .

(44) . .

(45) .

(46). .

(47) . . .

(48) . . .

(49) . . .

(50) .

(51)

(52). .

(53) . . .

(54) . . .

(55) . . .

(56) . . .

(57) . . .

(58) . .

(59) . . . . . . . . . . . . . .

(60) . . .

(61) . . .

(62) . . . . The data in Table 5 was converted from the column named Tournament in Table 4. This is nonstandard use of the term "tournament matrix." The latter terminology usually refers to matrix in which "1" denotes a victory and "-1" stands for a defeat. For example, to find the value of Price vs. Color in Table 5, we can read off the value of " Price is important than Color" of Table 4. The value of " Price is important than Color" is 1 (because we take the result of Strongly Agree(20) + Agree(14) > Strongly Disagree(0) + Disagree(5) as Price is more important than Color). The rule of 20. . Size. .

(63) . .

(64) .

(65) calculation for Tournament is that when the sum of tendency to agree (includes Strongly Agree and Agree) for the comparison attribute is greater than the sum of tendency to disagree (includes Strongly Disagree and Disagree), we will put "1" to the category of tendency to agree. Contrary to the case that category of tendency to disagree put "-1" to the category of tendency to disagree. We take the response category of Neither agree nor disagree as "neural" and put "0" to this category. Table 5--Response Matrix for Ordinal Gower Plot Color Shape Function Disposition Brand Price Price Color Shape Function Disposition Brand Purpose . Purpose . The following we will describe how to obtain the response matrix for Cardinal Gower plot. As the mentioned previously, we calculate the data of cardinal plot by using “Geometric mean” as a meaningful measures of average because the attributes are measured on a ratio scale. We take the value of " Price is important than Color" in Table 6 as the example. The value is 2.338 ( 40 0.250 × 0.55 × 11 × 214 × 4 20 =2.34).. Notice that the value of Price vs. Price in Table 5 is 1, because this is skew-symmetric matrix. The Response Matrix for Cardinal Gower Plot is an n × n response matrix with entries r jk > 0 satisfying rkj = 1 / r jk for all 1 ≤ j, k ≤ n .. Table 6 -- Response Matrix for Cardinal Gower Plot Color Shape Function Disposition Brand Purpose Price Price Color Shape

(66)

(67) Function

(68) Disposition

(69) Brand

(70) Purpose . The Ordinal Gower Plot of Overall condition displays as Figure 2. We can inferred the ranking can from the ordinal Gower plot of Figure 2, the preference ordering would 21.

(71) thus seem to be Function> Purpose > Disposition > Brand > Price > Shape > Color. That is, customers will focus their attention on Function of car. Marketing people would propose their marketing plan according to this preference ranking and develop a vehicle, which focuses on function improvement or innovation. On the other hand, marketing people can also find the market position of their own product and draft a proper marketing plan for product promotion. Figure 2-- Ordinal Gower Plot of Overall condition. As the result of Figure 2-- Ordinal Gower Plot of Overall condition, we know that the preference ordering would thus seem to be Function> Purpose > Disposition > Brand > Price > Shape > Color. However, while we check the cardinal ranking of Brand, Price and Disposition in Figure 4-Cardinal Gower plot of overall condition, we can find that the intensity of those three preferences turn into a ambiguous status. As we have mentioned in the previous article, the data will lack of collinearity in the display while cardinal inconsistency is not hold for which the identity r jl = r jk × rkl . We can take this as a rough indication of cardinal inconsistency in the respondent’s judgment. Although, it's lack of cardinal consistency, we still can take the result of ordinal Gower plot. In this case, we have found that ordinal Gower Plot has a great shape to support the result of preference ranking of the market: Function> Purpose > Disposition > Brand > Price > Shape > Color. Then, we can learn the specific meaning of those preferences in ambiguous status from cardinal Gower Plot. That is, there is only few data points to support the preference ranking of Disposition > Brand 22.

(72) > Price. It means that the preference (ex: Disposition win Brand) win by a narrow edge in Tournament. We can brief that lack of cardinal consistency is because the data points of one attribute would not exactly to be proportional to the data points of the other attribute. The result of lack of cardinal consistency did mean something to marketing but not much. As everyone knows, a company designs the product or provides the service is to satisfy the great majority of customers in the market. There is always exception. Therefore, ordinal Gower plot provides the general information of the market and cardinal Gower plot provides the specific information to check the detailing. Market people could make the judgment for their marketing plan while they have learned the information. They can focus on those preferences (ex: Function and Purpose) that have high degree of intention and ignore those preferences that are in ambiguous status for product design. Or, they can pay attention to the ambiguous preference but not much and make some change in their marketing plan. Whatever the actions the marketing people take, the actions are according to the market status. Ordinal and cardinal Gower plots did provide the solid result of the market status for sure. Figure 3- Diagrammatic explanation to Ordinal Gower plot of overall condition. Prefer. Half-plane determined. (0,0). by a line going through the origin 180 degree line through origin. Hate. 23.

(73) As we describe Figure3, the points are located on a circle centered at the origin and arranged counterclockwise in order of preference; the angle between two consecutive points is then equal to 180 / n degrees, so that all the points lie in a half-plane determined by a line going through the origin. The following we will display how we turn Table5 Matrix into Figure 2 Ordinal Gower Plot: r=[ 0. 1. 1. -1. -1. -1. -1. -1. 0. -1. -1. -1. -1. -1. -1. 1. 0. -1. -1. -1. -1. 1. 1. 1. 0. 1. 1. 1. 1. 1. 1. -1. 0. 1. -1. 1. -1. -1. 0. -1. 1. -1. 1. 1. 0]. 1 1. 1 1. a=r [u,s,v]=svd(a); The “svd” is singular value decomposition. The data for X-axis from [u]=U [-0.5345 -0.3333 -0.4816 0.1189 -0.3333 -0.4816 -0.1189] The data for Y-axis from [v]=V [0.0000 -0.4179 -0.2319 0.5211 0.4179 0.2319 0.5211]. 24.

(74) Figure 4-Cardinal Gower plot of overall condition. Figure 5 is a diagrammatic explanation that is base on the previous description of the method to judge if the cardinal Gower plot is cardinal consistency. When the agent’s responses are cardinally consistent, the construction yields a set of points on a straight line that does not cross the origin.. 25.

(75) Figure 5- Diagrammatic explanation to Cardinal Gower plot of overall condition. A straight line that does not cross the origin. The following we will display how we turn Table6 Matrix into Figure 4 Cardinal Gower Plot: r=[1. 2.34. 1.15. 0.71. 1.07. 0.93. 0.60;. 0.43. 1. 0.73. 0.45. 0.63. 0.47. 0.50;. 0.87. 1.37. 1. 0.68. 0.90. 0.73. 0.52;. 1.41. 2.22. 1.46. 1. 2.22. 2.18. 1.49;. 0.93. 1.60. 1.11. 1.07. 2.11. 1.65. 2.00. 1.37 1.93. 0.67. 0.45. 1. 1.13. 0.78;. 0.46. 0.89. 1. 0.73;. 1.27. 1.37. a=log(r) [u,s,v]=svd(a); The “svd” is singular value decomposition. The data for X-axis from [u]=U [-0.4074 -0.5223 -0.3656 0.0280 26. 1;].

(76) -0.3899 -0.4751 -0.2213] The data for Y-axis from [v]=V [0.1723 -0.5037 -0.0970 0.6534 0.1235 0.1976 0.4753]. 4.2 Response data sheet (Breakdown in certain criteria) The following we will introduce the Gower plot by data breakdown in certain criteria: Age, Sex and intention to buy a car within 2 years. In previous paragraph, we show the power of Gower plot in overall market status analysis. We hereby demonstrate the explanation of Gower plot for some certain group of customers. The calculation method is same as 4.1. The difference between 4.1 and 4.2 is that we should classify the data with the certain criteria and then calculate the classified data to come out the ordinal and cardinal Gower plots.. 27.

(77) 1. Table 7- Response data sheet of Age below 20 Strongly. Reponse Category disagree: Point Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Fuction Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. 0.25. Neither agree Strongly Disagree: nor disagree: Agree: Agree: 0.50. 1.00. 2.00. 0 . . 0 . . 0. . . . . . . Ordinal. Geomean Tournament. . . 0. 4.00. Cardinal. . Sample Size. . . . . . . . . .

(78) !. " . . 1 . . . #

(79) !. " . . 0 . . . . . . 0 . . . . . 0 . . . . . 0 . . 0 . . . . . . #

(80) !. " . . . . . . 0 . . 0 . . . . . . 0 . . . . . . #

(81) !. " . . . . 0 . . . . . 0 . . 1 . . . #

(82) !. " . . 1 . . . #

(83) !. " . . 1 . . 0 0 0 . 28. . . . . . . #

(84) !. " . #

(85) !. " . . #

(86) !. " . . . . . .

(87) 2. Table 8- Response data sheet of Age between 21-30 Strongly. Reponse Category disagree: Point Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Fuction Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. 0.25. Neither agree Strongly Disagree: nor disagree: Agree: Agree: 0.50. 1.00. 5 $. 0 ). 0 (. 1. 2.00. %. +. %&% ' .(.. * +. ++. /. %0+' ,1/. +. ++. %. .. (. %. 0 ,. %. 0 (. +. 0 ,. +. 0. 1. (. 1 ). +. 1 +. 5 ). 4. +&+' (/%. +. ++. ). (3+' /$(. +. ++. ). (3+' $)$. +. ++. + +. 1. ,. ,. (. ,. , (. -. +. 0 -. ) $. ,. ++. $. $. -. +. ++. %. 4. %0+' -).. ++. $. %. ++. +. -. (. ++. * +. (3+' 1/1. -. +. 0. * +. %&% ' 1,1. +. % +. %2% ' $1). %0+' (/%. $. -. 29. -. -. 0. 0. ++. ,. 1. Size. * +. (. (. Geomean Tournament. Sample. %&% ' ($). +. /. Ordinal. % ,. 2. 1. 4.00. %. Cardinal. +&+' %%%. %. %&% ' 1,1. * +. ++. +&+' ))-. +. ++. (3+' /$(. +. ++. %&% ' -4+ -. * +. ++. %&% ' )((. * +. ++. %&% ' $-(. * +. ++. %&% ' .(.. %. %0+' ,%1 (. ++. ,3+' ,%1. ++ + +. ++ ++.

(88) 3. Table 9- Response data sheet of Age between 31-40 Strongly. Reponse Category disagree: Point Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Fuction Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. 0.25. Neither agree Strongly Disagree: nor disagree: Agree: Agree: 0.50. 1.00. 13. 2.00. /. 4 .. 0 ). 3. % +. ). +. 3. ,. 1. ,. 1 (. 0 ,. +. 0. ,-. %5+' +,,. ,-. +. + -. )3+' /(+. +. ,-. + (. .&,' (++. +. ,-. ,%. ,3+' .-(. +. + $. (3+' $1,. + ,. +&+' +1.. +. + -. (3+' -4+ -. (. + . +. .. ). + 1. 4 $. 2 ). 0 .. -. ,. $. ( + (. +. ,-. +. +. 8. +. /6,' ++ .. 1. 30. ,-. +. + -. ,. 10. * +. (3+' )1/. +. +. 1. 0. %&% ' 1-+. + ). +&+' ,,-. -. 10. ++. ,3+' )1/. -. 0. ,-. + (. %. 0. * +. + /. -. 0. Size. %&% ' -,%. ,3+' ,,-. +. -. Geomean Tournament. Sample. + (. (. 2. Ordinal. (. +). -. 2. 4.00. +. Cardinal. ,-. +. ,+. ,-. +. ,+. ,+ +. ,,-. ,3+' 1(-. +. ,-. %&% ' )+ ). * +. ,-. %&% ' - 1$. * +. ,-. %&% ' ).). * +. ,-. .. ,7% ' ./,. +. ,-. + (. (3+' ((). +. ,-. /. )3+' ,$%. +. ,-.

(89) 4. Table 10- Response data sheet of Age between 41-50 Strongly. Reponse Category disagree: Point Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Fuction Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. 0.25. Neither agree Strongly Disagree: nor disagree: Agree: Agree: 0.50. 1.00. 2. 2.00. +. 1. +. ,. 2. %. +. %. 1. 4.00. %. %. +. ,. %. ,. Cardinal. Ordinal. Geomean Tournament. Sample Size. %&% ' )%%. * +. %&% ' ).). * +. -. %&% ' /%/. %. -. %2% ' 1-+. -. %. -. 1 %. %. ,. +&+' -4+ -. +. -. 0 %. %. (. +8,' (/1. +. -. 0 %. %. 0 %. %. 1 %. 0 %. %. 0 %. %. 1 %. %. ( % +. 0 +. 1 +. 0. +. +. 4 %. 0 %. 2 +. +&+' +1.. +. +. -. +. (&(' ($-. +. -. +. ,3+' $1,. +. -. +&+' -4+ +&+' %%% +. %. % %. % %. -. %. -. + %. 3. +. +. %. %. +. ,&,' 1,1. ,. +. +8,' (/1 -&- ' %%%. +. +. -. %. -. (&(' ($-. +. -. %&% ' ,./. * +. %&% ' ,)%. * +. %6,' %%%. +. %&% ' )%%. * +. -. 1 %. %. ,. +&+' -4+ -. +. -. 0 %. %. ,. ,&,' 1,1. +. -. 31.

(90) 5. Table 11- Response data sheet of Female Strongly. Reponse Category disagree: Point Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Fuction Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. 0.25. Neither agree Strongly Disagree: nor disagree: Agree: Agree: 0.50. 1.00. 11 ). 3 $. 2.00 %. + +. 2. /. (. 3. +. ). .. %. 3 (. 1 ,. 1 (. .. , + (. %. 0 (. %. 0 ,. %. 2. 2 -. + /. /6,' ()-. (. , (. ). + / +. + /. $. 17,' + /%. +. + /. + ,. (3+' .,%. +. + /. (3+' ((%. +. 1. +. $. $. +. (3+' + //. %. + %. + /. %0+' % 1). $. (. 0. +. )3+' )%(. (. 0. 32. + /. (. -. + /. +. + ). 3. + / +. +&+' ((%. +. 6. +. + %. + ,. /. ,3+' ,//. + /. 1. $. 8. + /. +. ,. 10. + /. * +. +. 1. %. * +. %&% ' 1+ $. +&+' )$$. +. +. %&% ' (-$. ,3+' % 1). -. 1. Size. /. /. ( -. Geomean Tournament. Sample. + (. +. 2. Ordinal. %2% ' .,,. , %. 0. 1. 4.00. Cardinal. + / +. + /. (3+' 1-(. +. + /. %&% ' (-$. * +. + /. %&% ' -%1. * +. + /. %&% ' )$). * +. + /. +9% ' .,,. %. + /. ,3+' )$$. +. + /. (3+' --(. +. + /.

(91) 6. Table 12- Response data sheet of Male Strongly. Reponse Category disagree: Point Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Fuction Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. 0.25. Neither agree Strongly Disagree: nor disagree: Agree: Agree: 0.50. 1.00. 9 2. 2.00. .. +. ). +. + %. +. 0 2 . -. ,. 1 ,. 1. .. +. 4. 4.00 -. +. -. ,. Cardinal. Ordinal. Geomean Tournament %&% ' )%%. * +. Sample Size ,(. +9% ' .+ -. * +. ,(. + ). ,3+' ),). +. ,(. ++. %2% ' .-,. + ,. +&+' %$,. + $. (3+' /,%. + ). +&+' (.(. + ,. 17,' +,-. +. ,(. %. ,( +. ,(. +. ,( +. ,(. 0 (. 0 (. + -. (3+' $$.. +. ,(. 0 ,. +. + -. $&,' %$+. +. ,(. 0 ,. +. + -. $&,' %$+. +. ,(. 0 ). +. + -. (3+' )/,. +. ,(. 0. % (. + % (. 0 ,. +. 1. , +. 9 1. + ) $. $. +. 7 ). +. ) + %. 1. 3. +. $. 0. + %. 33. ,3+' +,1 ,3+' ),). %. 11. 4. + % + ). %. +. +. ,(. +. ,(. )&,' %%%. +. ,(. %&% ' )-/. * +. ,(. %&% ' )%%. * +. ,(. %&% ' /$,. * +. ,(. .. +9% ' 1$%. * +. ,(. ++. ,3+' %.). +. ,(. /. $3+' (+ ,. +. ,(.

(92) 7. Table 13- Response data sheet of Plan to buy a car within 2 year Strongly. Reponse Category disagree: Point Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Fuction Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. 0.25. Neither agree Strongly Disagree: nor disagree: Agree: Agree: 0.50. 1.00. 10. + %. 2. + %. 2.00. +. 1 ). 2. +. $. +. 3 2. 0 -. 2 -. (. 1 $. 0. + ,. 2 (. 0. +. ,-. ++. 1:+' .-(. +. ,-. 34. ,-. + -. ,3+' ((). + (. .&,' (++. +. ,-. +. ,-. + $. -0+' 1(-. +. ,-. + ,. (3+' ((). +. ,-. + %. ,3+' %).. ,. + -. (3+' -)$. +. ,-. -0+' .-(. +. ,-. %&% ' -).. * +. ,-. + / +. (. + +. ) + %. -. . + -. ,-. ,. +. 1. ,-. +. 1. -. +. +&+' ((). %. $ -. ,-. (3+' )1/. +. 10. 3. ,-. +. + -. ,. 6. * +. ,3+' -4+ -. + $. + %. 7. %&% ' .+ /. +. %. 0 10. ++ + ). %5+' +)). ,-. +. ,. 0. * +. (3+' +)). +. ,. Size. %&% ' -).. ++. -. 0. Geomean Tournament. Sample. + ,. -. Ordinal. (. +). $. 1. 4.00. +. Cardinal. + -. ,-. %&% ' )+ ). * +. ,-. %&% ' /-.. * +. ,-. +9% ' 1$$. %. ,-. ,3+' ,$% /. %. )3+' ,./. +. ,+. ,-.

(93) 8. Table 14- Response data sheet of No plan to buy a car within 2 year Strongly. Reponse Category disagree: Point Price is important than Color Price is important than Shape Price is important than Function Price is important than Disposition Price is important than Brand Price is important than Purpose Color is important than Shape Color is important than Fuction Color is important than Disposition Color is important than Brand Color is important than Purpose Shape is important than Function Shape is important than Disposition Shape is important than Brand Shape is important than Purpose Function is important than Disposition Function is important than Brand Function is important than Purpose Disposition is important than Brand Disposition is important than Purpose Brand is important than Purpose. 0.25. Neither agree Strongly Disagree: nor disagree: Agree: Agree: 0.50. 1.00. 10. 2.00. -. 4.00. %. ,. Cardinal. Ordinal. Geomean Tournament. Sample Size. %&% ' (1$. * +. +$. 3 $. +. ). +9% ' 1%). * +. +$. 1 (. +. .. ,3+' -4+ -. +. +$. %2% ' $//. * +. +$. %&% ' .)1. +. +$. 3 1. %. 4. ). +. (. 1. 0 ,. %. + (. +&+' /)$. +. 1 (. %. ++. +&+' -4+ -. +. .. /6,' /%.. 0 %. %. +$ +$ +. +$. 0 ,. +. /. $&,' % 1.. +. +$. 0 (. +. /. )3+' 1(-. +. +$. 0 +. 1. %. ,. 1. % ,. 1. (3+' $1,. +. %0+' +1.. +. 1 ,. 1 +. 9 -. /. %. + % (. ). +. -. +. (. 1 1. 0 ). 0 ). (. %. $ ,. / +. +. +$ +$ +$. ,3+' ,-,. %. 9. 35. )&,' ,/1. + % ). -. 6. + %. $. +. +$. -0+' .+ ). +. +$. %&% ' -(.. * +. +$. %&% ' (1$. * +. +$. %&% ' )$.. * +. +$. +9% ' .+ /. * +. +$. ,3+' ,./. +. +$. -0+' -//. +. +$.

(94) Response Matrix for Ordinal Gower Plot 1. Table15- Response Matrix for Ordinal Gower Plot of Age below 20 . Price. Price Color Shape Function Disposition Brand Purpose. Color 01 -1 -1 -1 1 1 -1. Shape 1 01 -1 -1 -1 1 -1. Function 1 1 01 -1 -1 1 -1. Disposition Brand -1 -1 1 -1 1 -1 0-1 -1 0-1 1 1 1 1 1. Purpose 1 1 1 -1 -1 01 0 -1. 2. Table16- Response Matrix for Ordinal Gower Plot of Age between 21-30 . Price. Price Color Shape Function Disposition Brand Purpose. Color 0-1 1 1 -1 1 1 -1. Shape -1 01 -1 -1 -1 -1 -1. Function 1 1 00 0 1 -1 -1. Disposition Brand -1 -1 1 1 -1 1 0-1 -1 00 1 1 0 1 -1. Purpose 1 1 1 -1 1 01 0 -1. 3. Table17- Response Matrix for Ordinal Gower Plot of Age between 31-40 . Price Color Shape Function Disposition Brand Purpose. Price. Color 0-1 1 1 -1 -1 -1 -1. Shape -1 01 -1 -1 -1 -1 -1. Function 1 1 01 -1 -1 -1 -1. Disposition Brand 1 1 1 1 1 1 0-1 -1 01 1 1 -1 1 -1. 4. Table18- Response Matrix for Ordinal Gower Plot of Age between 41-50. 36. Purpose 1 1 1 -1 1 01 0 -1.