Multidimensional Scaling and Cluste Analysis - Multidimensional Scaling
Multidimensional Scaling and Cluste Analysis - Multidimensional Scaling
Multidimensional scaling basically involves two problems. First, the dimensions on which customers perceive or evaluate objects (organizations products, or brands) must be identified. For example, students must evaluate prospective colleges in terms of their quality, cost, distance from home and size. It would be convenient to work with only two dimensions, since the objects could then be portrayed graphically. However, it is not always possible to work with two dimensions, since additional dimensions sometimes are needed to represent customers' perceptions and evaluations. Second, objects need to be positioned with respect to these dimensions. The output of MDS is the location of the objects on the dimensions and is termed a perceptual map.
There are several approaches to multidimensional scaling. They differ in the assumptions they employ, the perspective taken, and the input data used. Figure 18-1 provides a categorization of the major approaches in terms of the input data. One set of approaches involves object attributes. I: the objects are colleges, the attributes could be faculty, prestige, facilities cost, and so on. MDS then will combine these attributes into dimensions such as quality. Another set of approaches bypasses attributes and considers similarity or preference relationships between objects directly. Thus two schools could be rated as to how similar they are or how much one is preferred over the other, without regard to any underlying attribute. The attribute-based approaches will be described first. A presentation of the nonattribute-based approaches will follow. Finally, the ideal-object concept will be discussed.
Attribute-Based Approaches
An important assumption of the attribute-based approaches is that we can identify attributes on which individuals' perceptions of objects are based. Let us start with a simple example. Suppose that the goal is to develop a perceptual map of the nonalcoholic beverage market.1 Suppose further that exploratory research has identified 14 beverages that seem relevant and nine attributes that are used by people to describe and evaluate these beverages. A group of respondents is asked to rate, on a seven-point scale, each of the beverages on the nine attributes. An average rating of the respondent group on each of the nine attributes, termed profile analysis in Chapter 9, would be of interest. However, it would be much more useful if the nine attributes could be combined into two or three dimensions or factors. Two approaches commonly are used to reduce the attributes to a small number of dimensions. The first is factor analysis, described in the previous chapter. A second is discriminant analysis.
Factor Analysis Since each respondent rates 14 beverages on nine attributes, he or she ultimately will have 14 factor scores on each of the emerging factors, one for each of the beverages. The position of each beverage in the perceptual space then will be the average factor score for that beverage. The perceptual map shown in Figure 18-2 illustrates this. Three factors, accounting for 77 percent of the variance, serve to summarize the nine attributes. Each of the beverages is then positioned on the attributes. Since three factors or dimensions are involved, two maps are required to portray the results. The first involves the first two factors, while the second includes the first and third factors. For convenience, the original attitudes also are shown in the maps as lines or vectors. The direction of the vectors indicates the factor with which each attribute is associated, and the length of the vector indicates the strength of association. Thus, in the left map, the "filling" attribute has little association with any factor, whereas in the right map, the "filling" attribute is strongly associated with the "refreshing" factor.
Discriminant Analysis Whereas the goal of factor analysis is to generate dimensions that maximize interpretability and explain variance, the goal of discriminant analysis is to generate dimensions (termed discriminant functions factors) that will discriminate or separate the objects as much as possible. As in factor analysis, each dimension is based on
a combination of the underlying attributes. However, in discriminant analysis, the extent to which an attribute will tend to be an important contributor toward a dimension depends on the extent to which there is a perceived difference among the objects on that attribute.
Comparing Factor and Discriminant Analysis Each of the approaches has advantages and disadvantages. Discriminant analysis identifies clusters of attributes on which objects differ. If all objects are perceived to be similar with respect to an attribute (such as the safety of an airline) then that attribute should not affect preference (such as the choice of an airline). Following that logic the discriminant analysis objective of selecting attributes that discriminate between objects seems sensible. A second useful characteristic of discriminant analysis is that it provides a test of statistical significance. The null hypothesis is that two objects actually are perceived identically. The test will determine the probability that the between-object distance was due simply to a statistical accident. A third quality of discriminant analysis is that it will identify a perceptual dimension even if it is represented by a single attribute.
In contrast, factor analysis groups attributes that are similar. If there are not several attributes representing a dimension, it will tend not to emerge in the factor analysis solution. Factor analysis is based on both perceived differences between objects and differences between people's perceptions of objects. Thus, it tends to provide a richer solution, use more of the attributes, and result in more dimensions. All perceptual dimensions are included, whether they discriminate between objects or not. Hauser and Koppelman conducted a study of shopping centers in which they compared several approaches to multidimensional scaling.2 They found that factor analysis dimensions provided more interpretive value than did those of discriminant analysis.
Introducing Importance Weights Both factor analysis and discriminant analysis ignore the relative importance of the individual attributes to customers. Myers and Tauber suggest that the attribute data be multiplied by importance weights and then be subjected to a factor analysis.3 As a result the attributes considered more important will have a greater tendency to be included in a factor analysis solution. They present a factor analysis perceptual map for snack food that included the dimensions of "convenience" and "nutrition." When that study was repeated, this time with importance weights introduced, a "child likes" dimension replaced the "convenience" dimension.
Correspondence Analysis In both factor analysis and discriminant analysis the variables are assumed to be interval scaled, continuous variables. A seven-point Likert scale (agree—disagree) would usually be used. However, often it is convenient to collect binary or zero-one data. Respondents might be asked to identify from an attribute list which ones desc: ibe a brand. The result will be a row of zeros and ones for each respondent. Or the respondent could be asked to pick three (or k) attributes that are associated with a brand or two (or k) use occasions that are most suitable for a brand. The result is again a row of zeros and ones for each respondent and each brand.
When the data consists of rows of zeros and ones reflecting the association of an attribute or other variable with a brand or other object, the appropriate MDS technique is termed correspondence analysis.4 Correspondence analysis generates as an output a perceptual map in which the elements of attributes and brands are both positioned.
Binary judgments are used in several contexts. First, if the number of attributes and objects is large, the task of scaling each object on each attribute may be excessive and unrealistic. Simply checking which attributes (or use occasions) apply to a given object may be a more appropriate task. Second, it may be useful to ask respondents to list all the attributes they can think of for a certain brand or to list all the objects or brands that would apply to a certain use occasion. For example, what snacks would you consider for a party to watch the Super Bowl? In that case, binary data would result and correspondence analysis would be the appropriate technique.
Nonattribute-Based Approaches
Attribute-based MDS has the advantage that attributes can have diagnostic and operational value and the dimensions can be interpreted in terms of their correlations with the attributes. Further, the Hauser and Koppelman study concluded that attribute data were easier for respondents to use and that dimensions based on attribute data predicted preference better than did dimensions based upon nonattribute data.5
However, attribute data also has several conceptual disadvantages. First, if the list of attributes is not accurate and complete, the study will suffer accordingly. The generation of an attribute list can be most difficult, especially when possible differences among people's perceptions are considered. Second, it may be that people simply do not perceive or evaluate objects in terms of underlying attributes. An object may be perceived or evaluated as a whole that is not decomposable in terms of attributes. Finally, attribute-based models may require more dimensions to represent them than would be needed by more flexible models, in part because of the linearity assumptions of factor analysis and discriminant analysis. These disadvantages lead us to the use of nonattribute data, namely, similarity and preference data.
Similarity Data Similarity measures simply reflect the perceived similarity of two objects in the eyes of the respondents. For example, each respondent may be asked to rate the degree of similarity of each pair of objects. The respondent is generally not told what criteria to use to determine similarity; thus, the respondent does not have an attribute list that implicitly suggests criteria to be included or excluded. In the following example, the respondent judged Stanford to be quite similar to Harvard.
The number of pairs to be judged for degree of similarity can be as many as n(n - 1) -=- 2, where n is the total number of objects. With 10 brands, there could be 45 pairs of brands to judge (although fewer could be used).
Although at least seven or eight objects should be judged, the approach is easier to illustrate if only four objects are considered. First, the results of the pairwise similarity judgments are summarized in a matrix, as shown in Figure 18-3. The numbers in the matrix represent the average similarity judgments for a sample of 50 respondents. Instead of similarity ratings, the respondents could be asked simply to rank the pairs from most to least similar. An average rank-order position then would replace the average similarity rating matrix. It should be noted, however, that rank ordering can be difficult if 10, or more objects are involved.
A perceptual map could be obtained from the average similarity ratings; however, it is also possible to use only the ordinal or "nonmetric" portion of the data. Thus, the knowledge that objects A and C in Figure 18-3
have an average similarity of 1.7 is replaced by the fact that objects A and C are the most similar pair. The conversion to rank-order information. Ordinal or nonmetric information is often preferred for several reasons. First, it actually contains about the same amount of information, in that the output usually is not affected by replacing interval! scaled or "metric" data with ordinal or nonmetric data. Second, the non-metric data often are thought to be more reliable.
The reader might be able to relocate the points differently and still satisfy the constraints so that the rankings of the distances in the map
Next, a computer program is employed to convert the rankings of similarity into distances in a map with a small number of dimensions, so thai similar objects are close together and vice versa. The computer will be programmed to locate the four objects in a space of two, three, or more dimensions, so that the shortest distance is between pair (A, C), the next shortest is pair (A, B), and the longest pair is (A, D). One possible solution that satisfied these constraints in two dimensions is the following: correspond to the rankings of the pairwise similarity judgments. This is because there are only a few points to move in the space and only six constraints to satisfy. With 10 objects and 45 constraints, the task of locating the points in a two-dimensional space is vastly more difficult, requiring a computer. Once a solution is found—the points are located in the space—it is unlikely that there will be a significantly different solution that still satisfies the constraints of the similarities matrix. Thus, we can argue that the intervally scaled nature of the distances between points really was hidden in the rank-order input data all the time.
The power of the technique lies in its ability to find the smallest number of dimensions for which there is a reasonably good fit between the input similarity rankings and the rankings of distance between objects in the resulting space. Usually, this means starting with two dimensions and, if this is not satisfactory, continuing to add dimensions until an acceptable fit is achieved. The determination of "acceptable" is a matter of judgment, although most analysts will trade off some degree of fit to stay with a two- or three-dimensional map because of the enormous advantages of visual interpretations. There are situations where more dimensions are necessary. This happened in a study of nine different types of sauces (mustard, catsup, relish, steak sauce, dressing, and so on). Most respondents perceived too many differences to be captured with two or three dimensions, in terms of either the types of foods the sauces would be used with or the physical characteristics of each sauce.6
A sample of 64 undergraduates provided similarity judgments for all 45 pairs of ten drinks including Coke, Diet Coke, 7-Up, Calistoga Natural Orange, and Slice. They were asked to rate the similarity of each pair such as Slice—Diet Coke on a nine-point scale. The two-dimensional solution is shown in Figure 18-4. Note that Slice is considered closer to Diet 7-Up than to 7-Up and Schweppes and Calistoga are separated even though they are very similar.
6James H. Myers and Edward Tauber, Market Structure Analysis (Chicago: American Marketing Association, 1977), p. 38.
The interpretation of the resulting dimension takes place "outside" the technique. Additional information must be introduced to decide why objects are located in their relative positions. Sometimes the location of the objects themselves can suggest dimensional interpretations. For example, in Figure 18-2 the location of the objects suggest dimension interpretations even without the attribute information. Thus the fruit punch vs. hot coffee object locations on the horizontal axis suggest a maturity dimension. In Figure 18-4, the objects on the horizontal axis indicate a cola—noncola dimension. The vertical axis seems to represent a diet/nondiet dimension because in both the cola group and in the noncola group the nondiet drinks tend to be lower than the diet drinks.
Ideal Objects
The concept of an ideal object in the space is an important one in MDS because it allows the analyst to relate object positioning to customer likes and dislikes. It also provides a means for segmenting customers according to their preferences for product attributes.
An ideal object is one the customer would prefer over all others, including objects that can be conceptualized in the space but do not actual!)' exist. It is a combination of all the customers' preferred attribute levels. Although the assumption that people have similar perceptions may be reasonable, their preferences are nearly always heterogeneous—their ideal objects will differ. One reason to locate ideal objects is to identify segments of customers who have similar ideal objects.
There are two types of ideal objects. The first lies within the perceptual map. For example, if a new cookie were rated on attribute scales such as
Very sweet Not at all sweet
Large, substantial Small, dainty
a respondent might well prefer a middle position on the scale.
The second type is illustrated by a different example. Suppose attributes of a proposed new car included then respondents would very likely prefer an end point on the scale. For instance, the car should be as inexpensive to buy and operate as possible. In that case, the ideal object would be represented by an ideal vector or direction rather than an ideal point in the space. The direction would depend on the relative desirability of the various attributes.
There are two approaches to obtaining ideal object locations. The first is simply to ask respondents to consider an ideal object as one of the objects to be rated or compared. The problem with this approach is that the conceptualization of an ideal object may not be natural for a respondent and the result may therefore be ambiguous and unreliable.
A second approach is indirect. For each individual, a rank-order preference among the objects is sought. Then, given a perceptual map, a program will locate the individual's ideal objects such that the distances to the objects have the same (or as close to the same as possible) rank order as the rank-order preference. The preferred object should be closest to the ideal. The second most preferred object should be farther from the ideal than the preferred but closer than the third most preferred, and so on. Often it is not possible to determine a location that will satisfy this requirement perfectly and still obtain a small number of dimensions with which an analyst would like to work. In that case, compromises are made and the computer program does as well as possible by maximizing some measure of "goodness-of-
fit."
Inexpensive to buy
Inexpensive to operate
Good handling
Expensive to buy Expensive to operate Bad handling
MDS Issues
Perceptual maps are good vehicles to summarize the position of brands and people in attribute space and, more generally, to portray the relationship between any variable or construct. It is particularly useful to portray the positioning of existing or new brands and the relationship of those positions to the relevant segments. There are a set of problems and issues in working with MDS, however:
1. When more than two or three dimensions are needed, the usefulness is reduced.
2. Perceptual mapping has not been shown to be reliable across different methods. Users rarely take the trouble to apply multiple approaches to a context to ensure that a map is not method specific.
3. Perceptual maps are static snapshots at a point of time. It is difficult from the model to know how they might be affected by market events.
4. The interpretation of dimensions can be difficult. Even when a dimension is clear it can involve several attributes and thus the implications for action can be ambiguous.
5. Maps are usually based on groups that are aggregated with respect to their familiarity with products, their usage level, and their attitude. The analysis can, of course, be done with subgroups created by grouping people according to their preferences or perception but with a procedure that is ad hoc at best.
6. There has been little study of whether a change in a brand's perception as reflected by a perceptual map will affect choice.
Multidimensional Scaling—A Summary
Application MDS is used to identify dimensions by which objects are perceived or evaluated, to position the objects with respect to those dimensions, and to make positioning decisions for new and old products.
Inputs Attribute-based data involve respondents rating the objects with respect to specified attributes. Similarity-based data involve a rank order c:~ between-object similarity that can be based on several methods of obtaining similarity information from respondents. Preference data also can provide the basis for similarity measures and generate perceptual maps from quite 4 different perspective.
Ideal points or directions are based upon either having the responden: conceptualize her or his ideal object or by generating rank-order preference data and using the data in a second stage of analysis to identify ideal points or directions.
Outputs The output will provide the location of each object on a limited number of dimensions. The selection of the number of dimensions is made on the basis of a goodness-of-fit measure (such as the percentage of variance explained in factor analysis) and on the basis of the interpretability of the dimensions. In attitude-based MDS, attribute vectors may be included tc help interpret the dimensions. Ideal points or directions may be an output in some programs.
Key Assumptions The overriding assumption is that the underlying data represent valid measures. Thus, we assume that respondents can compare objects with respect to similarity or preference or attributes. The meaning of the input data is rather straightforward; however, the ability and motivation of respondents to provide it is often questionable. A related assumption is that an appropriate context is used by the respondents. For some, a rank-order preference of beer could be based on the assumption that it was to be served to guests. Others might assume the beer was to be consumed privately.
With the attribute-based data, the assumption is made that the attribute list is relevant and complete. If individuals are grouped, the assumption that their perceptions are similar is made. The ideal object introduces additional conceptual problems.
Another basic assumption is that the interpoint distances generated by a perceptual map have conceptual meaning that is relevant to choice decisions.
Limitations A limitation of the attributed-based methods is that the attributes have to be generated. The analyst has the burden of making sure that respondents' perceptions and evaluations are represented by the attributes. With similarity and preference data, this task is eliminated, but the analyst then must interpret dimensions without the aid of such attributes, although attribute data could be generated independently and attribute-dimension correlations still obtained.
Multidimensional scaling basically involves two problems. First, the dimensions on which customers perceive or evaluate objects (organizations products, or brands) must be identified. For example, students must evaluate prospective colleges in terms of their quality, cost, distance from home and size. It would be convenient to work with only two dimensions, since the objects could then be portrayed graphically. However, it is not always possible to work with two dimensions, since additional dimensions sometimes are needed to represent customers' perceptions and evaluations. Second, objects need to be positioned with respect to these dimensions. The output of MDS is the location of the objects on the dimensions and is termed a perceptual map.
There are several approaches to multidimensional scaling. They differ in the assumptions they employ, the perspective taken, and the input data used. Figure 18-1 provides a categorization of the major approaches in terms of the input data. One set of approaches involves object attributes. I: the objects are colleges, the attributes could be faculty, prestige, facilities cost, and so on. MDS then will combine these attributes into dimensions such as quality. Another set of approaches bypasses attributes and considers similarity or preference relationships between objects directly. Thus two schools could be rated as to how similar they are or how much one is preferred over the other, without regard to any underlying attribute. The attribute-based approaches will be described first. A presentation of the nonattribute-based approaches will follow. Finally, the ideal-object concept will be discussed.
Attribute-Based Approaches
An important assumption of the attribute-based approaches is that we can identify attributes on which individuals' perceptions of objects are based. Let us start with a simple example. Suppose that the goal is to develop a perceptual map of the nonalcoholic beverage market.1 Suppose further that exploratory research has identified 14 beverages that seem relevant and nine attributes that are used by people to describe and evaluate these beverages. A group of respondents is asked to rate, on a seven-point scale, each of the beverages on the nine attributes. An average rating of the respondent group on each of the nine attributes, termed profile analysis in Chapter 9, would be of interest. However, it would be much more useful if the nine attributes could be combined into two or three dimensions or factors. Two approaches commonly are used to reduce the attributes to a small number of dimensions. The first is factor analysis, described in the previous chapter. A second is discriminant analysis.
Factor Analysis Since each respondent rates 14 beverages on nine attributes, he or she ultimately will have 14 factor scores on each of the emerging factors, one for each of the beverages. The position of each beverage in the perceptual space then will be the average factor score for that beverage. The perceptual map shown in Figure 18-2 illustrates this. Three factors, accounting for 77 percent of the variance, serve to summarize the nine attributes. Each of the beverages is then positioned on the attributes. Since three factors or dimensions are involved, two maps are required to portray the results. The first involves the first two factors, while the second includes the first and third factors. For convenience, the original attitudes also are shown in the maps as lines or vectors. The direction of the vectors indicates the factor with which each attribute is associated, and the length of the vector indicates the strength of association. Thus, in the left map, the "filling" attribute has little association with any factor, whereas in the right map, the "filling" attribute is strongly associated with the "refreshing" factor.
Discriminant Analysis Whereas the goal of factor analysis is to generate dimensions that maximize interpretability and explain variance, the goal of discriminant analysis is to generate dimensions (termed discriminant functions factors) that will discriminate or separate the objects as much as possible. As in factor analysis, each dimension is based on
a combination of the underlying attributes. However, in discriminant analysis, the extent to which an attribute will tend to be an important contributor toward a dimension depends on the extent to which there is a perceived difference among the objects on that attribute.
Comparing Factor and Discriminant Analysis Each of the approaches has advantages and disadvantages. Discriminant analysis identifies clusters of attributes on which objects differ. If all objects are perceived to be similar with respect to an attribute (such as the safety of an airline) then that attribute should not affect preference (such as the choice of an airline). Following that logic the discriminant analysis objective of selecting attributes that discriminate between objects seems sensible. A second useful characteristic of discriminant analysis is that it provides a test of statistical significance. The null hypothesis is that two objects actually are perceived identically. The test will determine the probability that the between-object distance was due simply to a statistical accident. A third quality of discriminant analysis is that it will identify a perceptual dimension even if it is represented by a single attribute.
In contrast, factor analysis groups attributes that are similar. If there are not several attributes representing a dimension, it will tend not to emerge in the factor analysis solution. Factor analysis is based on both perceived differences between objects and differences between people's perceptions of objects. Thus, it tends to provide a richer solution, use more of the attributes, and result in more dimensions. All perceptual dimensions are included, whether they discriminate between objects or not. Hauser and Koppelman conducted a study of shopping centers in which they compared several approaches to multidimensional scaling.2 They found that factor analysis dimensions provided more interpretive value than did those of discriminant analysis.
Introducing Importance Weights Both factor analysis and discriminant analysis ignore the relative importance of the individual attributes to customers. Myers and Tauber suggest that the attribute data be multiplied by importance weights and then be subjected to a factor analysis.3 As a result the attributes considered more important will have a greater tendency to be included in a factor analysis solution. They present a factor analysis perceptual map for snack food that included the dimensions of "convenience" and "nutrition." When that study was repeated, this time with importance weights introduced, a "child likes" dimension replaced the "convenience" dimension.
Correspondence Analysis In both factor analysis and discriminant analysis the variables are assumed to be interval scaled, continuous variables. A seven-point Likert scale (agree—disagree) would usually be used. However, often it is convenient to collect binary or zero-one data. Respondents might be asked to identify from an attribute list which ones desc: ibe a brand. The result will be a row of zeros and ones for each respondent. Or the respondent could be asked to pick three (or k) attributes that are associated with a brand or two (or k) use occasions that are most suitable for a brand. The result is again a row of zeros and ones for each respondent and each brand.
When the data consists of rows of zeros and ones reflecting the association of an attribute or other variable with a brand or other object, the appropriate MDS technique is termed correspondence analysis.4 Correspondence analysis generates as an output a perceptual map in which the elements of attributes and brands are both positioned.
Binary judgments are used in several contexts. First, if the number of attributes and objects is large, the task of scaling each object on each attribute may be excessive and unrealistic. Simply checking which attributes (or use occasions) apply to a given object may be a more appropriate task. Second, it may be useful to ask respondents to list all the attributes they can think of for a certain brand or to list all the objects or brands that would apply to a certain use occasion. For example, what snacks would you consider for a party to watch the Super Bowl? In that case, binary data would result and correspondence analysis would be the appropriate technique.
Nonattribute-Based Approaches
Attribute-based MDS has the advantage that attributes can have diagnostic and operational value and the dimensions can be interpreted in terms of their correlations with the attributes. Further, the Hauser and Koppelman study concluded that attribute data were easier for respondents to use and that dimensions based on attribute data predicted preference better than did dimensions based upon nonattribute data.5
However, attribute data also has several conceptual disadvantages. First, if the list of attributes is not accurate and complete, the study will suffer accordingly. The generation of an attribute list can be most difficult, especially when possible differences among people's perceptions are considered. Second, it may be that people simply do not perceive or evaluate objects in terms of underlying attributes. An object may be perceived or evaluated as a whole that is not decomposable in terms of attributes. Finally, attribute-based models may require more dimensions to represent them than would be needed by more flexible models, in part because of the linearity assumptions of factor analysis and discriminant analysis. These disadvantages lead us to the use of nonattribute data, namely, similarity and preference data.
Similarity Data Similarity measures simply reflect the perceived similarity of two objects in the eyes of the respondents. For example, each respondent may be asked to rate the degree of similarity of each pair of objects. The respondent is generally not told what criteria to use to determine similarity; thus, the respondent does not have an attribute list that implicitly suggests criteria to be included or excluded. In the following example, the respondent judged Stanford to be quite similar to Harvard.
The number of pairs to be judged for degree of similarity can be as many as n(n - 1) -=- 2, where n is the total number of objects. With 10 brands, there could be 45 pairs of brands to judge (although fewer could be used).
Although at least seven or eight objects should be judged, the approach is easier to illustrate if only four objects are considered. First, the results of the pairwise similarity judgments are summarized in a matrix, as shown in Figure 18-3. The numbers in the matrix represent the average similarity judgments for a sample of 50 respondents. Instead of similarity ratings, the respondents could be asked simply to rank the pairs from most to least similar. An average rank-order position then would replace the average similarity rating matrix. It should be noted, however, that rank ordering can be difficult if 10, or more objects are involved.
A perceptual map could be obtained from the average similarity ratings; however, it is also possible to use only the ordinal or "nonmetric" portion of the data. Thus, the knowledge that objects A and C in Figure 18-3
have an average similarity of 1.7 is replaced by the fact that objects A and C are the most similar pair. The conversion to rank-order information. Ordinal or nonmetric information is often preferred for several reasons. First, it actually contains about the same amount of information, in that the output usually is not affected by replacing interval! scaled or "metric" data with ordinal or nonmetric data. Second, the non-metric data often are thought to be more reliable.
The reader might be able to relocate the points differently and still satisfy the constraints so that the rankings of the distances in the map
Next, a computer program is employed to convert the rankings of similarity into distances in a map with a small number of dimensions, so thai similar objects are close together and vice versa. The computer will be programmed to locate the four objects in a space of two, three, or more dimensions, so that the shortest distance is between pair (A, C), the next shortest is pair (A, B), and the longest pair is (A, D). One possible solution that satisfied these constraints in two dimensions is the following: correspond to the rankings of the pairwise similarity judgments. This is because there are only a few points to move in the space and only six constraints to satisfy. With 10 objects and 45 constraints, the task of locating the points in a two-dimensional space is vastly more difficult, requiring a computer. Once a solution is found—the points are located in the space—it is unlikely that there will be a significantly different solution that still satisfies the constraints of the similarities matrix. Thus, we can argue that the intervally scaled nature of the distances between points really was hidden in the rank-order input data all the time.
The power of the technique lies in its ability to find the smallest number of dimensions for which there is a reasonably good fit between the input similarity rankings and the rankings of distance between objects in the resulting space. Usually, this means starting with two dimensions and, if this is not satisfactory, continuing to add dimensions until an acceptable fit is achieved. The determination of "acceptable" is a matter of judgment, although most analysts will trade off some degree of fit to stay with a two- or three-dimensional map because of the enormous advantages of visual interpretations. There are situations where more dimensions are necessary. This happened in a study of nine different types of sauces (mustard, catsup, relish, steak sauce, dressing, and so on). Most respondents perceived too many differences to be captured with two or three dimensions, in terms of either the types of foods the sauces would be used with or the physical characteristics of each sauce.6
A sample of 64 undergraduates provided similarity judgments for all 45 pairs of ten drinks including Coke, Diet Coke, 7-Up, Calistoga Natural Orange, and Slice. They were asked to rate the similarity of each pair such as Slice—Diet Coke on a nine-point scale. The two-dimensional solution is shown in Figure 18-4. Note that Slice is considered closer to Diet 7-Up than to 7-Up and Schweppes and Calistoga are separated even though they are very similar.
6James H. Myers and Edward Tauber, Market Structure Analysis (Chicago: American Marketing Association, 1977), p. 38.
The interpretation of the resulting dimension takes place "outside" the technique. Additional information must be introduced to decide why objects are located in their relative positions. Sometimes the location of the objects themselves can suggest dimensional interpretations. For example, in Figure 18-2 the location of the objects suggest dimension interpretations even without the attribute information. Thus the fruit punch vs. hot coffee object locations on the horizontal axis suggest a maturity dimension. In Figure 18-4, the objects on the horizontal axis indicate a cola—noncola dimension. The vertical axis seems to represent a diet/nondiet dimension because in both the cola group and in the noncola group the nondiet drinks tend to be lower than the diet drinks.
Ideal Objects
The concept of an ideal object in the space is an important one in MDS because it allows the analyst to relate object positioning to customer likes and dislikes. It also provides a means for segmenting customers according to their preferences for product attributes.
An ideal object is one the customer would prefer over all others, including objects that can be conceptualized in the space but do not actual!)' exist. It is a combination of all the customers' preferred attribute levels. Although the assumption that people have similar perceptions may be reasonable, their preferences are nearly always heterogeneous—their ideal objects will differ. One reason to locate ideal objects is to identify segments of customers who have similar ideal objects.
There are two types of ideal objects. The first lies within the perceptual map. For example, if a new cookie were rated on attribute scales such as
Very sweet Not at all sweet
Large, substantial Small, dainty
a respondent might well prefer a middle position on the scale.
The second type is illustrated by a different example. Suppose attributes of a proposed new car included then respondents would very likely prefer an end point on the scale. For instance, the car should be as inexpensive to buy and operate as possible. In that case, the ideal object would be represented by an ideal vector or direction rather than an ideal point in the space. The direction would depend on the relative desirability of the various attributes.
There are two approaches to obtaining ideal object locations. The first is simply to ask respondents to consider an ideal object as one of the objects to be rated or compared. The problem with this approach is that the conceptualization of an ideal object may not be natural for a respondent and the result may therefore be ambiguous and unreliable.
A second approach is indirect. For each individual, a rank-order preference among the objects is sought. Then, given a perceptual map, a program will locate the individual's ideal objects such that the distances to the objects have the same (or as close to the same as possible) rank order as the rank-order preference. The preferred object should be closest to the ideal. The second most preferred object should be farther from the ideal than the preferred but closer than the third most preferred, and so on. Often it is not possible to determine a location that will satisfy this requirement perfectly and still obtain a small number of dimensions with which an analyst would like to work. In that case, compromises are made and the computer program does as well as possible by maximizing some measure of "goodness-of-
fit."
Inexpensive to buy
Inexpensive to operate
Good handling
Expensive to buy Expensive to operate Bad handling
MDS Issues
Perceptual maps are good vehicles to summarize the position of brands and people in attribute space and, more generally, to portray the relationship between any variable or construct. It is particularly useful to portray the positioning of existing or new brands and the relationship of those positions to the relevant segments. There are a set of problems and issues in working with MDS, however:
1. When more than two or three dimensions are needed, the usefulness is reduced.
2. Perceptual mapping has not been shown to be reliable across different methods. Users rarely take the trouble to apply multiple approaches to a context to ensure that a map is not method specific.
3. Perceptual maps are static snapshots at a point of time. It is difficult from the model to know how they might be affected by market events.
4. The interpretation of dimensions can be difficult. Even when a dimension is clear it can involve several attributes and thus the implications for action can be ambiguous.
5. Maps are usually based on groups that are aggregated with respect to their familiarity with products, their usage level, and their attitude. The analysis can, of course, be done with subgroups created by grouping people according to their preferences or perception but with a procedure that is ad hoc at best.
6. There has been little study of whether a change in a brand's perception as reflected by a perceptual map will affect choice.
Multidimensional Scaling—A Summary
Application MDS is used to identify dimensions by which objects are perceived or evaluated, to position the objects with respect to those dimensions, and to make positioning decisions for new and old products.
Inputs Attribute-based data involve respondents rating the objects with respect to specified attributes. Similarity-based data involve a rank order c:~ between-object similarity that can be based on several methods of obtaining similarity information from respondents. Preference data also can provide the basis for similarity measures and generate perceptual maps from quite 4 different perspective.
Ideal points or directions are based upon either having the responden: conceptualize her or his ideal object or by generating rank-order preference data and using the data in a second stage of analysis to identify ideal points or directions.
Outputs The output will provide the location of each object on a limited number of dimensions. The selection of the number of dimensions is made on the basis of a goodness-of-fit measure (such as the percentage of variance explained in factor analysis) and on the basis of the interpretability of the dimensions. In attitude-based MDS, attribute vectors may be included tc help interpret the dimensions. Ideal points or directions may be an output in some programs.
Key Assumptions The overriding assumption is that the underlying data represent valid measures. Thus, we assume that respondents can compare objects with respect to similarity or preference or attributes. The meaning of the input data is rather straightforward; however, the ability and motivation of respondents to provide it is often questionable. A related assumption is that an appropriate context is used by the respondents. For some, a rank-order preference of beer could be based on the assumption that it was to be served to guests. Others might assume the beer was to be consumed privately.
With the attribute-based data, the assumption is made that the attribute list is relevant and complete. If individuals are grouped, the assumption that their perceptions are similar is made. The ideal object introduces additional conceptual problems.
Another basic assumption is that the interpoint distances generated by a perceptual map have conceptual meaning that is relevant to choice decisions.
Limitations A limitation of the attributed-based methods is that the attributes have to be generated. The analyst has the burden of making sure that respondents' perceptions and evaluations are represented by the attributes. With similarity and preference data, this task is eliminated, but the analyst then must interpret dimensions without the aid of such attributes, although attribute data could be generated independently and attribute-dimension correlations still obtained.
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