# Decide on the Number of Dimensions Marketing Research Help

The objective in MDS is to obtain a spatial map that best fits the input data in the smallest number of dimensions. However, spatial maps are computed in such a way that the fit improves as the number of dimensions increases. Therefore, a compromise has to be made. The fit of an MDS solution is commonly assessed by the stress measure. Stress is a lack-of-fit measure; higher values of stress indicate poorer fits. The following guidelines are suggested for determining the number of dimensions

1. A prior knowledge. Theory or past research may suggest a particular number of dimensions.
2. Interpret ability of the spatial map. Generally, it is difficult to interpret configurations or maps derived ill mere than three dimensions.
3. Elbow criterion. A plot of stress versus dimensional should be examined. The points in this plot usually form a convex pattern, as shown in Figure 21.3. The point at which an elbow or a sharp bend occurs indicates an appropriate number of dimensions. Increasing the number of dimensions beyond this point is usually not worth the
improvement in fit.
4. Ease of use. It is generally easier to work with two-dimensional maps or configurations than with those involving more dimensions.
5. Statistical approaches. For the sophisticated user, statistical approaches are also available for determining the dimensional.

Based on the plot of stress versus dimensional (Figure 21.3), interpret ability of the spatial map, and ease-of-use criteria. it was decided to retain a two-dimensional solution. This is shown in Figure 21.4.

Plot of Stress Versus Dimensionality

Assumptions and Limitations of MDS

It is worthwhile to point out some assumptions and limitations of MDS. It is assumed that the similarity of stimulus A to B is the same as the similarity of stimulus B to A. There are some instances where this assumption may be violated. For example, Mexico is perceived as more similar to the United States than the United Slates is to Mexico. MDS assumes that the distance (similarity) between two stimuli is some function of their partial similarities on each of several perceptual dimensions. Not much research has been done to test this assumption. When a spatial map is obtained, it is assumed that inter point distances are ratio scaled and that the axes of the map are multidimensional interval scaled. A limitation of MDS is that dimension interpretation relating physical changes in brands or stimuli to changes in the perceptual map is difficult at best. These limitations also apply to the scaling of preference data.

External Analysis of Preference Data

space. Two brands may be perceived to be similar (located close to each other in the perceptual space), yet one brand may be distinctly preferred over the other (i.e., the brands may be located apart in the preference space). These situations cannot be accounted for in internal analysis. In addition, internal analysis procedures are beset with computational difficulties.I?

We illustrate external analysis by scaling the preferences of our respondent into his spatial map. The respondent ranked the brands in the following order of preference (most preferred first): Colgate, Crest, Aim, Aqua-Fresh, Gleem, Pendent, Ultra Brite, Plus White, Close-Up, and Sensory. These preference rankings, along with the coordinates of the spatial map (Figure 215), were used as input into a preference scaling program to derive Figure 21.7. Notice the location of the ideal point. It is close to Colgate, Crest, Aim, and Aqua-Fresh, the four most preferred brands, and far from Close-Up and Sensory, the two least preferred brands. If a new brand were to be located in this space, its distance from the ideal point, relative to the distances of other brands from the ideal point, would determine the degree of preference for this brand. Another application is provided by the following example

Respondents Park in Different Spaces

A study conducted in 2008 examined consumer perceptions of automobiles by using multidimensional scales. Subjects rated several automobile attributes and the effect those attributes had on final product choice. Ratings were conducted using a 5-point scale, and each subject’s responses were summed across each dimension. The five highest scoring attributes overall were price, fuel economy, net horsepower, braking, and acceleration. The use of multidimensional scaling can help automakers better understand what attributes are most important to consumers, and they can use that knowledge to leverage their positioning in the industry. An illustrative MDS map of selected automobile brands derived from similarity data is shown. In this spatial representation, each brand is identified by its distance from the other brands. The closer TWO brands are (e.g., Volkswagen and Chrysler), the more similar they are perceived to be. The farther apart two brands are (e.g., Volkswagen and Mercedes), the less similar they are perceived to be. Small distance (i.e., similarity) may also indicate competition. To illustrate, Honda competes closely with Toyota but not with Continental or Porsche. The dimensions can be interpreted as economy/prestige and sportiness The position of each car on these dimensions can be easily determined

Joint Space Configuration of Automobile Brands and Consumer Preferences (Illustrative Output)

Such analysis can be done at the individual-respondent level, enabling the researcher to segment the market according to similarities in the respondents’ ideal points. Alternatively, the respondents can be clustered based on their similarity with respect to the original preference ranking and ideal points established for each segment

Attribute-Based Perceptual Map Using Discriminant Analysis

Discriminant analysis can also be used to develop attribute-based perceptual map of respondents against each identified discriminating function. This map facilitates marketers to identify attributes which best discriminate brands/groups in a perceptual map. Such spatial map can be obtained by plotting the brands on the basis of canonical discriminant functions evaluated at group means (group centroids) and attributes on the basis of standardized discriminant-function coefficients. Like other mapping techniques, a perceptual map drawn with the help of discriminant analysis can assist marketers in observing the complex market structure. It also helps in identifying the attribute-based positioning of groups/brands in the market and may provide an innovative insight for a new product development. However, while interpreting such map, marketers must be careful as this map provides only a partial explanation of consumers’ perception which is limited by a selected set of groups brands and attributes.

Procedure for Plotting Perceptual Map

Table 21.3 presents the results of discriminant analysis. It can be seen from the analysis that brand image, design, texture, and durability significantly discriminate suit brands while latest fashion does not significantly discriminate suit brands. Four brands and five attributes taken in our example imply that the maximum number of discriminating functions can be three. Looking at the canonical-discriminant functions’ descriptions in the result (Table 21.3), it is evident that three functions significantly discriminate brands. The next step is to plot attributes and brands on the map taking discriminating functions as dimensions. As we have identified three functions, we will have two perceptual maps of two dimensions: the first map with Function I (FI) and Function 2 (F2) as dimensions and ehe second map with Function I (FI) and Function 3 (F3) as dimensions. Standardized canonical-discriminant-function coefficients (Table 21.3) of attributes are used to plot attributes. and unstandardized canonical-discriminant functions, evaluated at group means (Table 21.3), are taken to plot brands on the map. For example, the standardized coefficients of brand-image attribute are -0.370 and -0.4703 on FI and F2 dimensions, respectively. Brand image (-0.370 and -0.4703) is positioned accordingly on the first map and an arrow is drawn from the origin to that point. This will be labeled . as brand-image vector (See Figure 21.8). Similarly, all other attribute vectors can be positioned in the perceptual map, one for each of the other four attributes-design,latest fashion, texture, and durability. To plot brand’s on the map, standardized canonical-discriminant functions’ evaluated at group means are taken. For example, for Van Heusen, the un standardized canonical-discriminant functions evaluated at group means from Table 21.3 are 0.016<>”and -0.7003 on FI and F2 respectively. Van Heusen is positioned accordingly on the first map. Similarly, other brands (Raymond, Blackberry, and Louis Philippe) are plotted on the map. The second perceptual map (with FI and F3 dimensions as axes) is drawn using the same procedure. The  maps shown in Figure 21.8 and Figure 21.9 have been drawn on an Excel sheet using the standardized canonical-discriminant-function coefficients and canonical-discriminant functions, evaluated at group-means value obtained from Table 21.3

Interpret Perceptual Plot

The perceptual maps (Figure 21.8 and Figure 21.9) show the relationship between brands and attributes. Attributes with- higher discriminant-function coefficient on a given dimension contribute more to discriminate brands in that dimension. The length of the arrow represents the relative effect of the respective attributes in discriminating on each dimension. Longer-attribute vectors in a given dimension which are closer to a given dimension are contributing more to

Sample Data Set

Results of Discriminant Analysis

interpretation of that dimension. As evident from the perceptual maps in Figure 21.8 and Figure 21.9, Dimension I is characterized by design and durability, Dimension 2 is primarily characterized by texture, and Dimension 3 is characterized by brand image. Latest fashion attribute is not useful in defining any of the dimensions, as its position is not close to any dimension in both the maps. It is also evident from Table 21.3 that the latest fashion does not significantly discriminate suit brands. The attribute-brand relationship can be indicated by seeing the proximity between the attribute points and any given group (brand) centroid. Longer arrows pointing more toward a group centroid (brand on the map) represent attributes strongly associated with the brand. Attribute vectors pointing the opposite direction from a given group (brand) centroid represent lower association of-the brand 011 that attribute. It can be noted from both the maps (Figure 21.8 and Figure 21.9) that Van Heusen scores high on F3 dimension. As Dimension 3 is primarily associated with brand image, one would expect that four brands are to be ordered on brand image in Dimension 3. Consumers who attach a higher importance on brand image would prefer Van Heusen as Van Heusen is perceived strong on brand image. Similarly, we can observe the positioning of other brands in the perceptual map. Raymond scored high on Dimension I indicating that it is perceived strong on design. Louis Philippe seems to be stronger in Dimension 2, indicating that this brand is perceived strong on texture attribute. Blackberry scores low on all three dimensions compared to competitors. It can also be noted that both Raymond as well as Blackberry brands ‘are uniquely positioned based on the attributes on both the maps, while Louis Phillipe and Van Heusen are distinctively positioned on Figure 21.8 alone, but are closely located in Figure 21.9, indicating that some attribute-perception similarity exists between these two brands.

Such interpretation of attribute-based positioning of brands in the market provides an insight for associated-marketing activities including a new product development

Perceptual Map

Perceptual Map of

The objective of discriminant analysis is to select attributes that discriminate between brands/groups. The major disadvantage of discriminant analysis-based perceptual map is that it exhibits a cluster of attributes on which brands/groups differ. In case the brands/groups are perceived to be similar with respect to an attribute (such as latest fashion in the above illustration) the attribute will not significantly discriminate brands/groups. Thus, there is little scope of proper interpretation of brands/groups on such attribute in the discriminant analysis-based perceptual map. Therefore, the researcher should be careful while selecting the attributes on which the consumer’s perception will be assessed in discriminant analysis-based perceptual map.

Sample Questionnaire (Part of the Questionnaire)

1. Which brand of business suit did you purchase or intend to purchase?

1. Van Heusen
2. Raymond
3. Blackberry
4. Louis Philippe