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• Análisis de la regresión
• Sistema de coordenadas cartesianas
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• Vehículo utilitario deportivo
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UVA-M-0665

METHODS FOR PRODUCING PERCEPTUAL MAPS FROM DATA

Introduction Graphics are instruments for reasoning about quantitative information. If carefully prepared, graphics can describe and explore complex data in a way that suggests the useful information. We often find that a picture of the data allows greater understanding and helps commit the important features of the data to memory.1 One very common graphic in marketing is a perceptual map. These maps are used to help managers visualize how their product(s) relate to other competitive offerings in the marketplace. Common terms in marketing such as “targeting” and “positioning” implicitly reference the idea that managers are or should be visualizing the perceptual differences between their product(s) and others in the marketplace. Perceptual maps are ubiquitous in marketing presentations to help the audience with visualizations of the competitive landscape. A simple example of a perceptual map is on the next page. It describes the market for sport utility vehicles.

1

E. R. Tufte, The Visual Display of Quantitative Information (Cheshire, CT: Graphics Press, 1983).

This note was prepared by Ronald T. Wilcox, Associate Professor of Business Administration, University of Virginia. It was written as a basis for class discussion rather than to illustrate effective or ineffective handling of an administrative situation. Copyright  2003 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to [email protected]. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of the Darden School Foundation. ◊

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Figure 1. The market for sport utility vehicles. Prestigious •

Cadillac Escalade •

•

Chevy Tahoe

• Nissan Jeep Pathfinder Grand Cherokee

•

•

Not Rugged •

Hummer H2

Toyota 4Runner

Ford Explorer

Rugged •

Jeep Cherokee

Not Prestigious In Figure 1, we can see that, among other things, Nissan Pathfinder, Toyota 4Runner, and Jeep Grand Cherokee are perceived to be fairly similar. Hummer H2 is perceived to be the most rugged SUV in this group and likewise Cadillac Escalade the most luxurious. Overall, this map allows us to compare each vehicle to all others on the map along the perceptual dimensions listed on the axis. This kind of analysis requires several assumptions. Among the most important of these assumptions are 1. Consumers evaluate all products in this category using the attributes listed along the axis of the perceptual map. 2. These attributes are the most important of all possible attributes a consumer might use to compare products in this category. 3. The points depicted on the map, representing products, are accurate representations of consumers’ view of the marketplace. We will now discuss how marketing researchers collect and use data on consumers’ perceptions to develop these maps. We will first consider the situation in which the analyst can make an initial hypothesis or guess as to what perceptual attributes consumers may be using to evaluate a product and then move on to a situation in which no such prior hypothesis is possible.

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Attribute Ratings Method Constructing the XY plot There are situations in which product managers may have some idea about which perceptual attributes their product(s) and those of their competitors are considered prior to purchase. A good example is the SUV market depicted in Figure 1. Automobiles are very high involvement purchases, usually requiring consumers to make mental trade-offs between price and a number of other attributes of competing products before a purchase decision is made. Over time, automobile manufacturers have developed a reasonably sophisticated understanding, often through marketing research, of the attributes consumers consider in this process. In order to develop a perceptual map, a manager must determine which two of the possible attributes are the most important in the decision process and how consumers’ evaluate each product in the competitive set with respect to these two attributes. The analytics of this method begin by collecting consumer ratings for each product, both the product of interest and other products in the competitive set, for each possible attribute that management believes consumers might reasonably consider. Perceptual attributes are not the same as engineering attributes. For example, “number of miles per gallon” is an engineering attribute. It has a definitive answer. “Fuel efficiency,” while probably highly related to this engineering attribute, is a perceptual attribute. Consumers are not being asked to report the quantitative answer to this question, but are being asked about their overall impression of the vehicle along this dimension. Likewise, “interior space” is an engineering attribute while “roominess” is a perceptual attribute. Measuring consumers’ perceptions is generally accomplished by using a Likert scale survey instrument.2 The general form of this question is [Insert vehicle name] is a [insert attribute] sport utility vehicle.

Agree

Strongly Agree

Neither Agree nor Disagree

Disagree

Strongly Disagree

For example, for one question we would insert the name “Toyota 4Runner” and the attribute “Rugged.” In general, if the researcher wanted to product a map with N competitive products and was testing M perceptual attributes, he or she would have to ask NxM Likert scale questions. Once a sufficient sample of consumers have provided this information, the researcher 2

For a more detailed discussion of using a Likert Scale to measure consumers’ perceptions of product attributes, see G. Urban and J. Hauser, Design and Marketing of New Products (Upper Saddle River, NJ: PrenticeHall, 1993), 196–99.

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can convert this ordinal scale to a cardinal scale by assigning the value “5” to “Strongly Agree,” “4” to “Agree” and so on and compute the arithmetic mean of each attribute score for each vehicle. The output of this process is a matrix that takes the general form in Table 1. Table 1. Average attribute rating for each vehicle. Vehicle 1 Attribute 1 Attribute 2 …………. Attribute M

Vehicle 2

………..

Vehicle N

[average score]

In addition to this information, the researcher should also ask a single overall preference question, using the same Likert scale, for each vehicle. The preference question takes the general form: “I believe that [insert vehicle name] is an excellent [insert product category].” So, continuing our example, one question would be “I believe that Toyota 4Runner is an excellent sport utility vehicle.” Thus, the total data set collected by the research includes NxM vehiclesspecific average attribute scores and N overall preference scores. Recall that the researcher now needs to accomplish two tasks. First he or she needs to determine which two of the M possible perceptual dimensions are the most important in the overall evaluation process. Once that is complete, the researcher then needs to place each product in the appropriate position on the map. To determine which attributes should define the axis of the perceptual map, the researcher can estimate a preference regression using a simple linear model. Specifically, the researcher should estimate: Overall Preferencein = α + β1 Attribute1in + β2 Attribute2in + …+βM AttributeMin + εin where Overall Preferencein is the preference of respondent i for product n, Attibute1in is respondent i’s rating on Attribute 1 for product n and so forth. Greek letters represent parameters to be estimated by the regression. To keep things simple, we will assume that εin follows the standard assumptions of the classical linear model so that Ordinary Least Squares, using a software package such as Excel, can be used to estimate this model. Estimating this regression will yield estimates of the parameters (coefficients) with additional information that will allow the researcher to determine whether the parameter estimates are statistically significant.3 3

A good refresher on the basic linear model and problems that can arise with this model can be found in Philip E. Pfeifer’s “Introduction to Least Squares Modeling,” (UVA-QA-0500) (Charlottesville, VA: Darden Business Publishing, 1996) and “Problems in Regression” UVA-QA-0416 (Charlottesville, VA: Darden Business Publishing, 1995).

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From this point, determining the attributes to use to define the axes is straightforward. The researcher should choose the two attributes that have estimated coefficients that are the greatest in absolute value subject to the restriction that are statistically significant. Practically speaking, if either of the two estimated coefficients that are the greatest in absolute value are not statistically significant this probably means that (1) the sample size is not large enough or; (2) the set of attributes tested by the researcher is not capturing the important determinants of product evaluation, and serious consideration should be given to rethinking the data collection from the beginning. The two coefficients that are the greatest in absolute value will indicate which attributes are most important in determining overall preference and hence the most appropriate attributes to define the axes of the perceptual map.4 The use of the absolute value is important because it is possible that some attributes may negatively impact product evaluation. For example, in our SUV example, if one perceptual attribute being tested was “luxurious,” it is entirely possible that the estimated coefficient of this attribute would be negative as some individuals would equate luxury with an SUV that is not sufficiently capable of off-road use. It is also possible that the researcher could define a perceptual attribute as a negative rather than a positive. The perceptual attribute “gas guzzling” would likely show up as a negative influence in the preference regression. Once these two attributes have been determined, the researcher need only to use the average attribute ratings for these two attributes, found in Table 1, to plot each vehicle on an XY axis. It is customary to use the most important attribute as the X axis and the second-most important as the Y axis although it actually makes little difference. Much like Figure 1, positive attributes define the positive portions of each axis, and their negative counterparts define the negative portions. When constructing the actual map itself, it is useful to define the origin, usually the point (0,0) as (3,3) instead. The reason is straightforward. We would like to use the entire space to plot the products. Because perceptual attributes are measured on a one to five scale, defining the origin as (0,0) would force us to plot all products in the first quadrant. This produces a map that is more difficult to interpret than a map that uses the entire space. Thus, centering the map at (3,3) makes the graphic clearer and hence more useful as a decision aid. Here is Figure 1 reproduced with hypothetical average perceptual attributes used as coordinates.

4

It is not generally the case that one can interpret coefficients in a linear regression in this way. But because all the independent variables use the exact same scale, we can interpret the magnitude of the estimated coefficient as an indicator of relative importance in determining the dependent variable.

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Figure 2. The market for sport utility vehicles, attribute scores included. Prestigious •

Cadillac Escalade (1.2, 4.7) •

•

Hummer H2 (4.5, 4.6)

• Nissan Chevy • Jeep Pathfinder (3.6, 4) Tahoe (1.9, 3.8) Grand Cherokee (2.5, 3.8) •

Not Rugged •

Toyota 4Runner (3.5, 3.3)

Ford Explorer (2.1, 2)

•

Rugged

Jeep Cherokee (4, 1.7)

Not Prestigious Constructing the ideal vector We have now constructed a perceptual map using perceptual attribute ratings. There is one more potentially valuable piece of information that can be gleaned from the data that has been collected to develop this map. Up to this point, this map contains no information about which product would be preferred by consumers. Some preference information does appear obvious. For example, it appears that, all else equal, Nissan Pathfinder would be preferred by most people to Ford Explorer because it is generally viewed as more rugged and more prestigious. But, it is less obvious whether consumers generally prefer the Jeep Cherokee or the Toyota 4Runner. The Toyota is more prestigious but the Jeep is more rugged. What are the tradeoffs consumers are willing to make? This kind of issue becomes even more interesting when different perceptual maps are developed for different potential target audiences. For example, the trade-offs men are willing to make among perceptual attributes may be quite different from those women are willing to make. This kind of information is extremely valuable for product design decisions as well as target selection and marketing communication strategies. Fortunately, the procedure described above already contains enough information to be able to develop stronger statements about consumers’ preferences. This is generally accomplished through the construction of an “ideal vector,” a vector depicted on the perceptual map, which illustrates these trade-offs. To construct the ideal vector, the researcher needs to use the estimated preference regression model. In particular, he or she needs to use the estimated coefficients from the perceptual attributes that define the axes of the map. The ratio of these two

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coefficients will determine the slope of the ideal vector, a vector which originates at the origin of the map. The best way to explain the details of this procedure is through a simple example. Suppose we estimated the preference regression for SUVs, and it yielded the following results for the two most important attributes: Overall Preference = -2.7 + 1.25 × Prestige + 2.5 × Ruggedness +……….. where Prestige and Ruggedness have been previously determined to be the two most important attributes in the decision process. Following earlier arguments, the estimated coefficients indicate that Ruggedness is more important than Prestige in determining overall preference. In particular, because 2.5 = 2 × 1.25, we can conclude that Ruggedness is twice as important as Prestige. The ideal vector is constructed as follows: 1. Take the ratio of the coefficient of the second-most important perceptual attribute to the most important. In this case, we would have 1.25 ÷ 2.5 = 0.5. 2. Plot a vector with slope defined by this ratio and whose beginning point is at the origin of the graph.5

Figure 3. Constructing the ideal vector. Prestigious •

Cadillac Escalade •

• Chevy Tahoe

• Nissan Jeep Pathfinder Grand Cherokee

•

•

Not Rugged •

Hummer H2

Toyota 4Runner

Ford Explorer

Rugged •

Jeep Cherokee

Not Prestigious

5

The one exception to this is if the researcher places the most important attribute on the Y-axis and the secondmost important on the X-axis. In that case, one would want to use the ratio of the most important attribute to the second-most important attribute in determining the slope of the vector.

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Notice that the ideal vector tilts toward the axis defined by the most important attribute. Preference increases as you move out along the vector. Points of equal preference, or what economists would call utility, can be determined by constructing lines that are perpendicular to the ideal vector. Two of these utility lines are depicted in Figure 4. Figure 4. Constructing linear utility functions. Prestigious •

Cadillac Escalade •

• Chevy Tahoe

• Nissan Jeep Pathfinder Grand Cherokee

•

•

Not Rugged •

Hummer H2

Toyota 4Runner

Ford Explorer

Rugged •

Jeep Cherokee

Not Prestigious The answer to the Toyota 4Runner versus Jeep Cherokee now becomes clear. Because utility lines that intersect the ideal vector further from the origin represent higher levels of preference or utility relative to those that intersect closer to the origin, we can definitely state that, on average, people prefer the Jeep Cherokee to the Toyota 4Runner. We could make similar preference comparisons between any two vehicles in this map and hence order all vehicles in terms of their overall preference. If we constructed separate maps for different segments of the population, we could use the above analysis not only to see how perceptions vary across different groups of people but also to observe how differences in overall preferences relate to these perceptions.

Overall Similarity Method For some product categories, it is virtually impossible to determine a priori what attributes consumers might be using to evaluate a product. Consider a movie production company like New Line Cinema or Hollywood Pictures that is trying to understand how consumers evaluate movies and make decisions on which ones to see at the theater. What perceptual attributes do consumers use to evaluate a movie? First guesses would include things like amount of action, romance, presence of a particular film star, and so on. But movies by their

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nature are not easily described by a precise group of attributes that can be easily specified. For example, the movie About Schmidt is by most accounts an excellent movie. But unlike the attribute ratings method discussed earlier, it would be very difficult to adequately capture what makes this movie excellent via a list of perceptual attributes. It is simply a more intangible evaluation process than, for example, the evaluation process for sport utility vehicles. For these types of products, the attribute ratings method is a poor way to generate a perceptual map. We must turn to another method, overall similarity, to try to generate a picture of the competitive landscape. Overall-similarity methods work by asking consumers a series of questions designed to determine how similar or dissimilar different pairs of products are to each other. Once the researcher has determined the perceived similarity among all possible products in the competitive set, she can generate a two-dimensional map that graphically reproduces the perceptual distance between and among the products in the set. To see this clearly, let’s begin with a concrete example focused on current movies. A company like New Line Cinema might begin this process with a list of movies whose preferences they are interested in probing. For the purposes of our example let’s consider the movies About Schmidt, Lord of the Rings: Two Towers, Gangs of New York, Maid in Manhattan, A Guy Thing, and Bowling for Columbine. The researcher would ask an appropriately specified sample of consumers to evaluate how similar each movie is to each other movie. In our example there are six movies and fifteen distinct movie pairs. For each movie pair, say About Schmidt and Bowling for Columbine, there would be a question that reads

About Schmidt and Bowling for Columbine

Very Different 1 [ ]

2 [ ]

3 [ ]

Very Similar 4 5 [ ] [ ]

with a corresponding question for every single product pair. Once this judgment data has been collected, the researcher can simply take the means of the similarity scores and use these means to fill in the perceived similarity matrix. A perceived similarity matrix might look like the following (Table 2):

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Table 2: Movie similarity matrix.

About Schmidt Lord of the Rings Gangs of New York Maid in Manhattan A Guy Thing Bowling for Columbine

About Schmidt 5.0 4.2 3.0 2.0 1.0 3.5

Gangs of New Maid in York Manhattan

Lord of the Rings 5.0 4.0 1.5 2.0 2.5

5.0 1.7 3.4 2.2

5.0 2.1 1.9

A Guy Thing

5.0 1.2

Bowling for Columbine

5.0

These data represent perceptual distances, really the inverse of distances, between the movies. A good perceptual map will reproduce these distances, as closely as possible, in twodimensional space. Reproducing these distances in two-dimensional space is not a simple task. To see this clearly, consider the seemingly unrelated analogy of a three-legged stool and a four-legged chair. Which one is more likely to wobble back and forth when you are sitting on it? The answer is obvious, three-legged stools never rocks back and forth but a four-legged chair might. The reason can be tied directly to geometry. Just as it takes two points to define a line, it takes three points (legs) to define a two-dimensional plane (the floor). If you try to match four points (legs) to a two-dimensional plane, there is a chance that they will not all sit perfectly on the surface of any single plane. In mathematics this is called an “overidentified” space. For our purposes, this means that, whenever we have more than three products whose perceptual distances we want to reproduce in two-dimensional space, we are unlikely to be able to do it perfectly. The best we can hope for is to find a solution that reproduces these distances as closely as possible. The process of uncovering this solution is called multidimensional scaling, and because the perceptual data we collect is cardinal rather than ordinal the specific technique presented here is termed metric multidimensional scaling. Much like linear regression, the way metric multidimensional scaling (MMDS) reproduces these perceptual distances is in a way that minimizes the sum of squared errors between the actual distances (given in Table 2) and the distances reproduced on the map. The function that the procedure seeks to minimize is commonly called a “stress function” and is given by the equation:

Stress =



i< j

(dˆij − d ij ) 2

where dˆij are the distances specified on the perceptual map and dij are the actual perceptual distances found in Table 2. The subscripts i and j indicate that the given distance is between product i and j. There are a number of algorithms that MMDS can use to minimize this function,

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but these algorithms are included as easy-to-execute procedures in common statistical software. A discussion of these algorithms is beyond the scope of this technical note. Inserting the data in Table 2 into a MMDS procedure produces the following twodimensional map: MDS for Movie Data Final Configuration, dimension 1 vs. dimension 2 1.0 MAID

0.8 BOWL

0.6

Dimension 2

0.4 0.2

SCHMIDT

0.0 -0.2 RINGS

-0.4 AGUY

-0.6 -0.8 -1.0

NYGANGS

-0.6

-0.2

0.2

0.6

1.0

1.4

Dimension 1

Two things stand out from this procedure. First, there are no axes. This is because the procedure only finds distances among the products, not their relationships to any axis. In fact, you can take the above map and turn it upside down, sideways, or rotate it in any way you wish, and it is still the solution to the minimization problem. Second, and as a direct consequence of the first point, this procedure does not identify the perceptual dimensions of the map. So, unlike the attribute-rating method, the researcher must now decide where to draw the axes and what the axes should be named. The most common way to specify the names and position of the axes is for the researcher to use her/his knowledge of the category to define them. This may seem arbitrary, and in some respects it is, but often the position of the products in the space suggests some perceptual dimensions that weren’t obvious to the researcher before the data collection began. We leave it to the reader of this note to speculate about the dimensions of this particular map.

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A Common Theme These two methods for producing perceptual maps, attribute-rating and overall-similarity, both have the ability to depict important relationships among competitive products in twodimensional space. If the researcher believes he or she knows the relevant attributes in consumers’ decision processes, then the attribute-rating method is generally preferred as it leads to a more easily interpretable perceptual map. If the researcher is not willing to specify a group of potentially important perceptual attributes, then the overall-similarity method can be used. The rationale behind making a choice between these methods is similar to the trite expression “pay me now or pay me later.” The attribute-rating method requires more managerial insight and more data collection up front but has the benefit of an easily interpretable map as an output. The overall-similarity method requires less initial insight and judgment and uses a significantly simpler data collection procedure. The cost of this ease is a map, which management must exercise considerable judgment to interpret.