Applications of Multiple Regression Analysis

There are many possible applications of multiple regression analysis in marketing research:

• Estimating the effects of various marketing mix variables on sales or market share.
• Estimating the relationship between various demographic or psychographic factors and the frequency with which certain service businesses are visited.
• Determining the relative influence of individual satisfaction elements on overall satisfaction.
• Quantifying the relationship between various classification variables, such as age and income, and overall attitude toward a product or service.
• Determining which variables are predictive of sales of a particular product or service.

Multiple regression analysis can serve one or a combination of two basic purposes: (1) predicting the level of the dependent variable, based on given levels of the independent variables, and (2) understanding the relationship between the independent variables and the dependent variable.

Multiple Regression Analysis Measures

In the discussion of bivariate regression in Chapter 17, a statistic referred to as the coefficient of determination, or R², was identified as one of the outputs of regression analysis. This statistic can assume values from 0 to 1 and provides a measure of the percentage of the variation in the dependent variable that is explained by variation in the independent variables. For example, if R² in a given regression analysis is calculated to be .75, this means that 75 percent of the variation in the dependent variable is explained by variation in the independent variables. The analyst would always like to have a calculated R² close to 1. Frequently, variables are added to a regression model to see what effect they have on the R² value.

As models get larger, more independent or predictor variables, it is wise to look at a variation of the R² statistic called adjusted R², as the measure of fit for a regression model. The standard R² value tends to increase with every predictor variable that is added to the model, regardless of whether that variable truly adds to the explanatory power of the model. The adjusted R² corrects the coefficient of determination based on the relationship between the number of predictor variables and the overall sample size, producing a more rational estimate of model fit when several independent variables are included. The adjusted R² will always be less than or equal to R², being similar to the standard measure when the amount of sample per independent variable is large and producing a negative result when the sample size is very small and there are many predictors included in the model.

The b values, or regression coefficients, are estimates of the effect of individual independent variables on the dependent variable. It is appropriate to determine the likelihood that each individual b value is the result of chance. This calculation is part of the output provided by virtually all statistical software packages. Typically, these packages compute the probability of incorrectly rejecting the null hypothesis of b_n = 0.

Dummy Variables

In some situations, the analyst needs to include nominally scaled independent variables such as gender, marital status, occupation, and race in a multiple regression analysis.

Dummy variables can be created for this purpose. Dichotomous, nominally scaled independent variables can be transformed into dummy variables by coding one value (e.g., female) as 0 and the other (e.g., male) as 1. For nominally scaled independent variables that can assume more than two values, a slightly different approach is required. If there are K categories, K − 1 dummy variables are needed to uniquely identify every category. (Including K categories would over identify the model since the last category is represented by “0s” on the previous K − 1 variables.) Consider a question regarding racial group with three possible answers: African American, Hispanic, or Caucasian. Binary or dummy variable coding of responses requires the use of two dummy variables, X₁ and X ₂, which might be coded as follows:

Potential Use and Interpretation Problems

The analyst must be sensitive to certain problems that may be encountered in the use and interpretation of multiple regression analysis results. These problems are summarized in the following sections.

Applications of Multiple Discriminant Analysis

Discriminant analysis can be used to answer many questions in marketing research:

• How are consumers who purchase various brands different from those who do not purchase those brands?
• How do we target likely buyers for a new product from our database of existing customers in order to conduct the most effective prelaunch marketing campaign?
• How do consumers who frequent one fast-food restaurant differ in demographic and lifestyle characteristics from those who frequent another fast-food restaurant?
• How do consumers who have chosen either indemnity insurance, HMO coverage, or PPO coverage differ from one another in regard to healthcare use, perceptions, and attitudes?

Procedures for Clustering

A number of different procedures (based on somewhat different mathematical and computer routines) are available for clustering people or objects. Some examples of clustering techniques include K-means, two-stage, nearest neighbor, decision trees, ensemble analysis, random forest, BIRCH, and self-organizing neural networks. However, the general approach underlying all of these procedures involves measuring the similarities among people or objects in regard to their values on the variables used for clustering.¹¹ Similarities among the people or objects being clustered are normally determined on the basis of some type of distance measure. This approach is best illustrated graphically. Suppose an analyst wants to group, or cluster, consumers on the basis of two variables: monthly frequency of eating out and monthly frequency of eating at fast-food restaurants. Observations on the two variables are plotted in a two-dimensional graph in Exhibit 18.2. Each dot indicates the position of one consumer in regard to the two variables. The distance between any pair of points is positively related to how similar the corresponding individuals are when the two variables are considered together (the closer the dots, the more similar the individuals). In Exhibit 18.2, consumer X is more like consumer Y than like either Z or W.

Inspection of Exhibit 18.2 suggests that three distinct clusters emerge on the basis of simultaneously considering frequency of eating out and frequency of eating at fast-food restaurants:

• Cluster 1 includes those people who do not frequently eat out or frequently eat at fast-food restaurants.
• Cluster 2 includes consumers who frequently eat out but seldom eat at fast-food restaurants.
• Cluster 3 includes people who frequently eat out and also frequently eat at fast-food restaurants.

The fast-food company can see that its best targets are to be found among those who, in general, eat out frequently and eat at fast-food restaurants specifically. To provide more insight for the client, the analyst should develop demographic, psychographic, and behavioral profiles of consumers in cluster 3.

As shown in Exhibit 18.2, clusters can be developed from scatterplots. However, this time-consuming, trial-and-error procedure becomes more tedious as the number of variables used to develop the clusters or the number of objects or persons being clustered increases. You can readily visualize a problem with two variables and fewer than 100 objects. Once the number of variables increases to three and the number of observations increases to 500 or more, visualization becomes impossible. Fortunately, computer algorithms are available to perform this more complex type of cluster analysis. The mechanics of these algorithms are complicated and beyond the scope of this discussion. The basic idea behind most of them is to start with some arbitrary cluster boundaries and modify the boundaries until a point is reached where the average interpoint distances within clusters are as small as possible relative to average distances between clusters.

Additional discussion of using cluster analysis for market segmentation is provided in the accompanying Practicing Marketing Research feature.

Factor Scores

Factor analysis produces one or more factors, or composite variables, when applied to a number of variables. A factor, technically defined, is a linear combination of variables. It is a weighted summary score of a set of related variables, similar to the composite derived by averaging the measures. However, in factor analysis, each measure is first weighted according to how much it contributes to the variation of each factor.

In factor analysis, a factor score is calculated on each factor for each subject in the data set. For example, in a factor analysis with two factors, the following equations might be used to determine factor scores:

With these formulas, two factor scores can be calculated for each respondent by substituting the ratings she or he gave on variables A₁ through A₄ into each equation. The coefficients in the equations are the factor scoring coefficients to be applied to each respondent’s ratings. For example, Bob’s factor scores (see Exhibit 18.4) are computed as follows:

In the first equation, the factor scoring coefficients, or weights, for A₁ and A₂ (.40 and .30) are large, whereas the weights for A₃ and A₄ are small. The small weights on A₃ and A₄ indicate that these variables contribute little to score variations on factor 1 (F₁). Regardless of the ratings a respondent gives to A₃ and A₄, they have little effect on his or her score on F₁. However, variables A₃ and A₄ make a large contribution to the second factor score (F₂), whereas A₁ and A₂ have little effect. These two equations show that variables A₁ and A₂ are relatively independent of A₃ and A₄ because each variable takes on large values in only one scoring equation.

The relative sizes of the scoring coefficients are also of interest. Variable A₁ (with a weight of .40) is a more important contributor to factor 1 variation than is A₂ (with a smaller weight of .30). This finding may be very important to the product designer when evaluating the implications of various design changes. For example, the product manager might want to improve the perceived luxury of the car through product redesign or advertising. The product manager may know, based on other research, that a certain expenditure on redesign will result in an improvement of the average rating on “smooth ride” from 4.3 to 4.8. This research may also show that the same expenditure will produce a half-point improvement in ratings on “quiet ride.” The factor analysis shows that perceived luxury will be enhanced to a greater extent by increasing ratings on “smooth ride” than by increasing ratings on “quiet ride” by the same amount.

Factor Loadings

The nature of the factors derived can be determined by examining the factor loadings. Using the scoring equations presented earlier, a pair of factor scores (F₁ and F₂) are calculated for each respondent. Factor loadings are determined by calculating the correlation (from −1 to +1) between each factor (F₁ and F₂) score and each of the original ratings variables. Each correlation coefficient represents the loading of the associated variable on the particular factor. If A₁ is closely associated with factor 1, the loading or correlation will be high, as shown for the sample problem in Exhibit 18.5. Because the loadings are correlation coefficients, values near −1 or +1 indicate a close positive or negative association. Variables A₁ and A₂ are closely associated (highly correlated) with scores on factor 1, and variables A₃ and A₄ are closely associated with scores on factor 2.

Stated another way, variables A₁ and A₂ have high loadings on factor 1 and serve to define the factor; variables A₃ and A₄ have high loadings on and define factor 2.

Naming Factors

Once each factor’s defining variables have been identified, the next step is to name the factors. This is a somewhat subjective step, combining intuition and knowledge of the variables with an inspection of the variables that have high loadings on each factor. Usually, a certain consistency exists among the variables that load highly on a given factor. For instance, it is not surprising to see that the ratings on “smooth ride” and “quiet ride” both load on the same factor. Although we have chosen to name this factor “luxury,” another analyst, looking at the same result, might decide to name the factor “prestige.”

Number of Factors to Retain

In factor analysis, the analyst is confronted with a decision regarding how many factors to retain. The final result can include from one factor to as many factors as there are variables. The decision is often made by looking at the percentage of the variation in the original data that is explained by each factor.

There are many different decision rules for choosing the number of factors to retain. Probably the most appropriate decision rule is to stop factoring when additional factors no longer make sense. The first factors extracted are likely to exhibit logical consistency; later factors are usually harder to interpret, for they are more likely to contain a large amount of random variation.

Example of Conjoint Analysis

Put yourself in the position of a product manager for Titleist, a major manufacturer of golf balls. From focus groups recently conducted, past research studies of various types, and your own personal experience as a golfer, you know that golfers tend to evaluate golf balls in terms of three important features or attributes: average driving distance, average ball life, and price.

You also recognize a range of feasible possibilities for each of these features or attributes, as follows:

1. Average driving distance
   • 10 yards more than the golfer’s average
   • Same as the golfer’s average
   • 10 yards less than the golfer’s average
2. Average ball life
   • 54 holes
   • 36 holes
   • 18 holes
3. Price per ball
   • $2.00
   • $2.50
   • $3.00

From the perspective of potential purchasers, these attributes have a natural order (i.e., longer distance and longer ball life are always preferred over shorter options), so we can easily identify the ideal configuration. This is not always the case when dealing with attributes such as brand, physical appearance, or color. For this example, the consumer’s ideal golf ball would have the following characteristics:

• Average driving distance—10 yards above average
• Average ball life—54 holes
• Price—$2.00

From the manufacturer’s perspective, which is based on manufacturing cost, the ideal golf ball would probably have these characteristics:

• Average driving distance—10 yards below average
• Average ball life—18 holes
• Price—$3.00

This golf ball profile is based on the fact that it costs less to produce a ball that travels a shorter distance and has a shorter life. The company confronts the eternal marketing dilemma: the company would sell a lot of golf balls, but would go broke if it produced and sold the ideal ball from the golfer’s perspective. However, the company would sell very few balls if it produced and sold the ideal ball from the manufacturer’s perspective. As always, the “best” golf ball from a business perspective lies somewhere between the two extremes.

A traditional approach to this problem might produce information of the type displayed in Exhibit 18.6. As you can see, this information does not provide new insights regarding which ball should be produced. The preferred driving distance is 10 yards above average and the preferred average ball life is 54 holes. These results are obvious without any additional research.

Considering Features Conjointly

In conjoint analysis, rather than having respondents evaluate features individually, the analyst asks them to evaluate features conjointly or in combination so that advantages for one attribute can only be chosen at the expense of another attribute. The results of asking two different golfers to rank different combinations of “average driving distance” and “average ball life” conjointly are shown in Exhibits 18.7 and 18.8.

As expected, both golfers agree on the most and least preferred balls. However, analysis of their second through eighth rankings makes it clear that the first golfer is willing to trade off ball life for distance (accept a shorter ball life for longer distance), while the second golfer is willing to trade off distance for longer ball life (accept shorter distance for a longer ball life).

This type of information is the essence of the special insight offered by conjoint analysis. The technique permits marketers to see which product attribute or feature potential customers are willing to trade off (accept less of) to obtain more of another attribute or feature. People make these kinds of purchasing decisions every day (e.g., they may choose to pay a higher price for a product at a local market for the convenience of shopping there).

Estimating Utilities

The next step is to calculate a set of values, or utilities, for the three levels of price, the three levels of driving distance, and the three levels of ball life in such a way that, when they are combined in a particular mix of price, ball life, and driving distance, they predict each golfer’s rank order for that combination. Estimated utilities for golfer 1 are shown in Exhibit 18.9. As you can readily see, this set of numbers perfectly predicts the original rankings. The relationship among these numbers or utilities is fixed, though there is some arbitrariness in their magnitude or scale. In other words, the utilities shown in Exhibit 18.9 can be multiplied or divided by any constant and the same relative results will be obtained. Utilities for this simple example can be computed using ordinary least squares regression, but the exact procedures for estimating utilities of more complex exercises are beyond the scope of this discussion. They are normally calculated by using procedures related to regression, analysis of variance, linear programming, logic, or hierarchical Bayes analysis.

The trade-offs that golfer 1 is willing to make between “ball life” and “price” are shown in Exhibit 18.10. This information can be used to estimate a set of utilities for “price” that can be added to those for “ball life” to predict the rankings for golfer 1, as shown in Exhibit 18.11.

This step produces a complete set of utilities for all levels of the three features or attributes that successfully capture golfer 1’s trade-offs. These utilities are shown in Exhibit 18.12.

Simulating Buyer Choice

For various reasons, the firm might be in a position to produce only 2 of the 27 golf balls that are possible with each of the three levels of the three attributes. The possibilities are shown in Exhibit 18.13. If the calculated utilities for golfer 1 are applied to the two golf balls the firm is able to make, then the results are the total utilities shown in Exhibit 18.14. These results indicate that golfer 1 will prefer the ball with the longer life over the one with the greater distance because it has a higher total utility. The analyst need to only repeat this process for a representative sample of golfers to estimate potential market shares for the two balls. In addition, the analysis can be extended to cover other golf ball combinations.

The three steps discussed here—collecting trade-off data, using the data to estimate buyer preference structures, and predicting choice—are the basis of any conjoint analysis application. Although the trade-off matrix approach is simple, useful for explaining conjoint analysis, and effective for problems with small numbers of attributes, it is seldom used in the real world.

One of the most common approaches to conducting conjoint analysis is the use of a discrete choice or choice-based conjoint exercise. Two or more products are shown side-by-side with details provided on each key attribute being tested. Respondents are asked to select a single product from among the options shown. The exercise is repeated multiple times in order to present a wide variety of product designs, but no individual sees more than a fraction of the sometimes thousands or even millions of possible product combinations.

Computer-driven exercises might further adapt the exercise to each respondent, based on prior answers and personal demographics, to spend more time on the factors that seem to be driving product choice. Menu-based conjoint analysis can replicate the choices consumers make when choosing between “value meals” and a la carte items from a restaurant menu. Other computer-driven exercises allow respondents to design their own product with appropriate design constraints and pricing factored in to each option chosen, much the way consumers configure their own Dell computer online or select upgrades for a new car. These and many other approaches can be used to capture the information needed for estimating respondent utilities when designed, executed, and analyzed properly.

As suggested earlier, there is much more to conjoint analysis than has been discussed in this section. However, if you understand this simple example, then you understand the basic concepts that underlie conjoint analysis.

Limitations of Conjoint Analysis

Like many research techniques, conjoint analysis suffers from a certain degree of artificiality. Respondents may be more deliberate in their choice processes in this context than in a real situation. The survey may provide more product information than respondents would get in a real market situation. If key attributes or popular options within key attributes are excluded from the study, demand estimates could be severely impacted. Testing too many attributes or features will diminish the amount of attention that can be given to each individual’s most desired features, reducing measurement precision. The presentation of information (e.g., the order in which attributes are listed; whether pictures are used for some attributes, but not others; how price is displayed; etc.) can greatly impact what features respondents focus on and, ultimately, how they make their decisions. It is important to either be as neutral as possible in the presentation of a conjoint exercise or else try to replicate how the product or service is actually evaluated and compared in the marketplace in order to avoid biasing results.

Finally, it is important to remember that the advertising and promotion of any new product or service can lead to consumer perceptions that are very different from those created via factual descriptions used in a survey. Also keep in mind that consumers can’t purchase something they don’t know exists, so conjoint analysis operates under the assumptions of full awareness, unrestricted access, and complete knowledge of all product features.

Using Predictive Analytics

Privacy Concerns and Ethics

Most believe that predictive modeling is ethically neutral. However, the ways in which data are collected for predictive modeling and the types of data acquired can raise questions regarding privacy, legality, and ethics. For example, monitoring telephone calls and Internet usage for national security or law enforcement purposes has raised privacy concerns.

Commercial Predictive Modeling Software and Applications

Database providers such as Oracle and Microsoft provide tools optimized for their platform. Popular Big Data platform, Hadoop, has a variety of open-source and commercial tools available. There are an increasing number of highly integrated packages for predictive modeling, including:

• Angoss KnowledgeSTUDIO
• Clarabridge
• RapidMiner
• SAS Enterprise Miner
• SPSS Modeler
• STATISTICA Data Miner