Multivariate Statistical Analysis

580

After studying this chapter, you should be able to

1. Understand what multivariate statistical analysis involves and know the two types of multivariate analysis

2. Interpret results from multiple regression analysis3. Interpret results from multivariate analysis of variance

(MANOVA)4. Interpret basic exploratory factor analysis results5. Know what multiple discriminant analysis can be used to do6. Understand how cluster analysis can identify market

segments

CHAPTER 24MULTIVARIATE STATISTICAL ANALYSIS

Chapter Vignette: Cow-A-Bunga Never Goes Out of StyleAs humans, we long to relive the past. This yearning to hold on to the past is a common psycho-logical experience.1 The psychology of consumption is of interest to many people who are not psychologists, however. The fact is, nostalgia sells, and business researchers are very interested

in understanding exactly what nostalgia is, who is most prone to react to it, and how it contributes to business success.

When a boomer or Gen Xer walks through the toy store, he or she is likely to feel right at home. Toy com-

panies like Hasbro have realized that adults buy toys for kids to enjoy. Grown-up consumers like to buy things

they feel good about. Thus, the toy shelves are filled with throwback versions of GI Joe, Barbie, and even the Teenage Mutant Ninja Turtles.2 The game shelves are filled

with classic versions of familiar games like Risk, Stratego, and Monopoly.3

Not to be outdone, other marketers are also counting on nostalgic consumers. Appliance companies have turned to retro designs with classic 1950s versions of toasters,

blenders, and even ovens.4 Advertisers are also using nostal-gia to produce more effective sales appeals. Among others, Coca-Cola has used nostalgic advertising to help consumers

relive the past.5

This trend is expected to continue, as pointed out by Janet Hsu, President of Sanrio Global Consumer Products,

“Inspirational products will be a major trend in 2009 as consum-ers gravitate towards purchasing products that provide comfort

and emotional connection. Evergreen and nostalgic products will continue to sell well. . . .”6

How can organizations better integrate nostalgia into their business plans? Researchers are working on numerous issues related to nostalgia:

• How can nostalgia be measured?• What emotions is nostalgia associated with?7

• Can market segments be defined based on the type and amount of nostalgia experienced?• What happens to consumers when they experience nostalgia?• What makes a nostalgic consumer different from one who does not experience nostalgia?• What are the positive outcomes for the business when consumer nostalgia increases?

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Chapter 24: Multivariate Statistical Analysis 581

Nostalgia is a complex experience involving multiple thoughts and feelings. The complexity makes nostalgia somewhat difficult to study. Multivariate research procedures can help address these ques-tions as they consider the effects of multiple variables simultaneously. However, it seems that nostal-gic thoughts mean good vibes for the marketer.8 Cow-a-bunga!

IntroductionIf only business problems were really as simple as most textbook examples! Most coursework involves solving problems that have a definite answer. They are usually relatively well-defined problems in which the information provided in the problem can be used to produce one correct solution.

Unfortunately, in the real world, most business problems are ill-defined. Not only do they not have a definite answer, but generally information needs to be generated and massaged before any solution can be obtained. Therefore, most business research studies involve many variables that must be organized for meaning. As researchers become increasingly aware of the complex nature of busi-ness problems, they gain a greater appreciation for more sophisticated approaches to data analysis. This chapter provides an introduction to some forms of what are known as multivariate data analysis.

What Is Multivariate Data Analysis?

The preceding chapters have addressed univariate and bivariate analyses. Research that involves three or more variables, or that is concerned with underlying dimensions among multiple variables, will involve multivariate statistical analysis. Multivariate statistical methods analyze multiple variables or even multiple sets of variables simultaneously. How do we know when someone has experienced nostalgia and whether or not the experience has altered behavior? Nostalgia itself is a latent con-struct that involves multiple indicators that together represent nostalgia. As such, the measurement and outcomes of nostalgia lend themselves well to multivariate analysis.9 Likewise, many other business problems involve multivariate data analysis including most employee motivation research, customer psychographic profiles, and research that seeks to identify viable market segments.

The “Variate” in Multivariate

Another distinguishing characteristic of multivariate analysis is the variate. The variate is a math-ematical way in which a set of variables can be represented with one equation. A variate is formed as a linear combination of variables, each contributing to the overall meaning of the variate based upon an empirically derived weight. Mathematically, the variate is a function of the measured variables involved in an analysis:

Vk = f (X

1, X

2, . . . , X

m)

Vk is the kth variate. Every analysis could involve multiple sets of variables, each represented by

a variate. X1 to X

m represent the measured variables.

Here is a simple illustration. Recall that constructs are distinguished from variables by the fact that multiple variables are needed to measure a construct. Let’s assume we measured nostalgia with five questions on our survey. With these five variables, a variate of the following form could be created:

Vk = L

1X

1 + L

2X

2 + L

3X

3 + L

4X

4 + L

5X

5

Vk represents the score for nostalgia, X

1 to X

5 represent the observed scores on the five scale items

(survey questions) that are expected to indicate nostalgia, and L1 to L

5 are parameter estimates much

like regression weights that suggest how highly related each variable is to the overall nostalgia score.

variate

A mathematical way in which a set of variables can be repre-sented with one equation.

T O T H E P O I N T

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.

—S. Gudder


582 Part 6: Data Analysis and Presentation

While this equation might appear a little intimidating, don’t worry! We do not have to manu-ally calculate these scores. We’ll rely on the computer to do the heavy lifting. However, this type of relationship is common to multivariate procedures.

Classifying Multivariate TechniquesExhibit 24.1 presents a very basic classification of multivariate data analysis procedures. Two basic groups of multivariate techniques are dependence methods and interdependence methods.

All multivariatemethods

Does at leastone dependent variable

exist?

Yes No

Interdependencemethods

Dependencemethods

EXHIBIT 24.1Which Multivariate Approach Is Appropriate?

Our survey includes data that can best be analyzed with multivariate techniques. Take a look at the survey ques-tions that deal with satisfaction with the business school experience.

When reading the chapter, consider these questions and how they fit with the techniques described. When you have finished the chapter:

1. Run a factor analysis on the 8 questions from “Teacher’s Knowledge of Topics” through “Your

Overall Academic Performance” a. How many factors are retained? b. What would you “name” these factors? c. Create summated scale for each factor.

2. Run a multiple regression analysis with the Overall Experience question as the dependent measure and the summed scale(s) as the independent measure(s). Also include sex of the respondent as an independent vari-able (dummy variable) in your regression. Interpret the results:

a. Is the overall model significant? b. Which of the independent variables are significant? c. How much variance in Overall Experience is

explained by the predictor variables? d. Which of the independent variables is most important

in determining satisfaction with the Overall Experience?

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Dependence Techniques

When hypotheses involve distinction between independent and dependent variables, dependence techniques are needed. For instance, when we hypothesize that nostalgia is related positively to purchase intentions, nostalgia takes on the character of an independent variable and purchase intentions take on the character of a dependent variable. Predicting the dependent variable “sales” on the basis of numerous independent variables is a problem frequently investigated with depen-dence techniques. Multiple regression analysis, multiple discriminant analysis, multivariate analysis of vari-ance, and structural equations modeling are all dependence methods.

Interdependence Techniques

When researchers examine questions that do not distinguish between independent and dependent variables, interdependence techniques are used. No one variable or variable subset is to be predicted from or explained by the others. The most common interdependence methods are factor analysis, cluster analysis, and multidimensional scaling. A manager might utilize these techniques to determine which employee motivation items tend to group together (factor analysis), to identify profitable customer market segments (cluster analysis), or to provide a perceptual map of cities being consid-ered for a new plant (multidimensional scaling).

Influence of Measurement Scales

As in other forms of data analysis, the nature of the measurement scales will determine which multivariate technique is appropriate for the data. Exhibits 24.2 and 24.3 on the next page show that selection of a multivariate technique requires consideration of the types of measures used for both independent and dependent sets of variables. These exhibits refer to nominal and ordinal scales as nonmetric and interval and ratio scales as metric.

dependence techniques

Multivariate statistical techniques that explain or predict one or more dependent variables.

interdependence techniques

Multivariate statistical techniques that give meaning to a set of variables or seek to group things together; no distinction is made between dependent and inde-pendent variables.

One dependentvariable

Multipleindependent

and dependentvariables

Severaldependentvariables

Metric Nonmetric

Multipleregressionanalysis

Multiplediscriminant

analysis

Metric Nonmetric

Conjointanalysis

Metric

How manyvariables aredependent?

Dependencemethods

Multivariateanalysis ofvariance

StructuralEquationModeling

EXHIBIT 24.2Which Multivariate Dependence Technique Should I Use?



Analysis of DependenceMultivariate dependence techniques are variants of the general linear model (GLM). Simply, the GLM is a way of modeling some process based on how different variables cause fluctuations from the average dependent variable. Fluctuations can come in the form of group means that differ from the overall mean as in ANOVA or in the form of a significant slope coefficient as in regression. The basic idea can be thought of as follows:

Yi = μ + ΔX + ΔF + ΔXF

Here, μ represents a constant, which can be thought of as the overall mean of the dependent variable, ΔX and ΔF represent changes due to main effect independent variables (such as experi-mental variables) and blocking independent variables (such as covariates or grouping variables), respectively, and ΔXF represents the change due to the combination (interaction effect) of those variables. Realize that Y

i in this case could represent multiple dependent variables, just as X and F

could represent multiple independent variables. Multiple regression analysis, n-way ANOVA, and MANOVA represent common forms that the GLM can take.

Multiple Regression Analysis

Multiple regression analysis is an extension of simple regression analysis allowing a metric depen-dent variable to be predicted by multiple independent variables. Chapter 23 illustrated simple linear regression analysis with an example explaining a construction dealer’s sales volume with the number of building permits issued. Thus, one dependent variable (sales volume) is explained by one independent variable (number of building permits). Yet reality is more complicated and several additional factors probably affect construction equipment sales. Other plausible indepen-dent variables include price, seasonality, interest rates, advertising intensity, consumer income, and other economic factors in the area. The simple regression equation can be expanded to represent multiple regression analysis:

Yi = b

0 + b

1X

1 + b

2X

2 + b

3X

3 + . . . + b

nX

n + e

i

general linear model (GLM)

A way of explaining and predict-ing a dependent variable based on fluctuations (variation) from

its mean. The fluctuations are due to changes in independent

variables.

multiple regression analysis

An analysis of association in which the effects of two or more

independent variables on a single, interval-scaled dependent

variable are investigated simultaneously.

Metric Nonmetric

Factoranalysis

Clusteranalysis

Metricmultidimensional

scaling

Nonmetricmultidimensional

scaling

Are inputsmetric?

Interdependencemethods

EXHIBIT 24.3Which Multivariate Interdependence Technique Should I Use?



Thus, as a form of the GLM, dependent variable predictions (Y ) are made by adjusting the constant

(bo), which would be equal to the mean if all slope coefficients are 0, based on the slope coefficients

associated with each independent variable (b1, b

2, . . . , b

n).10

Less-than interval (nonmetric) independent variables can be used in multiple regression. This can be done by implementing dummy variable coding. A dummy variable is a variable that uses a 0 and a 1 to code the different levels of dichotomous variable (for instance, residential or commercial building permit). Multiple dummy variables can be included in a regression model. For example, dummy cod-ing is appropriate when data from two countries are being compared. Suppose the average labor rate for automobile production is included in a sample taken from respondents in the United States and in South Korea. A response from the United States could be assigned a 0 and responses from South Korea could be assigned a 1 to create a country variable appropriate for use with multiple regression.

■ A SIMPLE EXAMPLE

Assume that a toy manufacturer wishes to explain store sales (dependent variable) using a sample of stores from Canada and Europe. Several hypotheses are offered:

• H1: Competitor’s sales are related negatively to our firm’s sales.• H2: Sales are higher in communities that have a sales office than when no sales office is

present.• H3: Grammar school enrollment in a community is related positively to sales.

Competitor’s sales is how much the primary competitor sold in the same stores over the same time period. Both the dependent variable and the competitor’s sales are ratio variables measured in euros (Canadian sales were converted to euros). The presence of a sales office is a categorical variable that can be represented with dummy coding (0 = no office in this particular community, 1 = office in this community). Grammar school enrollment is also a ratio variable simply represented by the number of students enrolled in elementary schools in each community (in thousands).11 A sample of 24 commu-nities is gathered and the data are entered into a regression program to produce the following results:

Regression equation: Y = 102.18 + 0.387X1 + 115.2X

2 + 6.73X

3

Coefficient of multiple determination (R2) = 0.845

F-value = 14.6; p < 0.05

Note that all the signs in the equation are positive. Thus, the regression equation indicates that sales are positively related to X

1, X

2, and X

3. The coefficients show the effect on the depen-

dent variable of a 1-unit increase in any of the independent variables. The value or weight, b1,

associated with X1 is 0.387. Thus, a one-unit increase ($1,000) in competitors’ sales volume (X

1)

in the community is actually associated with an increase of $387 in the toy manufacturer’s sales (0.387 � $1,000 = $387). The value of b

2 = 115.2, which indicates that an increase of $115,200

(115.2 thousand) in toy sales is expected with each additional unit of X2. Thus, it appears that

having a company sales office in a community is associated with a very positive effect on sales. Grammar school enrollments also may help predict sales. An increase of 1 unit of enrollment (1,000 students) indicates a sales increase of $6,730.

Because the effect associated with X1 is positive, H1 is not supported; as competitor sales increase,

our sales increase as well. The effects associated with H2 and H3 are also positive, which is in the hypoth-esized direction. Thus, if the coefficients are statistically significant, H2 and H3 will be supported.

■ REGRESSION COEFFICIENTS IN MULTIPLE REGRESSION

Recall that in simple regression, the coefficient b1 represents the slope of X on Y. Multiple regres-

sion involves multiple slope estimates, or regression weights. One challenge in regression models is to understand how one independent variable affects the dependent variable, considering the effect of other independent variables. When the independent variables are related to each other, the regression weight associated with one independent variable is affected by the regression weight of another. Regression coefficients are unaffected by each other only when independent variables are totally independent.

dummy variable

The way a dichotomous (two group) independent variable is represented in regression analysis by assigning a 0 to one group and a 1 to the other.



Conventional regression programs can provide standardized parameter estimates, β1, β

2, and so

on, that can be thought of as partial regression coefficients. The correlation between Y and X1, control-

ling for the correlation that X2 has with the Y, is called partial correlation. Consider a standardized

regression model with only two independent variables:12

Y = β1X

1 + β

2X

2 + e

i

The coefficients β1 and β

2 are partial regression coefficients, which express the relationship between

the independent variable and dependent variable taking into consideration that the other variable also is related to the dependent variable. As long as the correlation between independent variables is modest, partial regression coefficients adequately represent the relationships. When the correla-tion between two independent variables becomes high, the regression coefficients may not be reliable, as illustrated in the Research Snapshot on the next page.

When researchers want to know which independent variable is most predictive of the depen-dent variable, the standardized regression coefficient (β) is used. One huge advantage of β is that it provides a constant scale. In other words, the βs are directly comparable. Therefore, the greater the absolute value of the standardized regression coefficient, the more that particular independent vari-able is responsible for explaining the dependent variable. For example, suppose in the toy example above, the following standardized regression coefficients were found:

β1 = 0.10

β2 = 0.30

β3 = 0.10

The resulting standardized regression equation would be

Y = 0.10X1 + 0.30X

2 + 0.10X

3 + e

i

Using standardized coefficients, the researcher concludes that the relationship between competi-tor’s sales (X

1) and company sales (Y ) is the same strength as is the relationship between grammar

school enrollment (X3) and company sales. Perhaps more important, though, the conclusion can

also be reached that the relationship between having a sales office in the area (X2) and sales is three

times as strong as the other two relationships. Thus, management may wish to place more emphasis on locating sales offices in major markets.

■ R 2 IN MULTIPLE REGRESSION

The coefficient of multiple determination in multiple regression indicates the percentage of varia-tion in Y explained by the combination of all independent variables. For example, a value of R2 = 0.845 means that 84.5 percent of the variance in the dependent variable is explained by the independent variables. If two independent variables are truly independent (uncorrelated with each other), the R2 for a multiple regression model is equal to the separate R2 values that would result from two separate simple regression models. More typically, the independent variables are at least moderately related to one another, meaning that the model R2 from a multiple regression model will be less than the separate R2 values resulting from individual simple regression models. This reduction in R2 is proportionate to the extent to which the independent variables exhibit multicollinearity.

■ STATISTICAL SIGNIFICANCE IN MULTIPLE REGRESSION

Following from simple regression, an F-test is used to test statistical significance by comparing the variation explained by the regression equation to the residual error variation. The F-test allows for testing of the relative magnitudes of the sum of squares due to the regression (SSR) and the error sum of squares (SSE ).

F = (SSR)/k

______________ (SSE)/(n – k – 1) = MSR _____ MSE

partial correlation

The correlation between two variables after taking into

account the fact that they are correlated with other

variables too.



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Too Much of a Good Thing! Researchers often test hypotheses by

examining regression coefficients. Thus, we are often looking for correlations, sometimes in all the wrong places. Financial data can be

problematic to analyze. Consider the case of a financial manager trying to analyze gross margin (dependent variable = margin per employee) using the following independent variables:

● Average sales per square foot per quarter● Average labor costs per week

The F of 13.57 is highly significant (<0.001), so the variables explain a large portion of the variance in the dependent variable.

Even though the model results appear strong, only one inde-pendent variable is significant at a Type I error rate of 0.050 � sales. However, the � coefficients do not make sense. The � coefficients for both sales and labor are beyond the range that � should theoretically take (�1.0 to 1.0). Nothing can be correlated with something more than perfectly (which would be a correlation of 1.0 or �1.0). Notice also that the two VIF fac-tors for sales and labor are in the 50s. Generally, when multiple VIF factors approach 5 or greater, problems with multicollinear-ity can be expected. The high correlation between sales and

● Years of experience for the manager● Job performance rating for the previous year (100-point scale)

Regression results can be obtained in SPSS by clicking on ANALYZE, REGRESSION, and then LINEAR. The VIF column must be requested by clicking on STATISTICS and then checkingCOLLINEARITY DIAGNOSTICS. After doing so, the following results are obtained. For the overall model,

The model R2 is 0.61 also supporting this conclusion. The results for the independent variable tests show the following:

labor is a problem illustrating multicollinearity. Multicollinearity (sometimes just referred to as collinearity) in regression analysis refers to how strongly interrelated the independent variables in a model are. As often occurs with financial data, they can be difficult to use as independent variables. In this case, the researcher may wish to rerun the model after dropping one of the offending variables.

ANOVA(b)

Model Sum of Squares df Mean Square F Sig.

1 Regression 142566.5332 4 35641.6333 13.56899 .0000008

Residual 91934.43848 35 2626.698242

Total 234500.9717 39

A Predictors: (constant), performance, experience, labor, salesB Dependent Variable: margin

Coefficients(a)

ModelUnstandardized

Coefficients B Std. Error

Standardized Coefficients

Beta t Sig. VIF

1 (Constant) 171.242614 235.9374392 0.725797 0.47279

Sales 0.090784631 0.030835442 2.339759409 2.944165 0.00572 56.3836

Labor �0.070267446 0.035014493 �1.587938574 �2.00681 0.05254 55.8971

Experience �0.488078747 0.955764142 �0.054331204 �0.51067 0.61279 1.0105

Performance �1.856084354 3.034080822 �0.068978263 �0.61175 0.54466 1.1351

a Dependent Variable: margin

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where

k = number of independent variables n = number of observationsMSR = Mean Squares RegressionMSE = Mean Squares Error

Degrees of freedom for the F-test (df ) are:

df for the numerator = k

df for the denominator = n – k – 1

For our toy sales example,

df (numerator) = 3

df (denominator) = 24 – 3 – 1 = 20

In the toy example, we have 24 observations (different communities) and 3 independent variables (competitor sales, sales office, and school enrollment). A table of critical F-values shows that for 3 and 20 df, and a 0.05 Type I error rate, a value of 3.10 or more is necessary for the regression model to be considered significant, meaning that it explains a significant portion of the total variation in the dependent variable. In practice, statistical programs will report the p-value associated with the F-test directly. Similarly, the programs report the statistical test for each individual independent variable. Independent variables with p-values below the acceptable Type I error rate are considered significant predictors of the dependent variable.

■ STEPS IN INTERPRETING A MULTIPLE REGRESSION MODEL

Multiple regression models often are used to test some proposed theoretical model. For instance, a researcher may be asked to develop and test a model explaining business unit performance. Why do some business units outperform others? Multiple regression models can be interpreted using these steps:

1. Examine the model F-test. If the test result is not significant, the model should be dismissed and there is no need to proceed to further steps.

2. Examine the individual statistical tests for each parameter estimate. An independent variable with significant results can be considered a significant explanatory variable. If an independent variable is not significant, the model should be run again with nonsignificant predictors deleted. Often, it is best to eliminate predictor variables one at a time, then rerun the reduced model.

3. Examine the model R2. No cutoff values exist that can distinguish an acceptable amount of explained variation across all regression models. However, the absolute value of R2 is more important when the researcher is more interested in prediction than in explanation. In other words, the regression is run for pure forecasting purposes. When the model is more oriented toward explanation of which variables are most important in explaining the dependent vari-able, cutoff values for the model R2 are not really appropriate.

4. Examine collinearity diagnostics. Multicollinearity in regression analysis refers to how strongly interrelated the independent variables in a model are. When multicollinearity is too high, the individual parameter estimates become difficult to interpret. Most regression programs can compute variance inflation factors (VIF) for each variable. As a rule of thumb, VIF above 5.0 suggests problems with multicollinearity.13

Exhibit 24.4 illustrates these steps. The regression model explains business unit profitability for a sample of 28 business units for a Fortune 500 company. The independent variables are hours (average hours spent in training for the workforce), budget (the percentage of the promotional budget used), and state (a dummy variable indicating whether the business unit is in Arizona and coded 0, or in Ohio and coded 1). In this case, the researcher is using a maximum acceptable Type I error rate of 0.05. The conclusion reached from this analysis is that hours spent in training seem to pay off in increased business unit profitability as evidenced by the significant, positive regression coefficient (β = 0.55, p < 0.05).

multicollinearity

The extent to which independent variables in a multiple regression analysis are correlated with each other; high multicollinearity can

make interpreting parameter esti-mates difficult or impossible.



ANOVA (n-Way) and MANOVA

As discussed above, regression is a form of the GLM with a single continuous dependent measure and continuous independent measure(s). An ANOVA or MANOVA model also represents a form of the GLM. ANOVA can be extended beyond one-way ANOVA to predict a continuous dependent variable with multiple categorical independent variables. Multivariate analysis of variance (MANOVA), is a multivariate technique that predicts multiple continuous dependent variables with multiple independent variables. The independent variables are categorical, although a continuous control variable can be included in the form of a covariate. Statistical programs usually refer to any ANOVA with only one dependent variable as univariate analysis of variance or simply as ANOVA.

■ NWAY UNIVARIATE ANOVA

The interpretation of an n-way ANOVA model follows closely from the regression results described above. The steps involved are essentially the same with the addition of interpreting differences between means:

1. Examine the overall model F-test result. If significant, proceed.2. Examine individual F-tests for each individual independent variable.

multivariate analysis of variance (MANOVA)

A multivariate technique that predicts multiple continuous dependent variables with mul-tiple categorical independent variables.

The SAS System

The REG ProcedureModel: MODEL1

Dependent Variable: Paid

Number of Observations Read 28Number of Observations Used 28

Analysis of Variance

Sum of MeanSource DF Squares Square F Value Pr > F

Model 3 1770668 590223 19.38 <.0001Error 24 731035 30460Corrected Total 27 2501703

Root MSE 174.52738 R-Square 0.7078Dependent Mean 654.03571 Adj R-Sq 0.6713Coeff Var 26.68469

Parameter Estimates

Parameter Standard StandardizedVariable DF Estimate Error t Value Pr > |t| Estimate VIF

Intercept 1 �109.90538 217.46253 �0.51 0.6179 0Hours 1 0.99438 0.27688 3.59 0.0015 0.55433 1.96Budget 1 6.60121 3.54784 1.86 0.0751 0.28210 1.89State 1 �66.36397 84.82434 �0.78 0.4416 �0.11073 1.65

EXHIBIT 24.4 Interpreting Multiple Regression Results

3. The model R2 is interpreted. The IVs explain over 70% of variation in the dependent variable.

2. Interpret individual parameter estimates. In this case, only hours is significant based on a p-value below .05 (0.0015).

4. VIFs are checked for multicollinearity problems. None are indicated here.

1. Interpret model F-test. Test is significant (p < .05).



3. For each significant categorical independent variable, interpret the effect by examining the group means (see Chapter 12).

4. For each significant continuous variable (covariate), interpret the parameter estimate (b).5. For each significant interaction, interpret the means for each combination. A graphical repre-

sentation as illustrated in Chapter 12 can greatly assist in this interpretation.

■ INTERPRETING MANOVA

Compared to ANOVA, a MANOVA model produces an additional layer of testing. The first layer of testing involves the multivariate F-test, which is based on a statistic called Wilke’s Lambda (Λ). This test examines whether or not an independent variable explains significant variation among the dependent variables within the model. If this test is significant, then the F-test results from individual univariate regression models nested within the MANOVA model are interpreted. The rest of the interpretation results follow from the one-way ANOVA or multiple regression model results above. The Research Snapshot on the next page provides an example of how to run and interpret MANOVA.

Discriminant Analysis

Researchers often need to produce a classification of sampling units. This process may involve using a set of independent variables to decide if a sampling unit belongs in one group or another. A physi-cian might record a person’s blood pressure, weight, and blood cholesterol level and then categorize that person as having a high or low probability of a heart attack. A researcher interested in retailing failures might be able to group firms as to whether they eventually failed or did not fail on the basis of independent variables such as location, financial ratios, or management changes. A bank might want to discriminate between potentially successful and unsuccessful sites for electronic fund trans-fer system machines. A human resource manager might want to distinguish between applicants to hire and those not to hire. The challenge is to find the discriminating variables to use in a predictive equation that will produce better than chance assignment of the individuals to the correct group.

Discriminant analysis is a multivariate technique that predicts a categorical dependent vari-able (rather than a continuous, interval-scaled variable, as in multiple regression) based on a linear combination of independent variables. In each problem above, the researcher determines which variables explain why an observation falls into one of two or more groups. A linear combination of independent variables that explains group memberships is known as a discriminant function. Discriminant analysis is a statistical tool for determining such linear combinations. The researcher’s task is to derive the coefficients of the discriminant function (a straight line).

We will consider an example of the two-group discriminant analysis problem where the dependent variable, Y, is measured on a nominal scale. (Although n-way discriminant analysis is possible, it is beyond the scope of this discussion.) Suppose a personnel manager for an electrical wholesaler has been keeping records on successful versus unsuccessful sales employees. The per-sonnel manager believes it is possible to predict whether an applicant will succeed on the basis of age, sales aptitude test scores, and mechanical ability scores. As stated at the outset, the problem is to find a linear function of the independent variables that shows large differences in group means. The first task is to estimate the coefficients of the applicant’s discriminant function. To calculate the individuals’ discriminant scores, the following linear function is used:

Zi = b

1X

1i + b

2X

2i + · · · + b

nX

ni

where

Zi = ith applicant’s discriminant score

bn = discriminant coefficient for the nth variable

Xni = ith applicant’s value on the nth independent variable

Using scores for all the individuals in the sample, a discriminant function is determined based on the criterion that the groups be maximally differentiated on the set of independent variables.

discriminant analysis

A statistical technique for predict-ing the probability that an object will belong in one of two or more

mutually exclusive categories of the dependent variable, based

on several independent variables.


How to Get MANOVA Results A department store developer gathered

data looking at the effect of nostalgia on customer impressions. A field experiment was set up in which a key department was either given

a modern design or a retro design. It was hoped that the retro design would create feelings of nostalgia. Several hundred consum-

ers were interviewed. Since two related dependent variables are involved (Y1 = interest and Y2 = excitement), MANOVA is the appro-priate technique.

MANOVA can be conducted using SPSS by clicking on ANALYZE, then GENERAL LINEAR MODEL, and then MULTIVARIATE. (If only one dependent variable were involved, the choice would be UNIVARIATE.) This opens a dialog box as shown here:

R E S E A R C H S N A P S H O THow

A dedata

on customer iset up in whic

a modern d deesign or a retro ddesign would create feeling

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The dialog box includes places to enter dependent vari-ables, fixed factors (between-subjects categorical independent variables), and covariates. In this case, the fixed factors are

1. Experimental variable (0 = modern, 1 = retro)2. Respondent sex (0 = male, 1 = female)

Respondent age is included as a covariate or control variable (years).

SPSS provided output that can be summarized briefly:

1. Multivariate Results: a. Wilke’s Lambda = 0.964 b. Overall multivariate F = 9.6 with 2 and 510 df c. The p-value associated with this result is less than 0.001.

Thus the multivariate results are significant, so the research proceeds to interpret the individual univariate ANOVA results for each dependent variable (SPSS pro-vides these results automatically).

2. The univariate model F statistics for each dependent variable are both significant (p < 0.001) so the researcher moves on to the next step.

3. The individual effects associated with Y1 (interest) are inter-preted. For example, for the experimental variable, the result is:

a. F = 0.4, with 1 and 511 df for interest (p = 0.531).

b. Age is not significant. c. The interaction is not significant.4. The individual effects associated with Y2 (excitement) are inter-

preted. For example, for the experimental variable, the result is: a. F = 13.4, with 1 and 511 df for excitement (p < 0.001). b. Sex and age are both significant predictors too

(p < 0.001). c. The interaction of sex and the retro/modern experimental

variable is also significant.5. After carefully reviewing the means for each experimental

cell as well as the covariate results, the researcher reaches the following conclusions:

a. The retro look produced more excitement but not neces-sarily more interest.

b. Women are more interested and more excited about shopping.

c. The effect of the retro condition was stronger for men than for women. That is, the difference in means between the retro and modern condition is larger for men than for women.

d. Younger consumers are more excited about shopping.

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Returning to the example with three independent variables, let us suppose the personnel manager finds the standardized weights in the equation to be

Z = b1X

1 + b

2X

2 + b

3X

3

= 0.069X1 + 0.013X

2 + 0.0007X

3

This means that age (X1) is much more important than sales aptitude test scores (X

2). Mechanical

ability (X3) has relatively minor discriminating power.

In the computation of the linear discriminant function, weights are assigned to the variables to maximize the ratio of the difference between the means of the two groups to the standard deviation within groups. The standardized discriminant coefficients, or weights, provide information about the relative importance of each of these variables in discriminating between the two groups.

A major goal of discriminant analysis is to perform a classification function. The purpose of classification in our example is to predict which applicants will be successful and which will be unsuccessful based on their age, sales aptitude test score, and mechanical ability, and to group them accordingly. To determine whether the discriminant analysis can be used as a good predic-tor of applicant success, current employees with known characteristics are used in constructing the model. Each observation (current employee) is placed into one of the groups based on the independent variables. Some will be classified successfully, but some will not. This information is provided in the “confusion matrix,” which is similar to cross-tabulations we discussed earlier. Sup-pose the personnel manager has 40 successful and 45 unsuccessful employees in the sample. The confusion matrix shows that the number of correctly classified employees (72 out of 85) is much higher than would be expected by chance:

Confusion Matrix

Predicted Group

Actual Group Successful Unsuccessful

Successful 34 6 40

Unsuccessful 7 38 45

Again, similar to cross-tabs and χ2, tests can be performed to determine whether the rate of correct classification is statistically significant.

Exhibit 24.5 summarizes multivariate dependence techniques.

Technique Purpose

Number of Dependent

Variables

Number of Independent

Variables

Type of Measurement

Dependent Independent

Multiple regression To investigate simultane-ously the effects of several independent variables on a dependent variable

1 2 or more Interval Interval

Discriminant analysis

To predict the probability that an object or individual will belong in one of two or more mutually exclusive categories, based on several independent variables

1 2 or more Nominal Interval

MANOVA To determine simultane-ously whether statisti-cally significant mean differences occur between groups on several variables

2 or more 1 or more Interval Nominal

EXHIBIT 24.5 Multivariate Dependence Techniques Summary



Analysis of InterdependenceSuppose we wished to identify the factors that are associated with pleasant shopping experiences,14 identify factors that would allow better flexibility and control of logistics programs,15 or identify groups of students each associated with a unique learning style.16 Each of these are problems that have been addressed through the use of a multivariate interdependence technique. Rather than attempting to predict a variable or set of variables from a set of independent variables, we use techniques like factor analysis, cluster analysis, and multidimensional scaling to better understand the relationships and structure among a set of variables or objects.

Factor Analysis

Factor analysis is a prototypical multivariate, interdependence technique. Factor analysis is a tech-nique of statistically identifying a reduced number of factors from a larger number of measured variables. The factors themselves are not measured, but instead, they are identified by forming a variate using the measured variables. Factors are usually latent constructs like attitude or satisfaction, or an index like social class. A researcher need not distinguish between independent and dependent variables to conduct factor analysis. Factor analysis can be divided into two types:

1. Exploratory factor analysis (EFA)—performed when the researcher is uncertain about how many factors may exist among a set of variables. The discussion here concentrates primarily on EFA.

2. Confirmatory factor analysis (CFA)—performed when the researcher has strong theoretical expectations about the factor structure (number of factors and which variables relate to each factor) before performing the analysis. CFA is a good tool for assessing construct validity because it provides a test of how well the researcher’s “theory” about the factor structure fits the actual observations. Many books exist on CFA alone and the reader is advised to refer to any of those sources for more on CFA.

Exhibit 24.6 illustrates factor analysis graphically. Suppose a researcher is asked to examine how feelings of nostalgia in a restaurant influence customer loyalty. Three hundred fifty customers at themed restaurants around the country are interviewed and asked to respond to the following Likert scales (1 = Strongly Disagree to 7 = Strongly Agree):

X1—I feel a strong connection to the past when I am in this place.

X2—This place evokes memories of the past.

X3—I feel a yearning to relive past experiences when I dine here.

X4—This place looks like a page out of the past.

X5—I am willing to pay more to dine in this restaurant.

X6—I feel very loyal to this establishment.

X7—I would recommend this place to others.

X8—I will go out of my way to dine here.

Factor analysis can summarize the information in the eight variables in a smaller number of variables. How many dimensions, or groups of variables, are likely present in this case? More than one technique exists for estimating the variates that form the factors. However, the general idea is to mathematically produce variates that explain the greatest total variance among the set of variables being analyzed. In this example, the factor analysis indicates there are two dimensions, or factors, as shown in Exhibit 24.6. Thus, EFA provides two important pieces of information:

1. How many factors exist among a set of variables?2. What variables are related to or “load on” which factors?

■ HOW MANY FACTORS

One of the first questions the researcher asks is, “How many factors will exist among a large num-ber of variables?” While a detailed discussion is beyond the scope of this text, the question is usually addressed based on the eigenvalues for a factor solution. Eigenvalues are a measure of how much

factor analysis

A prototypical multivariate, interdependence technique that statistically identifies a reduced number of factors from a larger number of measured variables.



variance is explained by each factor. The most common rule—and the default for most statistical programs—is to base the number of factors on the number of eigenvalues greater than 1.0. The basic thought is that a factor with an eigenvalue of 1.0 has the same total variance as one variable. It usually does not make sense to have factors, which are a combination of variables, that have less information than a single variable. So, unless some other rule is specified, the number of factors shown in a factor solution is based on this rule.

■ FACTOR LOADINGS

Each arrow connecting a factor (represented by an oval in Exhibit 24.6) to a variable (represented by a box in Exhibit 24.6) is associated with a factor loading. A factor loading indicates how strongly correlated a measured variable is with that factor. In other words, to what extent does a variable “load” on a factor? EFA depends on the loadings for proper interpretation. A latent construct can be interpreted based on the pattern of loadings and the content of the variables. In this way, the latent construct is measured indirectly by the variables.

Loading estimates are provided by factor analysis programs. In Exhibit 24.6, the factor loading estimates are shown beneath the factor diagram. The thick arrows indicate high loading estimates and the thin dotted lines correspond to weak loading estimates. Factors are interpreted by examin-ing any patterns that emerge from the factor results. Here, a clear pattern emerges. The first four variables produce high loadings on factor 1 and the last four variables produce high loadings on factor 2.

When a clear pattern of factor loadings emerges, interpretation is easy. Because the first four variables all have content consistent with nostalgia and the second four variables all have content consistent with customer loyalty, the two factors can easily be labeled. Factor one represents the latent construct nostalgia and factor 2 represents the latent construct customer loyalty.

■ FACTOR ROTATION

Factor rotation is a mathematical way of simplifying factor results. The most common type of factor rotation is a process called varimax. A discussion of the technical aspects of the concept of factor rotation is far beyond the scope of this book. However, factor rotation involves creating new refer-ence axes for a given set of variables. An initial factor solution is often difficult to interpret. Rota-tion “clears things up” by producing more obvious patterns of loadings. Users can observe this by

factor loading

Indicates how strongly a mea-sured variable is correlated with

a factor.

factor rotation

A mathematical way of simplify-ing factor analysis results so as

to better identify which variables “load on” which factors; the most

common procedure is varimax.

EXHIBIT 24.6A Simple Illustration of Factor Analysis

Nostalgia

X1

Loyalty

X2 X3 X4 X5 X6 X7 X8

Factor Loading Estimates:Variable: Factor 1 Factor 2

X1 .90 − .02X2 .88 .10X3 .85 .12X4 .70 .31X5 − .10 .90X6 − .05 .90X7 .20 .75X8 .21 .72



looking at the unrotated and rotated solutions in the factor analysis output. An example of how to run factor analysis is provided in the Research Snapshot above.

■ DATA REDUCTION TECHNIQUE

Factor analysis is considered a data reduction technique. Data reduction techniques allow a researcher to summarize information from many variables into a reduced set of variates or composite vari-ables. Data reduction is advantageous for many reasons. In general, the rule of parsimony suggests that an explanation involving fewer components is better than one involving more. Factor analysis accomplishes data reduction by capturing variance from many variables with a single variate. Data reduction is also a way of identifying which variables among a large set might be important in some analysis. Thus, data reduction simplifies decision making.

In our example, the researcher can now form two composite factors representing the latent constructs nostalgia and customer loyalty. These can be formed using factor equations of this form:

Fk = L

1X

1 + L

2X

2 + L

3X

3 + L

4X

4 + L

5X

5 + L

6X

6 + L

7X

7 + L

8X

8

data reduction technique

Multivariate statistical approaches that summarize the informa-tion from many variables into a reduced set of variates formed as linear combinations of measured variables.

rule of parsimony

The rule of parsimony suggests that an explanation involving fewer components is better than one involving more.

Getting Factor Results with SAS or SPSS

Although researchers may choose to use a spreadsheet to produce simple or even multiple regression results, they will almost always turn to a

specialized program for procedures like factor analysis. As a way of familiarizing readers with the mechanics involved, here are some instructions for getting factor results in each program.

SAS is most typically interfaced by writing short computer programs. SAS can read Excel spreadsheets quite easily. The data simply need to be “imported” into SAS by using the File dialog box (click on File to begin this process—see SAS docu-mentation contained in the help files for more on how to do this). Once the data are set up, a factor program can be easily produced. Suppose we wished to run a factor program includ-ing a varimax rotation on eight variables labeled X1–X8. The program would be

proc factor rotate = v;

var X1–X8;

After we click “run,” the results appear in the output window.

In SPSS, the click-through sequence is as follows:

● ANALYZE● DATA REDUCTION● FACTOR ANALYSIS

This produces a dialog box. Now follow the steps below to get results that would match those above:

● Highlight variables X1 to X8 (either individually or in multiples).

● Click the ▶ to move them into the “Variables” window.● Click “ROTATION.”

● Select VARIMAX.● Optional: Click “OPTIONS.”

● Select “SORTED BY SIZE.”● Select “SUPPRESS ABSOLUTE VALUES LESS THAN.”

■ These two options make the output easier to read by organizing the output by the size of the loadings on each factor and by not showing loadings below some specified absolute value (0.1 by default). For factor analyses involving many variables, this is par-ticularly helpful.

● Click “CONTINUE.”● Click “OK.”

The results will appear in the output window.

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where

Fk is the factor score for the kth factor (in this case there are two factors)

L represents factor loadings (ith) 1 through 8 for the corresponding factor

X represents the value of the corresponding measured variable

Using this type of equation, the scores for variables X1–X

8 can be summarized by two scores,

one for factor 1 and one for factor 2. This provides an example of the rule of parsimony. If the researcher wanted to analyze the correlation among these variables, now all that needs to be done is to analyze the bivariate correlation between factor 1 (nostalgia) and factor 2 (loyalty). This should prove much easier than analyzing an 8 × 8 correlation matrix. Statistical programs like SPSS and SAS will produce factor scores automatically if requested.

We can see that because F1 is associated with high values for L

1 through L

4 (and low values for

L5, L

6, L

7, and L

8) and F

2 is associated with high values for L

5 through L

8 (and low for L

1, L

2, L

3, and

L4), F

1 is determined almost entirely by the nostalgia items and F

2 is determined almost entirely

by the customer loyalty items. The factor pattern of high and low loadings can be used to match measured variables to factors in this way.

■ CREATING COMPOSITE SCALES WITH FACTOR RESULTS

When a clear pattern of loadings exists as in this case, the researcher may take a simpler approach. F

1 could be created simply by summing the four variables with high loadings and creating a sum-

mated scale representing nostalgia. F2 could be created by summing the second four variables

(those loading highly on F2) and creating a second summated variable. While not necessary, it is

often wise to divide these summated scales by the number of items so the scale of the factor is the same as the original items. For example, F

1 would be

((X1 + X

2 + X

3 + X

4 )/4)

The result provides a composite score on the 1–7 scale. The composite score approach would introduce very little error given the pattern of loadings. In other words, very low loadings suggest a variable does not contribute much to the factor. The reliability of each summated scale can be tested by computing a coefficient alpha estimate. Then, the researcher could conduct a bivariate regression analysis that would test how much nostalgia contributed to loyalty.

■ COMMUNALITY

While factor loadings show the relationship between a variable and each factor, a researcher may also wish to know how much a single variable has in common with all factors. Communality is a measure of the percentage of a variable’s variation that is explained by the factors. A relatively high communality indicates that a variable has much in common with the other variables taken as a group. A low communality means that the variable does not have a strong relationship with the other variables. The item might not be part of one of the common factors or might represent a separate dimension. Communality for any variable is equal to the sum of the squared loadings for that variable. The communality for X

1 is

0.902 + 0.022 = 0.8104

Communality values are shown on factor analysis printouts.

■ TOTAL VARIANCE EXPLAINED

Along with the factor loadings, the percentage of total variance of original variables explained by the factors can be useful. Recall that common variance is correlation squared. Thus, if each loading is squared and totaled, that total divided by the number of factors provides an estimate of the variance in a set of variables explained by a factor. This explanation of variance is much the same as R2 in multiple regression. Again, these values are computed by the statistics program so there is seldom a need to



compute them manually. In this case, though, the variance accounted for among the eight variables by the nostalgia factor is 0.36 and the variance among the eight variables explained by the loyalty factor is 0.35. Thus, the two factors explain 71 percent of the variance in the eight variables:

0.36 + 0.35 = 0.71

In other words, the researcher has 71% of the information in two factors that are in the original eight items, another example of the rule of parsimony.

Cluster Analysis

Cluster analysis is a multivariate approach for identifying objects or individuals that are similar to one another in some respect. Cluster analysis classifies individuals or objects into a small number of mutually exclusive and exhaustive groups. Objects or individuals are assigned to groups so that there is great similarity within groups and much less similarity between groups. The cluster should have high internal (within-cluster) homogeneity and high external (between-cluster) heterogeneity.

Cluster analysis is an important tool for the business researcher. For example, an organization may want to group its employees based on their insurance or retirement needs, or on job per-formance dimensions. Similarly, a business may wish to identify market segments by identifying subjects or individuals who have similar needs, lifestyles, or responses to marketing promotions. Clusters, or subgroups, of recreational vehicle owners may be identified on the basis of their similarity with respect to recreational vehicle usage and the benefits they want from recreational vehicles. Alternatively, the researcher might use demographic or lifestyle variables to group indi-viduals into clusters identified as market segments.

We will illustrate cluster analysis with a hypothetical example relating to the types of vacations taken by 12 individuals. Vacation behavior is represented on two dimensions: number of vacation days and dollar expenditures on vacations during a given year. Exhibit 24.7 is a scatter diagram that represents the geometric distance between each individual in two-dimensional space. The diagram portrays three clear-cut clusters. The first subgroup—consisting of individuals L, H, and B—suggests a group of individuals who have many vacation days but do not spend much money on their vacations. The second cluster—represented by individuals A, I, K, G, and F—represents intermediate values on both variables: average amounts of vacation days and average dollar expenditures on vacations.

cluster analysis

A multivariate approach for grouping observations based on similarity among measured variables.

EXHIBIT 24.7Clusters of Individuals onTwo DimensionsL

H B

A I

K

F

G

CJ

ED

1

2

3

Num

ber

of V

acat

ion

Day

s

Dollars Spent on Vacations



The third group—individuals C, J, E, and D—consists of individuals who have relatively few vaca-tion days but spend large amounts on vacations.

In this example, individuals are grouped on the basis of their similarity or proximity to one another. The logic of cluster analysis is to group individuals or objects by their similarity to or distance from each other. The mathematical procedures for deriving clusters will not be dealt with here, as our purpose is only to introduce the technique.

A classic study provides a very pragmatic example of the use of cluster analysis.17 Managers fre-quently are interested in finding test-market cities that are very similar so that no extraneous varia-tion will cause differences between the experimental and control markets. In this study the objects to be clustered were cities. The characteristics of the cities, such as population, retail sales, number of retail outlets, and percentage of nonwhites, were used to identify the groups. Cities such as Omaha, Oklahoma City, Dayton, Columbus, and Fort Worth were similar and cities such as Newark, Cleveland, Pittsburgh, Buffalo, and Baltimore were similar, but individual cities within each group were dissimilar to those within other groups or clusters. (See Exhibit 24.8 for additional details.)

This example should help to clarify the difference between factor analysis and cluster analysis. In factor analysis the researcher might search for constructs that underlie the variables (population, retail sales, number of retail outlets); in cluster analysis the researcher would seek constructs that underlie the objects (cities). Cluster analysis differs from multiple discriminant analysis in that the

Cluster Cluster ClusterNumber City Number City Number City

1 Omaha 7 Sacramento 13 Allentown Oklahoma City San Bernardino Providence Dayton San Jose Jersey City Columbus Phoenix York Fort Worth Tucson Louisville

2 Peoria 8 Gary 14 Paterson Davenport Nashville Milwaukee Binghamton Jacksonville Cincinnati Harrisburg San Antonio Miami Worcester Knoxville Seattle

3 Canton 9 Indianapolis 15 San Diego Youngstown Kansas City Tacoma Toledo Dallas Norfolk Springfield Atlanta Charleston Albany Houston Fort Lauderdale

4 Bridgeport 10 Mobile 16 New Orleans Rochester Shreveport Richmond Hartford Birmingham Tampa New Haven Memphis Lancaster Syracuse Chattanooga Minneapolis

5 Wilmington 11 Newark 17 San Francisco Orlando Cleveland Detroit Tulsa Pittsburgh Boston Wichita Buffalo Philadelphia Grand Rapids Baltimore

6 Bakersfield 12 Albuquerque 18 Washington Fresno Salt Lake City St. Louis Flint Denver El Paso Charlotte Beaumont Portland

Note: Points not in a cluster—Honolulu, Wilkes-Barre.Source: Reprinted by permission, Paul E. Green, Ronald E. Frank, and Patrick J. Robinson, “Cluster Analysis in Test-Market Selection,” Management Science, Vol. 13, P.B393 (Table 2), April 1967. Copyright © 1967, the Institute for Operations Research and the Management Sciences (INFORMS), 7240 Parkway Drive, Suite 310, Hanover, MD 21076, USA.

EXHIBIT 24.8 Cluster Analysis of Test-Market Cities



groups are not predefined. The purpose of cluster analysis is to determine how many groups really exist and to define their composition.

Multidimensional Scaling

Multidimensional scaling provides a means for placing objects in multidimensional space on the basis of respondents’ judgments of the similarity of objects. The perceptual difference among objects is reflected in the relative distance among objects in the multidimensional space.

In the most common form of multidimensional scaling, subjects are asked to evaluate an object’s similarity to other objects. For example, a sports car study may ask respondents to rate the similarity of an Acura TSX to a Chevrolet Corvette, then an Acura NSX to the Corvette, followed by a Lotus Elise to the Corvette, a Mustang to the Corvette, and so forth. Then, the comparisons are rotated (i.e., Acura NSX to the TSX, Lotus Elise to the TSX, and so on until all pairs are exhausted). Multidimensional scaling would then generate a plot of the cars, and the analyst then attempts to explain the difference in objects on the basis of the plot. The interpretation of the plot is left to the researcher.

In one study MBA students were asked to provide their perceptions of relative similari-ties among six graduate schools. Next, the overall similarity scores for all possible pairs of objects were aggregated for all individ-ual respondents and arranged in a matrix. With the aid of a com-puter program, the judgments about similarity were statisti-cally transformed into distances by placing the graduate schools into a specified multidimensional space. The distance between sim-ilar objects on the perceptual map was small for similar objects; dissimilar objects were farther apart.

Exhibit 24.9 on the next page shows a perceptual map in two-dimensional space. Inspection of the map illustrates that Harvard and Stanford were perceived as quite similar to each other. MIT and Carnegie also were perceived as very similar. MIT and Harvard, on the other hand, were perceived as dissimilar. The researchers iden-tified the two axes as “quantitative versus qualitative curriculum” and “less versus more prestige.” The labeling of the dimension axes is a task of interpretation for the researcher and is not statistically determined. As with other multi-variate techniques in the analysis

multidimensional scaling

A statistical technique that mea-sures objects in multidimensional space on the basis of respon-dents’ judgments of the similarity of objects.

How similar are these cars? This is the input to multidimensional scaling.

© R

EBEC

CA C

OO

K/RE

UTE

RS/L

AN

DO

V©

REB

ECCA

CO

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REU

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/LA

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OV



EXHIBIT 24.9Perceptual Map of Six Graduate Business Schools: Simple Space

Qualitative curriculum

Quantitative curriculum

Moreprestige

Lessprestige

Chicago

Carnegie

Massachusetts Instituteof Technology

Harvard

StanfordWharton

Source: Green, P. E., Carmone F. J., and Robertson, P. J., “Nonmetric Scaling Methods: An Exposition and Overview,” The Wharton Quarterly, Vol. 2, 1968, pp. 159–173.

of interdependence, there are several alternative mathematical techniques for multidimensional scal-ing. Likewise, there are multiple ways of using multivariate procedures to generate a perceptual map. For example, factor scores resulting from factor analysis can be plotted along the factor dimensions. Such an approach may show the competitive positioning of several different firms along dimensions related to value and quality.

Exhibit 24.10 summarizes the multivariate techniques for analysis of interdependence.

600

● The analysis stage illustrates the interdependency of the busi-ness research steps. How we structured the questionnaire and the level of data we gathered heavily influences the anal-ysis we can conduct. If we know the type of analysis we want to do, then we must construct the survey instrument around this analysis.

● Flow charts—such as those presented in Exhibits 24.1, 24.2, and 24.3—are very useful tools for a business researcher when determining the appropriate analytical technique.

● Multiple regression is used for two purposes.● To predict something based on known information. For

example, consider a fast-food restaurant considering a new location. Information from current restaurants can be used to build a model showing the relationship between independent variables such as population density, traf-fic flow, average income, population age distributions, distance to competitive restaurants, and so forth. These factors can be regressed on the dependent variable, sales volume. By using the unstandardized coefficients, the bs from this model, we can predict sales at potential loca-tions under consideration.

● To explain the drivers of something. Consider the fast-food example above. Which of these factors is most important in determining sales? By examining the standardized coefficients, the βs from this model, we can directly compare the different independent variables. What is the most important driver of restaurant sales?

● Multicollinearity can be a big problem. It can cause the parameter estimates to take on unreasonable and unreli-able values. VIFs of 5 or over are indicative of problems with multicollinearity.

● In a typical data matrix, with the variables as columns and cases as rows, we can think of factor analysis as grouping together the variables (columns) while cluster analysis groups together the respondents (rows).

● After clusters are determined, the cluster members can be “profiled” by examining the makeup (such as attitudes and demographic characteristics) of the group members. This is eas-ily done by using the cluster membership as the factor variable in ANOVA and the variables of interest in the dependent list. ©

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nts, tly variables. What is the



Technique Purpose Type of Measurement

Factor analysis To summarize into a reduced number of factors the information contained in a large number of variables

Interval

Cluster analysis To classify individuals or objects into a small number of mutually exclusive and exhaustive groups, ensuring that there will be as much likeness within groups and as much difference among groups as possible

Interval

Multidimensional scaling

To measure objects in multidimensional space on the basis of respondents’ judgments of their similarity

Varies depending on technique

EXHIBIT 24.10 Summary of Multivariate Techniques for Analysis of Interdependence

Summary1. Understand what multivariate statistical analysis involves and know the two types of multi-variate analysis. Multivariate statistical methods analyze multiple variables or even multiple sets of variables simultaneously. They are particularly useful for identifying latent constructs using multiple individual measures. Multivariate techniques represent data through the use of variates. Variates are mathematical combinations of variables. The two major types of multivariate procedures are inter-dependence and dependence techniques. Interdependence techniques do not distinguish depen-dent and interdependent variables, whereas dependence techniques do make this distinction.

2. Interpret results from multiple regression analysis. Multiple regression analysis predicts a con-tinuous dependent variable with multiple independent variables. The independent variables can be either continuous or categorical. Categorical variables must be coded as dummy variables. Multiple regression results are analyzed by examining the significance of the overall model using the F-test results, the individual parameter estimates, the overall model R2, and the model col-linearity diagnostics. Standardized regression coefficients have the advantage of a common scale, making them comparable from model to model and variable to variable.

3. Interpret results from multivariate analysis of variance (MANOVA). MANOVA is an extension of ANOVA involving multiple related dependent variables. Thus, MANOVA represents a form of the GLM predicting that multiple categorical independent variables affect multiple, related depen-dent variables. Interpretation of a MANOVA model is similar to interpretation of a regression model. However, the multivariate F-test results associated with Wilke’s lambda (Λ) are interpreted first, followed by interpretation of the individual ANOVA results.

4. Interpret basic exploratory factor analysis results. EFA is a data reduction technique in which the variance in multiple variables is represented by a smaller number of factors. The factors gener-ally represent latent factors or indexes. The pattern of loadings suggests both the number of latent factors that may exist and indicates which variables are associated with each factor. Rotated factor solutions are useful in properly interpreting factor analysis results.

5. Know what multiple discriminant analysis can be used to do. Another dependence technique is discriminant analysis. Discriminant analysis uses multiple independent variables to classify obser-vations into one of a set of mutually exclusive categories. In other words, discriminant analysis predicts a categorical dependent variable with multiple independent variables.

6. Understand how cluster analysis can identify market segments. Cluster analysis classifies mul-tiple observations into a smaller number of mutually exclusive and exhaustive groups. These should have as much similarity within groups and as much difference between groups as possible. In cluster analysis the groups are not predefined. However, clusters can be used to represent market segments because market segments also represent consumers who are similar to each other within a segment, but who are different from consumers in other segments.

Key Terms and Conceptscluster analysis, 597data reduction technique, 595dependence techniques, 583discriminant analysis, 590dummy variable, 585factor analysis, 593

factor loading, 594factor rotation, 594general linear model (GLM), 584interdependence techniques, 583multicollinearity, 588multidimensional scaling, 599

multiple regression analysis, 584multivariate analysis of variance

(MANOVA), 589partial correlation, 586rule of parsimony, 595variate, 581



Questions for Review and Critical Thinking1. Define multivariate statistical analysis.2. What is the variate in multivariate? What is an example of a

variate in multiple regression and in factor analysis?3. What is the distinction between dependence techniques and interde-

pendence techniques?4. What is GLM? How can multiple regression and n-way

ANOVA be described as GLM approaches?5. What are the steps in interpreting a multiple regression analy-

sis result? Can the same steps be used to interpret a univariate ANOVA model?

6. A researcher dismisses a regression result because the model R2 was under 0.70. Do you think this was necessarily wise? Explain.

7. Return to the simple example of regression results for the toy company presented in the chapter. Since the data come equally from Europe and Canada, does this represent a potential source of variation that is not accounted for in the researcher’s model?

How could the researcher examine whether or not sales may be dependent upon country?

8. What is a factor loading? 9. How does factor analysis allow for data reduction?10. How is the number of factors decided in most EFA programs?11. What is multidimensional scaling? When might a researcher use

this technique?12. What is cluster analysis? When might a researcher use this

technique?13. Name at least two multivariate techniques that can be useful in

constructing perceptual maps.14. A researcher uses multiple regression to predict a client’s sales

volume based on gross domestic product, personal income, disposable personal income, unemployment, and the consumer price index. What problems might be anticipated with this mul-tiple regression model?

Research Activities1. Use the multistep process to interpret the regression results below. This model has been run by a researcher trying to explain customer

loyalty to a restaurant. The independent variables are customer perceptions of value, atmosphere, quality, and a location variable labeled center. This is a dummy variable that takes the value of 1 if the restaurant is in a shopping center and 0 if it is a stand-alone location. What substantive conclusions would you recommend to the restaurant company?

Model Summary

Model R R SquareAdjusted R Square

Std. Error of the

Estimate

1 0.176 0.031 0.027 0.996

DV � Loyalty

ANOVA(b)

Model Sum of Squares df Mean Square F Sig.

1 Regression 27.9731 4 6.9933 7.049 0.0000138

Residual 876.0469 883 0.9921

Total 904.0200 887

Coefficient s(a)

ModelUnstandardized

Coefficients BStd.

Error

Standardized Coefficients

Beta t Sig. VIF

1 (Constant) �0.306 0.229 �1.338 0.181

Value 0.104 0.036 0.099 2.877 0.004 1.087

Atmosphere 0.048 0.026 0.067 1.883 0.060 1.144

Quality 0.044 0.028 0.054 1.590 0.112 1.038

Center �0.250 0.071 �0.124 �3.508 0.000 1.132



2. Interpret the following GLM results. Following from an example in the chapter, Performance is the performance rating for a business unit manager. Sales is a measure of the average sales for that unit. Experience is the number of years the manager has been in the industry. The variable dummy has been added. This variable is 0 if the manager has no advanced college degree and a 1 if the manager has an MBA. Do you have any recommendations?

The SAS System 21:06 Wednesday, April 22, 2009 The GLM Procedure Dependent Variable: performance

Sum ofSource DF Squares Mean Square F Value Pr > F

Model 3 173.6381430 57.8793810 13.87 <.0001

Error 36 150.2341040 4.1731696

Corrected Total 39 323.8722470

R-Square Coeff Var Root MSE performance Mean

0.536132 2.514731 2.042834 81.23468

Source DF Type III SS Mean Square F Value Pr > F

dummy 1 136.9511200 136.9511200 32.82 <.0001sales 1 22.4950649 22.4950649 5.39 0.0260Experience 1 2.2356995 2.2356995 0.54 0.4689

Level of -------performance------- -----------sales---------- ----Experience--------

dummy N Mean Std Dev Mean Std Dev Mean Std Dev

0 22 79.4848842 1.78987031 15979.7723 2008.32604 23.8984087 8.273274851 18 83.3733171 2.50773844 16432.0080 2015.18863 20.6788050 8.96324112

3. Interpret the following regression results. All of the variables are the same as in number 2. These results are produced with a regression program instead of the GLM-univariate ANOVA program.a. What do you notice when the results are compared to those in number 2? Comment.b. List the independent variables in order from greatest to least in terms of how strong the relationship is with performance.c. When might one prefer to use an ANOVA program instead of a multiple regression program?

The SAS System 21:07 Wednesday, April 22, 2009The REG Procedure

Model: MODEL1Dependent Variable: performance

Number of observations Read 40Number of observations Used 40

Analysis of Variance

Sum of MeanSource DF Squares Square F Value Pr > F

Model 3 173.63814 57.87938 13.87 <.0001Error 36 150.23410 4.17317Corrected Total 39 323.87225

Root MSE 2.04283 R-Square 0.5361Dependent Mean 81.23468 Adj R-Sq 0.4975Coeff Var 2.51473

Parameter Estimates

Parameter Standard StandardizedVariable Label DF Estimate Error t Value Pr > |t| Estimate

Intercept Intercept 1 72.68459 2.88092 25.23 <.0001 0dummy dummy 1 3.80621 0.66442 5.73 <.0001 0.66546Sales Sales 1 0.00038324 0.00016507 2.32 0.0260 0.26578Experience Experience 1 0.02829 0.03866 0.73 0.4689 0.08475



4. Interpret the following factor analysis results. The variables represent sample results of self-reported emotions while viewing a film. Why are only two factors reported below? What would you name the two summated scales which could be produced based on these results?

Total Variance Explained

Component

Initial Eigenvalues

Total% of

Variance Cumulative %

Extraction Sums of Squared Loadings

Total% of

Variance Cumulative %

1 2.94 36.74 36.74 2.94 36.74 36.74

2 2.51 31.34 68.08 2.51 31.34 68.08

3 0.71 8.84 76.92

4 0.60 7.53 84.45

5 0.42 5.20 89.65

6 0.29 3.67 93.32

7 0.29 3.64 96.96

8 0.24 3.04 100.00

Extraction Method: Principal Component Analysis.

Component Matrix(a)

Factor 1 Factor 2

Interesting 0.664 �0.327

Anxious 0.444 0.511

Enthusiastic 0.842 �0.332

Worried 0.295 0.828

Exciting 0.812 �0.206

Tired 0.269 0.835

Happy 0.784 �0.383

Guilty 0.398 0.675

Extraction Method: Principal Component Analysis.A 2 components extracted.

Rotated Component Matrix(a)

Component Factor 1 Factor 2

Interesting 0.739 �0.024

Anxious 0.194 0.648

Enthusiastic 0.904 0.044

Worried �0.073 0.876

Exciting 0.825 0.147

Tired �0.100 0.872

Happy 0.872 �0.025

Guilty 0.084 0.779

Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser Normalization.A Rotation converged in 3 iterations.

5. ’NET Go to http://www.census.gov and examine some of the tables for your area. Cut and paste the table into a spreadsheet or statistical program. Run one dependence and one interdependence technique on the data. Interpret the results.

6. ’NET Use http://www.ask.com to find an F-ratio calculator that will return a p-value given a calculated F-ratio and the degrees of free-dom associated with the test.

7. ’NET The Federal Reserve Bank of St. Louis maintains a database called FRED (Federal Reserve Economic Data). Navigate to the FRED database at http://www.stls.frb.org/fred/index.html. Use the consumer price index, exchange rates, interest rates, and one other variable to predict the consumer price index for the same time period. The data can either be downloaded or cut and pasted into another file.



The Utah Jazz are interested in understanding the market for the National Basketball Association. A study is conducted as described here.

Data CollectionData came from a survey of adult residents of a large

western metropolitan area. Respondents were selected in accordance with a quota sample of the area that was based on the age and sex characteristics reported in the most recent census. Six age categories for both males and females were used to gain representation of these characteristics in the market. In addition, interviewers were assigned to various geographic regions to ensure representation of the market with respect to socioeconomic characteristics. A total of 225 respondents age 18 and over provided data for the study.

Interviews were conducted by trained interviewers using a self-completion questionnaire. The presence of the interviewers served to answer any questions that might arise as well as to ensure compli-ance with the instructions.

Measures for the variables in the three categories of AIO (atti-tudes, interest, and opinions) were obtained using six-point rating

scales. For example, the item for price proneness asked, “When you are buying a product such as food, clothing, and personal care items, how important is it to get the lowest price?” This item was anchored with “Not at all important” and “Extremely important.”

The broadly defined category of demographics included standard socioeconomic characteristics as well as media preferences and atten-dance at professional hockey matches and university basketball games. Demographics were obtained using a variety of forced-choice and free-response measures, the natures of which are indicated in the vari-able information presented in Case Exhibit 24.1–1. The categorical measures of type of dwelling and preferred type of radio programming were coded as dummy variables for analysis. The criterion measure of patronage came from an open-ended question asking how many NBA games the respondent had attended during the past season.

Data AnalysisThe distribution of responses to the attendance item was skewed, as might be expected. Thus, 57.3 percent of the respondents reported having attended none of the 41 possible games. Those who attended at least one game were recorded in accordance with specification

Case 24.1 The Utah Jazz

CASE EXHIBIT 24.11 Characteristics of the Market for Professional Basketball

Means

None Low High Loading

Variables (n � 129) (n � 47) (n � 49) F-Ratio p I II

Market Orientationa

Price proneness 3.99 4.04 3.63 1.31 .271 Quality proneness 4.95 4.74 4.82 .74 .480 Product awareness 4.45 4.02 4.00 3.71 .026 Product involvement 4.34 4.43 4.14 .66 .517 Prepurchase planning 4.21 3.85 3.82 2.03 .134 Brand loyalty 3.95 4.39 3.92 .96 .384 Information search 3.83 3.55 3.96 1.06 .347

Interests in Leisure Pursuitsb

Need for change from work routine 4.11 4.34 4.55 1.92 .150 .34 Need for independence in leisure choice 4.88 4.94 4.96 .09 .911 .08 Need for companionship during leisure 4.85 5.13 4.88 1.16 .317 .10 Preference for passive versus active pursuits 3.64 4.15 4.57 7.28 .001 .70 Self-image as athletic 3.67 4.38 4.47 5.89 .003 .60 Childhood attendance at sporting events 3.38 3.89 4.18 5.41 .005 .60 Pleasure from sporting events 3.14 3.66 4.27 10.62 .000 .84

Opinions about Professional Sportsc

Athletes as a reference group 3.51 3.64 4.18 3.90 .022 .30 –.19 Excitement from enthusiastic crowd 4.27 4.72 4.73 2.70 .069 .24 .20 Excitement from animosity between teams 3.29 3.28 4.27 6.94 .001 .36 –.41 Acceptance of alcoholic beverages at games 2.60 3.64 3.39 6.88 .001 .34 .46 Enjoyment from large crowds 3.91 3.85 4.49 3.22 .042 .23 –.32 Enjoyment when standing at games 3.37 3.44 3.90 2.25 .108 .22 –.17 Excitement of professional basketball 4.09 3.91 4.67 5.34 .005 .27 –.49 Satisfaction from professional basketball 3.17 3.70 4.80 24.98 .000 .78 –.26 Importance of a winning team 4.26 4.69 5.07 6.12 .003 .39 .02

Demographicsd

Years in local area (number of years) 24.47 23.51 19.04 2.02 .135 –.24 Sex (0 = female, 1 = male) .40 .55 .65 5.45 .006 .39


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(Continued)



CASE EXHIBIT 24.11 Characteristics of the Market for Professional Basketball (Continued)

of the light half and the heavy half of the market. This category of patrons was split as nearly as possible at the median, giving 20.9 per-cent who attended one or two games and 21.8 percent who attended three or more. The three patronage categories thus used for analysis were subsequently termed the none, low, and high attendance segments.

Given the categorical nature of the criterion measure and the continuous nature of the predictor variables, both univariate analysis of variance and discriminant analysis were employed for the survey. Each of the four categories of predictor variables was subjected to a separate discriminant analysis to test the multivariate hypothesis of relationship between patronage and the predictor set in question. The univariate ANOVAs were used to provide complementary information about the nature of the segments.

ResultsCase Exhibit 24.1–1 gives the results of the analyses conducted on the four sets of predictor variables. Each set produced at least one variable that was significant in univariate analysis. Three of the four discriminant analyses were significant.

The first predictor set involving AIOs, “marketing orientation,” provided only a single variable that ANOVA showed to differentiate

among the members of the three patronage segments. The discrimi-nant analysis was nonsignificant.

“Interests in leisure pursuits” emerged as more predictive. By univariate ANOVA, four variables were found significant at the 0.05 level. The discriminant analysis was significant at p = 0.004.

“Opinions about professional sports” provided significant pre-diction of patronage. Seven of the nine variables reached significance at the 0.05 level in univariate analysis. The discriminant analysis was significant beyond p = 0.001, and it produced two significant functions. The first significant function provided 79.8 percent of the explained variance, and the second function provided 20.2 percent.

Finally, the set “demographics” was also found to be related to patronage. Counting the four dummy-coded measures of dwelling type and the five similar preferences for radio programming as sepa-rate variables, 7 of the 22 demographics reached significance in uni-variate analysis. The discriminant analysis was significant at p = 0.004.

QuestionInterpret the managerial significance of the ANOVA and multiple discriminant analysis results.

Source: Courtesy of the American Marketing Association. Adapted from paper presented at AMA conference, 1984.


Demographicsd

Marital status (0 = single, 1 = married) .60 .62 .45 2.00 .138 –.21 Household size (number of persons) 3.13 3.27 3.14 .11 .896 .01 Rents apartment (0 = no, 1 = yes) .18 .32 .35 3.70 .026 .30 Rents a house (0/1) .09 .09 .08 .03 .967 –.03 Owns a house (0/1) .60 .49 .41 3.08 .048 –.29 Owns a condominium (0/1) .05 .02 .06 .50 .607 .01 Head of household (0/1) .52 .64 .67 2.19 .115 .24 Occupational prestige of self (NORC scale) 68.05 69.36 70.63 1.27 .284 .19 Job leaves evenings free for entertainment (0/1) .87 .85 .92 .57 .567 .10 Prefers easy-listening music radio programming (0/1) .39 .34 .29 .83 .438 –.15 Prefers contemporary popular music radio (0/1) .16 .28 .27 1.96 .143 .20 Prefers rock music radio (0/1) .14 .11 .27 2.76 .066 .23 Prefers country-western music radio (0/1) .15 .19 .08 1.22 .299 –.12 Prefers talk and news radio programming (0/1) .09 .04 .06 .52 .597 –.08 Education (years of schooling) 13.08 13.66 13.56 5.11 .007 .38 Age (years) 41.51 39.79 33.59 4.21 .016 –.34 Annual household income (7-point scale) 4.88 5.11 5.16 .65 .523 .13 Monthly personal expenditures on entertainment for household (dollars) 85.10 112.45 101.29 1.38 .254 .13 Attendance at university basketball (games last year) .92 1.89 4.14 15.29 .000 .66 Attendance at professional hockey (matches last year) .69 2.28 2.78 5.33 .006 .37

aCanonical discriminant analysis not significant at p = .189; therefore, no loadings are given.bCanonical discriminant analysis significant at p = .004, first function significant. Centroids for the market segment groups are as follows: none, –.29; low, .19; high, .59.cCanonical discriminant analysis significant at p = .000, both functions significant. Centroids for the market segment groups on the first function are as follows: none, –.47; low, .26; high, 1.00. Centroids on the second function are as follows: none, –.10; low, .57; high, –.27.dCanonical discriminant analysis significant at p = .004, first function significant. Centroids for the market segment groups are as follows: none, –.41; low, .14; high, .97.

Means

None Low High Loading

Variables (n � 129) (n � 47) (n � 49) F-Ratio p I II



Download the data sets for this case from http://www.thomsonedu.com/marketing/zikmund or request them from your instructor.

Use the data labeled profit for this case. The data go along with The Research Snapshot on page 587. In addition, management has collected

several semantic differential scales from the managers asking them to use emotions to describe the way they feel about their jobs. The emotions include

involved exciting fun satisfied happy pleased

The managers want to understand turnover. So, another variable is included that gives the likelihood a manger will quit within

12 months (labeled turnover in data). After running some initial regression models with eight independent variables predicting turnover, management was confused. They complained that there were too many variables to make sense of.

Thus, the researcher turned to a data reduction technique. Afterwards, a regression model with fewer independent variables gave some clear direction regarding emotions and turnover:1. Perform the appropriate multivariate technique to identify

underlying dimensions that may exist among the emotion ratings.

2. Create scales for any underlying dimensions.3. Use these scales as independent variables in a regression model.4. Interpret the results.

Case 24.2 How Do We Keep Them?


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Multivariate Statistical Analysis