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Chapter 10: Multivariate Methods of Marketing Research I: Factor, Cluster, and Discriminant

Analyses

TRUE/FALSE

1) In dependence models, no variable or variables are designated as being predicted by

others, because the researcher is interested in the interrelationships among all of the variables

taken together.

Answer: False

This is a description of interdependence methods.

2) Factor analysis tells us which variables are similar to one another and how they should be

grouped.

Answer: True

Factor analysis provides this information about variables, whereas cluster analysis tells us

which cases are similar and how they should be grouped.

3) Cluster analysis clarifies the underlying structure of multivariate data in a way that makes

it the best complement to regression analysis.

Answer: False

Factor analysis is the perfect complement because it will reduce a data set to a smaller set of

variables that are not highly correlated.

4) Factor analysis can be used for data transformations by identifying covariates that are

exactly uncorrelated and make good inputs for dependence methods.

Answer: True

Through factor analysis, a large group of variables can be reduced to a smaller set of factors

that are not correlated.

5) The principal .reason for rotating a factor analysis solution is to increase the amount of

“explained” variance in the variables.

Answer: False

The principal reason for rotating a factor analysis solution is to aid in the interpretation of the

factors.

6) The principal components methodology used in discriminant analysis determines the

values in the linear combination that explains as much variance between correlation matrices

as possible.

Answer: False

The principal components methodology is used with factor analysis, not discriminant

analysis, and it is to explain variance in the correlation matrix, not between correlation

matrices.

7) Varimax rotation involves an oblique rotation of the factors.

Answer: False

Varimax rotation involves an orthogonal rotation of the factors.

8) One of the primary problems with the principal components analysis of factor analysis is

that it can take most of the variance to explain the first factor and leave little variance for

other factors to explain.

Answer: True

Unfortunately, principal components analysis will try to explain as much of the variance as

possible in that first factor, thus sometimes leaving little for other factors.

9) If the eigenvalue of the first factor on a factor analysis is 5.3, this could be interpreted that

this first factor is doing the work of about 5.3 of the original variables.

Answer: True

The eigenvalue is the proportion of the variance explained relative to the average.

10) Varimax rotation in factor analysis will tell researchers the optimal number of factors they

should have in a data reduction situation.

Answer: False

Researchers have to use their own judgment to determine the number of factors, based on the

principal components analysis. The varimax rotation is used after the number of factors has

been decided.

11) The MinEigen criterion imposed by computer programs retains only those factors with

eigenvalues greater than 1.

Answer: True

The MinEigen criterion retains only those factors with eigenvalues greater than 1 (i.e., the

“bigger than average” factors).

12) Statisticians often call the clusters obtained through cluster analysis “latent” because they

need to be discerned via analysis and are not directly observable.

Answer: True

With cluster analysis, the clusters are formed through analysis and are not defined a priori.

13) Among the most common uses of cluster analysis in marketing are segmenting customers

and products.

Answer: True

Cluster analysis is excellent for segmentation projects, because the program is able to locate

clusters within a data set.

14) Unlike regression analysis, cluster analysis does not have any independent variables.

Answer: False

Cluster analysis does not have any dependent variables; all are independent variables.

15) In cluster analysis it is especially dangerous if the ranges of the variables are dramatically

different.

Answer: True

For cluster analysis to be effective, the ranges of the variables cannot be drastically different;

otherwise, the variables with the largest range will dominant the solution.

16) Variables that have drastically different ranges can be standardized for cluster analysis

through use of a z-transform, which gives all of the variables equal ranges.

Answer: True

The purpose of a z-transform is to convert all of the variables to scales with a mean of 0 and

standard deviation of 1.

17) To use cluster analysis, researchers must choose either a distance metric or a clustering

criterion.

Answer: False

Researchers must choose both a distance metric and a clustering technique.

18) Hierarchical clustering works well if a researcher needs a fixed number of clusters,

whereas nonhierarchical clustering is more useful when one needs to compare different

clustering solutions.

Answer: False

It is just the opposite; nonhierarchical clustering works well if a researcher needs a fixed

number of clusters, whereas hierarchical clustering is more useful when one needs to

compare different clustering solutions.

19) In hierarchical cluster analysis, once a group of objects is clustered together, they are

together for life, i.e. the program never separates them during the analysis.

Answer: True

This is one of the disadvantages of hierarchal clustering. It keeps an object in the original

group for the duration of the program, even though the cluster might be better off in another

cluster.

20) Discriminant analysis and factor analysis are examples of interdependence methods.

Answer: False

Although factor analysis is an interdependence method, discriminant analysis is a dependence

method.

21) Discriminant analysis can only be used for a binary nominal dependent variable.

Answer: False

It can be used for any k number of categories in nominal data.

22) Although discriminant analysis works well for discriminating among known groups of a

dependent variable, it cannot be used reliably for predicting into which of the pre-established

groups a new set of items will fall when the group membership of those items is not already

known.

Answer: False

One of the uses of discriminant analysis is classification/prediction. That is, given a new set

of items (e.g., customers, products, firms) whose group membership is not known, into which

of the pre-established groups are they likely to fall?

23) A discriminant function is a linear combination of the independent variables that makes

the predicted mean for each category as different as possible. This function is of the form

Answer: True

This is the definition of a discriminant function.

24) The v’s of a DF are estimated so that

is minimized.

Answer: False

The v’s of a DF are estimated so that

is maximized.

25) A confusion matrix crosstabulates the distances between discriminant functions to

facilitate graphical analysis.

Answer: False

A confusion matrix categorizes correct and incorrect predictions by crosstabulating the

dependent variable category that the discriminant function predicts a subject will be in with

the category that the subject is actually in.

MULTIPLE CHOICE

1) All of the following are underlying reasons for the increased use of multivariate analysis

techniques except

a. expanding global usage of marketing research

b. marketing problems are rarely completely described by one or two variables

c. dramatic increase in computer speed and advances in software packages

d. improved understanding of statistical concepts among marketing researchers

Answer: A

Although more marketing research is used globally, it is not an underlying reason for

increased usage of multivariate techniques.

2) In dependence methods

a. one or more variables is being predicted by a set of dependent variables

b. one or more variables is being predicted by a set of independent variables

c. the interrelationship of a set of variables, taken together, is studied

d. one variable is being predicted by a set of independent variables

Answer: B

With dependence methods one or more dependent variables is predicted by a set of

independent variables. It is interdependence methods that study the interrelationship of a set

of variables taken together.

3) Both _______________ analyses are classified as data reduction methods because they

take a great deal of data and summarize them with a much smaller set of quantities.

a. regression and factor

b. factor and discriminant

c. factor and cluster

d. cluster and discriminant

Answer: C

The data reduction methods are factor analysis and cluster analysis.

4) _______________ tells us which cases, or people, or objects are similar and how they

should be grouped.

a. Factor analysis

b. Cluster analysis

c. The principle components method

d. Discriminant analysis

Answer: B

Cluster analysis tells which cases are similar and how they can be grouped.

5) Factor analysis can be used for all of the following applications in marketing research

except

a. data reduction

b. structure identification

c. scaling

d. segmentation

Answer: D

Segmentation is an application for cluster analysis, not factor analysis.

6) Factor analysis can help determine coefficients in multiple regression, because

a. principle components make useful variables

b. factor loadings make useful coefficients

c. coefficients measure the effect of unit changes in a variable assuming all other variables

remain constant

d. all of the above

Answer: C

Coefficients in multiple regression measure the effects of unit changes in a variable assuming

all other variables remain constant, and factor analysis can provide uncorrelated factors.

7) A problem in developing a scale is in weighting the variables being combined in the scale.

_______________ can do this by using the _______________ as the weights.

a. factor analysis, factor loadings

b. cluster analysis, centroid distances

c. multiple regression, coefficients of the regression

d. discriminant analysis, coefficients of the DF

Answer: A

Factor analysis is an excellent method for determining the weights for a scale.

8) A principle component can be defined as

where the b values

Answer: A

z is the linear combination, called a principal component. The principal components

methodology determines values for {b1, b2, …, bm} that explain as much variance in the

correlation matrix as possible.

9) Factor analysis uses three steps in arriving at a solution. The first step is to

a. develop a set of correlations between all combinations of variables

b. sort the data by cases from lowest to highest

c. determine the optimum number of factors to explain the variables

d. rotate the factors

Answer: A

The steps are:

1) develop a set of correlations between all combinations of variables

2) extract an initial set of factors

3) rotate the initial factors

10) In factor analysis, the object of the initial extraction is to

a. develop weights for each variable

b. find a set of factors that are linear combinations of the variables in the correlation matrix

c. rotate the factors so that they will be uncorrelated with one another

d. rotate the factors so they can become correlated with one another

Answer: B

The purpose of extraction is to extract a smaller set of factors that are linear combinations of

the variables in the correlation matrix.

11) In factor analysis, the _______________ looks for the single best factor to explain all of

the variables in the data set, then it constructs a second factor that explains as much as what is

left over as possible, and then a third, and so on.

a. varimax rotation method

b. principal components method

c. eigenvalue

d. hierarchical clustering solution

Answer: B

This is the purpose of the principal components analysis and is the first step in running a

factor analysis.

12) The very purpose of _______________ is to gauge just how much redundancy there is in

a set of variables and to assess which questions or variables best align with others and then to

group them together.

a. factor analysis

b. cluster analysis

c. multiple regression

d. discriminant analysis

Answer: A

Factor analysis is excellent for reducing a large number of variables, or questions on a survey,

to a smaller set of constructs or variables.

13) In factor analysis, the _______________ represents how much variance a factor explains

relative to how much it would be expected to explain by chance alone, that is, on the average.

a. principal component

b. factor loading

c. eigenvalue

d. correlation matrix

Answer: C

By definition, the eigenvalue represents how much of the total variance is explained by a

factor relative to the average.

14) When interpreting a factor analysis, factors with eigenvalues slightly greater than 1

a. should be discarded

b. are suspect

c. should be assigned higher loadings

d. should be rotated again

Answer: B

Factors with eigenvalues below 1 should be discarded, those slightly above one considered

suspect, and those greater than 1 retained.

15) In terms of deciding whether to keep or discard a factor based on its eigenvalue, the

typical cutoff value is around

a. 0

b. 1

c. 2

d. 5

Answer: B

Typically, 1 is used as the cutoff, below which factors are discarded. How much greater than

1 an eigenvalue should be for a factor to be deemed acceptable is subject to the

experimenter’s judgment and needs.

16) In the factor analysis output, the proportion refers to

a. the total sum-of-squares error in the factor

b. the portion of the error that remains unexplained by that factor

c. how much incremental variance is accounted for by each successive factor

d. how much of the total variance is explained by all of the factors taken together up to that

factor

Answer: C

The proportion tells how much of incremental variance is explained by that factor.

17) A scree plot gives a simple, but fairly crude, visual representation of

a. how quickly the quality of the factors degrade in terms of incremental variance

b. the hierarchy of the clustering solution

c. which segments correspond to which clusters

d. the factor loadings for each of the factors

Answer: A

A scree plot is a visual representation of the incremental variances of each of the factors.

18) In examining factor loadings in a factor analysis, a high loading would have a score

a. above 0

b. above 3

c. close to either -1 or 1

d. with an absolute value approaching 5

Answer: C

Factor loadings range from -1 to +1 and scores close to either -1 or +1 are high loadings.

19) Suppose a factor analysis results in 8 factors and that the communality for variable 12 is

0.863. The 0.863

a. means about 86.3 percent of the variance of variable 12 is explained by the eight factors

b. means that 86.3 percent of the variance of variable 12 is explained by the factor on which it

loads

c. is the factor loading for variable 12

d. explains how much of the variable is error

Answer: A

Communalities explain how much variance a particular variable has explained by all of the

factors in the model.

20) _______________ refer(s) to how well all the factors in the solution taken together

explain each of the variables.

a. The overall eigenvalue

b. The F-test

c. Factor loadings

d. Communalities

Answer: D

Communalities refer to how well two things correlate; in the case of factor analysis, how well

the factors taken all together correlate with each variable.

21) After making a decision on how many factors to use in a factor analysis, a varimax

rotation is used because it will reorient the original factors so their loadings are near

a. 1

b. -1

c. 1 or -1

d. 1, 0, or -1

Answer: D

Varimax will reorient factor loadings near 1, 0, or -1 so they can be interpreted.

22) Loadings of -1 or 1 mean the factors are _______________ with those variables.

a. parallel

b. perpendicular

c. orthogonal

d. uncorrelated

Answer: A

Loadings of -1 or 1 mean the factors are “parallel” or perfectly correlated with those

variables; loadings of 0 mean the factors are “orthogonal,” or perpendicular, and

uncorrelated.

23) Knowing the value that someone scores on the principle factor after a varimax rotation

tells you _______________ their score on other factors.

a. the loading coefficient of

b. the linear combination for

c. nothing about

d. the further rotation required to calculate

Answer: C

Varimax rotation results in uncorrelated factors, so knowing the score for one factor tells you

nothing at all about their score on any other factor.

24) In simple terms, _______________ works on dividing up data points so that those in the

same group are close to one another while those in different groups are far away.

a. factor analysis

b. cluster analysis

c. principal components analysis

d. discriminant analysis

Answer: B

The goal of cluster analysis is to locate groups or clusters within a data set that are similar to

each other and different than the other groups, or data points.

25) The best analysis technique for discovering segments within a market is

a. factor analysis

b. cluster analysis

c. principal components analysis

d. discriminant analysis

Answer: B

Segmentation of a market is really just clustering its brands or its customer base, albeit often

informally.

26) In cluster analysis, if the range of the variables is drastically different, then the

a. standard error will be large

b. cluster program will not be able to arrive at a solution

c. variables with the largest ranges will dominate the solution

d. the distance metric will exceed the level of aggregation

Answer: C

When the variables are drastically different, then the variables with the larger ranges will

dominant the solution.

27) A z-transformation is done by

a. subtracting the mean from the data point and then dividing by the variable’s standard

deviation

b. taking the square root of the sum of the squares of the differences between the data points

c. dividing each data point by the variable’s mean

d. subtracting the variable’s standard error from the value of each data point

Answer: A

To perform a z-transformation, the variable’s mean is subtracted from each data value, and

the result is then divided by the variable’s standard deviation. This makes each variable have

a mean of 0 and a standard deviation of 1.

28) The first thing the clustering routine does is

a. rotate the data orthogonally

b. calculate the communalities

c. calculate the cluster loadings

d. standardize the raw data

Answer: D

The first thing the clustering routine does is take the raw data and standardize it.

29) In the distance or dissimilarity matrix for a cluster analysis using squared Euclidean

distance, the larger the number

a. the more unlike the pair

b. the more alike the pair

c. the tighter the cluster

d. the larger the loading

Answer: A

The table shows dissimilarities so the larger the value, the greater the distance between the

two points.

30) In hierarchical cluster analysis, the _______________ tells researchers what order

various clusters join with one another. It begins by assuming each object is in its own cluster

and then lets the various objects join up.

a. dissimilarity schedule

b. agglomeration schedule

c. squared Euclidean distance

d. scree plot

Answer: B

The agglomeration schedule begins with each object in its own cluster, then agglomerates

objects into like clusters.

NOTE: Use the following table from a cluster analysis to answer the following questions.

31) After the first 3 clusters are formed, there are 9 clusters remaining,

a. each containing a single brand

b. 9 clusters with a single brand and 3 clusters with two brands each

c. 3 clusters with a single brand and 3 clusters with two brands each

d. 6 clusters with a single brand and 3 clusters with two brands each

Answer: D

Based on the table, the first three clusters contain two brands each. Given the total of 12

brands, that leaves 6 brands that are in their own clusters.

32) At the point where 7 clusters remain and the centroid distance is 0.617, the Schlitz beer

joins which cluster of beers?

a. Coors and Hamm’s

b. Miller Lite and Schlitz Light

c. Stroh’s Bohemian and Heileman’s Old Style

d. Budweiser and Lowenbrau

Answer: D

At this point, Budweiser, Lowenbrau, and Schlitz are all in the same cluster.

33) If the researcher were to stop after the centroid distance of 0.683, how many clusters

would be left with a single brand in it?

a. one

b. two

c. three

d. four

Answer: C

The three single-brand clusters are Heineken, Budweiser Light, and Coor’s Light.

34) When researchers want to divide a set of items into a known number of clusters, the

clustering technique that should be used is _______________ clustering.

a. k-means

b. principal components

c. varimax

d. hierarchical

Answer: A

k-means clustering tells the cluster analysis program how many clusters the researcher wants.

35) When analyzing a nominal dependent variable in terms of several interval independent

variables, researchers use

a. discriminant analysis

b. the principle components method

c. factor analysis

d. cluster analysis

Answer: A

Discriminant analysis is appropriate for analyzing nominal dependent variables in terms of

several interval or binary independent variables.

36) The basic idea of _______________ is to find a linear combination of the independent

variables that makes the mean scores across categories of the dependent variable as different

as possible.

a. discriminant analysis

b. factor analysis

c. principle components

d. cluster analysis

Answer: A

The objective of discriminant analysis is to find differences between different categories of

the nominal dependent variable.

37) All of the following of real-world uses of discriminant analysis except

a. distinguishing known groups of a dependent variable

b. determining the underlying structure of a set of variables

c. determining which input variables are most important to prediction

d. classifying new customers

Answer: B

Determining the underlying structure of a set of variables is a use of factor analysis, not

discriminant analysis.

38) A discriminant function

a. is a linear combination of independent variables

b. can be written as

c. minimizes

d. all of the above

e. both a and b

Answer: E

39) A _______________ matrix categorizes correct and incorrect predictions.

a. distance

b. correlation

c. confusion

d. dissimilarity

Answer: C

A confusion matrix categorizes correct and incorrect predictions by crosstabulating the

dependent variable category that the discriminant function predicts a subject will be in with

the category that the subject is actually in.

40) _______________ is a collection of methods that test whether a set of means are the

same.

a. Discriminant analysis

b. Analysis of covariance

c. Analysis of variance

d. Regression

Answer: C

Analysis of variance (ANOVA) is a collection of methods, all of which are unified by tests

for whether a set of means are the same.

SHORT ANSWER

1) Identify the three steps in a factor analysis solution.

Answer: The three steps are:

1) develop a set of correlations between all combinations of variables

2) extract a set of initial factors from the correlation matrix

3) rotate the initial factors to find a final solution

NOTE: Use the following table from a cluster analysis to answer the following questions.

2) If the researcher were to stop after the centroid distance of 0.683 with 6 remaining clusters,

identify each cluster and which brands are in it.

Answer: CL10: Miller Lite, Schlitz Light

CL9: Stroh’s Bohemian, Heileman’s Old Style

CL6: Coors, Hamm’s, Budweiser, Lowenbrau, Schlitz

plus 3 clusters each containing single item:

Heineken

Budweiser Light

Coor’s Light

3) When Heineken joins with a cluster, identify all of the other brands already in the cluster it

joins.

Answer: Brands include Coors, Hamm’s, Budweiser, Lowenbrau, and Schlitz.

4) When Budweiser Light joins with a cluster, identify all of the other brands already in the

cluster it joins.

Answer: Miller Lite and Schlitz Light

5) When is discriminant analysis appropriate?

Answer: Discriminant analysis is appropriate when one seeks to understand a nominal

dependent variable in terms of several (interval or binary) independent variables.

ESSAY

1) Discuss the similarities and differences between factor and cluster analyses.

Answer: Both factor and cluster analyses could be characterized as “data reduction”

techniques. Factor analysis tells us which variables are similar to one another and how they

should be grouped; cluster analysis tells us which cases are similar and how they should be

grouped.

2) Discuss the possible applications in marketing research for factor analysis.

Answer: Applications include:

1) data reduction—reducing a large set of variables to a smaller set of underlying factors

2) structure identification—identify the basic structure underlying a set of measures

3) scaling—identify weights that can be used with a weighted scale

4) data transformation—transform correlated variables into a smaller set of uncorrelated

factors that can be used with dependence analyses methods

3) Discuss the meaning of “eigenvalue” and the use of these within factor analysis.

Answer: An eigenvalue is a measure of the ratio of the variance explained by a specific factor

to that by the average factor. Therefore the sum of the eigenvalues for a factor analysis

solution will equal the original number of variables. A typical approach is for factors with

eigenvalues less than 1 to be discarded; those with eigenvalues not too much greater than 1 to

be considered suspect; and those with “large” eigenvalues retained in the solution. How much

greater than 1 an eigenvalue should be for a factor to be deemed acceptable is a judgment

call.

4) What is a latent class, and how does it apply to segmentation analysis?

Answer: A latent class are clusters that are not directly observable but need to be identified

via analysis, for example customer or product segments. Cluster analysis is used to identify

latent classes for segmentation analyses.

5) What is a discriminant function, and how does it relate to cluster analysis and

segmentation?

Answer: A discriminant function is a linear combination of the independent variables that

maximally discriminates among categories of the dependent variable. Discriminant functions

can be used to separate segments, often following a cluster analysis. On a graph of two

dimensions of a clustering solution, a discriminant function can be drawn as a line dividing

segments.

Test Bank for Modern Marketing Research: Concepts, Methods, and Cases

Fred M. Feinberg, Thomas Kinnear, James R. Taylor

9781133188964, 9781133191025, 9780759391710