These notes contain an outline of the chapter.
1.0. "Scaling" refers to arranging objects on a "scale," i.e., assigning numbers to them. Such numbers permit the representation of the objects as points along a line, or in a plane, or in space.
1.1. Multidimensional scaling (MDS) is essentially an exploratory technique designed to identify the evaluative dimensions employed by respondents and represent the respondents' perception of objects spatially.
1.2. These visual representations are referred to as "spatial maps".
The primary strength of MDS is its decompositional nature. It does not require the specification of the attributes used in evaluation. Rather, it employs a global measure of evaluation, such as similarities between objects, and then infers the dimensions of evaluation that constitute the overall evaluation. In this way it "decomposes" the overall evaluation into dimensions.
MDS is best used as an exploratory tool in identifying the perceptual dimensions used in the evaluation of a set of objects. It use of only global judgments and its ability to be "attribute-free" provide the research with an analytical tool minimizing the potential bias from mis-specification of the attributes of the objects.
4.1. Objectives of MDS
4.2.2. The analyst must ensure that the objects selected for analysis do have some basis of comparison.
Merely asking respondents for comparative responses between objects does not mean that underlying evaluative dimensions exist.
4.2.3. The number of objects must be determined while balancing two issues: a greater number of objects to ensure adequate information for higher dimensional solutions versus the increased effort demanded of the respondent as the number of objects increases.
a. Rule of thumb: more than four objects for each derived evaluative dimension.
Thus at least five ojbects are required for a
one-dimensional perceptual map.
b. Violating the rule of thumb: having less than the
suggested number of objects for a given dimensionality
causes an inflated estimate of fit and may adversely impact
validity of the resulting perceptual maps.
Input measures of similarity may be metric or nonmetric. The results from both types are very similar, meaning that the researcher's choice is dependent mostly on the preferred mode of data collection.
4.2.5. Choice of using either similarity or preference data based on research objectives.
b. Preference data -- reflect the preference order across the
set of objects, but are not directly related to attributes,
since we are not able to demonstrate the correspondence
between attributes and choice.
i. Nature of preference data: arrange the stimuli in terms
of dominance relationships. The stimuli are ordered
according to the preference for some property of the
stimuli.
ii. Data collection modes:
(a) direct ranking -- objects are ranked from the most
preferred to least preferred
(b) paired comparisons (when presented with all possible
pair combinations, the preferred object in each pair
is indicated).
c. Combination approach: Methods are available for combining
the two approaches, but in each instance the analyst must
assume the inference can be made between attributes and
preference without being directly assessed.
4.3.1. Different respondents will perceive a set of stimuli to have different dimensionalities.
While the towing capacity of a vehicle may be the most important dimension for one person, another may not even know such a dimension exists.
4.3.2. Not all respondents will attach the same level of importance to a dimension, even if all respondents perceive the dimension.
Even if all respondents perceive a given dimension, they may attach different levels of importance to it: Two consumers may be aware of the towing capacity of a vehicle and only one of them attaches any importance to this dimension. The other consumer really pays no attention to this dimension and makes use of other dimensions.
4.3.3. Respondents' dimensions and importance-levels for the these dimensions will change over time.
While it would be very convenient for marketers if consumers always used the same decision process with the same stimuli dimensions, this is not the case. Over time, consumers will assign different levels of importance to the same dimensions of stimuli or may even completely change the dimensions that they evaluate. People change as they go through life. What is important to an 18 year old may not be of any concern to a 50 year old. This change in consumers' lives is reflected in their evaluation of stimuli.
4.4.1. How does MDS determine the optimal positioning of objects in perceptual space?
a. Most MDS programs follow a several-step procedure
which involves selection of a configuration, comparison
of fit measures, and reduction of dimensionality.
b. The primary criterion for determining an optimal position
is the preservaton of the ordered relationship between
the original rank data and the derived distances between
points.
c. Degenerate solutions: The researcher should be aware of
degenerate solutions, which are inaccurate perceptual
maps. Degenerate solutions may be identified by a
circular pattern of objects or a clustered pattern of
objects at two ends of a single dimension.
4.2.2. How is the number of dimensions to be included in the perceptual map to be determined?
a. Tradeoff of best fit with the smallest number of dimensions
possible. Interpretation of more than three dimensions
is difficult.
b. Three approaches for dtermining the number of dimensions:
i. Subjective evaluation: The researcher evaluates the
spatial maps and determines whether or not the resulting
configuration looks reasonable.
ii.Stress measurement: measures the proportion of variance
in the data that is not accounted for by the model. It is
the opposite (complement) of the fit index.
(a) Desire a low stress index, since stress is minimized
when the objects are placed in a configuration such
that the distances between the objects best matches
the original distances.
(b) Scree plot of stress index: Plot stress versus
number of dimensions.
iii.Overall fit index: a squared correlation index which
indicates the amount of variance in the data that can be
accounted for by the model. This is a measure of how well
the model fits the data. Desired levels of the fit index
are similar to those desired when using the squared
multiple correlation coefficient in regression.
c. Parsimony: Parsimony should be sought in selecting the
the number of dimensions. The stress measure and the
overall fit index react much the same as R-square in
regression. As you add dimensions, the fit index always
improves and stress always decreases. Thus, the analyst
must make a trade-off between the fit of the solution and
difficulty of interpretation due to the number of dimensions.
4.4.3. With preference data, three additional issues are 1) implicit
or explicit estimation of the ideal point, 2) use of an internal or
external analysis, and 3) portrayal of the ideal point.
a. Ideal point estimation: Ideal points (preferred combination
of perceived attributes) may be determined by explicit
or by implicit estimation procedures.
i. Explicit estimation: Respondents are asked to identify
or rate a hypothetical ideal combination of attributes.
ii. Implicit estimation: An ideal combination of attributes
is empirically determined from respondents' responses to
preference measure questions.
b. Internal versus external analysis
i. Internal analysis develops spatial maps solely from
preference data.
ii. External analysis fits ideal points based on preference
data to a stimulus space developed from similarities
data.
iii. Recommendation: External analysisis the choice, due to
computational difficulties with internal analysis and the
fact that perceptual space (preference) and evaluative
space (similarities) may not contain the same dimensions
with the same levels of importance.
c. Vector or point representation of ideal point -- ideal point
is the most preferred combination of dimensions.
i. Point representation is location of most preferred
combination of dimesnions from the consumer's standpoint.
ii. Vectors are lines extended from the origin of the graph
toward the point which represents the combination of
dimensions specified as ideal.
iii. Difference in representation: With a point
representation, deviance in any direction leads to a less
preferred object, while with a vector, less preferred
objects are those located in the direction opposite to
that in which the vector is pointing.
4.4.5. Interpreting the MDS Results
a. For decompositional methods, the analyst must identify
and describe the perceptual dimensions. Procedures for
identification of dimensions may be objective or
subjective.
i. Subjective procedures: The researcher or the respondent
visually inspects the perceptual map and identifies
the underlying dimensions. This is the best approach
when the dimensions are highly intangible or
affective/emotional.
ii. Objective procedures: Formal methods, such as PROFIT, are
used to empirically derive underlying dimensionality from
attribute ratings.
b. When using compositional methods, the analyst should compare
the perceptual map against other measures of perception for
interpretation.
Perceptual map positions are totally defined by the
attributes specified by the researcher.
4.4.6. Validating the MDS Results. Validation will help ensure generalizability across objects and to the population.
a. Split-samples or multi-samples may be utilized to compare
MDS results.
b. Only the relative positions of objects can be compared
across MDS analyses. Underlying dimensions cannot be
compared across analyses.
c. Bases of comparisons across analyses: Visual or based on
a simple correlation of coordinates.
d. Multi-approach method: Applying both decompositional and
compositional methods to the same sample and looking for
convergence.
Correspondence analysis (CA) is another form of perceptual mapping. CA involves the use of contingency or cross-tabulation data. It has been discussed earlier in our notes on Chapter 6, Canonical Correlation Analysis.
5.1. Objectives of CA
5.1.1. CA will accommodate both nonmetric data and nonlinear relationships.
a. Contingency tables are used to transform nonmetric data
to metric form.
b. Dimensional reduction is performed in a manner similar to
factor analysis.
c. CA yields a perceptual map in which both row and column
categories are represented in multi-dimensional space.
5.2. Research Design of CA
CA requires only a rectangular matrix of nonnegative data.
5.2.1. Rows and columns do not have predefined meanings, but represent responses to categorical variables.
5.2.2. The categories for a row or colmn may be a single variable or a set of variables (e.g., age/gender combinations).
5.3. Assumptions in CA
Unlike other multivariate techniques, CA does not have a strict set of assumptions. The researcher need only be concerned with including all relevant attributes.
5.4. Deriving the CA Results and Assessing Overall Fit
5.4.1. CA derives a single representation of categories (both rows and columns) in the same multi-dimensional space (sometimes called a "bi-plot").
5.4.2. The researcher must identify the number and importance of the dimensions.
Eigenvalues help determine the appropriate number of dimensions and help in evaluating the relative importance of each dimension.
5.5. Interpretation of CA Results
The degree of similarity among categories is directly proportional to the proximity of categories in perceptual space.
5.6. Validation of the Results
5.6.1. Generalizability of the results may be confirmed by split-sample or multi-sample analyses.
5.6.2. The researcher should also assess the sensitivity of the analysis to the addition or deletion of certain objects and/or attributes.