ALSCAL Program in SPSS Questions

Open the MDS data-set in SPSS.

a. Using the ALSCAL program, assume that the data are metric (interval) and compute the distance matrix between pairs of sports for the 50 respondents with two-dimensional MDS

b. Which sports are the most similar according to consumer perceptions? Which ones are the most different?

c. Save the final co-ordinates and compute the bivariate correlations with the expert panel evaluations

d. Try the three-dimensional configuration. Which solution works better according to the STRESS value and R 2 ?

e. Repeat the analysis, this time looking for similarities between respondents for the ten sports on two-dimensions

f. How many clusters of consumers appear from the graph?

STATISTICAL MEASUREMENTS,
ANALYSIS AND RESEARCH
Instructor: Amreeta Choudhury
M4:2A Lecture 11 Part 1
Session Objectives
• Discuss Preferences, perceptions and multidimensional scaling
• Run multidimensional scaling
• Summarize principles and applications of correspondence analysis
• Theory and techniques of correspondence analysis
• Running correspondence analysis
• Use in Marketing including the following:
– Translate preference orderings and consumer perceptions toward products
(objects) into graphical representations, Perceptual maps, etc.
• Introduce Final Project
Multidimensional scaling
• a set of statistical techniques which allow one to
1. translate consumer preferences or perceptions towards
products or brands into a reduced number of dimensions
(usually two or three)
2. Represent them graphically into a preference map or
perceptual map
• It is also possible to show both objects and subjects (the
consumers) in the same graph through multidimensional
unfolding (MDU)
• MDU is a technique which unfolds the coordinates for
consumers (or groups of consumers) on the basis of their
preferences or perceptions through an ideal point model
3
Chapter 13
Chapter 13
Common Space
0.75
London
Paris
0.50
Berlin
0.25
Dimension 2
Interpretation: How trendy is the city
Example of MDS output – holiday destinations in two
dimensions
Amsterdam
Rome
0.00
Madrid
-0.25
Athens
Stockholm
-0.50
Bruxelles
-0.75
-0.5
0.0
0.5
Dimension 1
May be interpreted as “climate”
4
• Each of the respondents is asked to
rank the cities, without necessarily
specifying why one city was preferred to
another
• Similarities in ranking across an
adequate number of respondents reflect
perceived similarities between cities (e.g.
London is more similar to Berlin than to
Athens)
• Graph distances reflect dissimilarities
• If the two dimensions can be labelled
according to some criterion, as for
principal component or factor analysis,
then it becomes possible to understand
the main perceived differences.
Chapter 13
Marketing applications
• Sensory evaluation and new product development
• Example, a company developing a low-salt soup
An evaluation panel is asked to assess a set of existing soup brands according to
several criteria concerning taste, smell, thickness, storage duration, perceived
healthiness and price
Consumers are asked to identify their ideal product in terms of the same
characteristics which may not coincide with one of the existing soups
The final output is a perceptual map displaying both consumer preferences (in
terms of their ideal products) and the current positioning of the existing brand
A concentration of consumers’ ideal points identify a segment (cluster analysis
might also be used as a tool to segment respondents)
if no brands appear in the neighbourhood of a segment then there is room for the
development of a new product in that area
If the perceptual dimensions have been clearly identified this also allows one to
choose the characteristics of the new products.
5
Example of brand positioning
The two dimensions are the output of some
reduction technique
– PCA or FA for interval (metric) data
– correspondence analysis for non-metric
data
coordinates for brands are obtained by
running PCA (or FA) on sensory
assessments (usually through a panel of
experts unless objective measures exists)
Consumer positions (as individuals or as
segments) can be defined in two ways
1) using their “ideal brand” characteristics
2) by translating preference ranking for
brands into coordinates through unfolding
6
Chapter 13
Chapter 13
Brand positioning
The product should be
healthy as both A & C
like that dimension.
The thicker it is, the
closer is to C
compared to A
Segment A
chooses
three but it
is not that
close
Consumer segment B
is close to Brand three
Brand 1 and 4 are
perceived as similar
7
There is room for a new
product for segment C
also close to sgm. A
Brand 5
Brand five survives
because of segment
C, but it is far from C’s
preferences
Consumer segment D
is happy with Brand 2
Brand repositioning. If brand five had this marketing research information, one could
improve one’s performance by enhancing the perceived healthiness of the product
(e.g. reducing the salt content and through a targeted advertising campaign). This
would move brand fivcloser to segment C
Chapter 13
Other applications of MDS
• If consumer perceptions are compared through MDS before and after
an advertising campaign aimed at changing perceptions it becomes
possible to measure the success of the advertising effort
• Finally, MDS could be exploited to simplify data interpretation and
provide some prior insight before running psycho-attitudinal surveys.
8
Chapter 13
Running MDS
• MDS is a container for statistical techniques to produce perceptual
or preference maps.
• There is a range of options and choices depending on the type of
MDS data.
• object of the analysis: it can be a product, a brand or any other
target of consumer behaviour, like tourism destinations in the initial
example. The object can be depicted as a set of characteristics,
represented through
• objective dimensions (e.g. salt content in grams)
• subjective dimensions as declared by respondents (subjects) in a
survey
9
Preferences and perceptions
• With subjective dimensions, consumer evaluations can be based on
preferences or perceptions
• Measurement through preferences (preference map)
• the subjects rank several objects according their overall evaluations (e.g.
ordering of soup brands).
• Measurement through perceptions (perceptual or subjective
dimensions, perceptual map)
• the respondent must attach a subjective value to an object’s feature (e.g. a
rating of the thickness of each soup brand)
• When individual attribute perceptions are measured, respondents
may be asked to state the combination of an object’s features that
correspond to their ideal object (to be translated into an ideal point
in the spatial map).
• The ideal point can alternatively (and preferably) be derived
through an unfolding statistical model.
10
Chapter 13
Measurement
• Preferences
• rank order scales
• Q-sorting
• other comparative scales.
• Perceptions
• non-comparative scales
Likert
Stapel
Semantic Differential Scales.
• Two types of variables for MDS
• Non-metric variables just reflect a ranking, so that it is not possible to assess whether
the distance between the first and second object is larger or smaller than the distance
between the second and the third.
• Metric variables reflect respondent perception of the distances
• Generally, preference rankings are classified as non-metric and perceptions and
objective dimensions are metric.
• This distinction can be very important, as it leads to two different MDS
approaches.
11
Chapter 13
Chapter 13
Non-metric vs. metric MDS
• The output of non-metric MDS aims to preserve the preference
ranking supplied by the respondents
• Metric MDS also takes into account the distances as measured by
perceptions or objective quantities.
• This distinction is often overcome by the use of techniques which
allow one to transform non-metric variables and treat them as if they
were metric, like the PRINQUAL procedure in SAS or correspondence
analysis (see lecture 14)
12
Chapter 13
Multidimensional scaling steps
1.
4.
Decide whether mapping is based on an aggregate evaluation of
the objects or on the evaluation of a set of attributes
(decompositional versus compositional methods)
Define the characteristics of the data collection step (number of
objects, metric versus non-metric variables)
Translate the survey or objective measurements into a similarity
or preference data matrix
Estimate the perceptual map
5.
6.
7.
Decide on the number of dimensions to be considered
Label the dimensions and the ideal points
Validate the analysis
2.
3.
13
Chapter 13
Decompositional vs. compositional MDS
• Decompositional (attribute-free) approach
• The spatial maps reflect the subject evaluations
• Comparisons of the objects in their integrity
• Advantages: respondent assessment is easier, it is possible to obtain a separate
perceptual map for each subject or for homogeneous groups of subjects
• Limits: no specific information on the determinants of the relative position of the
objects. It is not possible to plot both the objects and the subjects in the same map. It
is difficult to label the dimensions (labels are based on the researcher’s knowledge
about the objects)
• Compositional (attributed-based) approach
• Subject assess es a set of attributes (compositional or attribute-based approach).
• Preferred when it is relevant to describe the dimensions and explain the positioning of
objects and subjects in the perceptual map
• Requirements: all the relevant attributes must be considered while avoiding including
irrelevant ones; the combination of attributes must be adequate to reflect the overall
object evaluation.
• The method to be used depends on the chosen approach
14
Objects and variables
• The higher the number of objects the more accurate the output of MDS in statistical
terms
• However, data quality suffers because it might be difficult for subjects to provide
large number of comparisons.
• The number of objects required for the analysis increases with the number of
dimensions being considered
• For two-dimensional MDS it is advisable to have at least ten objects
• For three-dimensional MDS it is advisable to have about fifteen objects
• As the number of objects increases goodness-of-fit measures become less reliable).
• Measurement through metric or non-metric variables
• The starting matrix for MDS is different
• With non-metric data (ordinal variables or paired comparison data) the initial data matrix
only considers ranking and not the distance between the objects
• With metric variables the matrix preserves the distances observed in the subject
evaluations.
• Most of MDS methods can also deal with mixed data-set with both metric and non-metric
data
15
Chapter 13
Data matrix
• Data for MDS are similarities between objects or preference (ranking) of
objects
• Decompositional approach: a matrix for each subject exists, which translates
into a matrix comparing all objects
• Compositional approach is chosen, a matrix for each subject and attribute
exists and this translates into a matrix comparing all objects for each
attribute
• Similarity data: the subject compares all pairs of objects and ranks the pairs
in terms of their similarity (usually this leads to non-metric MDS)
• The similarity (or dissimilarity) matrix can be also computed from metric
evaluation (rating) of the objects
• Compositional approach: summarize (e.g. through averaging) the distances between the
objects across the subjects, assuming all subjects have the same weight
• Decompositional approach: a synthetic measure of similarity between objects is computed
for each subject (weights can be used if available and appropriate) then the similarity
matrix across the subjects is derived
16
Chapter 13
Estimation
• Estimation starts from a proximity or similarity matrix and produces
a set of n-dimensional coordinates
• Distances in this n-dimensional space reflect as closely as possible
the distances recorded by the proximity matrix
• Metric scaling is based on a proximity matrix derived from metric
data
• Non-metric scaling projects dissimilarities based on ranking (ordinal
variables) preserving the order emerging from the subjects’
preferences
• Non-metric scaling should also be applied to metric distances when
the researcher suspects that collected data might be affected by
relevant measurement errors (e.g. when respondents may
encounter difficulties in stating their perceptions with precision
while ordering can be regarded as more reliable)
17
Chapter 13
Chapter 13
Metric scaling
• With metric variables, one might apply FA (or PCA) to reduce the dimensions and
obtain the scores which represent the coordinates. However, there is a difference
• Coordinates obtained from PCA and FA are the best representation of the original data
matrix in terms of variability
• Metric scaling coordinates ensure that the distance between two points is as close as
possible to the distance as measured in the proximity matrix
• Classical MultiDimensional Scaling technique (CMDS) also known as principal
coordinate analysis
• Decompositional approach (unique similarity matrix comparing all objects)
• The proximity or similarity matrix is obtained by applying the Euclidean distance on the
data matrix (or other distance measures as those for cluster analysis).
• The objective of CMDS is to extract a a n-dimensional configuration of points whose
distances dij* are as close as possible to the original distances dij according to the
following quadratic equation
p
i −1
 (d − d )
i = 2 j =1
2
ij
*2
ij
18
Non-metric scaling
•
•
•
Ordinal variables (preference data)
coordinates are obtained through computational algorithms
Many procedures. The original method (Shepard-Kruskal) is as follows
•
•
•
•
•
The procedure can be extended to include a search for the optimal number of
dimensions n.
Other algorithms:
•
•
•
19
given a number of dimensions n, the p objects are represented through an arbitrary
initial set of coordinates
a function S is defined to measure how distant the current set of coordinates is
from the original ordering (monotonicity requirement)
using an iterative computer numerical algorithm the values that minimize S are
found
ALSCAL (SPSS & SAS)
Algorithms in the MDS procedure in SAS
INDSCAL
Chapter 13
Chapter 13
Goodness-of-fit and STRESS
• The STRESS measure (STandardized REsiduals Sum of
Squares) is a function of the original and derived
distances to evaluate the goodness-of-fit of a MDS
solution:
p −1
STRESS =
p
ˆ )2
(
d
−
d
  ij ij
i =1 j =i +1
p −1
p
2
d
  ij
i =1 j =i +1
• The smaller the stress function, the closer are the derived
distances to the original ones
20
STRESS and number of dimensions
• The STRESS value decreases as the number of dimensions increases
• The number of dimensions can be evaluated through a scree diagram
of STRESS against the number of dimensions (as for FA, PCA or cluster
analysis) where the optimal number corresponds to an elbow
• The preferred number of dimensions is usually two or three which
allows for graphical examination
• The search usually goes from one to five dimensions
• Identification of the optimal number within the metric and non-metric
iterative algorithm
• An additional step evaluates the STRESS function
• The algorithm stops when the addition of a further dimension does not reduce
the STRESS value to a perceptible extent
• With two dimensions a STRESS value below 0.05 is generally
considered to be satisfactory.
21
Chapter 13
Chapter 13
Labelling dimensions
• Interpretable dimensions (attaching a meaning to coordinates)
enhance the use of MDS maps (e.g. new product development)
• Interpretation may be difficult
• Compositional approaches (or attribute ratings are otherwise
available) allow for more objective methods based on the relative
weight of each attribute (something similar to factor loadings)
22
Chapter 13
Ideal points
• Objective: position ideal points (for each subject) and the
actual brand evaluations (the objects) within the same
map
• Ideal point: set of coordinates which represents the
stated optimal combination of attributes (under the
compositional approach)
• If no precise statement is made by the subject it is still
possible to locate the ideal point
• Indirect positioning of ideal points is based on a
procedure which ensures that distances of the objects
from a subject’s ideal point reflect the preference
ordering as much as possible
23
Chapter 13
Internal vs. external preference mapping
• Internal Preference Mapping (IPM)
• the proximity matrix for the objects is based on evaluations from consumers. The final map
shows:
products as they are perceived by the consumers
consumers according to their preferences.
• External data (i.e. objective measures or expert evaluations) can be used to interpret the
dimensions but not to draw the map
• External Preference Mapping (EPM)
• the proximity matrix contains objective (analytic) measures of product characteristics (or
evaluations from expert panels)
• The maps contain information external to the set of consumers which provide their evaluation
of the products
• The final map shows
products as they are evaluated by the external source
consumers according to their preferences
24
Chapter 13
Internal preference mapping
• The ideal point (or vector) for each subject is estimated from the
preference orderings through unfolding
• Example
• four brands (A, B, C and D) are evaluated.
• Consumer one states a preference ordering as D, B, C and A, where D is the
most preferred brand
• Consumer two states the ordering C, B, D, A
• The ideal point for Consumer one will be closer to D and far away from A,
while for Consumer two the ideal point will probably be still far away from A
but closer to C than to D.
• The distance of the ideal point from the objects in the product space should
reflect as much as possible the ordering of the consumer preferences
25
Chapter 13
IPM and unfolding
• The ideal product is not necessarily a precise point in the
preference map but could be represented as a line (or an arrow)
going from the least preferred objects towards the most preferred
ones
• Unfolding approach
• Decompose all preference orderings for a given set of objects (products) so
that the same products can be represented in a lower dimensional space
• Once the products are positioned on the preference map it is possible to see
where the subjects (consumers) are positioned
• While the dimensions reflect some product characteristics that are the same
for all consumers each consumer attaches a different weight to those
dimensions
• Consumers have different ideal points because they place a different weight
on the dimensions
26
External preference mapping
• EPM follows a different philosophy from IPM
– It strictly requires the use of perceptual (metric) data
– Evaluations of the product characteristics are on a measurement scale rather than their
simple ordering
– Measurements are usually based on analytic or objective evaluations or expert
evaluations (external to the set of consumers which provide their product evaluation).
• The input matrix contains the (quantitative) measurements of all
attributes for each product.
• A data reduction techniques (usually PCA) allows one to attach a
set of coordinates (the principal component scores) to each of the
products
• The principal components define the dimensions of the map and
can be interpreted (labelled) through the component loadings.
• An algorithm (e.g. PREFMAP) allows one to elicit the position of
subjects (consumers) in the map.
27
Chapter 13
Chapter 13
IPM or EPM?
• Both approaches can be applied to the same data set but they reflect
different philosophies
– a consumer very much likes red full-bodied wine and white sweet and sparkling
wine
– IPM: these two products share similar preferences and will be positioned next to
each other
– EPM: the product characteristics are very different they will look distant on the
perceptual map.
• The choice between IPM and EPM is mainly related to the choice of
prioritizing either the preferences of the subjects (IPM) or the product
characteristics (EPM).
• When many dimensions are chosen the two approaches produce
similar results but with reduced dimensions discrepancies are likely to
emerge
28
IPM vs. EPM
• IPM is better when
• Perceptual data are inadequate to reflect preferences as it is not necessarily
true that the combination of the product attributes is an adequate
representation for the product
• Physical attributes as they are perceived by the consumer are
processed into a number of perceived benefits and these benefits
are then translated into preferences
• Thus the relative weight of the physical attributes as compared to
the abstraction process could drive the choice between IPM and
EPM.
• For those goods where the cognitive evaluation is mainly based on
the objective attributes EPM seems to be preferable
• Goods where the connection between perceptions and
preferences is not so natural (and affective processes play a major
role) are better analyzed with IPM
29
Chapter 13
Chapter 13
MDS in SPSS – the data
• MDS data set
• Fifty individuals (the subjects or consumers) were asked to rank
ten sports (the objects or products) according to their preference
• a panel of expert sport journalists provided an evaluation of the
attributes of each sport (the product characteristics) in terms of
strategy, suspense, physicality and dynamicity
• The final data set (MDS.sav) has one row for each sport and one
column for each consumer plus four columns for the sports’
attributes
30
Chapter 13
The MDS data set
Ratings by consumers
31
Evaluations of product
characteristics by experts
Chapter 13
IPM & unfolding
32
Chapter 13
Unfolding
Proximities are defined
from the subjects’
preference rankings
This nominal variable
provides the labels for
the objects (sports)
When measures for the
same set of objects are
provided by different
sources (e.g. different
groups/scenarios) –
data should be stacked
Defines model
options
Allows to
place
restrictions
33
Defines
options for the
algorithm
Choose
plots
Displays and saves
additional output
Chapter 13
Unfolding options
Select identity
as data come
from a single
source
Rankings are
dissimilarities
and ordinal
data
Number of
dimensions to
be explored
34
Chapter 13
Options
Convergence criterion for
the STRESS function
Choose the
starting
configuration
The penalty term helps avoid
degenerative solutions (where
points can hardly be distinguished
from each other).
The weight of the penalty term
increases as the strength
becomes smaller.
When the penalty range is zero, no
correction is made to the Stress-I
criterion, while larger range values
lead to solutions where the
variability of the transformed
proximities is higher
35
Plots
The final
common
space shows
subjects and
objects on the
same plot
36
Chapter 13
Applies different
colors or markets to
different objects
Chapter 13
Outputs
Output tables can be
selected here
Output coordinates
(distances) can be saved
into a new file
37
Unfolding output
Chapter 13
Measures
Iterations
Final Function Value
Function Value
Parts
Badness of Fit
Goodness of Fit
Variation
Coefficients
Degeneracy Indices
38
Stress Part
Penalty Part
Normalized Stress
Kruskal’s Stress-I
Kruskal’s Stress-II
Young’s S-Stress-I
Young’s S-Stress-II
Dispersion Accounted For
Variance Accounted For
Recovered Preference
Orders
Spearman’s Rho
Kendall’s Tau-b
Variation Proximities
Variation Transformed
Proximities
Variation Distances
Sum-of-Squares of
DeSarbo’s
Intermixedness Indices
Shepard’s Rough
Nondegeneracy Index
992
.3835645
.0410912
3.5803705
.0016885
.0410908
.1905153
.0720164
.1781156
.9983115
.9666225
.8471837
.8617494
.7273984
.5043544
.3322572
.5071630
.4694185
.5609796
The final STRESS-I value of 0.04 is acceptable.
Other measures of “badness-of-fit” and “goodness-of-fit” are
provided and confirm that the results are acceptable.
The variation coefficient of the transformed proximities
can be used to check for the risk of degenerated solutions
(points are too close to each other). In this case, the
variation coefficient of the transformed proximities is 0.33 as
compared to the 0.50 of the original ones, which means that
most of the variability is retained after transformation.
Furthermore, the distances show a variability which is more
or less equal to the original one, indicating that the points in
space should be scattered enough to reflect the initial
distances.
The DeSarbo’s Intermixedness index and the Shepard’s
RNI also provide warning signals for degenerated solutions:
the former should be as close to zero as possible and the
latter as close to one as possible. There are no strong
signals for a degenerated solution
One may wish to try different parameters for the penalty
term to see whether these indicators improve.
Chapter 13
Plots
Plot of objects
39
Plot of subjects
Chapter 13
Joint plot
According to the sample, basketball,
baseball and cricket share
similarities in subjects’ perceptions
and so do American football, motor
sports and ice hockey.
A third “cluster” is provided by
handball, waterpolo and volleyball,
while football seems to be
equidistant from all other sports.
Consumers are also grouped in
clusters according to their
preferences and the joint
representation allows one to show
not only which sports (products) are
closer to the preferences of different
segments, but also which sports
need to be repositioned to attract
more public, like the cluster with
volleyball, waterpolo and handball.
40
Chapter 13
Repositioning
• If one can attach a meaning to dimensions one and two it
becomes possible to understand what characteristics of
the products should be changed
• A method to obtain an interpretation of the coordinates
consists in looking at the correlations betweens the
coordinates of the sports and the object characteristics
that can be measured objectively or through the
evaluation of expert panellists.
• The algorithm has created an output file coord.sav which
contains the two coordinates for each sport and
consumer and can be used to obtain the bivariate
correlations
41
Chapter 13
Labelling dimensions
DIM_1 DIM_2 Strategy Suspense Physicity Dinamicity
DIM_1
1.000
0.000
-0.839
-0.167
0.130
0.362
DIM_2
0.000
1.000
0.338
-0.180
0.330
0.168
Sports on the left side of the graph are likely to be more strategic, while those
on the right are more dynamic.
Considering the second dimension, as values move towards the top, the sports
are expected to become more physical and strategic, while negative values
seem to indicate a lack of suspense.
Ideally, those who want to bring people closer to volleyball, water-polo or
handball should try and move the points toward the top left area, thus trying to
persuade “consumers” that these sports are more strategic (especially),
dynamic and physical than currently thought.
42
Field Work
Field work
Complete the questions on NYU Home under our Class website. Submit online by next week!
STATISTICAL MEASUREMENTS,
ANALYSIS AND RESEARCH
Instructor: Amreeta Choudhury
M4:2B Lecture 11 Part 2
Session Objectives
• Discuss Preferences, perceptions and multidimensional scaling
• Run multidimensional scaling
• Summarize principles and applications of correspondence analysis
• Theory and techniques of correspondence analysis
• Running correspondence analysis
• Use in Marketing including the following:
– Translate preference orderings and consumer perceptions toward products
(objects) into graphical representations, Perceptual maps, etc.
• Introduce Final Project
Chapter 14
Correspondence analysis
• Multivariate statistical technique which looks into the association of
two or more categorical variables and display them jointly on a
bivariate graph
• It can be used to apply multidimensional scaling to categorical
variable.
3
Chapter 14
Correspondence analysis
and data reduction techniques
• Factor and principal component analyses are only applied to metric (interval or
ratio) quantitative variables
• Traditional multidimensional scaling deals with non-metric preference and
perceptual data when those are on an ordinal scale
• Correspondence analysis allows data reduction (and graphical representation of
dissimilarities) on non-metric nominal (categorical) variables
• The issue with categorical (non-ordinal) variables is how to measure distances
between two objects: Correspondence analysis exploits contingency tables and
association measures
4
Chapter 14
Example (Trust data)
• Do consumers with different jobs (q55) show preferences for some
specific type of chicken (q6)?
Correspondence Table
In a typical week, what type of fresh or frozen chicken do you buy for
your household’s home consumption?
If employed, what is your
‘Value’
‘Standard’
‘Organic’
‘Luxury’
occupation?
chicken
chicken
chicken
chicken
Active Margin
I am not employed
17
50
10
17
94
Non manual employee
11
74
14
28
127
Manual employee
6
19
4
8
37
Executive
0
7
6
14
27
Self employed
1
18
7
3
29
professional
Farmer / agricultural
1
1
1
0
3
worker
Employer / Entrepreneur
0
4
2
3
9
Other
11
31
1
1
44
Active Margin
47
204
45
74
370
5
Chapter 14
Independence
•
•
•
6
If the two characters are independent then the number in the cells
of the table should simply depend on the row and column totals
(lecture 9)
Measure the distance between the expected frequency in each cell
and the actual (observed) frequency
Compute a statistic (the Chi-square statistic) which allows one to
test whether the difference between the expected and actual value
is statistically significant
Chapter 14
Reducing the number of dimensions
• The elements composing the Chi-square statistic are
standardized metric values, one for each of the cells
• They become larger as the association between two
specific characters increases
• These elements can be interpreted as a metric measure
of distance
• The resulting matrix is similar to a covariance matrix
• A method similar to principal component analysis can be
applied to this matrix to reduce the number of
dimensions
7
Chapter 14
Coordinates
• The principal component scores provide standardized values that can
be used as coordinates
• One may apply the same data reduction technique
• first by rows (synthesizing occupation as a function of types of chicken)
• then by column (synthesizing types of chicken as a function of occupation)
• The first two components for each application generate a bivariate
plot which shows both the occupation and the type of chicken in the
same space
8
Chapter 14
Output from
Correspondence Analysis
Unemployed
are closer to
“Value” chicken
9
Executives prefer
“Luxury” chicken
Applications
• It is possible to represent on the same graph consumer
preferences for different brands and characteristics of a
specific product (e.g. car brands together with colour,
power, size, etc.)
• This allows one to explore brand choice in relation to
characteristics opening the way to product modifications
and innovations to meet consumer preferences
• Correspondence analysis is particularly useful when the
variables have many categories
• The application to metric (continuous) data is not ruled
out but data need to be categorized first
10
Chapter 14
Summary
• Correspondence analysis is a compositional technique which starts
from a set of product attributes to portrait the overall preference for
a brand
• This technique is very similar to PCA and can be employed for data
reduction purposes or to plot perceptual maps
• Because of the way it is constructed correspondence analysis can be
applied to either the row or the columns of the data matrix
• For example if rows represent brands and columns are different
attributes:
1. By applying the method by rows one obtains the coordinates for the brands
2. The application by columns allows one to represent the attributes in the same
graph
11
Chapter 14
Chapter 14
Steps to run correspondence analysis
•
•
•
•
•
12
Represent the data in a contingency table
Translate the frequencies of the contingency table into a matrix of
metric (continuous) distances through a set of Chi-square
association measures on the row and column profiles
Extract the dimensions (in a similar fashion to PCA)
Evaluate the explanatory power of the selected number of
dimensions
Plot row and column objects in the same co-ordinate space
Chapter 14
The frequency table
Categorical variable X (k categories)
Categorical variable Y (l categories)
x1
x2
…
xi
…
xk
y1 y2 …
f11 f12
f21 f22
yj
f1j
f2j
fi1
fij
fil
fk1 fj2
f01 f02
fkj
f0j
fkl
f0l
Column profile
13
…
yl
f1l
f2l
Row profile
f10
f20
…
fi0
…
fkl
1
Column masses
Row masses
Interpretation of coordinates
• The categories of the x variable can be seen as different
coordinates for the points identified by the y variable
• The categories of the y variable can be seen as different
coordinates for the points identified by the x variable
• Thus it is possible to represent the x and y categories as
points in space, imposing (as in multidimensional scaling)
that they respect some distance measure
14
Chapter 14
Representations
• Take the row profile (the categories of x) and plot the
categories in a bi-dimensional graph, using the
categories of y to define the distances
• This allows one to compare nominal categories within
the same variable: those categories of x which show
similar levels of association with a given category of y
can be considered as closer than those with very
different levels of association with the same category of
y
• The same procedure is carried out transposing the table
which means that the categories of y can be represented
using the categories of x to define the distances
15
Chapter 14
Computing the distances
•
When the coordinates are defined simultaneously for the categories of x and
y the Chi-square value can be computed for each cell as follows
•
Obtain the expected table frequencies
•
Where nij and fij are the absolute and relative frequencies, respectively, ni0 and n0j (or fi0 and f0j)
are the marginal totals for row i and column j (the row masses and column masses) respectively
and n00 is the sample size (hence the total relative frequency f00 equals one)
•
f =
n n
=
f f
= fi 0  f 0 j
i0
0j
i0
0j
*
ij now be computed for each cell (i,j)
The Chi-square value can
00
00
 ij2 =
16
Chapter 14
n
( f ij − f ij* ) 2
f ij*
f
These are the quadratic distances
between category i and category j
of the x variable
The distance matrix
• The matrix 2 measures all of the associations between the
categories of the first variable and those of the second one.
• A generalization of the multivariate case (MCA is possible by
stacking the matrix
• Stacking: compose a large matrix by blocks, where each block is the
contingency matrix for two variables (all possible associations are taken into
consideration)
• The stacked matrix is referred to as the Burt Table
• To obtain similarity values from the 2 matrix:
• compute the square root of the elemental Chi-square values
• use the the appropriate sign (the sign of the difference fij –fij*)
• large positive values correspond to strongly associated categories
• large negative values identify those categories where the association is strong
but negative indicating dissimilarity
17
Chapter 14
Estimation
• The resulting matrix D contains metric and continuous similarity
data
• It is possible to apply PCA to translate such a matrix into
coordinates for each of the categories first those of x then those of
y
• Before PCA can be applied some normalization is required so that
the input matrix becomes similar to a correlation matrix
• The use of the square root of the row masses (columns) for
normalizing the values in D represents the key difference from PCA
• The rest of the estimation process follows the results of the PCA
• As for PCA eigenvalues are computed, one for each dimension,
which can be used to evaluate the proportion of dissimilarity
maintained by that dimension
18
Chapter 14
Inertia
• Inertia is a measure of association between two categorical
variables based on the Chi-squared statistic.
• In correspondence analysis the proportion of inertia explained by
each of the dimensions can be regarded as a measure of goodnessof-fit because the effectiveness of correspondence analysis
depends on the degree of association between x and y
• Total inertia
– is a measure of the overall association between x and y
– is equal to the sum of the eigenvalues
– corresponds to the Chi-square value divided by the number of observations
– A total inertia above 0.20 is expected for adequate representations
• Inertia values can be computed for each of the dimensions and
represent the contribution of that dimension to the association
(Chi-square) between the two variables
19
Chapter 14
Chapter 14
SPSS example
• EFS data set:
• economic position of the
household reference
person (a093)
• type of tenure (a121)
• Their Pearson Chi-square
value is 274, which means
significant association at
the 99.9% confidence
level)
20
Chapter 14
Analysis
Define the range, i.e. the categories for each
variable that enter the analysis
Some categories
can be indicated as
supplementary:
they appear in the
graphical
representation, but
do not influence the
actual estimation of
the scores
21
Chapter 14
Model options
Choose the number of
dimensions to be
retained
Choice of
distance measure
Standardization (only for
Euclidean distance)
Normalization
Which variable
should be
privileged?
22
Chapter 14
Number of dimensions
• The maximum number of dimensions for the analysis is
equal to
• the number of rows minus one, or
• the number of columns minus one (whichever the smaller)
• In our example, the maximum number of dimensions
would be five which reduces to four due to missing values
in one row category.
• As shown later in this section one may then choose to
graphically represent only a sub-set of the extracted
dimensions (usually two or three) to make interpretation
easier
23
Chapter 14
Distance measure
• Chi-square distance (as discussed earlier)
• Euclidean distance
• uses the square root of the sum of squared differences between pairs of rows
and pairs of columns
• this also requires one to choose a method for centering the data (see the SPSS
manual for details)
• For this example standard correspondence analysis (with the Chisquare distance) does not require a standardization method.
24
Normalization method
• Defines how correspondence analysis is run: whether to give priority to comparisons
between the categories for x (row) or those for y (columns)
• This choice influence the way distances are summarized by the first dimensions
• Row principal normalization: the Euclidean distances in the final bivariate plot of x
and y are as close as possible to the Chi-square distances between the rows, that is
the categories of x
• The opposite is valid for the column principal method
• Symmetrical normalization: the distances on the graph resemble as much as
possible distances for both x and y by spreading the total inertia symmetrically
• Principal normalization: inertia is first spread over the scores for x, then y
• Weighted normalization: defines a weighting value between minus one and plus one
where minus one is the column principal zero is symmetrical and plus one is the row
principal
• EFS example: the row principal method is more appropriate as it is more relevant to
see how differences in socio-economic conditions impact on the tenure type than it
is by looking at distances between tenure types.
25
Chapter 14
Chapter 14
Additional statistics
Although CA is a
nonparametric method,
it is possible to compute
standard deviations and
correlations under the
assumption of
multinomial distribution
of the cell frequencies,
(when data are obtained
as a random sample
from a normally
distributed population)
26
Allows one to order the categories of x and y using scores
obtained from CA
E.g. the tenure types and the socio-economic conditions
might follow some ordering but cannot be defined with
sufficient precision to consider these variables as ordinal.
One can use the scores in the first dimension (or the first
two) to order the categories and produce a permutated
correspondence table.
Chapter 14
Plots
Three graphs:
•Biplot (both x & y)
• x only (rows)
• y only (columns)
One usually chooses to
represent only the first
two or three of the
extracted dimensions
27
Chapter 14
Output
The first dimensin explains 85%, the first two 93%
of total inertia. However, note that total inertia
does not correspond to total variability, but to the
variability of the extracted dimensions
The SV is the
square root of inertia
(the eigenvalue)
Summary
Proportion of Inertia
Dimension
1
2
3
4
Total
Singular
Value
.669
.209
.173
.072
a. 24 degrees of freedom
Usually a value of
total inertia above
0.2 is regarded as
acceptable
28
Inertia
.447
.044
.030
.005
.526
Chi Square
231.402
Sig.
.000a
Accounted for
.850
.083
.057
.010
1.000
The Chi-square stat
suggests strong and
significant association
Cumulative
.850
.933
.990
1.000
1.000
Confidence Singular Value
Standard
Deviation
.031
.055
.055
.053
2
.094
Correlation
3
-.032
.011
4
-.022
.081
-.042
These precision measures
are based on the
multinomial distribution
assumption
Row scores
Chapter 14
Score in Dimension
These categories have a higher relevance because
they are more important categories in the original
correspondence table. These two categories
(especially retirement) strongly contribute to
Overview Row Points
explaining the first dimensionContribution
The mass column shows
the relative weight of each
category on the sample
Economic position of
Household Reference
Person
Self-employed
Fulltime employee
Pt employee
Unemployed
Worka related govt train
prog
Ret unoc over min ni age
Active Total
b
Of Point to Inertia of Dimension
1
2
3
.016
.001
.496
.334
.030
.027
.010
.295
.318
.001
.622
.157
4
.407
.071
.300
.202
1
.290
.984
.156
.013
.
.000
.000
.000
.000
.
.
.
.
.
.288
.526
.639
1.000
.052
1.000
.002
1.000
.020
1.000
.992
.008
.000
.000
1.000
Mass
.080
.539
.077
.018
1
.296
.527
-.239
-.154
2
.025
.049
-.409
-1.223
3
.433
-.039
-.352
.509
4
-.164
.026
-.143
.241
Inertia
.024
.152
.028
.033
.000
.
.
.
.
.286
1.000
-.999
.089
.015
.019
Of Dimension to Inertia of Point
2
3
4
.002
.620
.089
.008
.005
.002
.453
.336
.055
.814
.141
.032
The second dimension is
characterized by unemployed and
part-time employees
Total
1.000
1.000
1.000
1.000
a. Supplementary point
b. Row Principal normalization
Scores are computed for each
category but the supplemental one,
provided there are no missing data
Scores are the coordinates for the
map
29
Shows how total inertia has been
distributed across rows (similar to
communalities)
Chapter 14
Column scores
• The same exercise is carried out on columns, however the
row principal method does not normalize by column
Overview Column Pointsb
Score in Dimension
Tenure – type
Local Authority rented
unfurnished
Housing association
Other rented unfurnished
Rented furnished
Owned with mortgage
Owned by rental
purchase
Owned outright
Rent freea
Active Total
Mass
1
2
3
4
.098
-.699
-1.993
.051
1.106
.039
.048
.388
.066
.050
.032
.457
-.781
.487
.531
.971
-1.263
-2.023
-1.098
.371
2.821
-2.190
-2.270
.233
-1.273
.891
-4.585
.133
.039
.022
.014
.196
.040
.012
.009
.431
.002
1.179
1.120
-1.287
5.002
.002
.295
.009
1.000
-1.244
-.957
.819
-1.039
-.382
-2.996
.018
-3.705
.214
.007
.526
a. Supplementary point
b. Row Principal normalization
30
Contribution
Inertia
1
Of Point to Inertia of Dimension
2
3
Of Dimension to Inertia of Point
2
3
4
4
1
.000
.120
.548
.436
.000
.016
1.000
.105
.205
.038
.063
.524
.240
.164
.025
.107
.040
.669
.008
.462
.245
.284
.982
.118
.413
.119
.014
.405
.333
.349
.004
.014
.010
.248
.000
1.000
1.000
1.000
1.000
.003
.003
.004
.057
.725
.064
.058
.153
1.000
.457
.000
1.000
.198
.000
1.000
.043
.000
1.000
.000
.000
1.000
.954
.512
.040
.059
.006
.338
.000
.090
1.000
1.000
By column the first dimension is especially related to the
“owned by mortgage” and “owned outright” categories
Total
Chapter 14
Bi-plot
Employed individuals are
closer to owned
accommodations
Retired individuals are
also close to owned
accommodations
Part-time employees and
unemployed individuals are closer
to rented accommodations and
other forms of accommodations
31
Chapter 14
Multiple Correspondence Analysis(MCA)
When all variables are multiple
nominal, then optimal scaling applies
MCA
32
Chapter 14
Plot with 3 variables
The analysis
now also
includes the
government
office region
33
Field Work
Field work
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