Tutorials in Quantitative Methods for Psychology
2011, Vol. 7(1), p. 5-14.
5
Correspondence Analysis
applied to psychological research
Laura Doey and Jessica Kurta
University of Ottawa
Correspondence analysis is an exploratory data technique used to analyze categorical
data (Benzecri, 1992). It is used in many areas such as marketing and ecology.
Correspondence analysis has been used less often in psychological research, although
it can be suitably applied. This article discusses the benefits of using correspondence
analysis in psychological research and provides a tutorial on how to perform
correspondence analysis using the Statistical Package for the Social Sciences (SPSS).
Correspondence analysis (CA) has become most popular
in fields such as ecology, where data is collected on the
abundance of various animal species in specific sampling
units/areas (terr Braak, 1985). The amount of data involved
in these samples makes it difficult to get a clear sense of the
data at a glance (Palmer, 1993). With the use of CA, an
exploratory data technique for categorical data, ecologists
have been able to transform these complicated tables into
straightforward graphical displays (Hoffman & Franke,
1986). Ecological data is multidimensional making
visualization of more than two dimensions difficult.
Correspondence Analysis is an ideal technique to analyze
this form of data because of its ability to extract the most
important dimensions, allowing simplification of the data
matrix (Palmer). However, this statistical technique merits
further attention within the field of psychological research.
In fact, the lack of awareness of this useful statistical method
puts psychological researchers at a disadvantage. To
demonstrate that CA can be suitably applied to
psychological research, this paper will use a research
question from community psychology. Additionally, due to
the gap in the literature on how to perform CA using
statistical software programs, the current paper will describe
how to perform CA using SPSS software.
History of CA
CA originated approximately 50 years ago and has been
referred to by a variety of names such as dual scaling,
method of reciprocal averages and categorical discriminant
analysis (Hoffman, & Franke, 1986). These names are
thought to stem from the fact that CA has been used to
analyze many different questions and has been given a
different name each time it answers a different question.
These differing names may also be attributed to the different
versions of CA that were being developed in several
countries simultaneously (Abdi, & Williams, 2010; StatSoft,
Inc., 2010).
Jean-Paul Benzecri, French linguist and data analyst, was
an important figure in the initiation of the modern
application of CA in the 1960s and 1970s, making CA
popular in France (Greenacre, 2007; StatSoft, Inc., 2010).
Before 1970, CA was relatively unknown in English
speaking countries with only one English publication on CA
in existence, written by Benzecri (Clausen, 1988). The
application of CA eventually spread to countries such as
Japan and England (Clausen, 1988). However, compared to
other statistical techniques little was published in English on
the topic of CA regardless of its inclusion in American
statistical packages such as SPSS in the 1980s (Clausen,
1988). Today CA is popular in some areas of the social
sciences, such as marketing and ecology (Hoffman, &
Franke, 1986; terr Braak, 1985).
What is CA?
Unlike the many statistical techniques that test
hypotheses that have been formed a priori, CA is an
exploratory data technique that explores categorical data for
which no specific hypotheses have been formed (Storti,
6
2010). More specifically, CA analyzes two-way or multi-way
tables with each row and column becoming a point on a
multidimensional graphical map, also called a biplot (Storti,
2010). This biplot typically consists of two or three
dimensions (StatSoft, Inc., 2010). Rows with comparable
patterns of counts will have points that are close together on
the biplot and columns with comparable patterns of counts
will also have points that are close together on the biplot
(SAS Institute Inc., 2010). The row and column points are
shown on the same graphical display allowing for easier
visualization of the associations among variables (Storti).
CA uses the chi-square statistic—a weighted Euclidean
distance—to measure the distance between points on the
biplot (see Clausen, 1988, pp. 12 for equation). In other
words, the chi-square distance measures the association
between variables. It is important to note that the chi-square
distance can be used to examine the associations between
categories of the same variable but not between variables of
different categories. For example, if the types of mental
health services (i.e., community outpatient centers, hospitals
etc.; row data) available in various provinces (column data),
with abundance as the entries, was examined, it would be
possible to obtain the chi-square differences between types
of mental health services and between provinces but not
between types of mental health services and provinces
(Clausen). Due to the fact that CA is a non-parametric
statistic, there is no theoretical distribution to which the
observed distances can be compared. Therefore, contrary to
the classical utilization of the chi-square test, when applied
to CA the chi-square test does not reveal whether the
association between variables is statistically significant. CA
does not support significance testing and is instead used
post-hoc as an exploratory method (StatSoft, Inc., 2010).
Dimensions
The goal of CA is to explain the most inertia, or variance,
in the model in the least number of dimensions. One way to
understand dimensions is that they are comparable to a
principal component in factor analysis, the association
between the categorical variables (Statsoft Inc., 2010). Some
researchers state that the maximum number of dimensions
needed to exactly represent the table is the number of rows
minus one, or the number of columns minus one (Greenacre,
1984, as cited by Moser, 1989). Other researchers use slightly
different rules of thumb when deciding how many
dimensions to retain (see Hair et al., 2007). The researcher
typically chooses enough dimensions to meet the research
objectives (usually two or three). Given the goal of this
paper as an introduction to the topic of CA, it will not
explore the mathematics involved in calculating dimensions
(see Clausen, 1988, p. 17 for additional details).
Comparison of CA to Other Statistical Techniques
CA is similar to a number of other statistical techniques.
For example, CA and factor analysis are both exploratory
methods that attempt to explain the variance in a model and
decompose this variance into a low-dimensional
representation. In other words, both these techniques
attempt to reduce the variability of a model by calculating
the minimum number of factors that can explain the most
variability in the model (Clausen, 1988; Statsoft Inc., 2010).
However, factor analysis determines which variables go
together to explain the most covariance between descriptors,
whereas CA determines which category variables are
associated with one another.
CA is also similar to Principal Component Analysis
(PCA). In fact, CA has been described as a “generalized” or
“weighted” PCA of a contingency table (Abdi, 2010; SAS
Institute Inc., 2010). CA and PCA both present data in a lowdimensional
plane that accounts for the model’s main
variance. The distances between the points in this lowdimensional
space closely resemble the original distances
from the high-dimensional space (Fellenberg, Hauser, Brors,
Neutzner, Hoheisel, Vingron, 2001). However, PCA extracts
which variables explain the largest amount of variance in
the data set, whereas the focus of CA is to examine the
associations among variables (Fellenberg et al., 2001).
In addition, both CA and cluster analysis are exploratory
methods which sort variables based on their degree of
correspondence to facilitate the analysis of data, but are not
appropriate methods to be used for significance testing
(StatSoft, Inc., 2010). Cluster analysis discovers whether
different variables are related to one another, whereas CA
goes a step further to explain how variables are related
(StatSoft, Inc.) Lastly, multidimensional scaling (MDS) is
similar to CA in that both methods examine the association
between categories of rows and columns, produce a map of
these associations and determine the dimensions that best fit
the model (SAS Institute Inc., 2010; StatSoft, Inc., 2010).
Additionally, both multidimensional scaling and CA have
few assumptions that must be met in order for the solution
to be accurate (SAS Institute Inc.).
Benefits of CA
One of the benefits of CA is that, as previously
mentioned, it can simplify complex data from a potentially
large table into a simpler display of categorical variables
while preserving all of the valuable information in the data
set. This is especially valuable when it would be
inappropriate to use a table to display the data because the
associations between variables would not be apparent due
to the size of the table. It is also important to note that most
7
other exploratory statistical techniques do not provide a plot
of associations among variables.
When other statistical techniques cannot be used to
analyze data because certain assumptions are not met, CA
becomes useful due to its flexible data requirements
(Hoffman & Franke). For example, when a Likert scale is
used to collect data, the spaces between descriptors (i.e.,
“almost never”, “sometimes” and “often”) are not
necessarily equivalent. For example, the distance between
“almost never” and “sometimes” is not necessarily
equivalent to the distance between “sometimes” and
“often”. In this type of scenario, CA is a useful technique
because it focuses mainly on how variables correspond to
one another and not whether there is a significant difference
between these variables.
If one wishes to analyze continuous data with CA, the
data can be categorized and subsequently analyzed as
discrete data. In addition, CA demonstrates how variables
are associated by the approximate distance of points to one
another on the biplot, and not simply that they are
associated. Another benefit of CA is that it can reveal
relationships that would not be identified using other nonmultivariate
statistical techniques, such as performing pairwise
comparisons. Moreover, CA presents data using two
dual displays—one display for the row data and one display
for the column data. This makes analysis of the data easier
compared to the many statistical techniques that do not
provide dual displays (Hoffman, & Franke, 1986). CA makes
it easy to add supplementary data points that may aid in the
interpretation of the model into the analysis post-hoc. In
other words, CA allows for the addition of row or column
points that carry zero inertia to the biplot after it has been
constructed. Lastly, CA is also good way to examine data
validity and facilitates the treatment of outliers (Fellenberg
et al., 2001; Hoffman, & Franke).
Assumptions of CA
Violation of the following assumptions may make the
conclusions drawn about the association among variables
inaccurate and the biplot a less valuable guide for analyzing
the data (Garson, 2008). Firstly, homogeneity of variance
across row and column variables must be met (Garson). CA
assumes that the statistical properties are similar across
rows and columns. For example, there must not be any
empty variables (i.e., variables for which all entries consist
of zeros). Secondly, CA assumes that the data being
analyzed is discrete; however, originally continuous
variables can be categorized into discrete variables. Third,
the data should be made up of several categories (typically
more than three); if CA is used to analyze only two or three
categories this analysis in unlikely to be more informative
than the original table itself (Garson). Fourth, all values in
the frequency table must be non-negative so that the
distances between the points on the biplot are always
positive (Garson). CA does not make any distributional
assumptions (i.e., assumptions of normality; Garson).
CA in Other Research Areas
CA is a valuable statistical technique in part because it
can be used by all researchers and professionals who
analyze categorical variables. Therefore, CA is used to
analyze research questions across many domains
(Greenacre, 1993).
It has been mentioned that CA has become more popular
in ecological research. CA is also popular in marketing
research because researchers in this area frequently collect
categorical data due to the simplicity of this collection
method. (Hoffman, & Franke, 1986). Along with ecology and
marketing, CA has also been applied in other areas such as
sociology, archaeology, geology and medicine (Greenacre,
2007). Due to the wide application of CA across other
domains, the benefits of CA and the simplicity of collecting
categorical data in psychological research, it is surprising
that CA is not more commonly used in psychology.
In fact, CA could be suitably applied to many different
domains within psychological research. CA would be
especially relevant for the many burgeoning areas in
psychology (i.e., positive psychology) in which it is
necessary to determine which questions could be asked or
which hypotheses could be formed, due to its exploratory
nature (Fellenberg et al., 2001). In addition, categorical data
is often easier and less time consuming to collect in
psychological research. For example, it is less complex to ask
individuals whether or not they have been mildly
depressed, moderately depressed or severely depressed
over the last six months compared to asking them to
describe the severity of their depressive symptoms. CA is
often used when a researcher wants to get a general idea of a
population before conducting a more complex study. In
social psychology, CA may be used to look at the relation
between the prevalence of the various sexual orientations in
each geographical region in a city. From these results,
particular support programs could be put in place to target
at-risk individuals who experience difficulty expressing
their sexual orientation. In developmental psychology, CA
could be used to look at the associations between attachment
styles and the types of play children engage in.
As seen above, CA merits more attention within the field
of psychological research. To illustrate how CA can be
applied within psychology, this paper will use a research
question from community psychology to explain how CA is
performed using SPSS. This research question will look at
8
the behaviours that young individuals are most likely to be
at risk for developing (i.e., substance abuse, dropping out,
violence, mental health issues depending on their age group
(i.e., 10-12 years old, 13-15 years old, 16-18 years old, and 19-
21 years old).
SPSS tutorial
Using our community psychology example with
fabricated data, we can input the data into SPSS following
these steps:
Step 1: Entering the Data
First, three variables will be created; variable 1 will be
Age, variable 2 will be Risk, and variable 3 will be
Frequency. It is necessary to label each variable depending
on the number of categories within the variable. For Age,
there are four different age groups; 10-12, 13-15, 16-18 and
19-21. In the “Values” tab in SPSS in Variable View, we will
give four different values to our Age variable. Value 1 will
be age group 10-12, value 2 will be age group 13-15, and so
on. The Risk variable also has four categories, and we will
label these in the same way. Value 1 will be the risk of
Substance Abuse, value 2 will be Drop Out, value 3 will be
Violence and value 4 will be Mental Health.
Data can be entered in this way based on the assumption
that the frequency for each variable has previously been
calculated. If the frequencies of each variable are not
calculated and only raw data is available, it is possible to run
the same analysis without using Step 2, described below.
However, analyzing raw data in this way is uncommon
because of the substantial volume of the file. In our example,
inputting raw data would produce a file of 4996 lines.
In the Data View spreadsheet of our SPSS file, we will
have 16 entries, such that each Age category is matched with
each Risk category. Our Frequency variable will give the
frequency of occurrence in our sample data of each age
group with each risk factor (see Figure 1). For example, we
can see that Age group 10-12 combined with the Risk factor
of Substance Abuse has a frequency of 156 out of our sample
of 4996 subjects.
Step 2: Weight the data
Once all of the data has been inputted into SPSS, the next
step is to weight the cases by frequency. To do this, click on
Data Weight Cases Weight Cases by: Frequency
Variable: Frequency OK (see Figure 2). This is done to
inform SPSS that the frequencies need not be compiled.
Step 3: Running the Analysis
Once the data is weighted by Frequency, it is now
possible to run the analysis in SPSS. To do this, click on
Analyze Dimension Reduction Correspondence
Analysis (see Figure 3). Next, insert each of the Age and Risk
variables into the Row and Column profiles respectively. It
does not matter which variable is on which axis when
running the analysis.
In order to run the full analysis, the range of the rows
and columns must be defined for each variable. Under Row,
click Define Range. In our Age variable, we only have four
categories. We will include all four categories here. Beside
Minimum value, enter the value 1; beside Maximum value,
enter the value 4; click Update and Continue.
Using CA, it is possible to run a preliminary analysis in
which only part of the data is analyzed. With large data sets,
Figure1.Entering data into SPSS. Figure 2. Weighting cases by frequency.
9
researchers may choose to include only part of their data in
the analysis initially, and later include all variables. This
may be useful if there are numerous categories within each
variable and the researcher wishes to reduce the noise and
focus on a particular association within the dataset before
looking at the global picture. In our example, all of the data
will be used for analysis.
It is also necessary to define the range for Risk, including
all four categories. Beside Minimum value, enter the value 1;
beside Maximum value, enter the value 4; click Update and
Continue. SPSS will now run a CA using all of our data. If
there were more than four categories for the row or column
variables, the maximum number would be entered when
defining the ranges in order to analyze the entire dataset.
By clicking on Model, the researcher can specify how
they would like SPSS to produce the results of the analysis.
The first option is choosing the number of dimensions to
include in the solution. In this example, the number of
dimensions was set at two. This is the default setting in SPSS
when running CA; however the dimensions can be
increased at the discretion of the researcher depending on
the type of research being done. The Distance Measure
should be set to Chi square, the Standardization Method
should be set to ‘Row and column means are removed’, and
the Standardization Method, depending on how you would
like to interpret your results, can be chosen (see Figure 4). In
this example, Symmetrical was chosen in order to be able to
compare rows to columns (other standardization methods
are described below); click Continue.
The next option, Statistics, allows the researcher to
choose which output tables to include in the output (see
Figure 5). By default in SPSS, the first three options are
chosen. In our example, we chose to also include Row and
Column Profiles, as well as Confidence Statistics for Row
Points and Column Points. Again, this is at the discretion of
the researcher whether or not to include certain tables; click
Continue. For more information on the options provided in
this step, click on Help in the bottom right corner of the
Correspondence Analysis: Statistics box.
The final option, Plots, allows the researcher to choose
how the analysis should be displayed graphically. This is the
most important part of the output (see Figure 6). Under
Scatterplots, the researcher can choose to display the biplot
graphically, as well as only row points and only column
points in a separate graph. Displaying Row points or
Column points in a scatterplot is useful when comparing
row points or column points in order to simplify the data
that would be produced in a biplot. Line plots can be used
to display row or column categories after standardization
and normalization has been performed. In this example, we
did not ask for line plots. Finally, Plot Dimensions allows
the researcher to choose whether or not to include all of the
dimensions that SPSS was asked to produce for the analysis,
or restrict the dimensions included in the graphical
representation of the data. In this example, SPSS was asked
to ‘Display all dimensions in the solution’; click Continue.
At this point, the parameters for CA have been set and
SPSS can now run the final analysis; click OK (see Figure 7).
Step 4: Interpreting the Output
SPSS will produce a tabulation table, shown in Table 1,
called a Correspondence Table. The data given here is based
Figure 3. Running correspondence analysis in SPSS. Figure 4. Setting the parameters of the model.
10
on the data entered for the age and risk variables in SPSS. It
will include the frequencies for each of the row and column
categories that were given and produce a summation for
each of the row categories and column categories called the
‘Active Margin’. For example, it can be seen below that all of
the frequencies for the age group 10-12 across each of the
Risk categories sum to 688. Similarly, all of the frequencies
for the risk factor Violence across each of the Age categories
sums to 1463.
Next in the SPSS output is a Row Profiles table as shown
in Table 2. This table gives the weighted frequency of each
of the row points, such that the total for the whole row will
sum to 1.
The row profiles are calculated by taking each row point
and dividing it by its respective Active Margin for that row.
For example, for the age group 10-12 and the risk factor
Substance Abuse, the frequency (as given by the
Correspondence Table) is 156. The Active Margin for that
row is 688. Therefore, 156/688 = .227. This is done for each
value in the table.
Similarly, SPSS produces a table called Column Profiles
(Table 3), and these are calculated in the same way as the
Row Profiles table.
From the Correspondence Table, we see that age group
16-18 with the risk factor Mental Health has a frequency of
437. The Active Margin for the column of Mental Health is
1589. Therefore, 437/1589 = .275.
The Summary table (Table 4) is the most important table
provided in the SPSS output for CA.
CA uses the chi-square statistic to test for total variance
explained, along with the associated probability. The chisquare
statistic is high when there is a high correspondence
between the rows and columns of a table (Fellenberg,
Hauser, Brors, Neutzner, Hoheisel, Vingron, 2001). The first
thing to look at in the summary table is whether or not the
model is significant. In this example, our model is highly
significant at the .000 level, with an alpha of .05 and a chisquare
value of 210.373. We also see that SPSS has generated
three dimensions to explain our model. In CA, SPSS only
produces dimensions that can be interpreted, rather than
including all dimensions that explain something about the
Figure 5. Choosing the statistics to be shown in Figure 6. Determining the graphic display
the SPSS output. of the biplot.
Figure 7. A view of the final window before running the
analysis.
11
model. For this reason, inertia does not always add up to
100%. The Inertia column gives the total variance explained
by each dimension in the model. In our model, the total
inertia (total variance explained) is 4.2%. This indicates that
for our model, knowing something about Age explains
around 4% of something about Risk and vice versa. This
association is weak, but still highly significant as indicated
by our chi square statistic.
Each dimension is listed according to the amount of
variance explained in the model. Dimension 1 will always
explain the most variance in the model, followed by
Dimension 2 and so on. In this example, Dimension 1
explains 3% of the total 4.2% of variance accounted for.
Furthermore, Dimension 2 explains 1.2% of the total 4.2% of
variance accounted for. Dimension 3 explains 0% of the total
variance accounted for, and would therefore be dropped
from further analysis. The Singular Value column gives the
square roots of the eigenvalues, which describes “the
maximum canonical correlation between the categories of
the variables in analysis for any given dimension” (Garson,
2008). In CA, eigenvalues and inertia are synonymous in
that, “each axis has an eigenvalue whose sum equals the
inertia of the cloud (mass of points; Benzecri, 1992).
The values in the Proportion of Inertia column give the
percent of variance that each dimension explains of the total
variance explained by the model. In this example,
Dimension 1 explains approximately 71% of the total 4.2% of
variance explained in the model. Furthermore, Dimension 2
explains approximately 29% of the 4.2% of variance
explained in the model. Dimension 3 explains too little of the
total variance explained to be kept for further analysis.
There is no “rule of thumb” or criteria for keeping or
rejecting dimensions for analysis based on proportion of
inertia; it depends on the research question and the
researcher decides what is clinically significant versus
statistically significant for any given case. In essence, this
example dictates that there are two dimensions that can
explain the most variance between risk factors and age
group. Some research questions may reveal that three
dimensions are necessary to explain most of the variance.
The Overview Row Points (Table 5) gives information on
how each of the row points is plotted in the final biplot. The
‘Mass’ column in this table indicates the proportion of each
age group with respect to all age groups in the analysis. The
column ‘Score in Dimension’ indicates the coordinates in
each dimension (1 and 2) where each row category will be
situated on the biplot. Inertia again reflects variance. The
‘Contribution’ column reflects how well each of the points
load onto each of the dimensions, as well as how well the
extraction of dimensions explains each of the points. In this
example, we see that the 10-12 age group loads heavily on
Dimension 1 (74%) and not heavily on Dimension 2 (~ 2%).
It can also be seen that the extraction of Dimension 1
explains 99% of the variance in the 10-12 age group across
risk factor, whereas the extraction of Dimension 2 only
explains around 1% of the variance in the 10-12 age group
across risk factor. As seen in Table 6, the Overview Column
Points gives the same information for the plotting of column
points on the biplot.
In this example, the risk factor for Drop Out loads well
onto Dimension 1 (65%), and not as well on Dimension 2 (~
18%). Furthermore, Dimension 1 explains around 90% of the
Table 1. Correspondence Table.
Table 2. Row Profile Table.
12
variance in Drop Out across age group, and Dimension 2
explains around 10% of the variance in Drop Out across age
group.
Tables 7 and 8, Confidence Row Points and Confidence
Column Points, provide the standard deviations of row and
column scores in each dimension, which is used to assess the
precision of the estimates of points on their axes, much like
confidence intervals are used in other statistical analyses.
Finally, SPSS produces a biplot, which provides a visual
display of each of the values in the dataset plotted with their
axes. This provides a global view of the trends within the
data. In this example, because only two dimensions were
extracted, SPSS can display the results in 2D in the form of a
biplot. In the event that three dimensions would be used, a
3D graph would be produced to represent each dimension.
When using a biplot, the chi-square statistic reveals the
strength of trends within the data, which is based on the
point distances of categories. The distance between any row
points or column points gives a measure of their similarity
(or dissimilarity). Points that are mapped close to one
another have similar profiles, whereas points mapped far
away from one another have very different profiles.
Distances between row and column points are interpreted
differently. Only general statements can be made about
observed trends; precise conclusions cannot be drawn.
Because we asked SPSS to standardize our data using
symmetrical normalization, we can compare rows to
columns in a general fashion. Standardization in CA allows
for a more evenly weighted distribution among large
differences and small differences in distances between
points, so that they can be compared without larger
differences skewing the data and overbearing the smaller
differences (Storti, 2010). Symmetrical normalization is a
technique used to standardize row and column data so as to
be able to make general comparisons between the two.
Other forms of standardization allow you to compare row
variable points or column variable points, or rows or
columns, but not rows to columns (see Garson, 2008 for
Table 3. Column Profile Table.
Table 4. Summary Table.
Table 5. Overview Row Points.
13
further information on other standardization techniques for
CA).
All of our data has been graphically represented by the
biplot above. With the caveat that this particular model only
explains ~4% of risk factors based on age, some general
trends can be seen. For example, we see that the age group
of 16-18 is particularly at risk for problems with Substance
Abuse. We also see that the age group 10-12 is not
particularly at risk for Drop Out, but is more at risk for
problems with Mental Health. Age group 19-21 appears to
be more prone risk factors such as Violence and Mental
Health rather than Substance Abuse and Drop Out. Age
group 13-15 appears to have a marginal risk for Violence,
Mental Health and Drop Out, but not as much for Substance
Abuse.
Conclusion
CA is a statistical technique that is used primarily by
social scientists and behavioural researchers to explore the
relations among multivariate categorical variables (de
Leeuw, 2005; Hoffman & Franke, 1986). CA is used less
frequently in psychological research than in other areas, but
could be suitably applied to various psychological research
questions. In fact, psychological researchers would be at a
disadvantage if they were not aware of the many benefits of
CA, especially the graphical map this statistical technique
provides, which facilitates the visualization of the
associations between the rows and columns of a table. The
goal of the current paper is to show how CA can be used to
examine psychological data using the example of the
associations between various age groups and degree of risk
for developing problematic behaviors (i.e., substance abuse,
dropping out, violence, mental health issues). This is only
one of the many research questions that could be explored
within the realm of psychology using CA.
Furthermore, with the results that are found using CA,
additional research can be done to answer more specific
research questions. For example, our data shows that 16-18
year olds are most at risk for substance abuse. With this
information, a social psychologist could explore what makes
16-18 year olds more susceptible to substance abuse or what
substances are most commonly abused in this age group.
This could facilitate prevention and intervention programs
in targeting at-risk individuals within this age group, as well
as discovering areas of resilience within this age group. In
general, CA provides an extremely useful general picture of
associations between variables, and follow-up statistics can
provide a more in-depth look at a particular research in
question.
References
Abdi, H., & Williams, L. J. (2010). Correspondence Analysis.
Encyclopedia of Research Design. Thousand Oaks, CA:
Sage
Benzecri, J.-P. (1992). Correspondence Analysis Handbook.
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Böckenholt, U., & Takane, Y. (1994). Linear constraints in
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Recent Developments and Applications (pp. 112-127).
London: Academic Press.
Clausen, S. E. (1988). Applied Correspondence Analysis: An
Introduction. Thousand Oaks, CA: Sage.
Table 6. Overview of Column Points.
Table 7. Confdence Row Points.
Table 8. Confdence Column Points.
14
De Leeuw, J. (2005). Review of Correspondence Analysis and
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Fellenberg, K., Hauser, N. C., Brors, B., Neutzner, A.,
Hoheisel, J. D., & Vingron, M. (2001). Correspondence
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Manuscript received 12 November 2010.
Manuscript accepted 1 February 2011.
Plot 1. A biplot displaying various risk factors among adolescents and how
they relate to specific age
groups on two dimensions.