1
Correspondence Analysis
Correspondence analysis is a descriptive/exploratory technique designed to analyse simple two-way and multi-way tables containing some measure of correspondence between the rows and columns.
The results provide information which is similar in nature to those produced by Factor Analysis techniques, and they allow one to explore the structure of categorical variables included in the table. The most common kind of table of this type is the two-way frequency cross-tabulation table.
Friday 21 April 2023 09:00 AM
2
Correspondence Analysis
Correspondence analysis (CA) may be defined as a special case of Principal Components Analysis (PCA) of the rows and columns of a table, especially applicable to a cross-tabulation.
However CA and PCA are used under different circumstances. Principal components analysis is used for tables consisting of continuous measurement, whereas correspondence analysis is applied to contingency tables (i.e. cross-tabulations). Its primary goal is to transform a table of numerical information into a graphical display, in which each row and each column is depicted as a point.
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Correspondence Analysis
In a typical correspondence analysis, a cross-tabulation table of frequencies is first standardised, so that the relative frequencies across all cells sum to 1.0.
One way to state the goal of a typical analysis is to represent the entries in the table of relative frequencies in terms of the distances between individual rows and/or columns in a low-dimensional space.
There are several parallels in interpretation between correspondence analysis and factor analysis.
4
Correspondence Analysis
Correspondence Analysis, with Special Attention to the Analysis of Panel Data and Event History Data
Peter G. M. van der Heijden and Jan de Leeuw
Sociological Methodology 1989 19 43-87
5
Correspondence Analysis
Correspondence Analysis Applied to Psychological Research
L. Doey and J. Kurta
Tutorials in Quantitative Methods for Psychology
2011, Vol. 7(1)7(1), p. 5-14.
6
Correspondence Analysis
An Introduction to Correspondence Analysis
P.M. Yelland
The Mathematica Journal 2010, Vol. 1212, p. 1-23.
7
Correspondence Analysis
Correspondence analysis is a useful tool to uncover the relationships among categorical variables
N. Sourial, C. Wolfson, B. Zhu, J. Quail, J. Fletcher, S. Karunananthan, K. Bandeen-Roche, F. Béland and H. Bergman
Journal of Clinical Epidemiology 2010
Volume 63, Issue 6, Pages 638-646
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Correspondence Analysis
The data summarises individuals political affiliation (1,…,5) and geographic region (1,…,4) .
1 Liberal
2 Tend Lib
3 Moderate
4 Tend Cons
5 Conservative
This document is loosely based on SPSS 10; Correspondence Analysis Output, Faculty of Social and Behavioural Sciences, Leiden University, Leiden-Netherland.
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Correspondence Analysis
The data summarises individuals political affiliation (1,…,5) and geographic region (1,…,4) .
1 Northeast
2 Midwest
3 South
4 West
10
Correspondence Analysis
The data (a) summarises individuals political affiliation (1,…,5) and geographic region (1,…,4) .
725 individuals, so 725 rows of data
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Correspondence Analysis
Analyze > Dimension Reduction > Correspondence Analysis
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Correspondence Analysis
Select row/column variables. And define the ranges.
Having defined the ranges. Use the buttons at the side of the screen to set desired parameters.
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Correspondence Analysis
Define Row Range. Select row bound, Update and then Continue
There are 4 regions.
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Correspondence Analysis
Define Column Range. Select column bound, Update and then Continue
There are 5 political affiliations.
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Correspondence Analysis
Finally
Use the buttons at the side of the screen to set desired parameters.
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Correspondence Analysis
Select Statistics
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Correspondence Analysis
Select Plots
Select continue to return to the main screen.
Finally use the OK button to run the analysis, or Paste to preserve the syntax.
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Correspondence Analysis
Syntax
CORRESPONDENCE
TABLE = region4(1 4) BY politics(1 5)
/DIMENSIONS = 2
/MEASURE = CHISQ
/STANDARDIZE = RCMEAN
/NORMALIZATION = SYMMETRICAL
/PRINT = TABLE RPOINTS CPOINTS RPROFILES CPROFILES RCONF CCONF
/PLOT = NDIM(1,MAX) BIPLOT(20) RPOINTS(20) CPOINTS(20) TRROWS(20)
TRCOLUMNS (20) .
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Correspondence Analysis
The Correspondence Table is simply the cross-tabulation of the row and column variables, including the row and column marginal totals, serving as input.
Correspondence Table
19 23 58 16 15 131
26 31 71 47 35 210
18 27 75 46 70 236
30 19 40 26 33 148
93 100 244 135 153 725
RegionNortheast
Midwest
South
West
Active Margin
Liberal Tend Lib Moderate Tend Cons Conservative Active Margin
Political Outlook
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Correspondence Analysis
The Row Profiles are the cell contents divided by their corresponding row total (eg. 19/131=0.145 for the first cell). This table also shows the column masses (column marginals as a percent of n) (eg. 93/725=0.128). These are intermediate calculations on the way toward computing distances between points. Note the column of 1’s.
Row Profiles
.145 .176 .443 .122 .115 1.000
.124 .148 .338 .224 .167 1.000
.076 .114 .318 .195 .297 1.000
.203 .128 .270 .176 .223 1.000
.128 .138 .337 .186 .211
RegionNortheast
Midwest
South
West
Mass
Liberal Tend Lib Moderate Tend Cons Conservative Active Margin
Political Outlook
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Correspondence Analysis
Column Profiles are the cell elements divided by the column marginals (ex. 19/103=0.204). This table also shows the row masses (row marginals as a percent of n) (ex. 131/725=0.181). These are intermediate calculations on the way toward computing distances between points. Note the row of 1’s.
Column Profiles
.204 .230 .238 .119 .098 .181
.280 .310 .291 .348 .229 .290
.194 .270 .307 .341 .458 .326
.323 .190 .164 .193 .216 .204
1.000 1.000 1.000 1.000 1.000
RegionNortheast
Midwest
South
West
Active Margin
Liberal Tend Lib Moderate Tend Cons Conservative Mass
Political Outlook
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Correspondence Analysis
In the Summary table, we first look at the chi‑square value and see that it is significant, justifying the assumption that the two variables are apparently related.
Summary
.189 .036 .627 .627 .035 -.043
.124 .015 .268 .895 .040
.078 .006 .105 1.000
.057 41.489 .000a 1.000 1.000
Dimension1
2
3
Total
SingularValue Inertia Chi Square Sig. Accounted for Cumulative
Proportion of Inertia
StandardDeviation 2
Correlation
Confidence SingularValue
12 degrees of freedoma.
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Correspondence Analysis
SPSS has computed the interpoint distances and subjected the distance matrix to principal components analysis, yielding in this case three dimensions.
Summary
.189 .036 .627 .627 .035 -.043
.124 .015 .268 .895 .040
.078 .006 .105 1.000
.057 41.489 .000a 1.000 1.000
Dimension1
2
3
Total
SingularValue Inertia Chi Square Sig. Accounted for Cumulative
Proportion of Inertia
StandardDeviation 2
Correlation
Confidence SingularValue
12 degrees of freedoma.
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Correspondence Analysis
Only the interpretable dimensions are reported, not the full solution, which is why the eigen values add to something less than 100% (labelled Inertia; these are the percent of variance explained by each dimension) - in this case only 0.057 = 5.7%. This reflects the fact that the correlation between region and political outlook, while significant, is weak.
Summary
.189 .036 .627 .627 .035 -.043
.124 .015 .268 .895 .040
.078 .006 .105 1.000
.057 41.489 .000a 1.000 1.000
Dimension1
2
3
Total
SingularValue Inertia Chi Square Sig. Accounted for Cumulative
Proportion of Inertia
StandardDeviation 2
Correlation
Confidence SingularValue
12 degrees of freedoma.
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Correspondence Analysis
The eigen values (called inertia here) reflect the relative importance of each dimension, with the first always being the most important, the next second most important, etc.
Summary
.189 .036 .627 .627 .035 -.043
.124 .015 .268 .895 .040
.078 .006 .105 1.000
.057 41.489 .000a 1.000 1.000
Dimension1
2
3
Total
SingularValue Inertia Chi Square Sig. Accounted for Cumulative
Proportion of Inertia
StandardDeviation 2
Correlation
Confidence SingularValue
12 degrees of freedoma.
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Correspondence Analysis
The singular values are simply the square roots of the eigen values. They are interpreted as the maximum canonical correlation between the categories of the variables in analysis for any given dimension.
Summary
.189 .036 .627 .627 .035 -.043
.124 .015 .268 .895 .040
.078 .006 .105 1.000
.057 41.489 .000a 1.000 1.000
Dimension1
2
3
Total
SingularValue Inertia Chi Square Sig. Accounted for Cumulative
Proportion of Inertia
StandardDeviation 2
Correlation
Confidence SingularValue
12 degrees of freedoma.
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Correspondence Analysis
Note that the "Proportion of Inertia" columns are the dimension eigen values divided by the total (table) eigen value. That is, they are the percent of variance each dimension explains of the variance explained: thus the first dimension explains 62.7% of the 5.7% of the variance explained by the model.
Summary
.189 .036 .627 .627 .035 -.043
.124 .015 .268 .895 .040
.078 .006 .105 1.000
.057 41.489 .000a 1.000 1.000
Dimension1
2
3
Total
SingularValue Inertia Chi Square Sig. Accounted for Cumulative
Proportion of Inertia
StandardDeviation 2
Correlation
Confidence SingularValue
12 degrees of freedoma.
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Correspondence Analysis
The standard deviation columns refer back to the singular values and helps the researcher assess the relative precision of each dimension.
Summary
.189 .036 .627 .627 .035 -.043
.124 .015 .268 .895 .040
.078 .006 .105 1.000
.057 41.489 .000a 1.000 1.000
Dimension1
2
3
Total
SingularValue Inertia Chi Square Sig. Accounted for Cumulative
Proportion of Inertia
StandardDeviation 2
Correlation
Confidence SingularValue
12 degrees of freedoma.
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Correspondence Analysis
Keyword interpretations
Mass: the marginal proportions of the row variable, used to weight the point profiles when computing point distance. This weighting has the effect of compensating for unequal numbers of cases.
Scores in dimension: scores used as coordinates for points when plotting the correspondence map. Each point has a score on each dimension.
Inertia: Variance
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Correspondence Analysis
Contribution of points to dimensions: as factor loadings are used in conventional factor analysis to ascribe meaning to dimensions, so "contribution of points to dimensions" is used to intuit the meaning of correspondence dimensions.
Contribution of dimensions to points: these are multiple correlations, which reflect how well the principal components model is explaining any given point (category).
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Correspondence Analysis
The Overview Row Points table, for each row point in the correspondence table, displays the mass, scores in dimension, inertia, contribution of the point to the inertia of the dimension, and contribution of the dimension to the inertia of the point.
Overview Row Pointsa
.181 -.702 .309 .020 .470 .139 .832 .105 .938
.290 -.130 .065 .005 .026 .010 .181 .030 .210
.326 .540 .194 .020 .501 .099 .901 .076 .977
.204 -.055 -.675 .012 .003 .752 .010 .970 .979
1.000 .057 1.000 1.000
RegionNortheast
Midwest
South
West
Active Total
Mass 1 2
Score in Dimension
Inertia 1 2
Of Point to Inertia ofDimension
1 2 Total
Of Dimension to Inertia of Point
Contribution
Symmetrical normalizationa.
Overview Row Pointsa
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Correspondence Analysis
The Overview Column Points table is similar to the previous one, except for the column variable (party rather than region) in the correspondence table.
Overview Column Pointsa
.128 -.491 -.800 .016 .163 .663 .363 .630 .993
.138 -.351 .124 .003 .090 .017 .921 .075 .995
.337 -.252 .334 .009 .113 .303 .448 .512 .960
.186 .237 -.037 .006 .055 .002 .308 .005 .313
.211 .721 -.094 .022 .579 .015 .940 .010 .950
1.000 .057 1.000 1.000
Political OutlookLiberal
Tend Lib
Moderate
Tend Cons
Conservative
Active Total
Mass 1 2
Score in Dimension
Inertia 1 2
Of Point to Inertia ofDimension
1 2 Total
Of Dimension to Inertia of Point
Contribution
Symmetrical normalizationa.
Overview Column Pointsa
33
Correspondence Analysis
The Confidence Row Points tables display the standard deviations of the row scores (the values used as coordinates to plot the correspondence map) and are used to assess their precision.
Confidence Row Points
.190 .307 .528
.169 .323 .066
.122 .206 -.685
.339 .148 -.026
RegionNortheast
Midwest
South
West
1 2
Standard Deviation inDimension
1-2
Correlation
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Correspondence Analysis
The Confidence Column Points tables display the standard deviations of the column scores (the values used as coordinates to plot the correspondence map) and are used to assess their precision.
Confidence Column Points
.387 .221 -.694
.072 .117 .801
.171 .122 .575
.215 .406 .095
.127 .302 .304
Political OutlookLiberal
Tend Lib
Moderate
Tend Cons
Conservative
1 2
Standard Deviation inDimension
1-2
Correlation
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Correspondence Analysis
The plots of transformed categories for dimensions display a plot of the transformation of the row category values and of column category values into scores in dimension, with one plot per dimension.
The x-axis has the category values and the y-axis has the corresponding dimension scores. Thus the category "Northeast" in the Overview Row Points table above had a score in dimension of -0.702, as shown on the plot.
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Correspondence Analysis
Refer back to “Overview Row Points” dimension 1Why join!
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Correspondence Analysis
Refer back to “Overview Row Points” dimension 2
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Correspondence Analysis
Refer back to “Overview Column Points” dimension 1
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Correspondence Analysis
Refer back to “Overview Column Points” dimension 2
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Correspondence Analysis
The uniplots for the row and column variables. Note that the origin of the axes is slightly different in the two plots.
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Correspondence Analysis
Refer back to “Overview Row Points” dimensions 1 & 2
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Correspondence Analysis
Refer back to “Overview Column Points” dimensions 1 & 2
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Correspondence Analysis
Finally the biplot correspondence map is obtained.
Note the axes now encompass the most extreme values of both of the uniplots.
Note that while some generalizations can be made about the association of categories (South more conservative, West more liberal). The researcher must keep firmly in mind that correspondence is not association. That is, the researcher should not allow the maps display of inter-category distances to obscure the fact that, for this example, the model only explains 5.7% of the variance in the correspondence table.
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Correspondence Analysis
Refer back to “Overview Row Points” dimensions 1 & 2 and “Overview Column Points” dimensions 1 & 2.
45
Correspondence AnalysisCare must be taken when interpreting the
previous plot. It must be remembered that distances between columns and rows are not defined.
Symmetrical normalization (via the model button slide) 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, 2012 for further information on other standardization techniques for correspondence analysis) also Doey and Kurta 2011 (slide).
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Correspondence Analysis
Input Of A Collated Data Matrix, so 5×4 rather than 725×2
An SPSS program that will do this operation is ANACOR, although since we are using data in table form, this has to be performed using the command syntax window.
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Correspondence Analysis
The data editor appear below. If you wish you may name the columns. These names will then appear in the final plots. Transpose the matrix for the row names to be employed.
Save the collated data matrix, “xls” or “sav”.
Note that we have only the matrix of interest in this view.
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Correspondence Analysis
You must employ the syntax
Either via File > Open > Syntax
49
Correspondence Analysis
With the prepared commands in an ascii file
ANACOR TABLE= ALL (5 , 4) /DIMENSION = 2 /NORMALIZATION = canonical /VARIANCES= COLUMNS /PLOT =NDIM (1 , 2)
Note the command "ALL" since we are providing the table
Note "5" for the number of rows
Note "4" for the number of columns
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Correspondence Analysis
Or via File > New > Syntax
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Correspondence Analysis
With the commands input into the Syntax Editor
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Correspondence Analysis
The solution is, of course, unchanged.
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Correspondence Analysis
Two more illustrative examples.
In the table, 14 topics are taken from the testbed of 1033 MEDLINE abstracts on biomedicine obtained from the National Library of Medicine. All the underlined words in the table denote keywords which are used as referents to the medical topics. The parsing rule used for this sample database required that keywords appear in more than one topic. Of course, alternative parsing strategies can increase or decrease the number of indexing keywords (or terms).
Computational Methods for Intelligent Information Access Michael W. Berry, Susan T. Dumais and Todd A. Letsche And apparently misreported byAlgorithms for Binary Factor Analysis Aleŝ Keprt
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Correspondence Analysis
Corresponding to the text in the previous table is the 18 by 14 term-document matrix shown here. The elements of this matrix are the frequencies in which a term occurs in a document or medical topic. For example, in medical topic M2, the second column of the term-document matrix, culture, discharge, and patients all occur once.
Can you discern any structure?
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Correspondence Analysis
210-1-2-3
2
1
0
-1
-2
-3
Component 1
Com
ponent
2
M14
M13M12
M11
M10
M9
M8
M7
M6M5
M4M3M2
M1
Row Plot
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Correspondence Analysis
210-1-2-3
2
1
0
-1
-2
-3
Component 1
Com
ponent
2
study
rise
respect
rats
pressure
patientsoestrogen
generation
fast
disease
dischargeddepressed
culture
close
bloodbehavior
age
abnormal
Column Plot
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Correspondence Analysis
Finally a set you might recognise!
Can you discern any structure?
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Correspondence Analysis
1.00.50.0-0.5-1.0-1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
Component 1
Com
ponent
2
Woods Wong
Wills
Widdrington Whitlock
Whelan
WebbWardWaller
WallaceVan
Tyrer
Townsend
Toward
ToddTithecott
Timoney
Thompson
Temple
Taylor
Taylor
TangSubhedar
Stevens
StapletonSparrow
Smith
Smith
Simpson
Shaikh
Scrivener
Scott
Sayer
Sams
RowleyRobertsRatcliffe
Randall
Preston
Pickard
Petersson
Pearson
Pearson
Pearson
Patel
Parkes
Papantoniou
Oliver
Norris
Nikoletsopoulou
Nichol
Newham
Negus
Moxham
Moss
Moores
Mccartney
Maunder
Maslen-Jones
Marsden
Macdonald
Lloyd
Lilley
LeslieLee
Lau
Lam
KiteJames
Irving
Hutton
HunterHuggins
HudsonHudson
Hudson
Holdsworth Hill
Hickford
Henly
Hawkins
Harrison
HarlandHarber
Halligan
Hall
Grencis
Grahamslaw
Grafton-Clarke
Gerrard
Gancarczyk
Gallagher
Froggatt
Fraser
FitzpatrickFerguson
Fairs
ElliottDowning
Douthwaite
Denton
Davis
Daley
Coward-WhittakerCooke
Cobb
CoatesClegg
Clarkson
Bushell
BrownBrennan Bolton
Bell
BarrettBamber
Ballard
Baker
BainbridgeAtlan
Appleby
Akhtar
Row Plot
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Correspondence Analysis
1.00.50.0-0.5-1.0-1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
Component 1
Com
ponent
2
PSY3097
PSY3031
PSY3030PSY3029
PSY3028
PSY3027
PSY3026
PSY3022
PSY3020
PSY3018
PSY3016
PSY3013
PSY3009
PSY3008
PSY3006
PSY3002
PSY3001
Column Plot
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SPSS Tips
Now you should go and try for yourself.
Each week our cluster (5.05) is booked for 2 hours after this session. This will enable you to come and go as you please.
Obviously other timetabled sessions for this module take precedence.