# Simple Correspondence Analysis of Nominal-Ordinal Contingency Tables

Authors: Eric J. Beh
Journal: Journal of Applied Mathematics and Decision Sciences
 (2008)

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## Abstract

The correspondence analysis of a two-way contingency table is now accepted as a very versatile tool for helping users to understand the structure of the association in their data. In cases where the variables consist of ordered categories, there are a number of approaches that can be employed and these generally involve an adaptation of singular value decomposition. Over the last few years, an alternative decomposition method has been used for cases where the row and column variables of a two-way contingency table have an ordinal structure. A version of this approach is also available for a two-way table where one variable has a nominal structure and the other variable has an ordinal structure. However, such an approach does not take into consideration the presence
of the nominal variable. This paper explores an approach to correspondence analysis using an amalgamation of singular value decomposition and bivariate moment decomposition. A benefit of this technique is that it combines the classical technique with the ordinal analysis by determining the structure of the variables in terms of singular values and location, dispersion and higher-order moments.


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## Body

## 1. Introduction

The analysis of categorical data is a very important
component in statistics, and the presence of ordered variables is a common
feature. Models and measures of association for ordinal categorical variables
have been extensively discussed in the literature, and are the subject of
classic texts including Agresti [1], Goodman [2], and Haberman [3].The visual description of the association between two
or more variables is a vital tool for the analyst since it can often provide a
more intuitive view of the nature of the association, or interaction, between
categorical variables than numerical summaries alone. One such tool is
correspondence analysis. However, except in a few cases ([4–7]), the classical approach
to correspondence analysis neglects the presence of ordinal categorical variables
when identifying the structure of their association. One way
to incorporate the ordinal structure of categorical variables in simple
correspondence analysis is to adopt the approach of Beh [8]. His method takes
into account the ordinal structure of one or both variables of a two-way
contingency table. At the heart of the procedure is the partition of the
Pearson chi-squared statistic described by Best and Rayner [9] and Rayner and
Best [10]. However, when there is only one ordered variable, Beh's [8] approach
to correspondence analysis does not consider the structure of the nominal
variable. This paper does consider the previously neglected nominal variable by
using the partition of the Pearson chi-squared statistic described by Beh [11].
The partition involves terms that summarize the association between the nominal
and ordinal variables using bivariate moments. These moments are calculated
using orthogonal polynomials for the ordered variable and generalized basis
vectors of a transformation of the contingency table for the nominal variable.The correspondence analysis approach described here,
referred to as singly ordered correspondence analysis, is shown to be
mathematically similar to the doubly ordered approach. The singly ordered and
doubly ordered approaches share many of the features that make the classical
approach popular. Details of classical correspondence analysis can be found by
referring to, for example, Beh [12], Benzécri [13], Greenacre [14], Hoffman and
Franke [15], and Lebart et al. [16]. A major benefit of singly
ordered correspondence analysis is that nominal row categories and ordinal
column categories can be simultaneously represented on a single correspondence
plot while ensuring that the structure of both variables is preserved.
Constructing such a joint plot for the singly ordered approach of Beh [8] is
not possible due to the scaling of coordinates considered in that paper. For
the technique described in this paper, the special properties linking the
bivariate moments and singular values provide the researcher with an
informative interpretation of the association in contingency tables. These
numerical summaries also allow, through mechanisms common to correspondence
analysis, a graphical interpretation of this association. Hybrid decomposition
has also been considered for the nonsymmetrical correspondence analysis of a
two-way contingency table by Lombardo et al. [17].This paper is divided into seven further sections.
Section2 defines the Pearson ratio and various ways in which it can be
decomposed to yield numerical and graphical summaries of association. The
decompositions considered are (a) singular value decomposition, used in
classical correspondence analysis, (b) bivariate moment decomposition, used for
the doubly ordered correspondence analysis approach of Beh [8], and (c) hybrid
decomposition. This latter technique amalgamates the two former procedures and
is important for the singly ordered correspondence analysis technique described
in this paper. Section 3 summarizes, by considering the hybrid decomposition of
the Pearson ratio, the coordinates needed to obtain a graphical summary of
association between the two categorical variables while Section 4 provides an
interpretation of the distance between the coordinates in the correspondence
plot. Section 5 defines the transition formulae which describe the relationship
between the coordinates of the two variables. Various properties of singly
ordered correspondence analysis are highlighted in Section 6. The features of
the technique are examined using a pedagogical example in Section 7 where it is
applied to the data described in Calimlin et al. [18]. Their contingency table summarizes the classification of
four analgesic drugs according to their effectiveness judged by 121 hospital
patients. The paper concludes with a brief discussion in Section 8.


## 2. Decomposing Pearson's Ratio

### 2.1. The Pearson Ratio

Consider a two-way contingency tableN that
cross-classifies n units/individuals according to I nominal row
categories and J ordered column
categories. Denote the (i,j)th element of N by nij, for i=1,2,…,I and j=1,2,…,J and the (i,j)th cell
relative frequency as pij=nij/n so that ∑i=1I∑j=1Jpij=1. Let the I×J matrix of these
values be denoted as P and let pi• be the ith row marginal
proportion of N so that ∑i=1Ipi•=1, and DI the I×I diagonal matrix
where the (i,i)th cell entry
is pi•. Similarly, let p•j be the jth column
marginal proportion so that ∑j=1Jp•j=1, and DJ the J×J diagonal matrix
where the (j,j)th cell entry
is p•j. Define pij/pi• as the ith row profile
and the (i,j)th element of DI−1P, and pij/p•j the jth column
profile and the (i,j)th element of DI−1PT.For the(i,j)th cell entry,
Goodman [19] described the measure of the departure from independence for row i and column j by the Pearson
ratio
(2.1)αij=pijpi•p•j.
In matrix
notation, the Pearson ratio αij is the (i,j)th cell value
of the matrix Δ, where
(2.2)Δ=DI−1PDJ−1.
Independence
between the I rows and J columns of N will occur when Δ=U, where U is the I×J unity matrix
where all the values are equal to 1. One can examine where independence does not
occur by identifying those Pearson ratios that are statistically significantly
different from 1.A more formal approach to determine whether there
exists an association between the row and column categories involves
decomposing the matrix of Pearson ratios,Δ. For the correspondence analysis of N, there are a variety of ways in
which the decomposition can be performed. Here we will
consider  three methods of decomposition: singular
value decomposition, bivariate moment decomposition, and hybrid decomposition.
It is the consideration of the third approach here that is important for the
method of correspondence analysis discussed in this paper. The use of hybrid
decomposition relies on some basic knowledge of singular value decomposition
and bivariate moment decomposition and so these will be described in the
following subsections.


### 2.2. Singular Value Decomposition

Classically, correspondence analysis involves
decomposing the matrix of Pearson ratios using singular value decomposition
(SVD) so that(2.3)Δ=A˜D˜λB˜T,
where A˜ and B˜ have the
property
(2.4)A˜TDIA˜=I,B˜TDJB˜=I,
respectively
where I is an identity matrix. Also, D˜λ=diag(1,λ1,…,λM*), where λm is the mth largest
singular value of αij, for m=1,…,M*.For the decomposition of (2.3), A˜ is an I×M matrix of left
generalized basic vectors, while B˜ is a J×M matrix of right
generalized basic vectors. In both cases, M=min(I,J) and the first
(trivial) singular vector of both matrices has all values equal to one. Let A and B be the matrices A˜ and B˜, respectively, with the trivial singular vector from
each is omitted. The matrix D˜λ is an M×M diagonal matrix
where the (m,m)th cell value
is the mth singular
value, λm, of Δ. These singular values have the property that they
are arranged in descending order so that 1=λ0≥λ1≥⋯≥λM*≥0, where M*=min(I,J)−1.Suppose we omit the trivial column vector fromA˜ and B˜ to give the I×M* matrix A and the J×M* matrix B, respectively. Also omit the first row and first
column from the matrix D˜λ (since the (1,1)th element of D˜λ is equal to 1),
obtaining the M*×M* matrix Dλ. Then the SVD of the Pearson ratio becomes the SVD of
(2.5)Δ−U=ADλBT
whose elements Goodman [19] referred to as Pearson
contingencies.The SVD of these contingencies leads to the Pearson
chi-squared statistic being expressed in terms of the sum of squares of the
singular values such that(2.6)X2=n∑m=1M*λm2=ntrace(Dλ2).


### 2.3. Bivariate Moment Decomposition

When a two-way contingency table consists of at least
one ordered variable, the ordinal structure of the variable needs to be taken
into consideration. Over the past few decades, there have been a number of
correspondence analysis procedures developed that take into account the ordinal
structure of the variables; see, for example, [4–7].
Generally, these procedures involve imposing ordinal constraints on the
singular vectors. Such a procedure therefore forces the position of the points
(along the first axis) of the plot to be ordered, thereby imposing what can
sometimes lead to unrealistic “correspondences” between row and column
categories. A way to overcome this problem is to consider using orthogonal
polynomials rather than imposing constraints on the columns of A and B considered in
the previous section.For a doubly ordered two-way contingency table, the
correspondence analysis approach of Beh [8] employs the bivariate moment
decomposition (BMD) of Pearson ratios so that
(2.7)Δ=A˜*Y˜B˜*T,
where
(2.8)A˜*TDIA˜*=I,B˜*TDJB˜*=I.For the decomposition of (2.7), A˜* is an I×I matrix of row
orthogonal polynomials, while B˜* is a J×J matrix of
column orthogonal polynomials. The (j,v)th element of B˜* may be
calculated by considering the recurrence relation
(2.9)bv(j)=Sv[(sJ(j)−Tv)bv−1(j)−Vvbv−2(j)],
where
(2.10)Tv=∑j=1Jp•jsJ(j)bv−12(j),Vv=∑j=1Jp•jsJ(j)bv−1(j)bv−2(j),Sv={∑j=1Jp•jsJ2(j)bv−12(j)−Tv2−Cv2}−1/2,
for v=0,1,…,J−1. These are based on the general recurrence relation
of Emerson [20] and depend on the jth score, sJ(j), assigned to reflect the structure of the column
variables. There are many different types of scores that can be considered and
Beh [21] discusses the impact of using four different scoring types (two
objectively and two subjectively chosen scores) on the orthogonal polynomials.
However, for reasons of simplicity and interpretability, we will be considering
the use of natural column scores sJ(j)=j, for j=1,2,…,J, and natural row scores in this paper. For both A˜* and B˜*, the first column vector is trivial, having values
equal to 1 so that b0(j)=1 and a0(i)=1. It is also assumed that b−1(j)=0 and a−1(i)=0, for all i and j.The matrixY˜ is of size I×J where the first
row and column have values all equal to 1. The nontrivial elements of this
matrix are referred to as bivariate moments, or generalized correlations, and
describe linear and nonlinear sources of association between the two
categorical variables. By omitting these trivial vectors, the decomposition of
(2.7) becomes
(2.11)Δ−U=A*YB*T,
where A* and B* are the row and
column orthogonal polynomials, respectively, with the first (trivial) column
vector omitted. The matrix Y has elements
which are the bivariate moments defined by
(2.12)Y=A*TPB*.By considering the BMD (2.11), the Pearson chi-squared
statistic can be partitioned into bivariate moments so that
(2.13)X2=n∑u=1I−1∑v=1J−1Yuv2=ntrace(YTY)=ntrace(YYT),
where the elements
of Y are
asymptotically standard normally distributed. Refer to Best and Rayner [22] and
Rayner and Best [23] for a full interpretation of (2.12) and (2.13). An advantage of
using BMD is that the (u,v)th element of Y, Yuv has a clear and simple interpretation; it is the (u,v)th bivariate
moment between the categories of the row and column variables. As a result,
Davy et al. [24] refer to these values as generalized correlations.
For example, the linear-by-linear relationship can be measured by
(2.14)Y11=∑i=1I∑j=1Jpij(sI(i)−μIσI)(sJ(j)−μJσJ),
where sI(i) and sJ(j) are the set of
row and column scores used to construct the orthogonal polynomials, and μJ=∑j=1JsJ(j)p•j and σJ2=∑j=1JsJ(j)2p•j−μJ2. The quantities μI and σI2 are similarly
defined. By decomposing the Pearson ratios using BMD when natural scores are
used to reflect the ordinal structure of both variables, Y11 is equivalent
to Pearson's product moment correlation; see Rayner and Best [23]. One can also
determine the mean (location) and spread (dispersion) of each of the nonordered
row categories across the ordered column categories by calculating μJ(i)=∑j=1JsJ(j)pij so that μJ=∑i=1IμJ(i) and σJ2(i)=∑j=1JsJ(j)2pij−μJ(i)2, respectively.


### 2.4. Hybrid Decomposition

Another type of decomposition, and one that was
briefly discussed by Beh [12], is what is referred to as hybrid decomposition
(HD). For a singly ordered contingency table, hybrid decomposition takes into
account the ordered variable and nominal variable by incorporating singular
vectors from SVD and orthogonal polynomials from BMD such that the Pearson
contingencies are decomposed by
(2.15)Δ−U=AZB*T.
The Z matrix of (2.15)
is defined as
(2.16)Z=ATPB*.
The M*×(J−1) matrix of Z values, {Z(u)v:u=1,2,…,M*,v=1,2,…,J−1}, can be derived by premultiplying (2.15) by ATDI and
postmultiplying it by DJB*T.If one considers the decomposition of the matrix of
Pearson contingencies using the hybrid decomposition of (2.15), then the partition
of the Pearson chi-squared statistic can be expressed in terms of the sum of
squares of the Z(u)v so that
(2.17)X2=n∑u=1M*∑v=1J−1Z(u)v2=ntrace(ZTZ)=ntrace(ZTZ),
where the
elements of Z are
asymptotically standard normal and independent. Refer to Beh [11] for more
details on (2.16) and (2.17).The effect of the column location component on the
two-way association in the contingency table is measured by∑u=1M*Z(u)12, while, in general, the vth-order column
component is ∑u=1M*Z(u)v2. The significance of these components can be compared
with the chi-squared with M* degrees of
freedom. Testing these column components allows for an examination of the trend
of the column categories, the trend being dictated by the vth orthogonal
polynomial. For example, the column location component determines if there is
any difference in the mean values of the column categories, while the column
dispersion component detects if there is any difference in the spread of the
columns.The first-order row location component on the two-way
association in the contingency table is measured by∑v=1J−1Z(1)v2, while in general, the uth-order row
component value is equivalent to ∑v=1J−1Z(u)v2. The row location component quantifies the variation
in the row categories due to the mean difference in the row categories.
Similarly, the row dispersion component quantifies the amount of variation that
is due to the spread in the row categories. Refer to Section 6 for more
informative details on the row components.Partitions of other measures of association using
orthogonal polynomials have also been considered. D'Ambra et al. [25]
considered the partition of the  Goodman-Kruskal tau index. For symmetrically
associated multiple categorical random variables, Beh and Davy [26, 27] considered
the partition of the Pearson chi-squared statistic, while for asymmetrically
associated variables Beh et al. [28] considered the partition
of the Marcotorchino index [29]. However, the application of extensions to
hybrid decomposition will not be considered here.


## 2.1. The Pearson Ratio

Consider a two-way contingency tableN that
cross-classifies n units/individuals according to I nominal row
categories and J ordered column
categories. Denote the (i,j)th element of N by nij, for i=1,2,…,I and j=1,2,…,J and the (i,j)th cell
relative frequency as pij=nij/n so that ∑i=1I∑j=1Jpij=1. Let the I×J matrix of these
values be denoted as P and let pi• be the ith row marginal
proportion of N so that ∑i=1Ipi•=1, and DI the I×I diagonal matrix
where the (i,i)th cell entry
is pi•. Similarly, let p•j be the jth column
marginal proportion so that ∑j=1Jp•j=1, and DJ the J×J diagonal matrix
where the (j,j)th cell entry
is p•j. Define pij/pi• as the ith row profile
and the (i,j)th element of DI−1P, and pij/p•j the jth column
profile and the (i,j)th element of DI−1PT.For the(i,j)th cell entry,
Goodman [19] described the measure of the departure from independence for row i and column j by the Pearson
ratio
(2.1)αij=pijpi•p•j.
In matrix
notation, the Pearson ratio αij is the (i,j)th cell value
of the matrix Δ, where
(2.2)Δ=DI−1PDJ−1.
Independence
between the I rows and J columns of N will occur when Δ=U, where U is the I×J unity matrix
where all the values are equal to 1. One can examine where independence does not
occur by identifying those Pearson ratios that are statistically significantly
different from 1.A more formal approach to determine whether there
exists an association between the row and column categories involves
decomposing the matrix of Pearson ratios,Δ. For the correspondence analysis of N, there are a variety of ways in
which the decomposition can be performed. Here we will
consider  three methods of decomposition: singular
value decomposition, bivariate moment decomposition, and hybrid decomposition.
It is the consideration of the third approach here that is important for the
method of correspondence analysis discussed in this paper. The use of hybrid
decomposition relies on some basic knowledge of singular value decomposition
and bivariate moment decomposition and so these will be described in the
following subsections.


## 2.2. Singular Value Decomposition

Classically, correspondence analysis involves
decomposing the matrix of Pearson ratios using singular value decomposition
(SVD) so that(2.3)Δ=A˜D˜λB˜T,
where A˜ and B˜ have the
property
(2.4)A˜TDIA˜=I,B˜TDJB˜=I,
respectively
where I is an identity matrix. Also, D˜λ=diag(1,λ1,…,λM*), where λm is the mth largest
singular value of αij, for m=1,…,M*.For the decomposition of (2.3), A˜ is an I×M matrix of left
generalized basic vectors, while B˜ is a J×M matrix of right
generalized basic vectors. In both cases, M=min(I,J) and the first
(trivial) singular vector of both matrices has all values equal to one. Let A and B be the matrices A˜ and B˜, respectively, with the trivial singular vector from
each is omitted. The matrix D˜λ is an M×M diagonal matrix
where the (m,m)th cell value
is the mth singular
value, λm, of Δ. These singular values have the property that they
are arranged in descending order so that 1=λ0≥λ1≥⋯≥λM*≥0, where M*=min(I,J)−1.Suppose we omit the trivial column vector fromA˜ and B˜ to give the I×M* matrix A and the J×M* matrix B, respectively. Also omit the first row and first
column from the matrix D˜λ (since the (1,1)th element of D˜λ is equal to 1),
obtaining the M*×M* matrix Dλ. Then the SVD of the Pearson ratio becomes the SVD of
(2.5)Δ−U=ADλBT
whose elements Goodman [19] referred to as Pearson
contingencies.The SVD of these contingencies leads to the Pearson
chi-squared statistic being expressed in terms of the sum of squares of the
singular values such that(2.6)X2=n∑m=1M*λm2=ntrace(Dλ2).


## 2.3. Bivariate Moment Decomposition

When a two-way contingency table consists of at least
one ordered variable, the ordinal structure of the variable needs to be taken
into consideration. Over the past few decades, there have been a number of
correspondence analysis procedures developed that take into account the ordinal
structure of the variables; see, for example, [4–7].
Generally, these procedures involve imposing ordinal constraints on the
singular vectors. Such a procedure therefore forces the position of the points
(along the first axis) of the plot to be ordered, thereby imposing what can
sometimes lead to unrealistic “correspondences” between row and column
categories. A way to overcome this problem is to consider using orthogonal
polynomials rather than imposing constraints on the columns of A and B considered in
the previous section.For a doubly ordered two-way contingency table, the
correspondence analysis approach of Beh [8] employs the bivariate moment
decomposition (BMD) of Pearson ratios so that
(2.7)Δ=A˜*Y˜B˜*T,
where
(2.8)A˜*TDIA˜*=I,B˜*TDJB˜*=I.For the decomposition of (2.7), A˜* is an I×I matrix of row
orthogonal polynomials, while B˜* is a J×J matrix of
column orthogonal polynomials. The (j,v)th element of B˜* may be
calculated by considering the recurrence relation
(2.9)bv(j)=Sv[(sJ(j)−Tv)bv−1(j)−Vvbv−2(j)],
where
(2.10)Tv=∑j=1Jp•jsJ(j)bv−12(j),Vv=∑j=1Jp•jsJ(j)bv−1(j)bv−2(j),Sv={∑j=1Jp•jsJ2(j)bv−12(j)−Tv2−Cv2}−1/2,
for v=0,1,…,J−1. These are based on the general recurrence relation
of Emerson [20] and depend on the jth score, sJ(j), assigned to reflect the structure of the column
variables. There are many different types of scores that can be considered and
Beh [21] discusses the impact of using four different scoring types (two
objectively and two subjectively chosen scores) on the orthogonal polynomials.
However, for reasons of simplicity and interpretability, we will be considering
the use of natural column scores sJ(j)=j, for j=1,2,…,J, and natural row scores in this paper. For both A˜* and B˜*, the first column vector is trivial, having values
equal to 1 so that b0(j)=1 and a0(i)=1. It is also assumed that b−1(j)=0 and a−1(i)=0, for all i and j.The matrixY˜ is of size I×J where the first
row and column have values all equal to 1. The nontrivial elements of this
matrix are referred to as bivariate moments, or generalized correlations, and
describe linear and nonlinear sources of association between the two
categorical variables. By omitting these trivial vectors, the decomposition of
(2.7) becomes
(2.11)Δ−U=A*YB*T,
where A* and B* are the row and
column orthogonal polynomials, respectively, with the first (trivial) column
vector omitted. The matrix Y has elements
which are the bivariate moments defined by
(2.12)Y=A*TPB*.By considering the BMD (2.11), the Pearson chi-squared
statistic can be partitioned into bivariate moments so that
(2.13)X2=n∑u=1I−1∑v=1J−1Yuv2=ntrace(YTY)=ntrace(YYT),
where the elements
of Y are
asymptotically standard normally distributed. Refer to Best and Rayner [22] and
Rayner and Best [23] for a full interpretation of (2.12) and (2.13). An advantage of
using BMD is that the (u,v)th element of Y, Yuv has a clear and simple interpretation; it is the (u,v)th bivariate
moment between the categories of the row and column variables. As a result,
Davy et al. [24] refer to these values as generalized correlations.
For example, the linear-by-linear relationship can be measured by
(2.14)Y11=∑i=1I∑j=1Jpij(sI(i)−μIσI)(sJ(j)−μJσJ),
where sI(i) and sJ(j) are the set of
row and column scores used to construct the orthogonal polynomials, and μJ=∑j=1JsJ(j)p•j and σJ2=∑j=1JsJ(j)2p•j−μJ2. The quantities μI and σI2 are similarly
defined. By decomposing the Pearson ratios using BMD when natural scores are
used to reflect the ordinal structure of both variables, Y11 is equivalent
to Pearson's product moment correlation; see Rayner and Best [23]. One can also
determine the mean (location) and spread (dispersion) of each of the nonordered
row categories across the ordered column categories by calculating μJ(i)=∑j=1JsJ(j)pij so that μJ=∑i=1IμJ(i) and σJ2(i)=∑j=1JsJ(j)2pij−μJ(i)2, respectively.


## 2.4. Hybrid Decomposition

Another type of decomposition, and one that was
briefly discussed by Beh [12], is what is referred to as hybrid decomposition
(HD). For a singly ordered contingency table, hybrid decomposition takes into
account the ordered variable and nominal variable by incorporating singular
vectors from SVD and orthogonal polynomials from BMD such that the Pearson
contingencies are decomposed by
(2.15)Δ−U=AZB*T.
The Z matrix of (2.15)
is defined as
(2.16)Z=ATPB*.
The M*×(J−1) matrix of Z values, {Z(u)v:u=1,2,…,M*,v=1,2,…,J−1}, can be derived by premultiplying (2.15) by ATDI and
postmultiplying it by DJB*T.If one considers the decomposition of the matrix of
Pearson contingencies using the hybrid decomposition of (2.15), then the partition
of the Pearson chi-squared statistic can be expressed in terms of the sum of
squares of the Z(u)v so that
(2.17)X2=n∑u=1M*∑v=1J−1Z(u)v2=ntrace(ZTZ)=ntrace(ZTZ),
where the
elements of Z are
asymptotically standard normal and independent. Refer to Beh [11] for more
details on (2.16) and (2.17).The effect of the column location component on the
two-way association in the contingency table is measured by∑u=1M*Z(u)12, while, in general, the vth-order column
component is ∑u=1M*Z(u)v2. The significance of these components can be compared
with the chi-squared with M* degrees of
freedom. Testing these column components allows for an examination of the trend
of the column categories, the trend being dictated by the vth orthogonal
polynomial. For example, the column location component determines if there is
any difference in the mean values of the column categories, while the column
dispersion component detects if there is any difference in the spread of the
columns.The first-order row location component on the two-way
association in the contingency table is measured by∑v=1J−1Z(1)v2, while in general, the uth-order row
component value is equivalent to ∑v=1J−1Z(u)v2. The row location component quantifies the variation
in the row categories due to the mean difference in the row categories.
Similarly, the row dispersion component quantifies the amount of variation that
is due to the spread in the row categories. Refer to Section 6 for more
informative details on the row components.Partitions of other measures of association using
orthogonal polynomials have also been considered. D'Ambra et al. [25]
considered the partition of the  Goodman-Kruskal tau index. For symmetrically
associated multiple categorical random variables, Beh and Davy [26, 27] considered
the partition of the Pearson chi-squared statistic, while for asymmetrically
associated variables Beh et al. [28] considered the partition
of the Marcotorchino index [29]. However, the application of extensions to
hybrid decomposition will not be considered here.


## 3. Profile Coordinates

One system of coordinates that could be used to
visualize the association between the row and column categories is to plot
along thekth axis {aik} for the ith row and {bk(j)} for the jth-ordered
column. Such coordinates are referred to as standard coordinates. These are
analogous to the set of standard coordinates considered by Greenacre [14, page
93].However, standard coordinates infer that each of the
axes is given an equal weight of 1. Thus, while the difference within the row
or column variables can be described by the difference between the points, they
will not graphically depict the association between the rows and columns.
Therefore, alternative plotting systems should be considered.Analogous to the derivation of profile coordinates in
Beh [8] using BMD, the row and column profile coordinates for singly ordered
correspondence analysis are defined by
(3.1)F=AZ,(3.2)G*=B*ZT,
respectively.
Therefore, by including the correlation quantities, the coordinates (3.1) and
(3.2) will graphically depict the linear and nonlinear associations that may
exist between the ordered column and nominal row categories.The relationship between the row (and column) profile
coordinates and the Pearson chi-squared statistic can be shown to be(3.3)X2=n∑i=1I∑v=1J−1pi•fiv2=n∑j=1J∑u=1M*p•j(gju*)2
by substituting
the elements of FTDIF and G*TDJG* into (2.17).
However, instead of using the Pearson chi-squared statistic as a measure of
association in a contingency table, correspondence analysis considers instead X2/n, referred to as the total inertia. By adopting X2/n as the measure
of association, (3.3) shows that when the profile coordinates are situated close
to the origin of the correspondence plot, X2/n will be
relatively small. Thus the hypothesis of independence between the rows and
columns will be strong. Profile coordinates far from the origin indicate that
the total inertia will be relatively large and the independence hypothesis
becomes weak. These conclusions may also be verified by considering the
Euclidean distance of a profile coordinate from the origin and other profile
coordinates in the correspondence plot; refer to Section 4 for more details.


## 4. Distances

### 4.1. Distance from the Origin

Consider theith row profile.
The squared Euclidean distance of this profile from the origin is
(4.1)dI2(i,0)=∑j=1J1p•j(pijpi•−p•j)2.
It can be shown
that by expressing this in terms of Pearson contingencies, and using (2.15) and
(2.4), this distance may be expressed in terms of the sum of squares of the ith row profile
coordinate such that
(4.2)dI2(i,0)=∑v=1J−1fiv2,
where fiv is the (i,v)th element of F. By substituting (4.2) into (3.3), the Pearson
chi-squared statistic can be expressed as
(4.3)X2=n∑i=1Ipi•dI2(i,0).
Therefore, row
profile coordinates close to the origin support the hypothesis of independence,
while those situated far from the origin support its rejection.
It can be shown
in a similar manner that
(4.4)X2=n∑j=1Jp•jdJ2(j,0),
where
(4.5)dJ2(j,0)=∑i=1I1pi•(pijp•j−pi•)2=∑u=1M*(gju*)2
is the squared
Euclidean distance of the jth column
profile from the origin and gju* is the (j,u)th element of
(3.2).


### 4.2. Within Variable Distances

The squared Euclidean distance between two row profile
coordinates,i and i′, can be measured by
(4.6)dI2(i,i′)=∑j=1J1p•j(pijpi•−pi′jpi′•)2.By considering the definition of the row profile
coordinates given by (3.1), the squared Euclidean distance between these two
profiles can be alternatively be written as
(4.7)dI2(i,i′)=∑v=1J−1(fiv−fi′v)2.Therefore, if two row profile coordinates have similar
profile, their position in the correspondence plot will be very similar. This
distance measure also shows that if two row categories have different profiles,
then the position of their coordinates in the correspondence plot will lie at a
distance from one another.Similarly, the squared Euclidean distance between two
column profiles,j and j′, can be measured by
(4.8)dJ2(j,j′)=∑u=1M*(gju*−gj′u*)2.These results verify the property of distributional
equivalence as stated by Lebart et al. [16, page 35], a
necessary property for the meaningful interpretation of the distance of
profiles in a correspondence plot.
(1)
If two profiles
having identical profiles are aggregated, then the distance between them
remains unchanged.(2)
If two profiles
having identical distribution profiles are aggregated, then the distance
between them remains unchanged.The interpretation of the distance between a
particular row profile coordinate and a column profile coordinate is a
contentious one and an issue that will not be described here, although a brief
account is given by Beh [12, page 269].


## 4.1. Distance from the Origin

Consider theith row profile.
The squared Euclidean distance of this profile from the origin is
(4.1)dI2(i,0)=∑j=1J1p•j(pijpi•−p•j)2.
It can be shown
that by expressing this in terms of Pearson contingencies, and using (2.15) and
(2.4), this distance may be expressed in terms of the sum of squares of the ith row profile
coordinate such that
(4.2)dI2(i,0)=∑v=1J−1fiv2,
where fiv is the (i,v)th element of F. By substituting (4.2) into (3.3), the Pearson
chi-squared statistic can be expressed as
(4.3)X2=n∑i=1Ipi•dI2(i,0).
Therefore, row
profile coordinates close to the origin support the hypothesis of independence,
while those situated far from the origin support its rejection.
It can be shown
in a similar manner that
(4.4)X2=n∑j=1Jp•jdJ2(j,0),
where
(4.5)dJ2(j,0)=∑i=1I1pi•(pijp•j−pi•)2=∑u=1M*(gju*)2
is the squared
Euclidean distance of the jth column
profile from the origin and gju* is the (j,u)th element of
(3.2).


## 4.2. Within Variable Distances

The squared Euclidean distance between two row profile
coordinates,i and i′, can be measured by
(4.6)dI2(i,i′)=∑j=1J1p•j(pijpi•−pi′jpi′•)2.By considering the definition of the row profile
coordinates given by (3.1), the squared Euclidean distance between these two
profiles can be alternatively be written as
(4.7)dI2(i,i′)=∑v=1J−1(fiv−fi′v)2.Therefore, if two row profile coordinates have similar
profile, their position in the correspondence plot will be very similar. This
distance measure also shows that if two row categories have different profiles,
then the position of their coordinates in the correspondence plot will lie at a
distance from one another.Similarly, the squared Euclidean distance between two
column profiles,j and j′, can be measured by
(4.8)dJ2(j,j′)=∑u=1M*(gju*−gj′u*)2.These results verify the property of distributional
equivalence as stated by Lebart et al. [16, page 35], a
necessary property for the meaningful interpretation of the distance of
profiles in a correspondence plot.
(1)
If two profiles
having identical profiles are aggregated, then the distance between them
remains unchanged.(2)
If two profiles
having identical distribution profiles are aggregated, then the distance
between them remains unchanged.The interpretation of the distance between a
particular row profile coordinate and a column profile coordinate is a
contentious one and an issue that will not be described here, although a brief
account is given by Beh [12, page 269].


## 5. Transition Formula

For the classical approach to correspondence analysis,
transition formulae allow for the profile coordinates of one variable to be
calculated when the profile coordinates of a second variable are known.To derive the transition formulae for a contingency
table with ordered columns and nonordered rows, postmultiply the left- and
right-hand sides of (3.1) by ZT. Doing so leads to
(5.1)FZT=AZZT=A(ATPB*)B*TDJG*,
upon
substituting (2.16) and (3.2). Based on the orthogonality properties (2.4) and (2.8),
the transition formula becomes
(5.2)FZT=DI−1PG*.
The transition
formula (5.2) allows for the row profile coordinates to be calculated when the
column profile coordinates are known.In a similar manner, it can be shown that(5.3)G*Z=DJ−1PTF.Beh [30] provided a description of the transition
formulae obtained for a doubly ordered correspondence analysis and the
configuration of the points in the correspondence plot. For singly ordered
correspondence analysis, similar descriptions can be obtained and are summarized
in the following propositions.
(i)
If the
positions of the row profile coordinates are dominated by the first principal
axis, thenZ(1)2≈0.(ii)
If the
positions of the row profile coordinates are dominated by the second principal
axis, thenZ(2)1≈0.(iii)
If the position
of the column profile coordinates are dominated by the first principal axis,
thenZ(2)1≈0.(iv)
If the
positions of the column profile coordinates are dominated by the second
principal axis, thenZ(1)2≈0.However, it is still possible thatZ(1)2 and/or Z(2)1 will be zero if
none of the row and column profile coordinates lie along a particular axis. For
such a case, it is not possible to determine when this will happen.For both classical and doubly ordered correspondence
analysis, when either the row or column profile positions is situated close to
the origin of the correspondence plot, then there is no association between the
rows and columns. This is indeed the case too for singly ordered correspondence
analysis as indicated by (3.3). The items summarized above show that, in this
case, Z(1)2≈0 and Z(2)1≈0. It can also be shown that Z(1)1≈0 and Z(2)2≈0.


## 6. Properties

The results above show that the mathematics and
characteristics of this approach to singly ordered correspondence analysis are
very similar to doubly ordered correspondence analysis and classical simple correspondence
analysis. However, there are properties of the singly ordered approach that
distinguish it from the other two techniques. This section provides an account
of these properties.Property 1.
The row component associated with themth principal
axis is equivalent to the square of the mth largest
singular value.To show this,
recall that the total inertia may be written in terms of bivariate moments and
in terms of the eigenvalues such that(6.1)X2n=∑u=1M*∑v=1J−1Z(u)v2=∑u=1M*λu2
which can be
obtained by equating the Pearson chi-squared partitions of (2.6) and (2.17). Therefore,
the square of the mth singular
value can be expressed by
(6.2)λm2=∑v=1J−1Z(m)v2,
where the
right-hand side of (6.2) is just the mth-order row
component. For example, the square of the largest singular value may be
partitioned so that
(6.3)λ12=Z(1)12+Z(1)22+⋯+Z(1)J−12.
Therefore, the
singly ordered correspondence approach using the hybrid decomposition of (2.16)
and (2.17) allows for a partition of the singular values of the Pearson
contingencies into components that reflect variation in the row categories in
terms of location, dispersion, and higher-order moments. That is, each singular
value can be partitioned so that information associated with differences in the
mean and spread of the row profiles can be identified. Higher-order moments can
also be determined from such a partition.Property 2.
The row component values are arranged in
descending order.This property
follows directly from Property1. Since the eigenvalues are arranged in a
descending order, so too are the row components.Property 3.
A singly ordered correspondence analysis allows
for the inertia associated with a particular axis of a simple correspondence
plot (called the principal inertia) to be partitioned in bivariate moments.Again, this
property follows directly from Property1, where the principal inertia of the mth axis is the
sum of squares of the bivariate moments when u=m.Property 4.
It is possible to identify which bivariate
moment contributes the most to a particular squared singular value and hence
its associated principal axis.This is readily seen from Property3.For classical correspondence analysis, the axes are
constructed so that the first axis accounts for most of the information in
variation in the categories, the second axis describes accounts for the second
most amount of variation, and so on. However, it is unclear what this variation
is, or whether it is easily identified as being statistically significant. By
considering the partition of the singular values, as described by (6.2), the
user is able to isolate important bivariate moments that include variation in
terms of location, dispersion, and higher-order components for each principal
axis. Therefore, there is more information that is able to be obtained from the
axes of the correspondence plot, and the proximity of the points on it, than
from a classical correspondence plot.


## 7. Example

Consider the contingency table given by Table1 which
was originally seen in Calimlin et al. [18] and analyzed by Beh [11]. The study
was aimed at testing four analgesic drugs (randomly assigned the labels A, B,
C, and D) and their effect on 121 hospital patients. The patients were given a
five-point scale consisting of the categories poor, fair, good, very good, and
excellent on which to make their decision.Table 1
Cross-classification of 121 hospital patients according to analgesic drug and its effect.
Analgesic drug effectPoorFairGoodVery GoodExcellentTotalDrug A51108630Drug
B53381231Drug
C106123031Drug
D71281129Total2721332019121If only a comparison of the drugs, in terms of the
mean value and spread across the different levels of effectiveness, was of
interest, attention would be focused on the quantitiesμJ(i) (and σJ(i)). These values
for Drug A, Drug B, Drug C, and Drug D are 3.3000 (1.2949), 3.6129 (1.4740),
2.2581 (1.0149), and 2.2069 (0.9606), respectively and were calculated using
natural scores for the column categories. Therefore, based on these quantities,
it is clear that Drug A and Drug B are very similar in terms of the two
components across the different levels of effectiveness. Therefore, these two
drugs have a similar effect on the patients. Also, these drugs are different to
Drug C and Drug D which are themselves quite similar in effectiveness. However,
the association between the Drugs and the different levels of effectiveness is
not evident from such measures. This is why correspondence analysis is a suitable
analytical tool to graphically depict and summarize the association. It can be
seen that Table 1 consists of ordered column categories and nonordered row
categories. Therefore, singly ordered correspondence analysis will be used to
analyze the effectiveness of the drugs.The Pearson chi-squared statistic of Table1 is
47.0712, and with a zero p-value,  it is highly statistically significant.
Therefore, with a total inertia of 0.3890, there is a significant association
between the drugs used and their effect on the patients.When a classical correspondence analysis is applied,
the squared singular values areλ12=0.30467, λ22=0.07734, and λ32=0.00701 and the
two-dimensional correspondence plot is given by Figure 1. Here, the first
principal axis accounts for 0.30467/0.3890 ×100=78.3% of
the total association between the two variables, and the second axis accounts
for 19.9%. Therefore, the two-dimensional plot of Figure 1 graphically depicts
98.2% of the association that exists between the analgesic drug being tested
and its level of effectiveness.Figure 1
Classical correspondence plot of Table1.Figure1 shows a clear association between the
analgesic drug being tested and the effectiveness of that drug. Drug B appears
to have an “excellent” effect on the patients that participated in the
study, Drug A was rated as “very good,” Drug D was deemed only “fair”
in its effectiveness and Drug C was judged “good” to “poor.” These
conclusions are also apparent when eyeballing the cell frequencies of Table 1.
However, it is unclear how the profile of each of the four drugs is different,
or where they may be similar. By adopting the methodology above, we can
determine how these comparisons may be made in terms of differences in
location, dispersion, and higher-order components.The component values that are associated with
explaining the variation in the position of the drug coordinates in Figure2
are ∑mZ(m)12=0.21034, ∑mZ(m)22=0.08418, ∑mZ(m)32=0.07268, and ∑mZ(m)42=0.02452. Therefore, Figure 2 is constructed using the first
(linear) principal axis with a principal inertia value of 0.21034=0.45863, and the second (dispersion) principal axis with a
principal inertia value of 0.08148=0.28545, for the four drugs. Together, these two axes
contribute to 75% of the variation of the drugs tested, compared with 98.2% of
the variation in the patients judgement of the drug. The third (cubic)
component contributes to 18.7% of this variation.Figure 2
Singly ordered correspondence plot of Table1.Applying singly ordered correspondence
analysis yieldsZ(1)1=−0.45648 and Z(1)2=−0.26016. Also, Z(1)3=0.16505 and Z(1)4=0.03696. Therefore, by considering  (6.3), we can see that
(7.1)0.3047=(−0.4565)2+(−0.2602)2+(0.1651)2+(0.0370)2.
That is, the
dominant source of the first (squared) singular value is due to the linear
component of the effectiveness of the drugs. Thus, the location component best
describes the variation of the profiles for the drug effectiveness levels along
the first principal axis of Figure 1 (68.4%).Figure2 shows the variation of these drugs in terms
of the linear and quadratic components. While Figure 1 indicates that the
effectiveness of Drug C and Drug D is different, Figure 2 shows that the
positions of Drug C and Drug D are similar across the column responses. This is
because the variation between the two drugs exists at moments higher than the
dispersion. It is also evident from Figure 2 that these two drugs have quite a
different effect than do Drug A and Drug B, which in themselves are different.
These conclusions are in agreement with the comments made earlier in the
example. Figure 2 also shows that by taking into account the ordinal nature of
the column categories, the variation between the drug effectiveness levels may
be explored. For example, “good” and “poor” share the very similar
first principal coordinate. However, there is slightly more variation (across
the drugs) for “good” than there is for “poor.”An important feature of Figure2 is that it depicts
the association between the drugs and the levels of effectiveness. It can be
seen from Figure 2, just as Figure 1 concluded, that Drug A and Drug B are more
effective in treating pain relief than Drug C and Drug D. However, because of
the use of hybrid decomposition, the position of the drug profile coordinates
have changed. Figure 1 concluded that Drug D was rated as “fair.” This is
primarily due to the relatively large cell frequency (with a value of 12) that
the two categories share; this feature is a common characteristic of classical
correspondence analysis. However, since the drug behaves in a similar manner
(in terms of location and spread) when compared with Drug C, its position has
shifted to the bottom right quadrant of the plot. Therefore, Drug D is
associated more with “poor” and “good” when focusing on these
components of the category.By observing the distance of each category from the
origin in Figure2, Drug B is the furthest away from the origin and so is less
likely than the other drugs to contribute to the independence between the drugs
and the patients effect. This is because Drug B contributes more to the row
location component (38.29%) than any of the other three drugs in the study,
while contributing to 67.79% of the variation in the dispersion component.
Further results on the dominance of the drugs to each of the axes in Figure 2
are summarized in Table 2. It shows the contribution, and relative contribution
of each drug to each of the two axes. Table 3 provides a similar summary, but
for the different effectiveness levels of the drugs.Table 2
Contribution of the drugs tested to each axis of Figure2.
Principal axis 1Principal axis 2Drug
testedContr'n% Contr'nContr'n% Contr'nDrug A0.0270512.860.000110.14Drug
B0.0805338.290.0552467.79Drug
C0.0488423.220.0122915.08Drug
D0.0539225.630.0138416.99Total0.210341000.08148100Table 3
Contribution of the effectiveness of the drugs tested to each axis of Figure2.
Principal axis 1Principal axis 2RatingContr'n% Contr'nContr'n% Contr'nPoor0.013564.460.001241.60Fair0.0750224.620.0357646.23Good0.019536.410.0243231.44Very
good0.0561518.430.004065.25Excellent0.1404046.080.0119715.48Total0.304661000.07735100Recall that Drug C and Drug D are positioned close to
one another in Figure2. Table 2 shows that they contribute roughly the same to
the location and dispersion components. Figure 2 also shows that
“excellent” is the most dominant of the drug effectiveness categories
along the first principal axis and this is reflected in Table 3, accounting
for nearly half (46.08%) of the principal inertia for its variable. The second
principal axis is dominated by the category “fair” which contributes to
46.23% of the second principal inertia.


## 8. Discussion

Correspondence analysis has become a very popular
method for analyzing categorical data, and has been shown to be applicable in a
large number of disciplines. It has long been applied in the analysis of
ecological disciplines, and recently in health care and nursing studies [31, 32], environmental management [33], and linguistics [34, 35]. It also has
developed into an analytic tool which can handle many data structures of
different types such as ranked data [30], time series data [36], and cohort data
[37].The aim of this paper has been to discuss new
developments of correspondence analysis for the application to singly ordered
two-way contingency tables. Applications of the classical approach to
correspondence analysis can be made, although the ordered structure of the
variables is often not always reflected in the output. When a two-way table
consists of one ordered variable, such as in sociological or health studies
where responses are rated according to a Likert scale, the ordinal structure of
this variable needs to be considered. The singly ordered correspondence
analysis procedure developed by Beh [8] is applicable to singly ordered
contingency tables. However, due to the nature of this procedure, only a
visualization of the association between the categories of the nonordered
variable can be made. Therefore, any between-variable interpretation is not
possible. The technique developed in this paper improves upon this singly
ordered approach by allowing for the simultaneous representation of the ordered
column and nonordered row categories.



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*Source: 218140-2008-02-14.xml*