INTRODUCTION
Canonical correspondence analysis is a widely used tool
for obtaining a graphical representation of the dependence structure between
the rows and columns of a contingency table (Benzecri, 1980; Greenacre,
1984; Lebart et al., 1984; Nishisato, 1980). This procedure can
also be used as a graphical analytic tool in sensory analysis. For example
McEwan and Schlich (1992) used correspondence analysis to describe the
consumer preference attributes of eight commercially available strawberry
jam. The graphical representation is achieved by assigning scores in the
form of coordinates to the row and column categories yielding a correspondence
plot. Generally simple correspondence analysis is performed by applying
a singular value decomposition to the standardized residuals of a twoway
contingency table. Many, including Greenacre (1984), discuss in detail
the theoretical, computational and application issues of the analysis.
This decomposition ensures that the maximum information regarding the
association between the two categorical variables are accounted for in
the first one or two dimensions of a correspondence plot. However, while
such a plot can identify those categories that are similar, for those
that are different, it is not as easy to clarify how the categories are
different.
The interpretation of the relationship between the categories
of a twoway, or multiway contingency table may be greatly simplified
if additional information concerning the row and column structure of the
table is available. By incorporating this external information through
linear constraints on the row and/or column scores a representation of
the data may be obtained that is not only more parsimonious but is also
easier to understand. In the classical analysis, Böckenholt and Böckenholt
(1990) considered this problem.
In order to overcome both of these remarks, the aim of
this study is to consider an extension of Simple correspondence analysis
of ordinal crossclassifications using orthogonal polynomials (Beh, 1997)
which takes into account external information through linear/ordinal constraints
on the row and/or column scores.
In Section 3 we analyze data that is derived from a sensory
survey. The survey was conducted on a panel of 300 consumers who were
asked to evaluate the organoleptic characteristics of a typical cheese
of south of Italy, “Mozzarella”. The results obtained shown
that by imposing constraints on ordinal correspondence analysis it is
possible to improve the interpretation of the relationship between the
categories.
A RESTRICTED APPROACH FOR SINGLE ORDINAL CORRESPONDENCE
ANALYSIS
Correspondence analysis is a popular graphical tool used
to analyze contingency tables. It is a technique that has, in the last
decade, experienced a boom in the diversity of applications and adaptations
and has been exposed to most disciplines. In the past, it has commonly
been performed by applying a singular value decomposition to a transformation
of the data in the contingency table.
Let P be an I x J twoway contingency table describing
the joint distribution of two categorical variables where the (i,j)`th
cell entry is given by the proportion p_{ij} = n_{ij}/n
for (I = 1, ...., I) and (j = 1, ..., J) with
and
Let D_{I} be the diagonal matrix whose elements are the row masses
and let D_{J} be the diagonal matrix whose elements are the column
masses .
Correspondence analysis is performed by applying singular value decomposition
to the matrix of standardized residuals .
According to the principle of Restricted Eigenvalue Problem
(Rao, 1973), Böckenholt and Böckenholt (1990) proposed a canonical
analysis of contingency tables (RCCA) which takes into account additional
information about the row and column categories of the table.
Let H and G be the matrices of external information (as
linear restrictions) of order (I x K) and (J x L) and ranks K and L, respectively,
such that H^{T}X = 0 and G^{T}Y = 0 where X and Y are
the restricted standardized row and column scores. Restricted CCA scores
(Böckenholt and Böckenholt, 1990) are obtained by a SVD of the
matrix
where, U^{T}U = I, V^{T}V = I and Θ
is a diagonal matrix with eigenvalues η in descending order. Standardized
row and column scores are given by
and
respectively, such that .
Constraints are often imposed by making use of orthogonal polynomials
which are suitable for subdividing total variation of the scores into
linear, quadratic, cubic, etc., components. For example, in order to obtain
a linear order for the standard scores, authors eliminate the effects
of the quadratic and cubic trend by including suitable constraint matrices.
The classical approach to correspondence analysis is
obtained when h = [D_{I} 1] and G[D_{J} 1] (absence
of external information). For deeper remarks on linear constraints in
correspondence analysis see also Böckenholt and Böckenholt (1990),
Takane et al. (1991), Böckenholt and Takane (1994) and Hwang
and Takane (2002).
It is possible to consider external information directly
on suitable matrices which reflect the different most important components
by taking into account the Pearson chisquared partition obtained by using
the orthogonal polynomial for contingency tables (Emerson, 1968). The
advantage of using these orthogonal polynomials relies in the fact that
the order information is considered in the analysis and the resulting
scoring scheme permits a clear interpretation of the linear, quadratic
or higher order trend components.
The correspondence analysis approach of Beh (1997) decomposes
the (i, j)`th Pearson ratio so that
For this method of decomposition, λ_{uv}
is the (u, v)`th generalised correlation (Davy et al., 2003) while
are the orthogonal polynomials for the i`th row and j`th column respectively.
The polynomials are calculated using the general recurrence relation of
Emerson (1968) and require a set of scores to reflect the nature of the
variables. For example, the polynomials for ordinal variables can be constructed
using natural scores.
This approach to correspondence analysis uses the Bivariate
Moment Decomposition (BMD) to identify linear (location), quadratic (dispersion)
and higher order moments for each of the ordinal variables. This feature
is not readily available by using classical singular value decomposition.
The first axis of the correspondence plot reflects differences in location
and the second axis reflects differences in dispersion. Higher order components
are reflected by higher dimensional axes.
As remarked before, the general recurrence relation of
Emerson`s (1968) can be used only when a set of scores for the ordered
column category has been considered. For sake of simplicity, only natural
column scores, {s_{J} (j) = j; j = 1,..., J}, will be considered
in this paper. Different scoring schemes can have distinct effects on
the orthogonal polynomials and simple correspondence analysis (Beh, 1998).
Suppose that P has nominal row and ordered column categories.
The set of column orthogonal polynomials are placed into a (J x J) matrix,
B, whose (v, j)`th element is denoted by b_{jv} such that B^{T}D_{J}B
= 1. Based on the recurrence relation b_{j0} is the first column
of B whose elements are all equal to 1 (trivial orthogonal polynomial)
and it is assumed that when v = 1, b_{jv} = 0.
There are various papers that deal with the partition
of the symmetric chisquared statistic with one or two ordered categories
(Best and Rayner, 1996). An alternative approach to partitioning the Pearson
chisquared statistic for a twoway contingency table, which is described
in Beh (2001), is to essentially combine the approach of orthogonal polynomials
for the ordered columns and singular vectors for the unordered rows, so
that
with M* ≤ I1 and where
which are asymptotically standard normally distributed
random variables. The parentheses around u indicates that (1)  (2) is
concerned with a nonordered set of row categories. Equation
2 can alternatively written in matrix form as
where, A is the I x (I1) matrix of left singular vectors,
while B is the J x (J1) matrix of the J1 nontrivial column orthogonal
polynomials. Previous equation can be rearranged so that
The value of Z_{(u)v} means that each principal
axis from a simple correspondence analysis can be partitioned into column
component values. In this way, the researcher can determine the dominant
source of variation of the ordered columns along a particular axis using
the simple correspondence analysis.
Therefore the Pearson ratio of the singly ordered analysis
is
where, Z_{(u)v} is defined by (2). The value
of a_{iu} is akin to the singular vector of simple correspondence
analysis calculated using a singular value decomposition and so is not
an orthogonal polynomial. It is associated with the row profiles, while
the set of orthogonal polynomials, {b_{v}(j)}, are associated
with the ordered columns.
Eliminating the trivial solution to (4) yields the Pearson
contingencies
leading to
where
is the matrix B without the first column b_{0}.
There are several properties that Beh (2001) derived
which show the relationship between the singular values and the bivariate
moments. Main properties are:
The row component associated with the m`th principal
axis is just the value of the m`th largest eigenvalue. In the same way,
total inertia may be written in terms of bivariate moments or as eigenvalues,
such that
so that
• 
This method allows for a decomposition of the singular
values into location, dispersion and higher order components. In this
way, it can be applied to a nonordered contingency table. That is,
each singular value can be partitioned so that information concerning
the mean difference and the spread of profiles can be found. Higher
order moments can also be determined from such a partition. 
• 
The row component values are arranged in descending order. 
• 
For such a singly ordered analysis, it allows for the principal
inertia of a simple correspondence plot to be partitioned into bivariate
moments. When the principal inertia of the m`th principal axis is
the sum of squares of the bivariate moments. For example, the largest
eigenvalue may be calculated by considering the linearbylinear,
linearbyquadratic and higher order bivariate moments, such that

• 
It is possible to identify which bivariate moment contributes
the most to a particular eigenvalue and hence principal axis. 
For other properties and an extension of this approach
to double ordered contingency tables see Beh (1998, 1999, 2001).
It is therefore possible to use these results extend
the Böckenholt and Böckenholt approach for a twoway crossclassification
with nominal and ordinal categorical variables.
Let
be the matrix identifying the most important linear (location), quadratic
(dispersion) or higher order moments by using the previous results.
Restricted Single Ordinal Correspondence Analysis (RSOCA)
scores are then obtained by performing a SVD of the matrix
where U^{T}U = I, V^{T}V = I and Θ
is a diagonal matrix with eigenvalues η that are arranged in descending
order.
Standardized row and column scores are given by
and
respectively, such that
We remark that Single Ordinary Correspondence Analysis
is obtained for H = [D_{I} 1] and G = [D_{J} 1] (where
there is an absence of external information).
By considering this approach it is possible to consider
external information directly on suitable matrices that reflect the most
important components. For example, researchers can impose external constraints
in order to restrict the spacing to be the same for some modalities without
considering those for ensuring linear order if they perform the restricted
analysis over
for v = 1. Moreover, they will be sure to eliminate the effects of the
others trends working only on the component of main importance and interest.
This approach can be easily extended to the other versions
of singly or doubly ordinal correspondence analysis (Beh, 1998, 1999,
2001)) as well as to the two and three way ordinal nonsymmetrical approaches
(D`Ambra et al., 2002; Lombardo et al., 2007; Beh et
al., 2007).
APPLICATION
To demonstrate the applicability of the technique described
above consider Table 1 . Three dairy farms in Campania
(Italy) participated in a study where 300 consumers were asked to evaluate
their Overall Satisfaction (OS) of a given piece of mozzarella cheese.
This was measured on a four point scale where the color (Milk, Cream,
Ivory) of the cheese was considered. The 300 independent individuals that
are classified into Table 1 . While it is apparent that
Table 1 may be considered as a 3x4x3 contingency table,
in the spirit of the analysis of Böckenholt and Böckenholt (1990)
we shall analyze it as a 9x4 table.
Table 2 summarizes the decomposition
of the total inertia of Table 1 into location, dispersion
and an error component. This error component consists of those measures
of moment higher than the dispersion. It shows that the most dominant
and only significant source of variation in the Satisfaction levels is
in terms of their location, while the spread of the profiles is not a
dominant source of variation.
To study the relationship between the two variables,
a singly ordered correspondence analysis is performed. To do so the overall
satisfaction categories are treated as ordinal and the row categories
are nominal.
The representation in Fig. 1 shows the relationship between
the modalities of the color (Fig. 1a) related to the three
dairyfarms, Salernitani (D1), La Baronia (D2), Prati del Volturno (D3) and
the modalities of the overall satisfaction (Fig. 1b). The
first factorial plane is obtained by taking into account the linear component,
ie first nontrivial orthogonal polynomial, from the decomposition (4). It explains
approximately the 90% of the total inertia of the Table 1
(61% axis 1 and 29% axis 2). The factorial plane shows that customers have expressed
a relatively low evaluation of the mozzarella product from the dairyfarm La
Baronia with respect to all three cheese colors. The milk colored cheeses at
the other two farms have a relatively high level of satisfaction. We also note
that there is a strong association between the satisfaction modalities 3 and
4. Following this consideration we impose a linear constraint on the column
modalities (overall satisfaction) with the aim to enforce a separation between
the low values of the overall satisfaction levels (1 and 2) and the high satisfaction
levels (3 and 4).
Table 1: 
Crossclassification of the evaluation of mozzarella cheese 

Table 2: 
Decomposition of the total inertia X^{2}/n 

Thus a constrained solution is computed by setting G = D_{I}1 and
H = (D_{c}1 G_{1}) with G_{1} = (0,0,1,1).
Under the restricted singly ordered correspondence analysis
approach Fig. 2 gives the plot of the coordinates for
the first two axes. The RSOCA plots allow one to better understand the
relationship between the two variables. The plot on lefthandside (Fig.
2a) shows that there is a strong association between the cream modalities
of the three dairyfarms and by observing their proximity with the points
in Fig. 2b, they are associated with the level OS 2
of overall satisfaction. Ivory colored cheese at Salernitani (D1) and
Prati del Volturno (D3) appear to be similarly distributed by observing
their proximity to one another. Their distance from any other point indicates
that their location is different to many other dairy farm/color combinations
This is true also for milk colored cheese at La Baronia.
The highest level of satisfaction is related to the Ivory
modalities of the mozzarella products by the dairyfarms D1 and D3 according
to the raw data of Table 1 . The plot shows also that
the linear constraint imposed allows the user to gain a better understand
the differences between the high and low levels of satisfaction relating
to the color of the cheese and the dairyfarms D1 and D3.
In order to better compare the ordinal and classical
(Böckenholt and Böckenholt, 1990) approaches to the constrained
analysis, restricted canonical correspondence analysis is performed with
the following column constrainst:

Fig. 1: 
Single ordinal correspondence analysis of mozzarella data 

Fig. 2: 
Restricted single ordinal correspondence analysis of mozzarella
data 

Fig. 3: 
Restricted (Böckenholt and Böckenholt) correspondence
analysis of mozzarella data 
While the first row of the G_{1}` matrix ensures
the linear spacing of the standardized columns (not necessary in RSOCA
because only the linear component is taken into account), the second row
aims to restrict the space between the low values of the overall satisfaction
variable (1 and 2) and the high evaluation (3 and 4), as before defined
for RSOCA.
By comparing Fig. 2 with Fig. 3, we can
appreciate the different interpretation between the two sets of axes. While
in RSOCA plot Ivory colored cheese at Salernitani (D1) and Prati del Volturno
(D3) appear to be similarly distributed by observing their proximity to one
another, they seem to be a little bit different from one another in Fig.
3, however with the classical restricted approach is not clear what the
cause of this difference is. By using RSOCA we know that the difference is not
in the first moment (location).
Figure 3 shows also that milk colored
mozzarella cheese from La Baronia is very different compared to other
colored cheeses from other dairy farms. RSOCA plot shows that this difference
is due to its location being very different from other areas.
Finally, Fig. 3 highlights that “Cream”
colored cheese at Salernitani (D1) and Prati del Volturno (D3) appear
to be associated with the level OS 3 of overall satisfaction while by
RSOCA we know that, according also to the raw data, they can be associated
to OS 2.
CONCLUSIONS
Several authors (Nishisato, 1980; Böckenholt and
Böckenholt, 1990) highlighted that introducing linear constraints
on the row and column coordinates of a correspondence analysis, representation
may be greatly simplify the interpretation of the data matrix. The alternative
approach of Beh for ordinal categorical data has been shown to be a more
useful and informative correspondence analysis method than the classical
technique commonly used. When Beh (1997) considered the partition of the
chisquared statistic for a twoway contingency table by combining the
approach of orthogonal polynomials for the ordered columns and singular
vectors for the unordered rows, he restricted the analysis to the integer
valued scores. The problem with such a scoring scheme is that it assumes
that the ordered categories are equally spaced. In general we know that
this may not be the case.
Following our proposal, suitable linear constraints can
be introduced in the combined Beh`s analysis in order to take into account
notequally spaced categories.
Moreover, in order to obtain a linear order for the standard
scores, Böckenholt and Böckenholt remove the effects of the
quadratic and cubic trend by including suitable constraint matrices even
if they do not know if they are statistically significant sources of variation.
The knowledge of their significant allows researcher to improve the interpretation
of the data matrix using suitable constraints for these significant components
otherwise ignored. Finally, we consider linear constraints directly on
suitable matrices that reflect only the most important components. Therefore,
the proposed technique, could be usefully applied when, analyzing contingency
table, the modality are non equispaced and there are the data are infected
by non linear sources of variability.