# Notes on Correspondence Analysis of Power Transformed Data Sets

Choulakian V., Université de Moncton, Canada<EMAIL_ADDRESS>
(January 2023)

###### Abstract

We prospect for a clear simple picture on CA of power transformed or the Box-
Cox transformed data initiated since 2009 by Grenacre. We distinguish two
types of data sets: strictly positive and with zeros; we concentrate on the
latter.

Key words: Contingency tables; compositional data; double centering; power
transformation; interactions; zeros; taxicab correspondence analysis.

AMS 2010 subject classifications: 62H25, 62H30

##  1 Introduction

Correspondence analysis (CA) and logratio analysis (LRA) are two popular
methods for the analysis and visualization of a two-way contingency table or a
compositional data set $\mathbf{N}=(n_{ij})$ of size $I\times J$ for
$i=1,...,I$ and $j=1,...,J.$ LRA is considered an ideal desirable method for
analysis-visualization of N, because it is row and column scales invariant;
but it can be applied only when $n_{ij}>0.$ While CA can be applied to data
when $n_{ij}\geq 0$, but it is NOT row and column scale invariant because it
is dependent on its row and column marginals.

The central topic of this paper is Greenacre’s (2010, Result 1), that we
rename it as theorem, and its application to $\mathbf{N}.$ Greenacre’s Theorem
concerns the convergence of CA of $\mathbf{N}^{(\alpha)}=(n_{ij}^{\alpha})$
for $\alpha\in(0,1]$ as $\alpha\rightarrow 0$ to uniformly weighted LRA of
$\mathbf{N}$ when $n_{ij}>0$. However, Greenacre (2011, 2022) attempts to show
its applicability - usefulness when $n_{ij}\geq 0$; that is, when the data set
$\mathbf{N}$ contains zero valued cells. We aim to shed further light on this
latter situation.

This paper is organized as follows: Section 2 presents preliminaries; section
3 presents Greenacre’s Theorem concerning CA of strictly positive data sets;
section 4 presents CA of nonnegative data sets having zero valued cells;
section 5 discusses three examples of real data sets with zero cells; finally
we conclude in section 6.

###  1.1 Some references

CA and LRA are based on three different principles: CA on Benzécri’s
distributional equivalence principle, RC association models on Yule’s scale
invariance principle, and CoDA on Aitchison’s subcompositional coherence
principle. A recent discussion of these three principles can be found in
Choulakian et al. (2023).

Benzécri (1973) is the reference book on CA. Beh and Lombardo (2014) present a
panoramic review of CA and its variants.

LRA includes two independently well developed methods: RC association models
for the analysis of contingency tables by Goodman (1979, 1981a, 1981b, 1991,
1996) and a set of compositional vectors (CoDA) by Aitchison (1986), see also
among others Greenacre (2018).

The relationship between CA and LRA has been discussed in many papers; see for
example, among others, Goodman (1996), Cuadras et al. (2006), Cuadras and
Cuadras (2015 ), Greenacre (2009, 2010), Beh and Lombardo (2022) and
Choulakian (2022).

##  2 Preliminaries on analysis of contingency tables

We consider a two-way contingency table $\mathbf{N}=(n_{ij})$ for $i=1,...,I$
and $j=1,...,J$, and $\mathbf{P=N/}n=(p_{ij})$ of size $I\times J$ the
associated correspondence matrix (probability table) of the contingency table
N. We define as usual $p_{i+}=\sum_{j=1}^{J}p_{ij}$ ,
$p_{+j}=\sum_{i=1}^{I}p_{ij},$ the vector
$\mathbf{r=(}p_{i+})\in\mathbb{R}^{I},$ the vector
$\mathbf{c=(}p_{+j})\in\mathbb{R}^{J}$, and $\mathbf{M}_{I}=Diag(\mathbf{r})$
the diagonal matrix having diagonal elements $p_{i+},$ and similarly
$\mathbf{M}_{J}=Diag(\mathbf{c}).$ We suppose that $\mathbf{M}_{I}$ and
$\mathbf{M}_{J}$ are positive definite metric matrices of size $I\times I$ and
$J\times J$, respectively; this means that the diagonal elements of
$\mathbf{M}_{I}$ and $\mathbf{M}_{J}$ are strictly positive.

###  2.1 Independence of the row and column categories

a) The $I$ row categories and the $J$ column categories are mutually
independent, then

$\sigma_{ij}=p_{ij}-p_{i+}p_{+j}=0$ (1)

for $i=1,...,I$ and $j=1,...,J$, and, where $\sigma_{ij}$ is the residual
matrix of $p_{ij}$ with respect to the independence model $p_{i+}p_{+j}.$

Remark 1: The contingency table $\mathbf{N}=(n_{ij})$ can also be represented
(coded) as an indicator matrix $\mathbf{Z=}\left[\mathbf{Z}_{I}\ \
\mathbf{Z}_{{}^{J}}\right]=\left[\mathbf{(}z_{\alpha i})\ \
\mathbf{(}z_{\alpha j})\right]\ $of size $n\times(I+J),$ where $z_{\alpha
i}=0$ if individual $\alpha$ does not have level $i$ of the row variable,
$z_{\alpha i}=1$ if individual $\alpha$ has level $i$ of the row variable;
$z_{\alpha j}=0$ if individual $\alpha$ does not have level $j$ of the column
variable, $z_{\alpha j}=1$ if individual $\alpha$ has level $j$ of the column
variable. Note that
$\mathbf{N}=\mathbf{Z}_{{}^{I}}{}^{\prime}\mathbf{Z}_{{}^{J}}\ $and
$\sigma_{ij}=p_{ij}-p_{i+}p_{+j}$ is the covariance between the $i$-th column
of $\mathbf{Z}_{{}^{I}}$ and the $j$-th column of $\mathbf{Z}_{{}^{J}}$.

b) The independence assumption $\sigma_{ij}=0$ can also be interpreted in
another way as

$\displaystyle\Delta_{ij}$ $\displaystyle=(\frac{p_{ij}}{p_{i+}p_{+j}}-1)=0$
(2) $\displaystyle=\frac{1}{p_{i+}}(\frac{p_{ij}}{p_{+j}}-p_{i+})$
$\displaystyle=\frac{1}{p_{+j}}(\frac{p_{ij}}{p_{i+}}-p_{+j});$

this is the column and row homogeneity model. Benzécri (1973, p.31) named the
conditional probability vector ($\frac{p_{ij}}{p_{+j}}$ for $i=1,...,I$ and
$j$ fixed) the profile of the $j$-th column. He also referred to the element
of $\frac{p_{ij}}{p_{i+}p_{+j}}$ as the density function of the probability
measure $(p_{ij})$ with respect to the product measure $p_{i+}p_{+j}$. The
element $\frac{p_{ij}}{p_{i+}p_{+j}}$ is named Pearson ratio in Goodman (1996)
and Beh and Lombardo (2014, p.123).

c) A third way to represent the independence assumption $\sigma_{ij}=0$ and
the row and column homogeneity models $\Delta_{ij}=0$ is via the ($w_{i}^{R}$,
$w_{j}^{C})$ weighted loglinear formulation, equation (3), assuming $p_{ij}>0$
and defining $G_{ij}=\log(p_{ij}),$

$\lambda(p_{ij},w_{i}^{R},w_{j}^{C})=G_{ij}-G_{i+}-G_{+j}+G_{++}=0,$ (3)

where $G_{i+}=\sum_{j=1}^{J}G_{ij}w_{j}^{C},$
$G_{+j}=\sum_{i=1}^{I}G_{ij}w_{i}^{R}$ and
$G_{++}=\sum_{j=1}^{J}\sum_{i=1}^{I}G_{ij}w_{j}^{C}w_{i}^{R}$; $w_{j}^{C}>0$
and $w_{i}^{R}>0,$ satisfying
$\sum_{j=1}^{J}w_{j}^{C}=\sum_{i=1}^{I}w_{i}^{R}\ =1,$ are a priori fixed or
data dependent probability weights. Two popular weights are marginal
($w_{i}^{R}=p_{i+},\ w_{j}^{C}=p_{+j})$ and uniform ($w_{i}^{R}=1/I,\
w_{j}^{C}=1/J).$ This is implicit in equation 7 in Goodman (1996) or equation
2.2.6 in Goodman (1991); and explicit in Egozcue et al. (2015).

Equation (3) is equivalent to the logratios

$\log(\frac{p_{ij}p_{i_{1}j_{1}}}{p_{ij_{1}}p_{i_{1}j}})=0\text{ \ \ for
}i\neq i_{1}\text{ and }j\neq j_{1},$

which Goodman (1979, equation 2.2) refers to as the ”null association” model.

Equation (3) is also equivalent to

$p_{ij}=\frac{\exp(G_{i+})\exp(G_{+j})}{\exp(G_{++})},$

from which we deduce that : under the independence assumption the marginal row
probability vector ($p_{i+})$ is proportional to the vector of weighted
geometric means ($\exp(G_{i+}));$ and a similar property is true also for the
columns; see for instance Egozcue et al. (2015).

###  2.2 Interaction factorization

Suppose the independence-homogeneity-null association models are not true,
then each of the three equivalent model formulations (1), (2), (3) can be
generalized to explain the nonindependence-nonhomogeneity-association, named
interaction, among the $I$ rows and the $J$ columns by adding $k$ bilinear
terms, where $k=rank(\mathbf{N)-}1$.

We designate any one of the interaction indices (1), (2), (3) by $\tau_{ij}.$

Benzécri (1973, Vol.2, p. 31-32) emphasized the importance of row and column
weights or metrics in multidimensional data analysis; this is the reason in
the French data analysis circles any study starts with a triplet
$\mathbf{(X,M}_{I},\mathbf{M}_{J}\mathbf{)}$, where X represents the data set,
$\mathbf{M}_{I}=(Diag(m_{i}^{r}))$ is the metric matrix defined on the rows
and $\mathbf{M}_{J}=(Diag(m_{j}^{c}))$ the metric matrix defined on the
columns. We follow the same procedure but $\mathbf{X}$ is the preprocessed
data, where:

a) In covariance analysis, $\mathbf{X}=(\tau_{ij}\mathbf{)=(}IJ\sigma_{ij})$
and

$\ \ \ \ \ \ \ \ \ \ (\mathbf{M}_{I},\mathbf{M}_{J})=(Diag(1/I),Diag(1/J)).$

b) In CA, $\mathbf{X}=(\tau_{ij}\mathbf{)=(}\Delta_{ij})$ and
$(\mathbf{M}_{I},\mathbf{M}_{J})=(Diag(p_{i+}),Diag(p_{+j}));$

c) In LRA,
$\mathbf{X}=(\tau_{ij}\mathbf{)=(}\lambda(p_{ij},w_{i}^{R},w_{j}^{C}))$ and
$(\mathbf{M}_{I},\mathbf{M}_{J})=(Diag(w_{i}^{R}),Diag(w_{j}^{C}))$ with
$\sum_{j=1}^{J}w_{j}^{C}=\sum_{i=1}^{I}w_{i}^{R}=1.$

Note that in a) we have multiplied $\sigma_{ij}$ by $IJ$ to unify
theoretically the covariance analysis with the CA analysis, see the subsection
on double centering.

We factorize the interactions in (1), (2) and (3) by singular value
decomposition (SVD) or taxicab SVD (TSVD) as

$\tau_{ij}=\sum_{\alpha=1}^{k}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha},$ (4)

where $f_{\alpha}(i)$ is the $i$th row principal coordinate and
$g_{\alpha}(j)$ is the $j$th column principal coordinate along the $\alpha$th
principal dimension. Also $\delta_{\alpha}$ is the dispersion measure of the
$\alpha$th principal axis.

Remark 2:

a) In the SVD case the parameters
$(f_{\alpha}(i),g_{\alpha}(j),\delta_{\alpha})$ satisfy the conditions: for
$\alpha,\beta=1,...,k$

$\delta_{\alpha}^{2}=\sum_{i=1}^{I}f_{\alpha}^{\ \
2}(i)m_{i}^{r}=\sum_{j=1}^{J}g_{\alpha}^{2}(j)m_{j}^{c}$
$0=\sum_{i=1}^{I}f_{\alpha}(i)m_{i}^{r}=\sum_{j=1}^{J}g_{\alpha}(j)m_{j}^{c}$
$0=\sum_{i=1}^{I}f_{\alpha}(i)f_{\beta}(i)m_{i}^{r}=\sum_{j=1}^{J}g_{\alpha}(j)g_{\beta}(j)m_{j}^{c}\text{\
\ for }\alpha\neq\beta.$

b) In the TSVD case the parameters
$(f_{\alpha}(i),g_{\alpha}(j),\delta_{\alpha})$ satisfy the conditions: for
$\alpha,\beta=1,...,k$

$\delta_{\alpha}=\sum_{i=1}^{I}|f_{\alpha}^{\
}|(i)m_{i}^{r}=\sum_{j=1}^{J}|g_{\alpha}(j)|m_{j}^{c}$
$0=\sum_{i=1}^{I}f_{\alpha}(i)m_{i}^{r}=\sum_{j=1}^{J}g_{\alpha}(j)m_{j}^{c}$
$0=\sum_{i=1}^{I}f_{\alpha}(i)\
sign(f_{\beta}(i))m_{i}^{r}=\sum_{j=1}^{J}g_{\alpha}(j)\
sign(g_{\beta}(j))m_{j}^{c}\text{\ \ for }\alpha>\beta.$

A description of TSVD can be found in Choulakian (2006, 2016).

Remark 3

a) In the case $(\tau_{ij}\mathbf{)=(}IJ\sigma_{ij})$, the bilinear
decomposition (4) is also named interbattery analysis first proposed by Tucker
(1958); later on, Tenenhaus and Augendre (1996) reintroduced it within
correspondence analysis circles, where they showed that the Tucker
decomposition by SVD produced on some correspondence tables more interesting
(interpretable) structure than CA.

b) In the case $(\tau_{ij}\mathbf{)=(}\Delta_{ij})$, the CA decomposition has
many interpretations. Essentially, for data analysis, purposes Benzécri (1973)
interpreted it as weighted principal components analysis of row and column
profiles. Another useful interpretation of CA, comparable to Tucker
interbattery analysis, is Hotelling(1936)’s canonical correlation analysis,
see Lancaster (1958) and Goodman (1991, 1996).

c) In the case $(\tau_{ij}\mathbf{)=(}\lambda(p_{ij},w_{i}^{R},w_{j}^{C}))$
and $(\mathbf{M}_{I},\mathbf{M}_{J})=(Diag(w_{i}^{R}),Diag(w_{j}^{C})),$ where
$(w_{i}^{R},w_{j}^{C})$ are prespecified; we note this case by TLRA or LRA;
for an example of general prespecified weights see Egozcue and Pawlowsky-Glahn
(2016). For the important particular case
$\lambda(p_{ij},w_{i}^{R},w_{j}^{C})=\lambda(p_{ij},1/I,1/J),$ we get
uniformly-weighted (or taxicab) logratio analysis uwLRA (or uwTLRA).

d) We are concerned with the property of scale dependence or independence of
the three interaction indices (1), (2) and (3). The three indices depend on
$p_{ij}$ where $p_{ij}=n_{ij}/\sum_{i,j}n_{ij}.$ To emphasize this dependence,
we express any one of of the three interaction indices by
$\tau_{ij}(n_{ij})=\tau(p_{ij},m_{i}^{R},m_{j}^{C}).$ Following Yule (1912),
we state the following

Definition 1: An interaction index $\tau_{ij}(n_{ij})$ is scale invariant if
$\tau_{ij}(n_{ij})=\tau_{ij}(a_{i}n_{ij}b_{j})$ for arbitrary scales $a_{i}>0$
and $b_{j}>0$.

It is evident that neither of the three indices with data dependent marginal
weights $(p_{i+},p_{+j})$ are scale invariant.

Concerning the association index (3) we have the following lemma, see
Choulakian (2022) or Choulakian et al. (2023).

Lemma 1: a) The association index (3) with prespecified weights
($w_{i}^{R},w_{j}^{C})$ is scale invariant. That is,

$\lambda(n_{ij},w_{i}^{R},w_{j}^{C})=\lambda(a_{i}n_{ij}b_{j}/n^{\ast},w_{i}^{R},w_{j}^{C}),$

for arbitrary scales $a_{i}>0$ and $b_{j}>0$ and
$n^{\ast}=\sum_{i,j}a_{i}n_{ij}b_{j}.$

b) To a first-order approximation,
$\lambda(n_{ij},w_{i}^{R},w_{j}^{C})\approx(\frac{p_{ij}}{w_{j}^{C}w_{i}^{R}}-\frac{p_{i+}}{w_{i}^{R}}-\frac{p_{+j}}{w_{j}^{C}}+1).$

###  2.3 Double centering

The interaction matrix $(\tau_{ij}\mathbf{)}$ is double centered, that is:

$\displaystyle 0$ $\displaystyle=\sum_{i=1}^{I}m_{i}^{r}m_{j}^{c}\tau_{ij}$
$\displaystyle=\sum_{j=1}^{J}m_{i}^{r}m_{j}^{c}\tau_{ij}.$ (5)

According to Tukey (1977, chapter 10), there are two kinds of double centering
row-PLUS-column and row-TIMES-column; we name them additive and
multiplicative, that we describe.

We consider the triplet $\mathbf{(Y,M}_{I},\mathbf{M}_{J}\mathbf{)}$, where
$Y_{ij}=h(p_{ij})$ where $h$ is a function and
$(\mathbf{M}_{I},\mathbf{M}_{J})=(Diag(m_{i}^{r}),Diag(m_{j}^{c})).$ We
compute the three means, two marginals and the total:$\
Y_{i+}=\sum_{j=1}^{J}Y_{ij}m_{j}^{c},$ $Y_{+j}=\sum_{i=1}^{I}Y_{ij}m_{i}^{r}$
and $Y_{++}=\sum_{j=1}^{J}\sum_{i=1}^{I}Y_{ij}m_{i}^{r}m_{j}^{c}.$

The multiplicative double centering is

$\tau_{ij}=Y_{ij}-\frac{Y_{i+}Y_{+j}}{Y_{++}},$ (6)

where rank($\tau_{ij})=\ $rank$(Y_{ij})-1.$

The additive double centering is

$\tau_{ij}=Y_{ij}-Y_{i+}-Y_{+j}+Y_{++},$ (7)

where rank($\tau_{ij})=\ $rank$(Y_{ij})-1$ or rank$(Y_{ij})-2.$

An evident difference between the two types of double centering is that (7) is
invariant to a row or column additive constants; Definition 1 and Lemma1a
become a corollary to this fact.

We consider two functional forms of $Y_{ij}=h(p_{ij}).$

a) $Y_{ij}=h(p_{ij})=\frac{p_{ij}}{m_{i}^{r}m_{j}^{c}}$ is the density
function of the joint probability measure $p_{ij}$ with respect to the product
measure $m_{i}^{r}m_{j}^{c}.$ First, the covariance interaction
$(\tau_{ij}\mathbf{)=(}IJ\sigma_{ij})$ is obtained from (6), when
$m_{i}^{r}=1/I$ and $m_{j}^{c}=1/J.$ So the rank$\mathbf{(}IJ\sigma_{ij})=\
$rank($IJp_{ij})-1.$ Second, the CA interaction
$(\tau_{ij}\mathbf{)=(}\Delta_{ij})$ is obtained either from (6) or from (7),
when $m_{i}^{r}=p_{i+}$ and $m_{j}^{c}=p_{+j}.$ So the
rank$\mathbf{(}\Delta_{ij})=\ $rank($\frac{p_{ij}}{p_{i+}p_{+j}})-1.$ Third,
the right-hand side of Lemma 1b is obtained from (7) when
$m_{i}^{r}=w_{i}^{R}$ and $m_{j}^{c}=w_{j}^{C}.$

b) $Y_{ij}=h(p_{ij})=\log p_{ij}$ for $p_{ij}>0.$ First, the log interaction
$(\tau_{ij}\mathbf{)=(}\lambda(p_{ij},w_{i}^{R},w_{j}^{C}))$ is obtained from
(7), when $\ m_{i}^{r}=w_{i}^{R}$ and $m_{j}^{c}=w_{j}^{C}.$ So the rank
$\mathbf{(}\lambda(p_{ij},w_{i}^{R},w_{j}^{C}))=$ rank($\log p_{ij})-1,$ as in
Goodman (1991, Table 11); or rank($\log p_{ij})-2,$ as in Goodman (1991, Table
10). Second, to our knowledge the log interaction obtained from (6) has not
been applied yet, most probably due to the fact that it is not row and column
scale invariant; see also subsection 3.1.

Choulakian et al. (2023) used the quality of the sign of the residuals index,
QSR, to choose an optimal case among the different cases mentioned above.

##  3 CA of power transformed strictly positive data

Here, we state Greenacre’s Theorem and provide a mathematical proof in the
appendix.

Let ($p_{ij}^{(\alpha)}=\frac{n_{ij}^{\alpha}}{\sum_{i,j}n_{ij}^{\alpha}})$
for $i=1,...,I$ and $j=1,...,J\ $be the correspondence table of the power
transformed strictly positive data, and
($\Delta_{ij}^{(\alpha)}=\frac{p_{ij}^{(\alpha)}-p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}{p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}})$
the CA interaction matrix.

Theorem (Greenacre (2010, Result 1)

Under the assumption $n_{ij}>0,$ for $\alpha\rightarrow 0$ and $\alpha>0$

$\displaystyle\Delta_{ij}^{(\alpha)}$
$\displaystyle=\frac{p_{ij}^{(\alpha)}-p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}{p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}$
$\displaystyle\approx\alpha\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)$
$\displaystyle=\alpha(\log p_{ij}+\frac{1}{IJ}\sum_{i,j}\log
p_{ij}-\frac{1}{I}\sum_{i}\log p_{ij}-\frac{1}{J}\sum_{j}\log p_{ij})$

Furthermore, $p_{i+}^{(\alpha)}\approx 1/I$ for $i=1,...,I$ and
$p_{+j}^{(\alpha)}\approx 1/J$ for $j=1,...,J.\vskip 12.0pt plus 4.0pt minus
4.0pt$

A way to see the theorem is to look at the following sequence of
approximations using Lemma1b, and, the fact that $p_{i+}^{(\alpha)}\approx
1/I$ for $i=1,...,I$ and $p_{+j}^{(\alpha)}\approx 1/J$ for $j=1,...,J$ as
$\alpha\rightarrow 0.$

$\displaystyle\lambda(p_{ij}^{(\alpha)},w_{i}^{R}$
$\displaystyle=1/I,w_{j}^{C}=1/J)=\alpha\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)$
$\displaystyle\approx(IJp_{ij}^{(\alpha)}+1-Ip_{i+}^{(\alpha)}-Jp_{+j}^{(\alpha)})$
$\displaystyle\approx(IJp_{ij}^{(\alpha)}-1)$
$\displaystyle\approx\Delta_{ij}^{(\alpha)}.$

###  3.1  Remarks

a) The Theorem is interesting, but not useful in empirical contexts, because
one can directly compute the log interactions
$\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J).$

b) In the above Theorem, the assumption $n_{ij}>0$ is fundamental: if
$\alpha=0,$ then $n_{ij}^{\alpha}=n_{ij}^{0}=1$ for all $(i,j);$ that is, the
rank of the matrix ($n_{ij}^{\alpha}=n_{ij}^{0}=1)$ is 1; thus both sides of
the equation in the Theorem are zero.

c) In the proof of the above Theorem in the appendix, one sees that

$\lim_{\alpha\rightarrow
0}\frac{\Delta_{ij}^{(\alpha)}}{\alpha}=\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J).$

d) Assume $n_{ij}\geq 1;$ then another power transformation is the logarithmic
transformation, also known as the Box-Cox transformation,
$\mathbf{L}=(logn_{ij}=\lim_{\alpha\rightarrow
0}\frac{1}{\alpha}(n_{ij}^{\alpha}-1))$. We have not seen any application of
CA to L, see the subsection 2.3 on double centering. Note that the assumption
in Greenacre’s Theorem $n_{ij}>0$ is weaker than the assumption $n_{ij}\geq 1$
for the use of the log transformation. Goodman’s RC model, based on the log
transformation, has been developed for the analysis of contingency tables
(tables with strictly positive counts), and not for compositional data where
often data in % have strictly positive values less than 1, besides having
positive values larger than 1; see for instance, the archeological
compositional CUPS data in Greenacre and Lewi (2008).

e) There is some kind of kinship between Greenacre’s Theorem and Goodman’s
marginal free CA (mfCA), see Goodman (1996, equation (46)). In mfCA of a
probability table $\mathbf{P=(}p_{ij})$ with row and column marginals
($p_{i+})$ and $(p_{+j})$ respectively, CA is applied to a matrix
$\mathbf{Q}=(q_{ij}=a_{i}p_{ij}b_{j})$ related to P in two steps:

Step 1: by the scale invariance property of the log interaction index, see
Lemma1a, under the assumption $n_{ij}>0$ there exists a unique probability
matrix $\mathbf{Q=}(q_{ij})$ which is related to $\mathbf{P=(}p_{ij})$ via the
strictly positive scales $(a_{i},b_{j}),$ that keeps Yule’s association
between the $i$-th row and the $j$-th column unchanged; that is,

$\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)=\lambda(q_{ij}=a_{i}p_{ij}b_{j},w_{i}^{R}=q_{i+}=1/I,w_{j}^{C}=q_{+j}=1/J).$

Note that Q has uniform row and column marginal weights similar to
($p_{ij}^{(\alpha)})$.

Step 2, mfCA of P is CA representation of Q

$IJq_{ij}-1=\sum_{\alpha=1}^{k}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha}.$

For further details, see Choulakian (2022).

##  4 CA of power transformed data set having zero valued cells

It is quite common that large compositional data sets or contingency tables
have zero valued cells. For this case, let $m$ be the number of zero valued
cells; so $IJ-m$ is the number of strictly positive valued cells in the table.
We see that for such tables the log transformation discussed in Remark b is
not valid, while the simple power transformation $n_{ij}^{\alpha}$ for
$\alpha\in[0,1]$ is valid. That is, the matrix $(lim\ n_{ij}^{\alpha})$ as
$\alpha\rightarrow 0$ converges to an indicator matrix $\mathbf{Z}=(z_{ij}),$
where $z_{ij}=1$ if $n_{ij}>0$ and $z_{ij}=0$ if $n_{ij}=0.$ A comparison of
this case with the observation in Remark a) under the assumption $n_{ij}>0$ is
insightful, where the indicator matrix becomes $\mathbf{Z}=(z_{ij}=1)$ of rank
1. Here, it is insightful to consider two distinct cases: Sparse tables
containing multiple zeros in at least two nonproportional rows or
nonproportional columns, and tables with exactly one column (or row) having at
least one zero valued cell. In the latter case the zero entries exhibit a
dominating influence very similar to the cases of a heavyweight cell and a
heavyweight column (or row) discussed by Benzécri (1979) and Lebart (1979) in
CA, and by Choulakian (2008) in TCA.

###  4.1 Tables with one column having m zero-valued cells

Here we discuss the case where there are $m$ for $m=1,...,I-1$ zero valued
cells in one column of the indicator matrix $\mathbf{Z}$ of size $I\times J.$
By Benzécri’s principle of distributional equivalence property which states
that in CA (or TCA) proportional rows or columns can be merged, CA (or TCA) of
Z is identical to CA (or TCA) of the $2\times 2$ contingency table

$\mathbf{R}=\left[\begin{tabular}[c]{ll}$0$&$m(J-1)$\\\
$(I-m)$&$(I-m)(J-1)$\end{tabular}\right],$ (8)

which has only one CA (or TCA) principal dimension.

Lemma 2

a) In CA of R the dispersion, named inertia, $\rho_{1}^{2}=\frac{m}{IJ.}.$

b) In TCA of R the taxicab dispersion
$\delta_{1}=\frac{4m(J-1)(I-m)}{(IJ-m)^{2}}.$

Proof of a): By equation 2.1.4 in Goodman (1996), for a $2\times 2$
contingency table the ”ninindependence” based on the correlation coefficient
is

$\rho_{1}^{2}=\frac{(p_{11}p_{22}-p_{12}p_{21})^{2}}{p_{1+}p_{2+}p_{+1}p_{+2}}.$
(9)

By replacing the probability values of the elements of R in (9), we get

$\displaystyle\rho_{1}^{2}$
$\displaystyle=\frac{m^{2}(J-1)^{2}(I-m)^{2}}{(I-m)m(J-1)(J-1)I(I-m)J}$
$\displaystyle=\frac{m}{IJ}.$

Proof of b): We have to calculate the cross-covariance values
$\sigma_{ij}=p_{ij}-p_{i+}p_{+j}$ for $i=1,2$ and $j=1,2$ of R

$\displaystyle\mathbf{\Sigma}$
$\displaystyle=\left[\begin{tabular}[c]{ll}$\sigma_{11}$&$\sigma_{12}$\\\
$\sigma_{21}$&$\sigma_{22}$\end{tabular}\right]$ (3)
$\displaystyle=\left[\begin{tabular}[c]{ll}$\sigma_{11}$&$-\sigma_{11}$\\\
$-\sigma_{11}$&$\sigma_{11}$\end{tabular}\right],$ (10)

because ($\sigma_{ij}=p_{ij}-p_{i+}p_{+j})$ is row and column centered.

There is one principal axis $\mathbf{u}_{1}^{\prime}=(1\ -1).$ So,

$\displaystyle\delta_{1}$ $\displaystyle=||\mathbf{\Sigma u}_{1}||_{1}$
$\displaystyle=4|\sigma_{11}|$
$\displaystyle=4\frac{m(J-1)(I-m)}{(IJ-m)^{2}}.$

###  4.2 Sparse tables

Suppose $\mathbf{Z}=(z_{ij})$ is an incidence matrix, a presence-absence data
set, where $z_{ij}=0$ means level $j$ is absent in the $i$-th individual,
$z_{ij}=1$ means level $j$ is present in the $i$-th individual. CA (or TCA) is
a popular method for the analysis of such tables, see for an example
Choulakian and Abou-Samra (2020). Putting $p_{ij}=z_{ij}/\sum_{i,j}z_{ij}$ and
supposing that the marginals $p_{+j}>0$ and $p_{i+}>0$, CA (or TCA) data
reconstruction formula is

$p_{ij}=p_{+j}p_{i+}(1+\sum_{\alpha=1}^{k}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha}).$
(11)

Now suppose we apply uniformly weighted LRA (or TLRA); observing
$\log_{2}(z_{ij}+1)=z_{ij}$, then the data reconstruction formula becomes

$p_{ij}=p_{+j}/I+p_{i+}/J-1/(IJ)+\sum_{\alpha=1}^{k}f_{\alpha}(i)g_{\alpha}(j)/\delta_{\alpha};$
(12)

a familiar one known as a FANOVA ( factor analysis and analysis of variance),
see Mandel (1971).

To choose the ”best” between (11) and (12) we use the quality of the signs of
the residuals index, (QSR), within Taxicab framework, see Choulakian(2021) and
Choulakian et al. (2023).

##  5 Examples

Here we analyze three publicly available contingency tables and provide
summary results. For the computations we use the two R packages ca and
TaxicabCA.

###  5.1 Example 1: Author data set

Greenacre and Lewi (2009) discussed the author contingency table that has one
zero valued count. This data set is of size $12\times 26$ and is included in
the correspondence analysis ca in R package by Greenacre et al. (2022).

We consider the power transformed data set
$\mathbf{N}^{(\alpha)}=(n_{ij}^{\alpha})$ for $\alpha=10\char 94\relax(-4);$
this value of $\alpha$ is used by Greenacre (2022). By the ca in R package the
first two dispersion-inertia values of $\mathbf{N}^{(\alpha)}$ are:

$0.003204,\ \ \ \ 0;$

while by Lemma2a, via the merged indicator matrix R of size $2\times 2,$ with
$m=1,\ I=12$ and $J=26$, we get the value of the first principal inertia:
$1/(12\ast 26)=0.003205$.

By the TaxicabCA in R package the first two dispersion values of
$\mathbf{N}^{(\alpha)}$ are:

$0.011369,\ \ \ 9e-06;$

while by Lemma2b, via the R matrix of size $2\times 2,$ with $m=1,\ I=12$ and
$J=26$, we get the value of the first principal taxicab dispersion:
$0.011373$.

This example shows the dominant influence of one zero-valued cell in
$\mathbf{N}^{(\alpha)}=(n_{ij}^{\alpha})$ for $\alpha=10\char 94\relax(-4).$
The next example shows that similar result is also obtained for multiple zero-
valued cells in one column.

###  5.2 Example 2: RBGlass1 data set

RBGlass1 is a compositional data set of size $105\times 11$ that has $m=26$
zeros in the variable Sb column. It is found in the R package archdata by
Carlson and Roth (2022).

We consider the power transformed data set
$\mathbf{N}^{(\alpha)}=(n_{ij}^{\alpha})$ for $\alpha=10\char 94\relax(-4).\
$By the ca in R package the first two dispersion-inertia values of
$\mathbf{N}^{(\alpha)}$ are:

$0.0225,\ 0;$

while by Lemma2a, via the merged indicator matrix R of size $2\times 2,$ with
$m=26,\ I=105$ and $J=11$, we get the value of the first principal inertia:
$26/(11\ast 105)=0.022511$.

By the TaxicabCA in R package the first two dispersion values of
$\mathbf{N}^{(\alpha)}$ are:

$0.0645,\ \ \ 7e-06;$

while by Lemma2b, via the R matrix of size $2\times 2,$ with $m=26,\ I=105$
and $J=11$, we get the value of the first principal taxicab dispersion:
$0.0645$.

###  5.3 Example 3: Rodent abundance data

We consider the rodent data set of size 28 by 9 found in the R package
TaxicabCA. This is an abundance data set of 9 species of rats in 28 cities in
California. Choulakian (2017) analyzed it by comparing the CA and TCA maps;
Choulakian (2021) showed that it has quasi-2-blocks diagonal structure;
furthermore Choulakian (2022) analyzed it by Goodman’s marginal-free CA and
marginal-free TCA methods.

Let N be the original data set of size $28\times 9$; the apparent percentage
of zero counts in N is 66.27%. The function ”CombineCollinearRowsCols” in the
package TaxicabCA in R merges the rows and the columns of N, which are
proportional; we see that the size of the Nmerged is $21\times 9$. So within
the CA framework the real percentage of zero counts in N is the percentage of
zero counts in Nmerged, which is 58.73%. Similarly the real size of the
indicator matrix Z is the size of the Zmerged, which is $14\times 9,$ whose %
of zeros is 60.32%.

We consider the power transformed data set
$\mathbf{N}^{(\alpha)}=(n_{ij}^{\alpha})$ for $\alpha=10\char 94\relax(-4)$.
The ca in R package produced exactly the same maps applied to
$\mathbf{N}^{(\alpha)}$ and Zmerged. Similarly, the TaxicabCA in R package
produced exactly the same maps applied to $\mathbf{N}^{(\alpha)}$ and Zmerged.

The four maps can be seen by applying the R code in the appendix.

##  6 Conclusion

We attempted to have a simple clear picture on CA of power transformed or the
Box-Cox transformed data initiated since 2009 by Grenacre. We distinguished
two types of data sets: strictly positive and with zeros. In particular, we
showed the dominant influence of the zero entries in the CA of power
transformed data when the power goes to zero; in this case the power
transformed data set becomes almost a 0-1 indicator matrix.

An alternative approach is to add a positive constant to any zero-valued cell,
then use the log transformation as in the development of (11); see, among
others, Lubbe et al. (2021) and Choulakian et al. (2023).

    Acknowledgements.

Choulakian’s research has been supported by NSERC of Canada.

    Appendix 1: The R code

The execution of the R code will produce the numerical results and the maps
discussed in the paper. The R code uses the following three packages.

a) The ca package, by Greenacre et al. (2022), does CA and produces the CA
map.

b) The TaxicabCA package, by Allard and Choulakian (2019), does TCA and
produces the TCA map.

c) The archdata package, by Carlson and Roth (2022) for the RBGlasse1 dataset.

# install packages

install.packages(c(”ca”, ”TaxicabCA”, ”archdata”))

library(TaxicabCA)

library(ca)

library(archdata)

#Choose a data set

dataMatrix = as.matrix(rodent)

dataMatrix $<$\- t(author)

data(RBGlass1)

dataMatrix $<$\- RBGlass1[, -1]

dim(dataMatrix)

#Compute dataMerged and IndicatorM

dataMerged $<$\- CombineCollinearRowsCols(dataMatrix, rows = T, cols = T)

dim(dataMerged)

IndicatorM = 1-(dataMatrix == 0)

IndicMerged $<$\- CombineCollinearRowsCols(IndicatorM, rows = T, cols = T)

dim(IndicMerged)

#Compute dataPowered and its marginals

alpha $<$\- 10^(-4)

dataPowered $<$\- dataMatrix^alpha

sum(dataPowered)

apply(dataPowered, 2, function(x) sum(x))

apply(dataPowered, 1, function(x) sum(x))

#CA map of rodent dataset

plot(ca(dataMatrix))

plot(ca(dataPowered))

plot(ca(IndicatorM))

# TCA maps

tca.Data $<$\- tca(dataMatrix, nAxes=2,algorithm = ”exhaustive”)

plot(tca.Data, axes = c(1, 2),labels.rc = c(2, 2))

tca.Data $<$\- tca(dataPowered, nAxes=2,algorithm = ”exhaustive”)

plot(tca.Data, axes = c(1, 2),labels.rc = c(2, 2))

tca.Data $<$\- tca(IndicatorM, nAxes=2,algorithm = ”exhaustive”)

plot(tca.Data, axes = c(1, 2),labels.rc = c(2, 2))

    Appendix 2:

Theorem (Greenacre (2010, Result 1)

Under the assumption $n_{ij}>0,$

$\displaystyle\Delta_{ij}^{(\alpha)}$
$\displaystyle=\frac{p_{ij}^{(\alpha)}-p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}{p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}$
$\displaystyle\approx\alpha\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)$
$\displaystyle=\alpha(\log p_{ij}+\frac{1}{IJ}\sum_{i,j}\log
p_{ij}-\frac{1}{I}\sum_{i}\log p_{ij}-\frac{1}{J}\sum_{j}\log p_{ij})$

for $\alpha\rightarrow 0$ and $\alpha>0.$

Proof: By Maclaurin-Taylor series expansion, in the neighborhood of
$\alpha=0,$ $n_{ij}^{\alpha}=\exp(\alpha\log n_{ij})=1+\alpha\log
x+O(\alpha^{2}),$ where $r(\alpha)=O(h(\alpha))$ means
$\lim_{\alpha\rightarrow 0}|\frac{r(\alpha)}{h(\alpha)}|=constant>0.$ So for
$i=1,...,I\ \ $and $j=1,...,J$

$\displaystyle p_{ij}^{(\alpha)}$
$\displaystyle=\frac{n_{ij}^{\alpha}}{\sum_{i,j}n_{ij}^{\alpha}}$
$\displaystyle=\frac{1+\alpha\log
n_{ij}+O(\alpha^{2})}{IJ+\sum_{i,j}\alpha\log n_{ij}+O(\alpha^{2})}$ (A1)
$\displaystyle=\frac{\left[1+\alpha\log
n_{ij}+O(\alpha^{2})\right]\left[IJ+\alpha\sum_{i,j}\log
n_{ij}+O(\alpha^{2})\right]}{\left[IJ+\alpha\sum_{i,j}\log
n_{ij}+O(\alpha^{2})\right]^{2}}$ $\displaystyle=\frac{IJ+\alpha IJ\log
n_{ij}+\alpha\sum_{i,j}\log
n_{ij}+O(\alpha^{2})}{\left[IJ+O(\alpha)\right]^{2}}.$ (A2)

By (5a) we have

$\displaystyle p_{i+}^{(\alpha)}$ $\displaystyle=\sum_{j}p_{ij}^{(\alpha)}$
$\displaystyle=\frac{J+\alpha\sum_{j}\log
n_{ij}+O(\alpha^{2})}{IJ+O(\alpha)};$ (A3)
$p_{+j}^{(\alpha)}=\frac{I+\alpha\sum_{i}\log
n_{ij}+O(\alpha^{2})}{IJ+O(\alpha)};$ (A4)
$p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}=\frac{IJ+\alpha J\sum_{i}\log
n_{ij}+\alpha I\sum_{j}\log
n_{ij}+O(\alpha^{2})}{\left[IJ+O(\alpha)\right]^{2}}.$

By (A1) and (A4), we have

$\displaystyle\Delta_{ij}^{(\alpha)}$
$\displaystyle=\frac{p_{ij}^{(\alpha)}-p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}{p_{i+}^{(\alpha)}p_{+j}^{(\alpha)}}$
$\displaystyle=\frac{\left[IJ+\alpha IJ\log n_{ij}+\alpha\sum_{i,j}\log
n_{ij}+O(\alpha^{2})\right]-\left[IJ+\alpha J\sum_{i}\log n_{ij}+\alpha
I\sum_{j}\log n_{ij}+O(\alpha^{2})\right]}{IJ+\alpha J\sum_{i}\log
n_{ij}+\alpha I\sum_{j}\log n_{ij}+O(\alpha^{2})}$
$\displaystyle=(\frac{\alpha IJ}{IJ})\frac{\log
n_{ij}+\frac{1}{IJ}\sum_{i,j}\log n_{ij}-\frac{1}{I}\sum_{i}\log
n_{ij}-\frac{1}{J}\sum_{j}\log
n_{ij}+O(\alpha)}{1+\alpha\frac{1}{I}\sum_{i}\log
n_{ij}+\alpha\frac{1}{J}\sum_{j}\log n_{ij}+O(\alpha^{2})}$
$\displaystyle=\alpha\frac{\lambda(p_{ij},w_{i}^{R}=1/I,w_{j}^{C}=1/J)+O(\alpha)}{1+O(\alpha)};$

which is the required result by Lemma 1.

Corollary

By (A3) and (A4), we see that $p_{i+}^{(\alpha)}=1/I+O(\alpha)$ for
$i=1,...,I$ and $p_{+j}^{(\alpha)}=1/J+O(\alpha)$ for $j=1,...,J.\vskip 12.0pt
plus 4.0pt minus 4.0pt$

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