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Statistica Applicata - Italian Journal of Applied Statistics Vol. 29 (2-3) 123

1. INTRODUCTION

In the present article, we study the variance of eigenvalues in PCA and

MCA, i.e., the mean of squared deviations from eigenvalues to their mean.

The variance of eigenvalues can be seen as an index of departure from

sphericity. We examine the relationship between the variance of eigenvalues

and the correlations between variables in PCA or the contingency mean

square coeﬃcients (usually denoted Φ2) between categorical variables in

MCA. This article develops the properties presented in Durand (1998).

There are few studies on the variance of eigenvalues. However, we can

ﬁnd the expression of the sum of squares of eigenvalues in studies on the

LINKAGE INDEX OF VARIABLES AND ITS RELATIONSHIP

WITH VARIANCE OF EIGENVALUES IN PCA AND MCA

Jean-Luc Durand1

Laboratoire d’Ethologie Expérimentale et Comparée, LEEC EA4443, Université

Paris 13 (Sorbonne Paris Cité), Villetaneuse, France

Brigitte Le Roux

MAP5, UMR 8145, Université Paris Descartes (Sorbonne Paris Cité), Paris

CEVIPOF, UMR 7048, Sciences Po, Paris, France

Abstract. In the present article, we show that, in principal component analysis (PCA) on

correlation matrix as well as in multiple correspondence analysis (MCA), the strength of

the relationship between variables is linked to the variance of the eigenvalues, and indicates

the axes to which the variables contribute the most. In PCA, we define the linkage index of

a variable as the mean of the squared correlations between this variable and the others. We

prove that the variance of eigenvalues is proportional to the mean linkage index and that,

for each variable, the variance of eigenvalues weighted by the contributions of the variable

to axes is proportional to the linkage index of the variable. In MCA, similar properties are

proven regarding both categorical variables and categories. We illustrate these properties

using two datasets coming from classical articles by Spearman (1904) for PCA and Burt

(1950) for MCA.

Keywords: PCA, MCA, Variance of eigenvalues, Contributions to axes.

1Corresponding author: Jean-Luc Durand, email: jean-luc.durand@univ-paris13.fr

doi.org/10.26398/IJAS.0029-006

124 Durand, J.-L., Le Roux, B.

number of axes to be used for interpretation, or on conﬁdence interval (see

e.g. Saporta, 2003; Karlis et al., 2003). The variance of eigenvalues is also

used in applications, especially in biological studies (Pavlicev et al., 2009).

2. PRINCIPAL COMPONENT ANALYSIS

In this section we study the variance of eigenvalues in the case of PCA on

correlation matrix.

2.1. BASIC PROPERTIES AND NOTATIONS

Let Idenote a set of nindividuals, Ka set indexing p(p>1) non–constant

variables on I; the k–th variable is denoted xk=xi

k, its mean xkand its

variance vk.

Let rkkbe the correlation between variables xkand xkand R=[rkk]

the correlation matrix between the pvariables. The calculation method of

standard PCA is based on the diagonalization of the correlation matrix R.

Let Lbe a set indexing the nonnull eigenvalues. If Λ=[λ]denotes the

diagonal matrix of eigenvalues (λ)∈Land A=[ak]the matrix of eigenvec-

tors, then the PCA of variables xkwrites RA =AΛ with AA=I. In the

simple linear regression of the standardized initial variable (xk−xk)/√vk

on the –th principal variable with variance 1, the regression coeﬃcient is

equal to the correlation coeﬃcient rk between the k–th initial variable and

the –th principal variable. We have the properties:

rk =λak and

∈L

(rk)2=1.

The contribution of variable xkto the variance λof axis , denoted

Ctr

k, is equal to r2

k/λ,with∀∈L,

k∈K

Ctr

k=1.

If the correlation matrix has full rank (all eigenvalues are strictly posi-

tive), one has the property:

∀k∈K,

∈L

Ctr

k=1.(1)

This property comes from Ctr

k=r2

k/λ=a2

k,with

∈L

a2

k =1since

matrix Ais orthogonal (AA=AA=I).

Linkage Index of Variables and Its Relationship with Variance of Eigenvalues … 125

2.2. VARIANCE OF EIGENVALUES

Theorem 2.1. The variance of eigenvalues, denoted V(λ), is such that

V(λ)=

1

p

k∈K

k

∈K

k

=k

r

2

kk

Proof.

The mean of eigenvalues is λ=1, the variance is V(λ)=

1

p

∈L

λ

2

−1,

∈L

λ

2

being equal to the trace of matrix R

2

. The entries of R

2

are equal to

k

∈K

r

kk

r

k

k

and the diagonal entries are (

k

∈K

r

2

kk

)

k∈K

, hence the trace of R

2

is equal to

k∈K

1+

k

=k

r

2

kk

=p+

k∈K

k

∈K

k

=k

r

2

kk

.

This expression of the variance invites us to consider the mean of the

squared correlations between one variable and the others. As we will see

later on, this quantity is an index of the strength of the link between one

variable and the others. Hence the following deﬁnition:

Deﬁnition 2.1 (Linkage index of a variable).We call linkage index of

variable x

k

,denotedLI

k

, the mean of the squared correlations between the

variable x

k

and the (p−1) others:

LI

k

=

1

p−1

k

∈K

k

=k

r

2

kk

Note that the linkage index of a variable is between zero and one.

Proposition 1. The mean linkage index of the pvariables, denoted LI,is

equal to the variance of eigenvalues divided by (p-1):

LI =

1

p−1

V(λ)

Comment. The mean linkage index is a measure of both the global mag-

nitude of correlations and of the departure from sphericity. In the particu-

lar case of an equicorrelation matrix, all oﬀ–diagonal elements of which are

equal to r(Morrison, 1976, p. 331), the mean linkage index is equal to r

2

.

.

.

.

126 Durand, J.-L., Le Roux, B.

2.3. VARIABLES AND EIGENVALUES

From now on, we assume that the correlation matrix has full rank.

We will now express the linkage index of a variable as a function of the

eigenvalues and the contributions of this variable to axes.

Considering the contributions of the initial variable x

kto axes (namely

(Ctr

k)∈Lwith

∈L

Ctr

k=1, see Equation 1), we deﬁne the weighted mean

(

λk) and the weighted variance (

Vk(λ)) of eigenvalues:

λk=

∈L

Ctr

kλand

Vk(λ)=

∈L

Ctr

k(λ−

λ)2

The following two properties can easily be shown:

∀k∈K,

λk=1 and

Vk(λ)=

∈L

λr2

k −1

Theorem 2.2. The linkage index of variable xkis proportional to

Vk(λ):

LIk=1

p−1

Vk(λ)

Proof.

By the reconstitution formula of the correlation matrix (see Le Roux and

Rouanet, 2004, p. 153), one has R=AΛA

, hence R

2

=AΛ

2

A

.Thek–th

diagonal entry of R

2

is equal to

k

∈K

r

2

kk

=1+

k

=k

r

2

kk

(see the proof of Theorem

2.1) and also to

∈L

λ

2

a

2

k

=

λ

r

2

k

(since r

k

=√λ

a

k

). Hence

∈L

λ

r

2

k

=

V

k

(λ)+1=1+

k

∈K

k

=k

r

2

kk

.

Corollary 2.2.1. The ratio of the linkage index of variable x

kand the mean

linkage index is equal to the ratio of the weighted variance of eigenvalues

Vk(λ)to the variance of eigenvalues V(λ):

LIk

LI =

Vk(λ)

V(λ)

Comments

1) The more the linkage index of a variable is superior to the mean, the

more this variable contributes to extreme axes (ﬁrst and last axes).

2) The more the linkage index of a variable is inferior to the mean, the

more this variable contributes to central axes (axes with variance near 1).

.

.

.

.

Linkage Index of Variables and Its Relationship with Variance of Eigenvalues … 127

2.4. APPLICATION TO SPEARMAN’S DATA

Table 1 (see Spearman, 1904, p.291) gives the correlations between perfor-

mance variables of English pupils in the following subjects: Classics (k1),

French (k2), English (k3), Mathematics (k4), Pitch Discrimination (k5),

and Music (k6).

Table 1: Correlations, linkage indexes LI

k

and ratios LI

k

/LI.

k1k2k3k4k5k6LIkLIk/LI

k1Classics 1 0.83 0.78 0.70 0.66 0.63 0.524 1.34

k2French 0.83 1 0.67 0.67 0.65 0.57 0.467 1.20

k3English 0.78 0.67 1 0.64 0.54 0.51 0.404 1.04

k4Mathematics 0.70 0.67 0.64 1 0.45 0.51 0.362 0.93

k5Pitch discrim. 0.66 0.65 0.54 0.45 1 0.40 0.302 0.78

k6Music 0.63 0.57 0.51 0.51 0.40 1 0.280 0.72

LI = 0.390

We notice (see Table 1) that Classics is the most correlated with other

variables with a linkage index equal to 0.524, which is 34% higher than the

average. As we can see in Table 2, this variable is the one that contributes

the most to axes 1and 6, for which variances are the farthest from 1

(“extreme” axes). In sharp contrast, Music is the least correlated with other

variables (linkage index equal to 0.280) and contributes heavily to axes 2

and 3, for which variances are the closest to 1 (“central” axes).

Tabl e 2 : Eigenvalues λ, contributions of variables to axes (in %),

variance of eigenvalues weighted by contributions

Vk(λ)and variance

ratios

Vk(λ)/V (λ).

123456

λ4.103 0.619 0.512 0.357 0.270 0.139

Vk(λ)

Vk(λ)/V (λ)

k1210027702.62 1.34

k21910554202.33 1.20

k317 0 12 59 4 9 2.02 1.04

k416 6313214 01.81 0.93

k513 51 15 2 18 0 1.51 0.78

k613 41 42 0 3 1 1.40 0.72

V(λ)=1.95

Table 1: Correlations, linkage indexes LIk and ratios LIk/LI.

128 Durand, J.-L., Le Roux, B.

3. MULTIPLE CORRESPONDENCE ANALYSIS

We will now adopt the same approach for MCA.

3.1. BASIC PROPERTIES AND NOTATIONS

Let Idenote the set of nindividuals and Qthe set of categorical variables

(questions). The table analyzed by mca is an I×Qtable such that the

entry in cell (i, q)is the category of variable qchosen by individual i.The

set of categories of variable qis denoted by Kqand its cardinal by Kq; the

overall set of categories is denoted by Kand its cardinal by K.

The number of individuals who have chosen category kis denoted by

nk(with nk>0) and the corresponding relative frequency by fk=nk/n.

Multiple correspondence on I×K.Let us denote δIK =(δik)i∈I,k∈K

the multiple correspondence on I×Kdeﬁned by

δ

ik

=1

0

if individual ihas chosen category k

if not

Performing the mca of the I×Qtable is equivalent to proceeding to

Correspondence Analysis of the I×Ktable δIK (Benzécri, 1977; Greenacre,

1984). The solution is given by the diagonalization of the symmetric matrix

S=[skk]with skk=1

Q

nkk−nknk/n

√nknk/n (nkkis the number of individuals who

have chosen both categories kand k).

We denote Lthe set indexing the K−Qnonnull eigenvalues and (yk

)∈L

the principal coordinates of the category point k. The sum of eigenvalues

(λ)∈Lis equal to (K−Q)/Q, hence the mean is λ=1/Q.

Burt table and mean square contingency coeﬃcients. The Burt

table associated with δIK is the symmetric K×Ktable deﬁned by:

b

kk

=

i∈I

δ

ik

δ

ik

=

n

k

if k=k

0ifk=k

with k, k

∈K

q

n

kk

if k∈K

q

and k

∈K

q

with q=q

Denoting Φ2

qqthe mean square contingency coeﬃcient of the contin-

gency table crossing variables qand q, one has: Φ2

qq =Kq−1and for q=q,

Φ2

qq=

k∈Kq

k∈Kq

(fkk−fkfk)2

fkfk.

The Φ2of the Burt table, denoted Φ2

Burt, is the average of the Φ2of

the Q2subtables of the Burt table. Denoting Φ2the mean of the Φ2of the

.

.

Linkage Index of Variables and Its Relationship with Variance of Eigenvalues … 129

Q(Q−1) non–diagonal subtables, one has:

Φ

2

Burt

=

1

Q

2

q∈Q

q

∈Q

Φ

2

qq

=

1

Q

K−Q

Q

+

1

Q

q∈Q

q

∈Q

q

=q

Φ

2

qq

=

1

Q

K−Q

Q

+

Q−1

Q

Φ

2

Contributions of categories and of variables. The squared distance

between the category point kand the mean point of the cloud is equal to

1−f

k

f

k

=

∈L

(y

k

)

2

.

The contribution of category kto axis , denoted Ctr

k

, is equal to

f

k

Q

(y

k

)

2

λ

. We have the two following properties:

∀∈L,

k∈K

Ctr

k

=1 and ∀k∈K,

∈L

Ctr

k

=1−f

k

The ﬁrst property follows the deﬁnition of contribution to axes. The second

one can be proven as follows: The -th unit eigenvector of Sassociated

with nonnull eigenvalue λ

is (c

k

)

k∈K

with c

k

=f

k

/Q(y

k

/√λ

)and the

Qones associated with null eigenvalue are (c

kq

)

q∈Q

with c

kq

=√f

k

for

k∈K

q

and 0 for k/∈K

q

. Hence

∈L

c

2

k

+

q∈Q

c

2

kq

=

f

k

Q

∈L

(y

k

)

2

λ

+f

k

=1.

By deﬁnition, the contribution of a variable to axis is the sum of

the contributions of its categories: Ctr

q

=

k∈K

q

Ctr

k

, and we have the two

following properties:

∀∈L,

q∈Q

Ctr

q

=1 and ∀q∈Q,

∈L

Ctr

q

=K

q

−1

Burt cloud. The mean point of the subcloud of individuals who have

chosen category kis called category mean point. Its proﬁle (obtained from

the Burt table) is equal to

1

Q

(f

kk

/f

k

)

k

∈K

; its squared distance to the

mean point is equal to

k

∈K

1

Q

2

(f

kk

/f

k

−f

k

)

2

f

k

/Q

=

1

Qf

k

k

∈K

(f

kk

−f

k

f

k

)

2

f

k

f

k

.Let-

ting φ

2

q

(k)=

k

∈K

q

(f

kk

−f

k

f

k

)

2

f

k

f

k

if k∈K

q

and q

=q, the squared distance

writes:

1

Qf

k

(1 −f

k

)+

1

Qf

k

q

∈Q

q

=q

φ

2

q

(k),with Φ

2

qq

=

k∈K

q

φ

2

q

(k)(2)

.

.

.

.

130 Durand, J.-L., Le Roux, B.

The Kcategory mean points deﬁne the Burt cloud (see Le Roux and

Rouanet, 2004, pp. 199-200). The principal coordinates of the category

mean point kon axis are equal to y

k

/√λ

, hence its squared distance to

the mean point is also equal to

∈L

(y

k

)

2

/λ

.

The eigenvalues verify the following property:

L

=1

λ

2

=Φ

2

Burt

.

3.2. VARIANCE OF EIGENVALUES

Theorem 3.1. The variance of eigenvalues, denoted V(λ), is such that:

V(λ)=

1

K−Q

Q−1

Q

Φ

2

Proof.

1

K−Q

∈L

(λ

−λ)

2

=

1

K−Q

Φ

2

Burt

−λ

2

with λ=1/Q. Hence the variance

is V(λ)=

1

K−Q

1

Q

K−Q

Q

+

Q−1

Q

Φ

2

−

1

Q

2.

This expression of the variance of eigenvalues leads us to consider the

mean of the Φ

2

between one categorical variable and the others, that is, it

leads us to the following deﬁnition.

Deﬁnition 3.1 (Linkage index of categorical variable).The linkage index

of categorical variable q,denotedLI

q

, is such that:

LI

q

=

1

K

q

−1

1

Q−1

q

∈Q,q

=q

Φ

2

qq

Note that the linkage index of a categorical variable is between zero

and one, since Φ

2

qq

≤K

q

−1.

Property 1 (Mean linkage index).The mean linkage index of the Qcate-

gorical variables weighted by (K

q

−1)

q∈Q

,denotedLI, is such that:

LI =

Q

2

Q−1

V(λ)=Φ

2

/(

K−Q

Q

)

Proof.

q∈Q

(K

q

−1) = K−Q, hence the weighted mean of linkage indexes is

1

K−Q

q∈Q

(K

q

−1)LI

q

=

1

K−Q

q∈Q

1

Q−1

q

∈Q,q

=q

Φ

2

qq

=

Q

2

Q−1

V(λ).

.

.

.

Linkage Index of Variables and Its Relationship with Variance of Eigenvalues … 131

Deﬁnition 3.2 (Linkage index of category).Given a category kof the

categorical variable q, the linkage index of category k,denotedLI

k

, is deﬁned

as follows:

LI

k

=

1

1−f

k

×

1

Q−1

q

∈Q,q

=q

φ

2

q

(k)

with for q

=q,φ

2

q

(k)=

k

∈K

q

(f

kk

−f

k

f

k

)

2

f

k

f

k

.

One deduces from

k∈K

q

φ

2

q

(k)=Φ

2

qq

that the linkage index of variable

qis equal to the mean of the linkage indexes of its categories weighted by

(1 −f

k

)

k∈K

q

:LI

q

=

1

K

q

−1

k∈K

q

(1 −f

k

)LI

k

.

3.3. CATEGORIES, CATEGORICAL VARIABLES AND EIGENVAL-

UES

In order to explain the link between categorical variables or categories and

eigenvalues, we will now express the linkage indexes in terms of eigenvalues

weighted by contributions to axes.

Lemma 3.1. The mean of eigenvalues weighted by the contributions of

category kto axes, denoted

λ

k

,isequalto1/Q.

Proof.

λ

k

=

∈L

Ctr

k

λ

/(

∈L

Ctr

k

)=

1

1−f

k

f

k

Q

∈L

(y

k

)

2

=

1

1−f

k

f

k

Q

1−f

k

f

k

=

1

Q

.

Lemma 3.2. The variance of eigenvalues weighted by the contributions of

category kof variable qto axes is denoted

V

k

(λ)and called k–variance of

eigenvalues; it is equal to

1

Q

2

(1−f

k

)

q

∈Q,q

=q

φ

2

q

(k).

Proof. The weighted sum of the squared eigenvalues is equal to

∈L

f

k

Q

(y

k

)

2

λ

=

∈L

f

k

Q

y

k

√λ

2

=

f

k

Q

1

Qf

k

(1 −f

k

)+

1

Qf

k

q

=q

φ

2

q

(k)(Equation 2). Hence

V

k

(λ)=

1

Q

2

(1 −f

k

)+

q

=q

φ

2

q

(k)/(1 −f

k

)−

1

Q

2

=

1

Q

2

(1−f

k

)

q

=q

φ

2

q

(k).

Theorem 3.2. The linkage index of category kis proportional to the vari-

ance of eigenvalues weighted by contributions of kto axes.

LI

k

=

Q

2

Q−1

V

k

(λ)

.

132 Durand, J.-L., Le Roux, B.

Proof.

V

k

(λ)=

Q−1

Q

2

1

(Q−1)(1−f

k

)

q

=q

φ

2

q

(k)=

K

q

−1

Q

2LI

k

.

From Property 1 and Property 3.2 we deduce that:

Corollary 3.2.1. The ratio of the linkage index of category kto the mean

linkage index is equal to the ratio of the k–variance of eigenvalues to the

variance of eigenvalues:

LI

k

LI =

V

k

(λ)

V(λ)

The properties about categorical variables follows the property of aver-

age of linkage indexes of categories (LI

q

=

1

K

q

−1

k∈K

q

(1 −f

k

)LI

k

) and of the

property of sum of contributions (Ctr

q

=

k∈K

q

Ctr

k

). We denote

V

q

(λ)the

q–variance of eigenvalues (variance of eigenvalues weighted by the contribu-

tions of variable q):

V

q

(λ)=

1

K

q

−1

∈L

Ctr

q

(λ

−

1

Q

)

2

. One has the following

property: LI

q

=

Q

2

Q−1

V

q

(λ). Hence:

LI

q

LI =

V

q

(λ)

V(λ)

Comments

1) The more the linkage index of a category (or a categorical variable) is

superior to the mean, the more this category (or this variable) contributes

to extreme axes (ﬁrst and last axes).

2) The more the linkage index of a category (or a categorical variable)

is inferior to the mean, the more this category (or this variable) contributes

to central axes (axes with variance near 1/Q).

3.4. APPLICATION TO BURT’SDATA

Burt’s data (Table 3), reproduced from Burt (1950, p.171), gives, for 100

individuals (men living in Liverpool), the observed response patterns and

their absolute frequencies for four attributes (categorical variables), that

is, AHair (a1:fair,a2:red,a3:dark ), BEyes (b1:light,b2:mixed,b3:

brown), CHead (c1:narrow,c2:wide), DStature (d1:tall,d2:short).

As we can see in Table 4, the categories having the highest linkage in-

dexes are the category b1(light)ofEyes and the two categories of Stature.

.

.

Linkage Index of Variables and Its Relationship with Variance of Eigenvalues … 133

Tabl e 3 : Observed response patterns with their absolute frequencies.

Abs.freq

a1b1c1d18

a1b1c1d24

a1b1c2d12

a1b2c1d11

a1b2c1d21

a1b2c2d12

a1b2c2d22

a1b3c2d22

Abs.freq

a2b1c1d16

a2b1c2d12

a2b2c1d12

a2b2c1d21

a2b2c2d22

a2b3c1d22

a1a2a3b1b2b3c1c2d1d2

22 15 63 33 36 31 69 31 43 57

Abs.freq

a3b1c1d19

a3b1c2d12

a3b2c1d13

a3b2c1d212

a3b2c2d12

a3b2c2d28

a3b3c1d11

a3b3c1d219

a3b3c2d13

a3b3c2d24

Tabl e 4 : Eigenvalues λ

, linkage index LI

k

,ratioLI

k

/LI and contribu-

tions of categories to axes (in %).

123456

λ

0.489 0.299 0.254 0.206 0.179 0.073

LI

k

LI

k

/LI

a1fair .054 0.63 95361235

a2red .027 0.31 60554200

a3dark .099 1.14 9100252

b1light .211 2.42 26210136

b2mixed .037 0.43 326 524 0 5

b3brown .093 1.07 12 16 2 23 3 14

c1narrow .020 0.23 015 014 0 1

c2wide .020 0.23 134 031 1 2

d1tall .170 1.95 20 0 0 2 16 20

d2short .170 1.95 15 0 0 1 12 15

LI = 0.087

Tabl e 5 : Eigenvalues λ

, linkage index LI

q

,ratioLI

q

/LI and contribu-

tions of categorical variables to axes (in %).

123456

λ

0.489 0.299 0.254 0.206 0.179 0.073

LI

q

LI

q

/LI

q1Hair .051 0.59 23 6 91 5 67 7

q2Eyes .115 1.32 41 45 8 47 4 55

q3Head .020 0.23 14904513

q4Stature .170 1.95 34 0 0 3 28 35

LI = 0.087

134 Durand, J.-L., Le Roux, B.

Their linkage indexes are about twice the mean (LI

b1

/LI = 2.42 and LI

d1

/LI =

LI

d2

/LI = 1.95). These three categories contribute heavily to “extreme”

axes 1and 6(together, they account for 61% of axis 1 and 71% of

axis 2). In contrast, both categories of Head and the category a2(red )

of Hair have the smallest linkage indexes, less than a third of the mean

(LI

c1

/LI = LI

c2

/LI = 0.23 and LI

a2

/LI = 0.31). The contributions of these

three categories to both “extreme” axes (1and 6) are very small (7% and

3%, respectively) but they contribute heavily to “central” axes 2,3and

4(49%, 55% and 50%, respectively).

In Table 5, we see that Head has a very small linkage index; this variable

does not contribute to the ﬁrst axis (neither to the 5th and the 6th axes).

4. CONCLUSION

In this paper, we emphasize that the higher the mean of the linkage indexes

of (numerical or categorical) variables, the higher the variance of eigenval-

ues, that is, the larger the departure of clouds from sphericity.

In addition, further analysis shows that the more the linkage index of

a variable is superior to the mean, the more this variable contributes to

extreme axes (ﬁrst and last axes); otherwise this variable contributes to

central axes (whose variances are close to the mean). So, if the range of

linkage indexes of variables is large, one can predict that the variables with

the greatest linkage indexes will play a preponderant role in the interpreta-

tion of ﬁrst axes. Then, if we decide to reduce the number of active variables

in the analysis, linkage indexes will be a useful tool: if a variable with a

weak linkage index is discarded, the proportion of variance associated with

the ﬁrst axes will increase and the interpretation will remain the same.

REFERENCES

Benzécri, J.-P. (1977). Sur l’analyse des tableaux binaires associés à une correspondance multiple.

Les cahiers de l’analyse des données, 2(1): 55–71, from a mimeographed note of 1972.

Burt, C. (1950). The factorial analysis of qualitative data. British Journal of Statistical Psychology,

3: 166–185.

Durand, J.-L. (1998). Taux de dispersion des valeurs propres en ACP, AC et ACM. Mathématiques

Informatique et Sciences humaines, 144: 15–28.

Greenacre, M. (1984). Theory and Applications of Correspondence Analysis. London: Academic

press.

Karlis, D., Saporta, G. and Spinakis, A. (2003). A simple rule for the selection of principal

components. Communications in Statistics-Theory and Methods, 32(3): 643–666.

Linkage Index of Variables and Its Relationship with Variance of Eigenvalues … 135

Le Roux, B. and Rouanet, H. (2004). Geometric Data Analysis. From Correspondence Analysis to

Structured Data Analysis. Dordrecht: Kluwer.

Morrison, D. (1976). Multivariate Statistical Methods. New York: McGraw-Hill Publ. Co.

Pavlicev, M ., Cheverud, J. and Wagner, G . (2009). M easuring morphological integration using

eigenvalue variance. Evolutionary Biology, 36(1): 157–170.

Saporta, G. (2003). A control chart approach to select eigenvalues in principal component and

correspondence analysis. 54th Session of the International Statistical Institute-Berlin.

Spearman, C. (1904). ‘General intelligence’, objectively determined and measured. American

Journal of Psychology, 15: 201–292.