Multiple Correspondence Analysisfactominer.free.fr/course/doc/MCA_course_slides.pdf · Data -...

Data - issues Studying the individuals Studying the categories Interpretation aids

Multiple Correspondence Analysis

François Husson

Applied Mathematics Department - AGROCAMPUS OUEST

[email protected]

1 / 38


Plan

1 Data - issues

2 Studying the individuals

3 Studying the categories

4 Interpretation aids

2 / 38


The data

21

Categorical variables

I

1

i

j

Individuals

1 J

vij

I individualsJ qualitative variables

vij : category of the j-th variablepossessed by the i-th individual

Example : survey where I peoplereply to J multiple-choice questions

3 / 38


The data

17

i’ 0 1 0 0 0

Kj = 5

I

1

i

j

Ind

ivid

ua

ls

1 J

i’

I

1

i

j

Ind

ivid

ua

ls

1 J

yik

Indicator Matrix

1 k Kj K Σ

J

IJIpk

I

Categorical variablesCategories

vij

married

4 / 38


Goals

1 Studying the individualsOne individual = one row of the CDT = set of categoriesSimilarity of individuals – Inter-individual variabilityPrincipal axes of the inter-individual variability

(in relation to the categories)2 Studying the variables

Links between qualitative variables(in relation to the categories)

Visualization of the set of associations between categoriesSynthetic variables

(quantitative indicators based on the qualitative variables)

⇒ Similar problem to PCA

5 / 38


Leisure activity data

• Extract from 2003 INSEE survey on identity construction,called the “history of life” survey

• 8403 individuals• 2 sorts of variables :

• Which of the following leisure activities do you practiceregularly : Reading, Listening to music, Cinema, Shows,Exhibitions, Computer, Sport, Walking, Travel, Playing amusical instrument, Collecting, Voluntary work, Homeimprovement, Gardening, Knitting, Cooking, Fishing, Numberof hours of TV per day on average

• supplementary variables (4 questions) : sex, gender, profession,marital status

6 / 38



Hobbies Number Sex Female 4616

Listening music 5947 Male 3787

Reading 5646 Age [15,25] 857

Walking 4175 (25,35] 1302

Cooking 3686 (35,45] 1646

Mechanic 3539 (45,55] 1837

Travelling 3363 (55,65] 1257

Cinema 3359 (65,75] 937

Gardening 3356 (75,85] 482

Computer 3158 (85,100] 85

Sport 3095 Marital Divorcee 792

Exhibition 2595 status Married 4333

Show 2425 Remarried 404

Playing music 1460 Single 2140

Knitting 1413 Widower 734

Volunteering 1285 Profession

foreman 735Fishing 945

management 1052Collecting 862

employee 2552

Number of hours watching TV 0 1017

unskilled worker 792

1 1223

manual labourer 1161

2 2156

technician 401

3 1775 other 212

4 2232 No answer 1498

Sociodemographic

variablesHobbies

7 / 38



Respondent

1

1

8403

18 1 4

Hobbies Sociodemographic

= yes

or no

MCA 1 : active = leisure activity, then use supplementary data forinterpretation• 1 individual = vector of leisure activities• Principal axes of variability of leisure vectors• Links between these axes and the supplementary variables

MCA 2 : active = supplementary variables, leisure activities assupplementary informationMCA 3 : active = BOTH 8 / 38


Transforming the complete disjunctive table

An individual’s weight is1I

yik = 1 if the i-th individual is in k-th category of the j-th variable(for each pk)

= 0 otherwise

Idea : xik = yik/pk

∑Ii=1 xikI

=1I

∑Ii=1 yikpk

=1I

I × pkpk

= 1

Centering : xik = yik/pk − 1

9 / 38


Plan

1 Data - issues

2 Studying the individuals

3 Studying the categories

4 Interpretation aids

10 / 38


Point cloud of individuals

category k

IN

)',( iid

kix '

KRI

iM

OGI =

'iM (weight = 1/I)

(weight = )Jpk

ikx

1

i

I

k1 K

Ind

ivid

ua

lsCategories

Indicator matrix

ikx

d2i,i ′ =K∑

k=1

pkJ

(xik − xi ′k)2 =K∑

k=1

pkJ

(yikpk− yi

′k

pk

)2=

1J

K∑k=1

1pk

(yik−yi ′k)2

• 2 individuals with same categories : distance = 0• 2 individuals with many shared categories : small distance• 2 individuals, only 1 with a rare category : large distance to indicate this• 2 individuals share rare category : small distance to indicate this shared

specificity

d(i ,GI )2 =

K∑k=1

pkJ(xik)

2 =K∑

k=1

pkJ

(yikpk− 1)2

=1J

K∑k=1

yijpk− 1

Inertia(NI ) =I∑

i=1

1Id2(i ,O)︸︷︷︸

inertia of i

=I∑

i=1

(1IJ

K∑k=1

yikpk− 1

I

)=

K

J− 1

11 / 38


Point cloud of individuals

category k

IN

)',( iid

kix '

KRI

iM

OGI =

'iM (weight = 1/I)

(weight = )Jpk

ikx

1

i

I

k1 K

Ind

ivid

ua

lsCategories

Indicator matrix

ikx

d2i,i ′ =K∑

k=1

pkJ

(xik − xi ′k)2 =K∑

k=1

pkJ

(yikpk− yi

′k

pk

)2=

1J

K∑k=1

1pk

(yik−yi ′k)2

d(i ,GI )2 =

K∑k=1

pkJ(xik)

2 =K∑

k=1

pkJ

(yikpk− 1)2

=1J

K∑k=1

yijpk− 1

Inertia(NI ) =I∑

i=1

1Id2(i ,O)︸︷︷︸

inertia of i

=I∑

i=1

(1IJ

K∑k=1

yikpk− 1

I

)=

K

J− 1

11 / 38


Building the point cloud of individuals

Getting factor axes, as usual, like for all factor analysis methods

Sequential construction : look for the axis maximizing the inertiaand orthogonal to previous axes

12 / 38



• Extract from 2003 INSEE survey on identity construction,called the “history of life” survey

• 8403 individuals• 2 sorts of variables :

• Which of the following leisure activities do you practiceregularly : Reading, Listening to music, Cinema, Shows,Exhibitions, Computer, Sport, Walking, Travel, Playing amusical instrument, Collecting, Voluntary work, Homeimprovement, Gardening, Knitting, Cooking, Fishing, Numberof hours of TV per day on average

• supplementary variables (4 questions) : sex, gender, profession,marital status

13 / 38


Diagram showing the inertia

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Eigenvalues

Dimension

Eig

enva

lues

0.00

0.05

0.10

0.15

14 / 38


Representation of the point cloud of individuals

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

Dim 1 (16.95%)

Dim

2 (6

.91%

)

15 / 38


Representation of the point cloud of individuals

What kind of pattern might we see ?

The Guttman effect

16 / 38


Individuals shown in terms of the gardening variable

Idea : use the catego-ries and variables tointerpret the plot of theindividuals

Put a category atthe barycenter of theindividuals in it

●

−1.0 −0.5 0.0 0.5 1.0

−1.

0−

0.5

0.0

0.5

1.0

Dim 1 (16.95%)

Dim

2 (

6.91

%)

Gardening_0Gardening_1

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Multiple Correspondence Analysisfactominer.free.fr/course/doc/MCA_course_slides.pdf · Data -...

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Transcript of Multiple Correspondence Analysisfactominer.free.fr/course/doc/MCA_course_slides.pdf · Data -...