Multiple Correspondence Analysis 1 Supplementary points Z1...

Multiple categorical variablesMultiple Correspondence Analysis

Supplementary pointsMissing and “middle” responses

Supplementary points

1

J1

K

1 J 1 KBurt matrix

B

Suppose J substantive categories and Kdemographic groups

1 J 1 K

Z1 Z2

Z1TZ1

Z2TZ1

Z1TZ2

Z2TZ2

Supplementary points

1

J1

K

1 J 1 KBurt matrix

BSuppose Jsubstantive

categories andK demographic

groups

Indicatormatrix ofindividual

respondent(case) data

Z2TZ1

Z1

shows each case as a point, at the average of his/her responses

shows eachdemographic category, at the average of thecases in this category

Z1TZ1

CA of B (or adjusted): standard coordinatesthe same

Data set “women94”Substantive variables: Do you strongly agree/ agree/ neither…nor…/ disagree/ strongly disagree to these statements…A: a working mother can establish a warm relationship with her childB: a pre-school child suffers if his or her mother worksC: when a woman works the family life suffersD: what women really want is a home and kidsE: running a household is just as satisfying as a paid jobF: work is best for a woman’s independenceG: a man’s job is to work; a woman’s job is the householdH: working women should get paid maternity leaveDemographic variablesg: gender (1=male, 2=female)m: marital status (1=married/living as married, 2=widowed, 3=divorced, 4=separated, but married, 5=single, never married)e: education (0=no formal education, 1=lowest education, 2= abovelowest education, 3=higher secondary completed, 4=above higher secondary level, below full university, 5=university degree completeda: age (1=16-25 years, 2= 26-35, 3=36-45, 4=46-55, 5=56-65, 6=66 and older) Sample: Spanish sample (year 2002); N=2471 (including missing values)

ISSP 1994 survey on Family and ChangingGender Roles

A [+] A working mother can establish just as warm and secure a relationship with her children as a mother who does not work

B [–] A pre-school child is likely to suffer if his or her mother works C [–] All in all, family life suffers when the woman has a full-time jobD [–] A job is all right, but what most women really want is a home and children E [?] Being a housewife is just as fulfilling as working for payF [+] Having a job is the best way for a woman to be an independent personG [?] Most women have to work these days to support their familiesH [+] Both the man and woman should contribute to the household incomeI [–] A man’s job is to earn money; a woman’s job is to look after the home and

familyJ [?] It is not good if the man stays at home and cares for the children and the

woman goes out to workK [?] Family life often suffers because men concentrate too much on their work

24 countries (N=33,590)

+ in favour of working women – against women working ? not clear

Data – middle alternative

Common to both surveys is the measurement scale, a 5-point bipolar scale:

strongly agree neither agree disagree stronglyagree nor disagree disagree

We are particularly interested in the middle alternative, how itassociates

– with other middle alternatives

– with other response categories

– with the demographic covariates

Of course, there are also missing values, but since we will analyse the data at the nominal level, a missing value is just an additional category

extreme categories

moderate categories

missing

Background & previous researchPresser & Schuman (1980)

The measurement of a middle position in attitude surveysThe Public Opinion Quarterly.Revised version in Schuman & Presser, Questions & Answers in Attitude Surveys. Experiments on Question Form, Wording, and Context. Sage, 1996

Use contingency tables and 2 tests in 5 split-ballot experiments.

They assess the consequences of offering/omitting “a logical middle position, for example whether one is liberal or conservative could be answered by ‘middle-of-the-road’ ”…. “Although there is a very slight decrease in the proportion of spontaneous ‘don't know’ responses when the middle alternative is offered, almost all the change in the middle position comes from a decline in the polar positions.”

Background & previous research

Andrich, de Jong & Sheridan (1997)

Diagnostic opportunities with the Rasch model for ordered response categories.In: Applications of latent trait and latent class models in the social sciences

Use Rasch modelling.

“…the middle category designated as Neutral, Not Sure or Undecided in the Likert-style response format … should not be treated as an attitude more or less somewhere between a negativeand a positive attitude.”

Background & previous researchGonzález-Romá and Espejo (2003)

Testing the middle response categories «Not sure», «In between» and «?» in polytomous items.Psicotema

Use Bock’s Nominal Model to verify ordering.

They use different wordings for the middle category, and find that the ordering is verified only for the wording “in between”.

Hernández, Espejo and González-Romá (2006)

The functioning of central categories Middle Level and Sometimes in graded response scales: Does the label matter?Psicotema

Background & previous researchO’Muircheartaigh, Krosnick & Helic (2000)Middle alternatives, acquiescence, and the quality of questionnaire data.Working paper posted on web.

Use response counts and structural equation modelling

‘Approximately half the respondents … were asked to select an answer from among the following five alternatives: strongly agree, agree to some extent, neither agree nor disagree, disagree to some extent, strongly disagree. These responses were coded 5, 4, 3, 2, and 1,respectively. The other half of the respondents … were asked to select an answer from a set of four, omitting the “neither agree nor disagree”option. Answers were coded 5, 3.66, 2.33, and 1, respectively.’

…“We also found evidence of acquiescence response bias in answers to the agree/disagree items...”

MethodsWe use correspondence analysis (CA) and several of its variants to visualize and interpret the relative positions of the response categories, at a country level, a respondent level and demographic-subgroup level.

1. Simple CA provides maps of aggregated or count data, for example proportions of question responses for the set of countries.

2. Subset correspondence analysis permits focusing on a particular set of response categories so that we can eliminate missing responses, for example, or restrict our attention to specific categories such as the middle responses.

3. Multiple correspondence analysis (MCA) and its refinement joint correspondence analysis (JCA) provide maps of individual-level data, concentrating on the two-way relationships between the questions. In this respect MCA functions like an exploratory factor analysis for categorical data (on nominal scales). Usually we are not interested in individual case points but in mean points for the demographic and other external variables (e.g., age, level of interest…)

4. Canonical correspondence analysis (CCA) focuses on (or partials out) external variables – thus we can explore variation in the responses that is focused on interest, for example, or eliminate aquiescence effects.

Question B: A pre-school child is likely to suffer if his or her mother works

B-agree B-MX B-disagreeAU 0.497 0.149 0.354DW 0.684 0.141 0.175DE 0.325 0.186 0.490GB 0.377 0.215 0.408NI 0.382 0.187 0.431US 0.406 0.142 0.452A 0.720 0.106 0.174H 0.727 0.160 0.113I 0.671 0.152 0.177IR 0.472 0.120 0.407NL 0.439 0.225 0.336N 0.359 0.218 0.424S 0.272 0.267 0.461CZ 0.487 0.197 0.315SL 0.584 0.177 0.238PL 0.664 0.115 0.220BG 0.682 0.178 0.140RU 0.717 0.130 0.153NZ 0.498 0.182 0.320CD 0.306 0.183 0.511RP 0.564 0.144 0.292IL 0.434 0.193 0.373J 0.373 0.238 0.389E 0.516 0.137 0.346ave 0.514 0.171 0.316 Ternary coordinates: Euclidean distance

B: A pre-school child is likely to suffer if his or her mother works


B-agree B-MX B-disagreeAU 0.497 0.149 0.354DW 0.684 0.141 0.175DE 0.325 0.186 0.490GB 0.377 0.215 0.408NI 0.382 0.187 0.431US 0.406 0.142 0.452A 0.720 0.106 0.174H 0.727 0.160 0.113I 0.671 0.152 0.177IR 0.472 0.120 0.407NL 0.439 0.225 0.336N 0.359 0.218 0.424S 0.272 0.267 0.461CZ 0.487 0.197 0.315SL 0.584 0.177 0.238PL 0.664 0.115 0.220BG 0.682 0.178 0.140RU 0.717 0.130 0.153NZ 0.498 0.182 0.320CD 0.306 0.183 0.511RP 0.564 0.144 0.292IL 0.434 0.193 0.373J 0.373 0.238 0.389E 0.516 0.137 0.346ave 0.514 0.171 0.316 Stretched ternary coordinates: chi2 distance

B: A pre-school child is likely to suffer if his or her mother works Animation

achieved bysaving 101

frames in R of the ternary

diagram as themetric

smoothlychanges from

equal weightedEuclidean

distance to differentially

weighted chi-square

distance, thenframes savedinto a GIF file


From Euclidean to chi2 distance(regular to irregular simplex)

B-agree B-M B-disagree B-XAU 0.497 0.140 0.354 0.009DW 0.684 0.104 0.175 0.037DE 0.325 0.150 0.490 0.036GB 0.377 0.176 0.408 0.040NI 0.382 0.151 0.431 0.036US 0.406 0.121 0.452 0.021A 0.720 0.088 0.174 0.018H 0.727 0.146 0.113 0.014I 0.671 0.138 0.177 0.015IR 0.472 0.082 0.407 0.038NL 0.439 0.197 0.336 0.028N 0.359 0.179 0.424 0.039S 0.272 0.220 0.461 0.047CZ 0.487 0.178 0.315 0.020SL 0.584 0.144 0.238 0.033PL 0.664 0.068 0.220 0.047BG 0.682 0.067 0.140 0.110RU 0.717 0.086 0.153 0.044NZ 0.498 0.157 0.320 0.026CD 0.306 0.160 0.511 0.022RP 0.564 0.143 0.292 0.001IL 0.434 0.174 0.373 0.019J 0.373 0.199 0.389 0.039E 0.516 0.090 0.346 0.047ave 0.514 0.138 0.316 0.033


From 3 to 4 categories…

Middle response (M) and missing

response (X)separated

Rotation in three

dimensions of the country

profiles withina tetrahedron,

starting and ending withmiddle (M)

and missing(X) categories lined up (i.e.,

the two-dimensional

map seenpreviously.

Chi2 distanceevens out thecontributions

by thecategories


irregular simplex in higher dimensions

principal axes of CA

E

JILRP

CD NZRU

BG

PL

SLCZ

S N

NL

IR

I HAUS

NIGBDEDW

AU

B-disagree

B-M

B-agree

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1

0.0875 (80.4 %)

0.0142 (13.0 %)

B-X

Two-dimensionalCA map, with“asymmetricscaling”, i.e.

• rows (countries) in principal coordinates as theprojections of theprofiles

• columns(responsecategories) in standardcoordinates as theprojections of thecorners of thesimplex

93.4% inertiaexplained

1994: response proportions, 24 x 66 table

e.g., Spain14.1% strongly agree to Qu.A

(average across countries: 24.9%)7.8% strongly agree to Qu.K

(average across countries: 13.6%)

Question A“A working mother can establish just as warm and secure a relationship with her

children as a mother who does not work”

A1 A2 AM A4 A5 AXAU 0.183 0.349 0.092 0.282 0.085 0.009DW 0.369 0.356 0.040 0.158 0.040 0.037DE 0.627 0.275 0.017 0.050 0.008 0.022GB 0.176 0.443 0.120 0.188 0.043 0.030NI 0.155 0.468 0.085 0.201 0.054 0.037US 0.289 0.409 0.050 0.189 0.048 0.015A 0.524 0.230 0.033 0.150 0.041 0.021H 0.327 0.197 0.194 0.161 0.107 0.013I 0.215 0.409 0.100 0.179 0.094 0.003IR 0.183 0.428 0.057 0.215 0.100 0.017NL 0.208 0.490 0.098 0.155 0.026 0.023N 0.118 0.415 0.129 0.254 0.052 0.032S 0.222 0.432 0.131 0.152 0.034 0.030CZ 0.212 0.261 0.096 0.263 0.157 0.012SL 0.169 0.430 0.077 0.268 0.038 0.018PL 0.173 0.319 0.065 0.327 0.074 0.041BG 0.293 0.218 0.071 0.139 0.207 0.072RU 0.250 0.406 0.069 0.185 0.039 0.050NZ 0.134 0.402 0.093 0.285 0.059 0.028CD 0.308 0.412 0.069 0.160 0.035 0.015RP 0.054 0.568 0.145 0.213 0.018 0.001IL 0.232 0.429 0.092 0.166 0.064 0.016J 0.515 0.158 0.147 0.073 0.077 0.030E 0.141 0.406 0.036 0.337 0.053 0.027

K1 K2 KM K4 K5 KX0.127 0.608 0.138 0.112 0.013 0.0030.117 0.488 0.148 0.142 0.028 0.0770.099 0.427 0.180 0.180 0.042 0.0720.058 0.544 0.180 0.172 0.021 0.0250.068 0.495 0.181 0.212 0.005 0.0400.084 0.480 0.198 0.168 0.028 0.0410.259 0.418 0.126 0.127 0.039 0.0320.384 0.349 0.171 0.055 0.025 0.0170.117 0.542 0.154 0.145 0.028 0.0140.135 0.596 0.071 0.132 0.033 0.0320.049 0.531 0.217 0.155 0.012 0.0360.090 0.570 0.179 0.107 0.012 0.0430.067 0.352 0.286 0.189 0.041 0.0650.196 0.386 0.220 0.140 0.040 0.0190.098 0.523 0.181 0.130 0.019 0.0480.098 0.519 0.122 0.147 0.023 0.0910.477 0.253 0.083 0.045 0.036 0.1050.111 0.295 0.243 0.224 0.050 0.0770.107 0.606 0.138 0.119 0.011 0.0180.099 0.476 0.210 0.163 0.031 0.0200.046 0.468 0.213 0.251 0.021 0.0020.200 0.494 0.148 0.103 0.023 0.0320.245 0.264 0.138 0.081 0.235 0.0370.078 0.510 0.088 0.226 0.021 0.076

······

Question K“Family life often suffers because men concentrate too much on their work”······

“Symmetric” CA map of response proportions1 strongly

agree2 agreeM middle4 disagree5 strongly

disagreeX missing

23-dimensional

Total inertia= 0.2453

60.9% inertiaexplained in 2-d CA map

KX

K5

K4

KM

K2

K1

JX

J5

J4

JM

J2

J1IX

I5

I4IM

I2

I1

HX

H5

H4

HM

H2

H1GX

G5

G4

GM

G2

G1

FX

F5

F4

FM

F2

F1EX

E5

E4

EM

E2

E1DX

D5

D4 DM

D2

D1CX

C5

C4 CM

C2

C1

CX

B5

B4

BM

B2B1

AX

A5

A4

AM

A2

A1

E

J

IL

RP

CD

NZ

RU

BG

PL

SL

CZ

SNNL

IR

I

H

AUS

NI

GB

DE

DWAU

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0.1021 (41.6%)

0.0472 (19.3%)

extreme responses

moderate& middle

responses

traditional

liberal

traditional

liberal

Qu.I“Man’s job is to earnmoney; woman’s jobis to lookafter home and family”

Decomposition of inertia across categories

Inertia contributions

12M45X

Missings (X)

Middles (M)

Category Inertia1 0.09212 0.0400M 0.01974 0.03945 0.0418X 0.0123

total 0.2453

middles & missings account for 0.03200 of the total inertia of 0.2453, i.e. , 13.0%; “extreme” responses account for 0.1339, i.e. 54.6%.

MethodsWe use correspondence analysis (CA) and several of its variants to visualize and interpret data at a country level as well as respondent and demographic-subgroup level.


2. Subset correspondence analysis permits focusing on a particular set of response categories so that we can eliminate missing responses, for example, or restrict our attention to specific categories such as the middle responses (Greenacre & Pardo, SMR, 2006)



CA of proportions of response categories

KX

K5

K4

KM

K2

K1

JX

J5

J4

JM

J2

J1IX

I5

I4IM

I2

I1

HX

H5

H4

HM

H2

H1GX

G5

G4

GM

G2

G1

FX

F5

F4

FM

F2

F1EX

E5

E4

EM

E2

E1DX

D5

D4 DM

D2

D1CX

C5

C4 CM

C2

C1

CX

B5

B4

BM

B2B1

AX

A5

A4

AM

A2

A1

E

J

IL

RP

CD

NZ

RU

BG

PL

SL

CZ

SNNL

IR

I

H

AUS

NI

GB

DE

DWAU

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0.1021 (41.6%)

0.0472 (19.3%)

extreme responses

moderate & middle

responses

missings

middles

Animation achieved by changing weightThe animations which link methods are achieved by either reducing the mass of certain points or transferring mass between points.

For example, to show the missings and all moderate responses in this example) we multiply these dummy variables by where starts at 1 (i.e., the regular MCA) and decreases in small steps of 0.01 until 0 (i.e., the subset MCA). At each step a hybrid of a regular MCA and a subset MCA is performed, maintaining the margins of the table constant.

= 1, 0.99, 0.98, 0.97, …………, 0.04, 0.03, 0.02, 0.01, 0

…………

CA tosubset

CAData:

WOMEN WORKING,

stackedfrequencies,

N=33590

Method:Contributionof categoriesnot in subset reduced by

factor , from1 (CA) to

limiting case of 0 (subset CA), always

usingoriginal

masses forcentring andweighting)

Total inertia

Principal inertiasin 2-d soln.

CA tosubset

CAData:

WOMEN WORKING,

stackedfrequencies,

N=33590




usingoriginal


Total inertia

Principal inertiasin 2-d soln.

Subset CA of proportions of middles and missings

KMJM

IM

HM

GM

FMEMDM

CMBMAM

KXJX IX

HXGX FXEX DX CXBX

AX

E

J

IL

RP

CDNZ

RU

BGPL

SLCZ

S

NNL

IRI

H

A

US NIGB

DE

DW

AU

-0.2

0

0.2

0.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

0.0144 (45.0%)

0.0065 (20.3%)

Of the (small – 13.0% – part of the) inertia of the M- and X-percentages, which is contained in a 22-dimensional space, 65.3% is explained in this map. The fact that all the M’s are together and all the X’s together and separate, does not reflect category associations at an individual level. For example, the proximity of Spain (E) and Russia (RU) means that their percentages of M- and X-responses are similar (less M’s than average, more X’s than average)






Looking at respondent-level dataTo investigate response behaviour at individual respondent level we pass from CA to multiple correspondence analysis (MCA). The classic definition of MCA is the CA of the data coded in an indicator matrix, i.e., as dummy variables, one variable for each response category

Dimensionality is 66 less the 11 linear restrictions on the columns, i.e. 55

A B C D E F G H I J K1 3 3 5 5 1 1 1 5 5 4

9 2 2 2 2 4 4 4 2 2 2...

Original responses (Q = 11)

A1 A2 AM A4 A5 AX B1 B2 BM ...1 0 0 0 0 0 0 0 1 ...

0 0 0 0 0 1 0 1 0 ......

Dummy variables (J = 66)

MCA of all response categories1 strongly

agree2 agreeM middle4 disagree5 strongly

disagreeX missing

Total inertia= 0.6626


Adjusted value:


KX

K5

K4KMK2

K1

JX

J5

J4 JM

J2

J1

IX

I5

I4

IMI2

I1

HX

H5

H4

HM

H2

H1

GX

G5

G4GM

G2

G1

FX

F5

F4FM

F2

F1

EX

E5

E4 EM

E2

E1

DX

D5

D4 DM

D2

D1

CX

C5

C4 CM

C2

C1

BX

B5

B4BM

B2

B1

AX

A5

A4AMA2

A1

-2.5

-2

-1.5

-1

-0.5

0

0.5

-0.5 0 0.5 1

0.1213 (18.3%)

0.1071 (16.2%)

allmissings

allextremes

allmoderates& middles

Decomposition of inertia across categories in MCA

middles & missings account for 0.2366 of the total inertia of 0.6626, i.e. , 35.7% ; missings account for largest part of inertia

Inertia contributions

12M45X

Missings (X)

Middles (M)

Category Inertia1 0.12972 0.0827M 0.08814 0.09285 0.1207X 0.1485

total 0.6626

MCA tosubset MCAData:

WOMEN WORKING, Burt matrix,

N=33590




usingoriginal


Total inertia

Principal inertia in 2-d soln.

middles

Data: WOMEN WORKING,

stackedfrequencies,

N=33590

Using capackage in R.Also showingthe country

points in “symmetric”

scaling so theyare more

spread out in the

visualizationfor ease of

interpretation

Reference: Nenadić, O. & Greenacre, M.J. (2007). Correspondence analysis in R, with two- and three-dimensional graphics: the ca package. Journal of StatisticalSoftware 20(3). URL http://www.jstatsoft.org/v20/i03/.

Rotating the subset MCA solution(excluding missings)

What does a “perfect unidimensional model”look like in an MCA?

In first two dimensions:

Parabola: the “horseshoe”/ “arch”/

“Guttman” effect

In dimensions 1 and 3:

Cubic

In dimensions 1 and 4:

Quartic, etc....

Data that follow a perfect traditional-to-liberal scale were generated, reversing the scales of oppositely worded statements, and randomlyadding 3% missing responses. Here are different MCA maps of the data:

-1.0 -0.5 0.0 0.5 1.0 1.5

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0.5

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A1

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AM

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H9

I1

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IM I4

I5

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JM

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-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

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I1

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IM

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IX

A1

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AM

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AX

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HX

-2 -1 0 1 2

-3-2

-10

1

B1

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BX

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FM

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FX

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CM

C4C5 CX

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DM

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DX

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EM

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I1

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IM

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IX

A1

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AM

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H5HX

What does a “perfect unidimensional model”look like in an MCA?

Looking by country: Spanish and WestGerman data

• Previous MCA maps based on all 33590 respondents from 24 countries.

• We have already seen that there are inter-country differences in theiroverall levels of middle and missing responses.

• Our aim is to investigate how respondents use the middle responsesand if there any associations with demographic variables. To avoidinter-country differences in our results we concentrate (separately) on two countries: Spain (N = 2494) and West Germany (N = 2324) –as we shall see, they present contrasting results.

• We will also introduce the following demographic variables into ourstudy:

Gender (2 categories)

Age (6 categories)

Marital status (5 categories)

Education (7 categories)

(Education not available in Spanish 1994 sample)

MCA of Spanish “women working” data (N = 2494)

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.

A1

A2AMA4

A5

AX

B1

B2BMB4

B5BX

C1

C2CM

C4

C5CX

D1

D2DMD4

D5DX

E1

E2EME4

E5

EX

F1

F2FMF4

F5FX

G1

G2

GMG4

G5

GX

H1

H2

HMH4

H5

HX

I1

I2IMI4

I5

IX

J1

J2JMJ4

J5

JX

K1

K2KM

K4

K5KX

Missing categories

Extreme categories (stronglyagree and strongly disagree)

Moderate and middlecategories

Subset MCA of Spanish “women

working” data(N = 2494)

missing responsesexcluded from subset

middles amongst themoderate responses in

this two-dimensionalview

-4 -3 -2 -1 0 1

-20

24

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A1

A2

AM

A4

A5B1

B2

BM

B4

B5

C1

C2

CM

C4

C5

D1

D2

DM

D4

D5

E1

E2

EME4

E5

F1

F2FM

F4

F5

G1G2GM

G4

G5

H1

H2

HM

H4

H5

I1

I2

IM

I4

I5

J1

J2

JMJ4

J5

K1

K2

KM

K4

K5

AXBXCXDXEXFXGXHXIXJXKX

traditional

liberal

middles

moderate& middle

strong Rotating thesubset MCA

solution (subset excludingmissings)Data: WOMEN

WORKING, Spanish data, subset MCA,

N=2494

Method:Rotation about

2nd axis, recorded using

plot3d function in capackage for R

KX

K5K4KM

K2

K1

JX

J5J4

JM J2

J1

IX

I5 I4

IMI2

I1

HX

H5

H4HM

H2

H1

GX

G5

G4GM

G2

G1

FX

F5

F4FM

F2

F1

EX

E5E4

EME2

E1

DX

D5D4

DM D2

D1

CX

C5C4 CM

C2

C1

BX

B5B4BM

B2

B1

AX

A5

A4AM

A2

A1

-3

-2

-1

0

1

2

3

4

5

6

-2 -1 0 1 2 3 4

0.318 (33.1%)

0.288 (24.9%)

(B5) Stronglydisagree that pre-school child willsuffer becausemother works

LIBERAL

MCA of W.GERMANDATA

N=2324

TRADITIONAL

Missingcategories

(H5) Stronglydisagree that

man & womenshould both

contribute to household

income

-3

-2

-1

0

1

2

3

4

5

6

-2 -1 0 1 2 3 4

0.318 (33.1%)

0.288 (24.9%)

MCA tosubset MCA


W.German sample, N=2324

Method:Contribution of

categories not in subset reduced by

factor , from 1 (CA) to limiting case of 0 (subset CA), always

using original masses for centring

and weighting)

Total inertia

Principal inertia in 2-d soln.

SubsetMCA of W German“womenworking”data(N = 2324)missingresponsesexcluded fromsubset

K5

K4

KMK2

K1

J5

J4 JM

J2

J1

I5

I4

IMI2

I1

H5

H4HMH2

H1 G5

G4GMG2

G1

F5

F4FMF2

F1

E5

E4EM

E2

E1

D5

D4DM

D2

D1

C5

C4

CM

C2

C1

B5

B4

BM

B2

B1

A5

A4AM

A2

A1

-2

-1

0

1

2

3

4

5

-2 -1 0 1 2 3 4

traditional

liberal

“strongly disagree” that family lifeoften suffers because menconcentrate too much on their work


W.German data, subset MCA,

N=2324


2nd axis, recorded usingrgl.snapshotin rgl packageand plot3d.cain ca package

for R

Rotating thesubset MCA solution(excludingmissings)


W.German data, subset MCA,

N=2324


1st axis, recorded usingrgl.snapshotin rgl packageand plot3d.cain ca package

for R

Rotating thesubset MCA solution(excludingmissings)

Response sets in Spanish sampleLooking more closely at the individual Spanish data, we discover the following response sets (figures for W.Germany for comparison):

(WG)“strongly agree” to all questions – 4 respondents (2)“agree” to all questions – 20 respondents (6)“neither/nor” to all questions – 6 respondents (0)“disagree” to all questions – 2 respondents (0)“strongly disagree” to all questions – 0 (0)missing values for all questions – 18 respondents (6)

50 out of the 2494 respondents, i.e. 2% (½%)

The “middle” and “missing” response sets accentuate the association within these categoriesThe “strongly agree” and “agree” response sets (categories 1 and 2) to questions which have reverse orientations will tend to bring the opposite poles closer than they would otherwise have been.We now remove all these response sets, all the features previously seen are still there, just their ordering on the principal axes changes: e.g., the group of “missings” is now on the 3rd dimension and the “middles” on the 5th.

MCA ofSpanish data

withoutresponse

sets

Data:WOMEN

WORKING, Spanishsample, N=2444

(without 50 response

sets)

Dimensions 1 and 2

KX

K5

K4

KM

K2

K1

JX

J5

J4

JM

J2

J1

IX

I5

I4

IM

I2

I1HX

H5

H4

HM

H2

H1

GXG5

G4

GMG2

G1

FX

F5

F4

FMF2

F1

EX

E5

E4EM

E2

E1

DX

D5

D4

DM

D2

D1

CX

C5

C4

CM

C2

C1

BX

B5

B4

BM

B2

B1

AX A5

A4

AM

A2

A1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-1 -0.5 0 0.5 1 1.5 2 2.5

0.102 (41.0%)

0.059 (23.8%)

MCA ofSpanish data

withoutresponse

sets

Data:WOMEN


(without 50 response

sets)

Dimensions1, 3 and 5

Focusing the display on the number of middle responses

11 questions (A,B,C,D,E,F,G,H,I,J,K) from ISSP survey

Pass the weight smoothly from therespondents to the group centroids:

Xn

respon-dents

m0 m1 ... m6+

Z

A1 A2 AM A4 A5 B1 B2 BM B4 B5 ... K1 K2 KM K4 K5

0 0 1 0 0 0 1 0 0 0 ... 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 ... 0 1 0 0 0

0 1 ... 01 0 ... 0

Classifica-tion in terms of number of middles

XTZ

1/n1/n.......

1/n

r

0

0

1w )1(

)(1

n

g groups

# m

10

r1r2.rg

Count of middles

MCA MCA-Discriminant analysis (middle groups): Spain

MCA MCA-Discriminant

analysis(middlegroups):

W. Germany






Partialling out acquiescence effects

O’Muircheartaigh, Krosnick & Helic (2000)

‘We estimated a model in which all items were allowed to load on the same latent factor representing attitude toward science, plus a second latent factor intended to represent acquiescence. All items were constrained to load equally on this latter factor, an assumption required to identify the model. This is reasonable, because acquiescence is defined as a tendency to agree with any item regardless of its content, so it should account for the same amount of variance in responses to all the items. The acquiescence factor was constrained to be uncorrelated with the factor representing attitudes toward science, another assumption required in order to identify the model.’

Identifying or partialling out “acquiescence effects”11 questions (A,B,C,D,E,F,G,H,I,J,K) from

ISSP Family and Changing Role survey II (1994).

X

1 2 M 4 5 X

Z

A1 A2 AM A4 A5 AX B1 B2 BM B4 B5 BX ... K1 K2 KM K4 K5 KX

0 0 1 0 0 0 0 1 0 0 0 0 ... 0 0 0 0 1 01 0 0 0 0 0 0 0 0 0 1 0 ... 0 0 0 0 0 1

2 3 1 3 1 13 1 0 2 2 3

counts of numberof responses in each category

The counts in the matrix X are variables which quantify tendenciesto use the same response category. CCA can restrict the MCA solution to be linearly related to any subset of these externalvariables. This is a projection in the MCA space. We can alsopartial out acquiescence effects by looking in the space orthogonalto the restricted space (this is an alternative strategy to subset CA).

MCA

Data:WOMEN


-3 -2 -1 0 1

-3-2

-10

1

CA

2A1

A2AMA4

A5

AX

B1

B2BMB4

B5

BX

C1

C2

CMC4

C5CX

D1

D2DMD4

D5

DX

E1

E2EM

E4

E5

EX

F1

F2FMF4

F5

FX

G1

G2

GMG4

G5

GX

H1

H2

HMH4

H5

HX

I1

I2

IM

I4

I5

IX

J1

J2JMJ4

J5

JX

K1

K2

KMK4

K5KX

missings

CCAcounts of

missings as external variable which

restricts thesolution

Data:WOMEN


missings

0 1 2 3 4

-2-1

01

CCA1

CA

1

A1

A2AMA4

A5

AX

B1

B2

BMB4

B5

BX

C1

C2

CMC4

C5

CX

D1

D2

DMD4

D5

DX

E1

E2

EME4

E5

EX

F1

F2

FMF4

F5

FX

G1

G2

GM

G4

G5

GX

H1

H2

HMH4

H5

HX

I1

I2

IMI4

I5

IX

J1

J2

JMJ4

J5

JX

K1

K2

KM

K4

K5

KX

Partial CCAsolution

orthogonal toone that

restricts thesolution to be

linearlyrelated tomissingcounts

Data:WOMEN


missings

-3 -2 -1 0 1

-2-1

01

CA1

CA

2

A1

A2

AM

A4

A5

AX

B1

B2

BM

B4

B5

BX

C1

C2

CM

C4

C5

CX

D1

D2

DM

D4

D5

DX

E1

E2

EME4

E5

EX

F1

F2FM

F4

F5

FX

G1G2

GM

G4

G5

GX

H1

H2

HM

H4

H5

HX

I1

I2

IM

I4

I5

IX

J1

J2

JMJ4

J5

JX

K1

K2

KM

K4

K5

KX

CCAcounts ofmissings

and middlesas external

variable which

restricts thesolution

Data:WOMEN


missings

0 1 2 3 4

-2-1

01

CCA1

CC

A2

A1A2

AM

A4A5

AX

B1B2

BM

B4B5

BX

C1C2

CM

C4C5CX

D1D2

DM

D4D5

DX

E1E2

EM

E4E5EX

F1F2

FM

F4F5

FX

G1G2

GM

G4G5

GX

H1H2

HM

H4

H5

HXI1I2

IM

I4I5 IX

J1

J2

JM

J4J5

JXK1K2

KM

K4K5

KX

middles

Data:WOMEN


CCA solution

partiallingout the

middles andmissings

andrestricted to

theextremes

andmoderates

Partial CCAsolution orthogonal to one that restricts the solution to be

linearly related to all acquiescence effectsData: WOMEN WORKING,

Spanish sample, N=2494

A1

A2

AM

A4

A5

AX

B1

B2

BMB4

B5

BX

C1

C2CM

C4

C5

CX

D1

D2DM

D4

D5

DXE1

E2EME4

E5EX

F1

F2

FM

F4

F5

FXG1

G2

GM

G4

G5

GXH1

H2HM

H4

H5

HX

I1

I2

IMI4

I5

IX

J1

J2

JMJ4

J5JXK1K2KM

K4

K5

KX

CCA of Spanish “women working” data:categories

-4 -2 0 2

-10

12

34

A1

A2

AM

A4

A5

AX

B1

B2

BMB4

B5

BX

C1

C2

CM

C4

C5

CX

D1

D2DM

D4

D5

DXE1

E2EME4

E5EX

F1

F2

FM

F4

F5

FX

G1

G2

GM

G4

G5

GXH1

H2

HM

H4

H5

HX

I1

I2

IM

I4

I5

IX

J1

J2

JMJ4

J5

JXK1K2KM

K4

K5

KX

conservative, extremeresponses

conservative

conservative, moderate responses

liberal, extremeresponses

liberal

liberal, moderate responses

all middles and missings at centre:

CCA of Spanish “women working”:cases

-8 -6 -4 -2 0 2 4

-20

24

6

+

+

+

+

+

+

+

+

+ +

+

++

++

+

+

+

+

+ ++

+

+++ +

+

++

+

+

+

+

+

++

+

++

+

+ +

++

+

+

+

+

+

++

++

+

+

++

+++++

+

+

+

CCA of W.German “women working”:categories

-4 -3 -2 -1 0 1 2

-10

12

34

A1

A2

AM

A4

A5

AXB1

B2

BMB4

B5

BXC1

C2

CM

C4

C5

CX

D1

D2 DM

D4

D5DX

E1

E2

EME4

E5

EX

F1

F2

FMF4

F5

FXG1

G2

GM

G4

G5

GX

H1

H2

HM

H4

H5

HX

I1

I2

IM

I4

I5IX

J1

J2

JM

J4

J5JX

K1K2

KMK4

K5

KXconservative

conservative, moderate responses

liberal, extremeresponses

liberal

liberal, moderate responses

conservative, extremeresponses

CCA of W.German “women working”:cases

-6 -4 -2 0 2 4

-20

24

6

Demographic categories

• Gender : g1 (male), g2 (female)

• Age 6 groups: a1 (up to 25), a2 (26-35), a3 (36-45)a4 (46-55), a5 (56-65), a6 (66 and over)

• Marital status 5 groups: m1 (married), m2 (widowed), m3 (divorced), m4 (separated), m5 (single)

• Education 7 groups: e1 (none), e2 (incomplete primary), e3 (primary), e4 (incomplete secondary), e5 (secondary), e6 (incomplete tertiary), e7 (tertiary)

(Education not available for Spanish sample in 1994)

Subset MCA of middle categories –“women working”, Spain

0 1 2 3

AM

BMCM

DM

EM FMGM

HM

IMJM

KM

exactly thosewordednegativelytowardswomen

0 1 2 3

-1

0

1

2

exactly thosewordedpositivelytowardswomen

Demographic averages on subset MCA map ofmiddle responses – “women working”, Spain

-0.10 -0.05 0.00 0.05 0.10 0.15

-0.1

0-0

.05

0.00

0.05

0.10

g1

g2a1

a2

a3a4 a5

a6

m1

m2

m3

m4

m5

widowed

separated

56-65yrs

divorced 66+yrs

moremiddles

statementspositive aboutwomen working

lessmiddles

statementsnegative

aboutwomen

working

16-25 yrs

M

F

Checking the MCA results on the data:calculating averages per respondent

Average per Respondent (Spain'94)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1 2 3 4 5 6

Age Group

MiddlesMissings

(M4)separated 0.654

(M2) widowed 0.308 (not significant)

(significant P<0.0001)

AGE GROUPSMARITAL GROUPS

(not significant) widowed &

separated groupssmall samples

Multiple Correspondence Analysis 1 Supplementary points Z1...

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