The unfolding modelas an alternative explanation

for finding two factors for a one dimensional concept

Wijbrandt H. van Schuur University of Groningen, The Netherlands

Seminar 1: “Four different reasons why one will find two factors for a one dimensional concept

“New developments in Survey Methodology” Seminar Series Research and Expertise Centre for Survey Methodology

Universitat Pombreu Fabra, Barcelona, Spain October 29, 2010

Dominance• In the dominance model: order of questions is represented

in terms of less to more ‘popular’ responses

• Cumulative scale: IF the ‘positive’ or ‘high’ answer is given to an impopular question, THEN the ‘positive’ or ‘high’ answer is given to all more popular questions

• Examples given in Intro to seminar: “The higher level of competence always requires the lower competences but with some extra capability”.

Survey questions• Q.1a Are you taller than 1.70m? yes/no Q.1b Are you

taller than 1.80m?

• Q.2a 2+2 = ? correct/incorrect Q.2b 15.72* √0.49 = ?

• Q.3a Believe in heaven? Agree/disagree Q.3b Believe in hell

• Q.4a Do you own a cd-player? yes/no Q.4b Do you own a dish washer?

Data MatrixA B C D E0 0 0 0 01 0 0 0 01 1 0 0 01 1 1 0 01 1 1 1 01 1 1 1 1

Correlation Matrix A B C D E A 1.00 0.63 0.45 0.32 0.20B 0.63 1.00 0.71 0.50 0.32C 0.45 0.71 1.00 0.71 0.45D 0.32 0.50 0.71 1.00 0.63E 0.20 0.32 0.45 0.63 1.00

Eigenvalues: 3.0, 1.0, 0.5, 0.3, 0.2 Two large ones (Principal Components Analysis, PCA)

Example 2

• Component Matrix Rotated Component Matrix Factor 1 2 Factor 1

2 A .66 .60 A -.89 B .83 - B

- .85 C .88 -C .62 .62 D .83 -

D .85 - E .66 -.60 E .89 - - : factor loading < .40

Polytomous itemsA B C D E F weight1 1 1 1 1 1 102 1 1 1 1 1 502 2 1 1 1 1 203 2 2 1 1 1 203 3 2 2 1 1 104 3 3 2 1 1 305 4 3 2 2 1 3205 4 4 2 2 1 805 4 4 3 2 1 305 5 4 3 2 1 2005 5 5 4 2 1 2005 5 5 5 4 2 205 5 5 5 5 2 10

Correlations and factor loadingsA B C D E F

A 1.00 0.89 0.75 0.55 0.63 0.07 B 0.89 1.00 0.90 0.80 0.64 0.14 C 0.75 0.90 1.00 0.92 0.60 0.22 D 0.55 0.80 0.92 1.00 0.64 0.42 E 0.63 0.64 0.60 0.64 1.00 0.77 F 0.07 0.14 0.22 0.42 0.77 1.00

Eigenvalues: 4.1, 1.2, 0.51, 0.06, 0.04, 0.02

Unrotated VARIMAX rotated Factor 1 2 Factor 1 2 A

0.83 -0.35 0.90 - B0.93 -0.32 0.97 - C0.93 - 0.93 - D0.89 - 0.79 0.41 E0.83 0.45 0.54 0.79 F0.45 0.88 - 0.99

Dominance model

A B C D E ─┴──┬──┴──┬───┴───┴─┬──┴── low S1 S2 S3 high

Dominance: item E dominates item D, C, B, and ASubject S3 dominates subject S2 and subject S1

Item E dominates Subjects S1, S2, and S3,

Subject S2 dominates items B and A, and Subject S1 dominates item A

Subject dominates item: positive or high response Item dominates subject: negative or low response

Proximity questions

• Survey questions Q.1a Do you like tea without sugar? yes/no Q.1b Do you like tea with 1 lump of sugar? Q.1c Do you like tea with 2 lumps of sugar?

• Q.2a Would you vote for leftist party? yes/no Q.2b Would you vote for centrist party? Q.2c

Would you vote for rightist party?

Proximity model

A B C D E ─┴──┬──┴────┬─┴───┴─┬──┴── low S1 S2 S3 high

Proximity: Subject S1 is close to (agrees with) items A and BSubject S2 is close to (agrees with) items B, C, and D

Subject S3 is close to (agrees with) items D and E

Positive response: agrees (is close) Negative response: disagrees (is distant)

Dichotomous datasetA B C D E

S1 1 1 0 0 0 S2 0 1 1 1 0 S3 0 0 0 1 1

1 1 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 pick 2/n pick 3/n pick any/n

Proximity dataset (hypothetical)Person A B C D E F

1 4 3 3 2 21 2 4 4 3 32 2 3 5 4 43 3 2

4 5 5 4 4 3 3 5 4 5 5 4 4 3 6 4 4 5 5 4 4 7 3 4 4 5 5 4 8 3 3 4 4 5 5 9 2 3 3 4 4 510 2 2 3 3 4 411 1 2 2 3 3 4

Correlation Matrix

A B C D E FA 1.00 .81 .63 .04 -.31

-.63 B .81 1.00 .75 .44-.03 -.31 C .63 .75 1.00 .65 .44 .04 D .04 .44 .65 1.00 .75 .63 E -.31-.03 .44 .75 1.00 .81 F-.63 -.31 .04 .63 .81 1.00

Eigenvalues: 2.83, 2.68, 0.27, 0.09, 0.08, 0.06

Factor loadings

• Unrotated Rotated (VARIMAX) Factor 1 2 Factor 1 2

A - -.89 A .90 -B .68 -.67 B .95 -C .90 - C .86 .41D .90 - D .41 .86E .68 .67 E - .95F - .89 F - .90

Electoral compass

• 36 statements with 5 response categories: completely agree (5) – tend to agree (4) – neutral (3) – tend to disagree (2) – completely disagree (1)

• Respondents are asked to give their opinion. These are then compared with the opinions of Obama, Clinton, Richardson, Edwards, McCain, Huckabee, Romney and Thomson

• Electoral advice: vote for candidate with whom you agree the most

More survey questionsPeople should have a background check and obtain a license before they can buy a gun Same sex marriages should be made legal US law should obligate all companies to provide health care insurances for their workers The new president should begin to bring home all US troops from Iraq immediately The tax cuts for people with a higher income should be reversed

All illegal immigrants without criminal record should be given the right to stay in the US legally The US should reduce its financial contribution to the UN An additional carbon tax on fuel will effectively reduce carbon emission The US had every right to invade IraqThe death penalty helps deter crimeBetter teachers should be paid higher wages than their colleagues For each crime there should be a fixed minimum sentence Iraq is just one front in a broader fight against Islamic terrorism Abortion should be made completely illegal Creationism should be taught in science classes in school The effects of global warming are grossly exaggerated Some form of torture is acceptable if it can prevent terrorist attacks The US should never sign international treaties on climate change that limit economic growth

Factor loadings (PCA)

Unrotated Rotated (VARIMAX) Factor 1 2Factor 1 2 Obama -.89 -

- .81 Clinton -.72 .51 - .87 Richardson -.53 - - .48 Edwards -.74 .49

- .87 McCain .68 .48.82 - Huckabee .78 -

.76 - Romney .73 .41 .81 - Thomson .86 - .81 -.40 (eigenvalues: 4.4, 1.2, 0.99, 0.45, 0.36, 0.29, 0.17, 0.16)

Localism- Cosmopolitanism

How interested are you in news about AThe world BEurope CYour country DYour province EYour local community 1:

not interested; 4: neutral; 7: very interested

Correlation matrix / Loadings A B C D E

A (World) 1.00 .76 .61 .27 .20 B (Europe) .76 1.00 .58 .48 .30 C (country) .61 .58 1.00 .44 .47 D (province) .27 .48 .44 1.00 .64 E (community) .20 .30 .47 .64 1.00

Eigenvalues: 2.9, 1.1, 0.5, 0.3, 0.2

Unrotated Rotated (VARIMAX) Factor 1 2 Factor 1 2 A World .76 -.55 .94 - B Europe .84 -.35 .87 - C country .82 - .71 .43 D province .72 .51 - .85 E community .65 .64 - .91

WARNING:

Any two (randomly chosen) cumulative scales joined together form an artificial proximity (unfolding) scaleImpopular – popular popular – impopular A B C D E F G H 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 1 1

Joined together, this looks like a pick 4/8 unfolding dataset

Lithmus test for unfolding analysisThe negative response must be ambiguous:

It can be given for two opposite reasons: the respondent is represented either too much to the left of the item, or the respondent is represented too much to the right of the item (Why not one lump: either no sugar, or more than one;

Why not vote for center party: either more to the left or more to the right)See Van Schuur & Kiers (1994). Why factor analysis is the incorrect model for analyzing bipolar concepts and what model to use instead. Applied Psychological Measurement, 18, 97-110.

Item-Response Theory

Does not rely on correlations (assumption: all items have the same distribution)It uses the fact that items are not meant to be replications of each other, but they have their own characteristicsExtensive software to apply to the dominance model (Rasch model, Mokken model) and the proximity or unfolding model (GGUM, MUDFOLD)

THANK YOU

THE END h.van.schuur@rug.nl