The unfolding model as 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 * √0.49 = ? Q.3aBelieve in heaven? Agree/disagree Q.3bBelieve in hell Q.4aDo you own a cd-player?yes/no Q.4bDo you own a dish washer?

Data Matrix A B C D E Correlation Matrix A B C D E A B C D E Eigenvalues: 3.0, 1.0, 0.5, 0.3, 0.2 Two large ones (Principal Components Analysis, PCA)

Example 2 Component MatrixRotated Component Matrix Factor 1 2 Factor 12 A.66.60A -.89 B.83 -B -.85 C.88 -C D.83 -D.85- E E : factor loading <.40

Polytomous items ABCDEFweight

Correlations and factor loadings ABCDEF A B C D E F Eigenvalues: 4.1, 1.2, 0.51, 0.06, 0.04, 0.02 UnrotatedVARIMAX rotated Factor 1 2 Factor 1 2 A B C D E F

Dominance model A B C D E ─┴──┬──┴──┬───┴───┴─┬──┴── l ow S 1 S 2 S 3 high Dominance: item E dominates item D, C, B, and A Subject S 3 dominates subject S 2 and subject S 1 Item E dominates Subjects S 1, S 2, and S 3, Subject S 2 dominates items B and A, and Subject S 1 dominates item A Subject dominates item: positive or high response Item dominates subject: negative or low response

Proximity questions Survey questions Q.1aDo you like tea without sugar? yes/no Q.1bDo you like tea with 1 lump of sugar? Q.1cDo you like tea with 2 lumps of sugar? Q.2aWould you vote for leftist party? yes/no Q.2bWould you vote for centrist party? Q.2cWould you vote for rightist party?

Proximity model A B C D E ─┴──┬──┴────┬─┴───┴─┬──┴── low S 1 S 2 S 3 high Proximity: Subject S 1 is close to (agrees with) items A and B Subject S 2 is close to (agrees with) items B, C, and D Subject S 3 is close to (agrees with) items D and E Positive response: agrees (is close) Negative response: disagrees (is distant)

Dichotomous dataset ABCDE S S S pick 2/npick 3/npick any/n

Proximity dataset (hypothetical) PersonABCDEF

Correlation Matrix ABCDEF A B C D E F Eigenvalues: 2.83, 2.68, 0.27, 0.09, 0.08, 0.06

Factor loadings UnrotatedRotated (VARIMAX) Factor12Factor12 A--.89A.90- B B.95- C.90-C D.90-D E.68.67E-.95 F-.89F-.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 questions People 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 Iraq The death penalty helps deter crime Better 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) UnrotatedRotated (VARIMAX) Factor12Factor12 Obama Clinton Richardson Edwards McCain Huckabee Romney Thomson (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) B (Europe) C (country) D (province) E (community) Eigenvalues: 2.9, 1.1, 0.5, 0.3, 0.2 UnrotatedRotated (VARIMAX) Factor 12Factor12 A World B Europe C country D province E community

WARNING: Any two (randomly chosen) cumulative scales joined together form an artificial proximity (unfolding) scale Impopular – popular popular – impopular ABCDEFGH Joined together, this looks like a pick 4/8 unfolding dataset

Lithmus test for unfolding analysis The 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,

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 characteristics Extensive software to apply to the dominance model (Rasch model, Mokken model) and the proximity or unfolding model (GGUM, MUDFOLD)

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