Linear Algebra and Geometric Approches to Meaning 4a. Question Order Effects

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Linear Algebra and Geometric Approches to Meaning

4a. Question Order Effects

Reinhard Blutner

Universiteit van Amsterdam

ESSLLI Summer School 2011, Ljubljana

August 1 – August 7, 2011

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1.Factual questions & attitude questions

2.Four types of question ordering effects

3.Modelling ordering effects

4.One-qubit realization

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Factual questions and attitude questions

1. a. What is your name?b. Where do you live?c. In what year did you first have an episode of back

pain lasting a week or more?

2. a. Do you think the use of marijuana should be made

legal, or not?b. Would you be for or against sex education in the

public schools?c. On the average, (Blacks/African-Americans) have worse jobs, income, and housing than white people. Do you think these differences are mainly due to discrimination?

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Importance of attitude questions

• Survey reports– Schuman, H. & Presser, S. (1981). Questions and answers

in attitude surveys: Experiments in question form, wording, and context. (New York: Academic Press)

– Tourangeau, R., Rips, L. J. & Rasinski, K. (2000). The Psychology of Survey Response. (Cambridge: Cambridge University Press)

– Moore, D. W. (2002). Measuring New Types of Question-Order Effects. Public Opinion Quarterly 66: 80-91.

• Personality theories– The Myers-Briggs type indicator (MBTI)– The big five personality test– Keirsey Temperament Sorter– The Singer-Loomis Inventory of Personality (SLIP)

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2

1.Factual questions & attitude questions

2.Four types of question ordering effects

3.Modelling ordering effects

4.One-qubit realization

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NORC 1948

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Schumann & Presser 1981

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Assimilation

assimilation

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Contrast

contrast

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Additive

additive

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Subtractive

subtractive

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3

1.Factual questions & attitude questions

2.Four types of question ordering effects

3.Modelling ordering effects

4.One-qubit realization

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Conditional probabilities

• Classical definition for propositions a and b :– (b/a) = (ab)/(a)

• Gerd Niestegge´s generalisation: (b/a) = (aba)/(a)

• Sequence of two yes/no-questions (a;b) = (a) (b/a) = (aba)

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Two-dimensional example (A) = .3 (B; A) = .6

(B) = .9

(A; B) = .2

A

(A; B)

B

s (B; A)

assimilation

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Question order effects (yes/no-questions)

Definition: a(b) = (a;b) + (a´;b) (b) (*)

Fact: a(b) = 2(a; b) 2((a) (b))½ cos, when con-sidering a pure state ; is a phase shift parameter

Proof: (a;b) + (a´;b) (b) =

baba + ba’ ba’ (ba+ ba’)(ba+ ba’) =ba ba’ ba’ ba =

2 Re(baba’) = 2 Re(ba(bba)) =

2 baba 2 Re(ab)=2(a; b) 2((a) (b))½ cos.

(*) Note: we write a´ instead of a

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Explaining order effects

Wang & Busemeyer (2010): Quantum Model of Question Order Effects. Submitted for publication

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Niestegge’s theorem

(a; b)(b; a) = (a’; b’)(b’; a’)

Proof: Using the fact that (a;b) = (a) (b/a) = (aba),

the proof reduces to showing that aba+a’b’a’ = bab+b’a’b’.

Using a’=1-a and b’=1-b this equality can be shown by elementary algebraic manipulations.

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Empirical test• Wang & Busemeyer (2010) found empirical

evidence that the constraint expressed by Niestegge’s theorem is satisfied in 3 of 4 cases:

a(b) = -.0031 assimilation -.0031 contrast-.0189 additive 0.1514 subtractive (sign.

different of 0)

• The prediction for the quantum model was based on the assumption that only the question order influences the question context. This assumption is violated in the Rose/Jackson data set. Respondents lacked sufficient knowledge about the baseball players. Thus, additional information was given. Hence, the context for the second question was changed not only by answering the first question, but also by the additional background information on the player in the second question.

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4

1.Factual questions & attitude questions

2.Four types of question ordering effects

3.Modelling ordering effects

4.One-qubit realization

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Qubit states

• A bit is the basic unit of information in classical computation referring to a choice between two discrete states, say {0, 1}.

• A qubit is the basic of information in quantum computing referring to a choice between the unit-vectors in a two-dimensional Hilbert space.

• The orthogonal states and can be taken

to represent true and false, the vectors in

between are appropriate for modeling

vagueness.

0

1

1

0

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Bloch spheres

Real Hilbert Space:

1

02sin

0

12cos

1

02sin

0

12cos ie

Complex Hilbert Space

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• 3 dimensions– Introverted vs. Extraverted– Thinking vs. Feeling– Sensation vs. iNtuition

• 8 basic types

C.G. Jung’s theory of personality

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Introverted iNtuitive Thinker

Sherlock Holmes

Shadow

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Diagnostic Questions

• When the phone rings, do you hasten to get to it first, or do you hope someone else will answer? (E/I)

• In order to follow other people do you need reason, or do you need trust? (T/F)

• c. Are you more attracted to sensible people or imaginative people? (S/N)

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Predictions of the model

• Real Hilbert space:

• Complex Hilbert space

1)()( 22 SETE

1)()( 22 SETE

1)()( 22 SETE

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Predictions of question order effects

• The qubit model forbids subtraction and contrast effects

• Assimilation and addition effects are possible.

cos()=0 cos()=1

T S

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Conclusions

• Descriptive framework for describing all four types of question order effects

• Restrictive framework: there are general constraints on order effects which can be empirically tested

• Explanatory framework? Why do we find certain ordering effects for certain pairs of questions? What does the phase parameter mean?