Linear Algebra and Geometric Approches to Meaning
4a. Question Order Effects
ESSLLI Summer School 2011, Ljubljana
August 1 – August 7, 2011
Reinhard Blutner
Universiteit van Amsterdam
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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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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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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
 (B; A)
B
s

(A) = .3

(B; A) = .6

(B) = .9

(A; B) = .2
 (A; B)
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 considering a pure state ;  is a phase shift parameter
Proof:
(a;b) + (a´;b)  (b) =
baba + ba’ ba’ (ba+ ba’)(ba+ ba’) =
ba ba’  ba’ ba =
2 Re(baba’) =
2 Re(ba(bba)) =
2 baba 2 Re(ab)=
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
-.0031
-.0189
0.1514
•
The
assimilation
contrast
additive
subtractive (sign. different of 0)
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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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 unitvectors in a two-dimensional Hilbert space.
• The orthogonal states
1
 
0
and
0
 
1
can be taken
to represent true and false, the vectors in
between are appropriate for modeling vagueness.
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Bloch spheres
Real Hilbert Space:
  cos 
1
0

    sin
 
2 0
2 1
 
 
Complex Hilbert Space
  cos 
1
i
    sin   e
2 0
2
 
0
  
1

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C.G. Jung’s theory of
personality
• 3 dimensions
– Introverted vs. Extraverted
– Thinking vs. Feeling
– Sensation vs. iNtuition
• 8 basic types
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Sherlock Holmes
Introverted iNtuitive Thinker
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:


2
2
E (T )  E ( S )  1
22
E(T)S
E()1
• Complex Hilbert space
2 2
E
(
T
)

E
(
S
)

1
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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?
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