Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Outline A Beta Unfolding Model for Continuous Bounded Responses1
The continuous response format
1
Continuous Response Scales
2
Beta Response Models The interpolation response mechanism The Cumulative Beta Response Model
Yvonnick Noel
The Beta Unfolding Model University of Brittany at Rennes (France) 3
Emergence of bimodality
4
Application: Attitude toward abortion
Durham, NH, June 30th 2008
1
These slides are available for download at: http://yvonnick.noel.free.fr/papiers
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
A Beta Unfolding Model for Continuous Bounded Responses The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
The interpolation mechanism
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
A Beta Unfolding Model for Continuous Bounded Responses The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
Justi cation
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
A Beta Unfolding Model for Continuous Bounded Responses The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
A random model The psychological values are assumed to be
quantities.
The only reference points in a visual analogue scale are the
labels at the segment boudaries. It is assumed that some
latent values, v0 et v1 , are granted
to both extreme responses, graphically located at
λ1 = 1
λ0 = 0
and
It is then assumed (Noël & Dauvier, 2007) that subjects
interpolate their agreement response x λ0 v0 + λ1 v1 = v0 + v1
Yvonnick Noel
This construction guarantees that the response lies
within [0; 1].
choice models in psychology (Luce, 1959), or discrete choice models in econometrics (McFadden, 1974), or to the matching law in reinforcement learning (Herrnstein, 1961).
It is assumed that subjects sample the values from two
Gamma densities with dierent shape parameters but a common scale parameter:
This hypothetical mechanism is very similar to classical
(in arbitrary units).
x=
Remark:
following :
v1 v0 + v1
A Beta Unfolding Model for Continuous Bounded Responses
non negative
v0 v1
∼ Γ(n, s )
∼ Γ(m, s )
It is known that (Kotz & Johnson, 1982, p.229):
X Yvonnick Noel
A Beta Unfolding Model for Continuous Bounded Responses
=
v1 v0 + v1
Yvonnick Noel
∼ β(m, n)
A Beta Unfolding Model for Continuous Bounded Responses
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
The beta density f (x ; m, n) =
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
A Beta Response Model
Γ(m + n) m−1 x (1 − x )n−1 Γ(m)Γ(n)
for
x ∈ [0; 1], m, n > 0
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
A structural model Distribution parameters are linked to
beta response model on the response Xij from subject i i = 1, ..., N ) to item j (j = 1, ..., p) may be written as:
item-speci c parameters by posing:
A
mij
3.0
(
Γ(mij + nij ) m f (xij ; mij , nij ) = x Γ(mij )Γ(nij ) ij
1.5
mij
et
nij
ij
may be interpreted as
−1
(1−xij )n
ij −1
with
nij
xij ∈ [0; 1]
acceptance and refusal
parameters, respectively.
0.5
1.0
f(x)
2.0
2.5
m = 1, n = 3 m = 1.5, n = 2.5 m = 2, n = 2 m = 6, n = 3 m = 0.5, n = 0.5
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
where
θi
et
δj
=
exp
subject and
θi − δj
2
θi − δj = exp − 2
are attitude and di culty parameters,
respectively. The acceptance parameter varies as a monotone increasing
0.0
function of attitude and a monotone decreasing function of 0.0
0.2
0.4
0.6
0.8
item di culty, and conversely for the refusal parameter.
1.0
x
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
A Beta Unfolding Model for Continuous Bounded Responses The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
E (Xij ; θi , δj )
= exp
=
1
exp
θi −δj 2
θi −δj
θ −δ + exp − 2 i
j
A Beta Unfolding Model for Continuous Bounded Responses
A Beta Unfolding Model for Continuous Bounded Responses The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
1.0
attitude towards abortion
0.6
questionnaire (Roberts, Donoghue and Laughlin, 2000): I cannot whole-heartedly support either side of the abortion debate. This item may be
refused for two dierent reasons (Andrich &
Luo, 1993):
2
exp(θi − δj ) + exp(θi − δj )
Yvonnick Noel
Response intensity
expected response function has a familiar form:
A sample item from the
0.8
the
0.2
The
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Ambivalent items
0.0
β(mij , nij )
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
Response function
expectation reads: mij µij = E (Xij ; mij , nij ) = mij + nij
In a beta model
A Beta Unfolding Model for Continuous Bounded Responses
0.4
Expected Response Function
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
−4
−2
0
θi − δj
2
4
because you are in favor of an unconditional right to abortion, because your are a convinced opponent of abortion.
Noel, Y, & Dauvier, B. (2007). A beta response model for continuous bounded responses, Applied Psychological Measurement, 31, 47-73. Yvonnick Noel
A Beta Unfolding Model for Continuous Bounded Responses
Yvonnick Noel
A Beta Unfolding Model for Continuous Bounded Responses
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
A beta model A
global disagree value may be de ned as:
For such items, we consider
=
+ exp(θi − δj ) + exp(δj − θi ) exp λj exp λj + 2 cosh(θi − δj )
−4
−2
Parameter interpretation:
λj aects the global level of item acceptation, αj is a discrimination parameter, τj aects response variance.
−4
0
2
A Beta Unfolding Model for Continuous Bounded Responses
Yvonnick Noel
A Beta Unfolding Model for Continuous Bounded Responses
0.8 0.6
4
−4
−2
0
θi − δj
θi − δ j
λ = 1.5 τ = 0
λ = 1.5 τ = 1.5
−2
0
2
4
θi − δj
Yvonnick Noel
0.4
+ τj ) − δj ) + τj } + exp {−αj (θi − δj ) + τj }
exp {αj (θi
Response intensity
exp(λj
0.0
= =
2
4
2
4
1.0
exp λj
mj nij
0.2
1.0
exp λj
λ = 0 τ = 1.5 1.0
λ=0τ=0
A more exible 4-parameter model is obtained by posing:
mj + nij mj mj + pij + qij
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
Response expectation and variance
expected response function is:
mj
A Beta Unfolding Model for Continuous Bounded Responses
0.8
=
A generalized model
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
0.6
=
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
nij is the sum of two refusal sources in opposite directions.
The refusal parameter operating
0.4
=
A Beta Unfolding Model for Continuous Bounded Responses
an item-speci c acceptance parameter.
Response intensity
= µij
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
∼ β(mij , pij + qij )
λj
0.0
E (Xij |θi )
vij(A) = (D ) vij 1 + vij(D2 ) + vij(A)
with
Response intensity
For this basic model, the
v (A) Xij = (D ) ij (A) vij + vij
A Beta Unfolding Model for Continuous Bounded Responses
Expected response function
interpolation response mechanism:
− δj ) + exp(δj − θi )
0.2
∼ Γ(qij , s )
nij = pij + qij
0.0
∼ Γ(pij , s )
Then from the
with
pij + qij
exp(θi
0.8
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
∼ Γ(mij , s )
∼ Γ(nij , s )
exp λj
0.6
vij vij(D1 ) vij(D2 )
vij
= = =
0.4
(A)
(D )
mj nij
0.2
j , are assumed to follow:
conditional independence of the latent values, we have:
0.0
and item
properties of the Gamma, and assuming
1.0
i
From the
attitude (θi ) and
Distribution parameters are connected to
item location (δj ) parameters by:
Response intensity
subject
vij(A) , vij(D1 ) and vij(D2 ) , for
vij(D ) = vij(D1 ) + vij(D2 )
0.8
The corresponding latent values
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
A structural model
0.6
three latent evaluations, for: Agree (A), Disagree for the rst reason (D1 ), Disagree for the second reason (D2 ).
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
0.4
A latent random model
The interpolation response mechanism The Cumulative Beta Response Model The Beta Unfolding Model
0.2
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Yvonnick Noel
−4
−2
0 θi − δ j
A Beta Unfolding Model for Continuous Bounded Responses
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Conditions of bimodality
Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Unimodal and bimodal items
EM algorithm for nite mixtures discretized to a set Θ of xed θk k = 1, ..., K ) values, with probabilities πk (Woodruf &
The attitude variable is
12
Hanson, 1996).
5
10
The density of an
0.8
U
U
B
8
4
R 0.2
2
0
0 −3
−2
−1
0
1
2
3
Attitude
Yvonnick Noel
A Beta Unfolding Model for Continuous Bounded Responses
0
1
1.0 0.6 0.4 0.2 0.0 −2
−1
0
1
2
−2
−1
0
1
0
1
−2
−1
0
1
2
0.8 0.0
0.2
0.4
xij
0.6
0.8 0.6 0.0
0.2
0.4
xij
2
−2
−1
0
1
2
−2
−1
0
1
θ^i
θ^i
θ^i
Item 5
Item 4
Item 6
2
0.8
xij
0.6
0.8 0.6
0.8 0.6 1
−2
−1
0
1
2
θ^i
Yvonnick Noel
2
1.0
θ^i Item 7
0
2
1.0
θ^i Item 27
1.0
θ^i Item 29
xij −1
xij
2
θ^i
0.4
−1
0.8
1.0 0.6 0.4 0.2 0.0 −1
Item 25
θ^i
A Beta Unfolding Model for Continuous Bounded Responses
−2
0.4 −2
Yvonnick Noel
xij
2
1.0
−2
'C': Constrained to unimodality, 'U': Unconstrained
0.8
1.0 0.8 0.6
xij 0.4 0.2 0.0 1
θ^i
xij
j
0
Item 30
0.4
j
j
j
−1
0.2
j
j
j
and item
Item 28
0.0
j
λ ,δ ,τ ,α
j
−2
1.0
j
j
λ ,δ ,τ ,α
17025.12 17025.12 -28968.59 -39736.15 -31750.56 -40419.47*
0.8
λ ,δ ,τ
j
1 1 21 1 12 2
0.6
j
j
128 128 158 178 217 227
Item 40
xij
λ ,δ ,τ
-8384.559 -8384.559 14642.29 20046.07 16092.28 20436.73
Item 47
0.4
j
πk
)
A Beta Unfolding Model for Continuous Bounded Responses
0.2
j
j
Item 39
0.2
4
j
λ ,δ
AIC
0.0
3
Abortion is a threat to our society (6) I nd myself agreeing with arguments both for and against abortion (27) I cannot whole-heartedly support either side of the abortion debate (30) Abortion should be an accepted mechanism for family planning (48)
λ ,δ
#Param. #Constr.
0.0
2
CBUM-2 UBUM-2 CBUM-3 UBUM-3 CBUM-4 UBUM-4
Loglik.
i =1
#
f (xi , θi |∆, π) |X, ∆(s ) , π(s )
Data and model plots
1.0
Sample items:
Parameters
ln
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
xij
was submitted to 443 subjects, on a web interface.
1
A Beta Unfolding Model for Continuous Bounded Responses
0.4
Donoghue and Laughlin, 2000; Roberts, Lin & Laughlin, 2001)
Model
( "N Y
parameters.
Concurrent BUM models
A 50-item questionnaire on attitude toward abortion (Roberts,
Eθ
=
is computed and maximized with respect to the
0.2
Attitude toward abortion
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
Q (∆, π|∆(s ) , π(s ) )
0.0
3
f (xi |θk , ∆)πk
EM algorithm, the expectation:
1.0
2
1.0
1
of the
0.8
0
s
0.6
−1
xij
−2
At each step
0.4
−3
'U': Unimodal density, 'B': Bimodal density A Beta Unfolding Model for Continuous Bounded Responses
k
4
0.2
0.0
R
0.0
1
A
1.0
0.2
Attitude
Yvonnick Noel Continuous Response Scales Beta Response Models Emergence of bimodality Application: Attitude toward abortion
X
=
0.4 2
0.8
λ∗j < 0 τj < − ln [2 cosh (αj (θi − δj ))]
0.4
f (xi |∆, π)
6
0.6
:
0.6
xij
= λj + τj )
3
oberved response vector xi , given the
of item parameters, is:
0.4
∗
0.6
∆
whole set
0.2
0.8
6
0.0
1.0
little value is granted to
... that is, when we have simultaneoulsy (λj
λ = 1.2 τ = −2 α = 1.8
λ = 2 τ = 0 α = 1.1
nij < 1.
0.2
both acceptation and refusal.
et
0.0
Response density is bimodal when
