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Bayesian Methods for Testing
Axioms of Measurement
George Karabatsos
University of Illinois-Chicago
University of Minnesota
Quantitative/Psychometric Methods Area
Department of Psychology
April 3, 2015, Friday.
Supported by NSF-MMS Research Grants SES-0242030 and SES-1156372
Outline I. Introduction: Axioms of Measurement.
A. Relations of Axioms to IRT models.
B. Rasch, 2PL, Monotone Homogeneity and
Double-Monotone IRT models.
II. General Bayesian Model for Axiom Testing
A. Model Estimation (MCMC).
B. Axiom Testing Procedures
III. Empirical Illustrations of Bayesian Axiom Testing.
a) Convict data (orig. analyzed by Perline Wright & Wainer, 1979, APM).
b) NAEP reading test data
IV. Dealing with axiom violations:
A Bayesian Nonparametric outlier-robust IRT model
with application to teacher preparation survey from PIRLS.
V. Extensions of the Bayesian axiom testing model.
VI. Conclusions 2
George Karabatsos, 4/3/2015
I. Introduction • IRT models aim to represent, via model parameters,
persons (examinees) and items on ordinal or interval
scales of measurement.
• In IRT practice, such measurement scales are assumed
for the parameters.
• The ability to represent persons and items on ordinal or interval
scales depends on the data satisfying a set of key
cancellation axioms (Luce & Tukey, 1964, JMP).
• These axioms are deterministic, but we can state these
axioms in more probabilistic terms, as follows.
• We first briefly consider the deterministic case,
to motivate the probabilistic approach. 3
George Karabatsos, 4/3/2015
4
I. (Deterministic) Axioms of Measurement
George Karabatsos, 4/3/2015
Levels of the column variable j = 1 2 3 4 5 6
Levels of the row variable
i = 0
Y(0,1)
Y(0,2)
Y(0,3)
Y(0,4)
Y(0,5)
Y(0,6)
1
Y(1,1)
Y(1,2)
Y(1,3)
Y(1,4)
Y(1,5)
Y(1,6)
2
Y(2,1)
Y(2,2)
Y(2,3)
Y(2,4)
Y(2,5)
Y(2,6)
3
Y(3,1)
Y(3,2)
Y(3,3)
Y(3,4)
Y(3,5)
Y(3,6)
4
Y(4,1)
Y(4,2)
Y(4,3)
Y(4,4)
Y(4,5)
Y(4,6)
5
Y(5,1)
Y(5,2)
Y(5,3)
Y(5,4)
Y(5,5)
Y(5,6)
6
Y(6,1)
Y(6,2)
Y(6,3)
Y(6,4)
Y(6,5)
Y(6,6)
5
I. Deterministic Single Cancellation Axiom
George Karabatsos, 4/3/2015
Levels of the column variable j = 1 2 3 4 5 6
Levels of the row variable
i = 0
Y(0,1)
Y(0,2)
Y(0,3)
Y(0,4)
Y(0,5)
Y(0,6)
1
Y(1,1)
Y(1,2)
Y(1,3)
Y(1,4)
Y(1,5)
Y(1,6)
2
Y(2,1)
Y(2,2)
Y(2,3)
Y(2,4)
Y(2,5)
Y(2,6)
3
Y(3,1)
Y(3,2)
Y(3,3)
Y(3,4)
Y(3,5)
Y(3,6)
4
Y(4,1)
Y(4,2)
Y(4,3)
Y(4,4)
Y(4,5)
Y(4,6)
5
Y(5,1)
Y(5,2)
Y(5,3)
Y(5,4)
Y(5,5)
Y(5,6)
6
Y(6,1)
Y(6,2)
Y(6,3)
Y(6,4)
Y(6,5)
Y(6,6)
Each
column:
Premise
Implication
Each row:
Premise
Implication
Like a
“Guttman
scale”
(1950)
6
I. Probabilistic Measurement Theory
George Karabatsos, 4/3/2015
Define:
ij
Probability
that person
with score
level i
answers
item j
correctly.
Test Items in easiness order j = 1 2 3 4 5 6
Ability Level (test score)
i = 0
01
02
03
04
05
06
1
11
12
13
14
15
16
2
21
22
23
24
25
26
3
31
32
33
34
35
36
4
41
42
43
44
45
46
5
51
52
53
54
55
56
6
61
62
63
64
65
66
Test Items in easiness order j = 1 2 3 4 5 6
Ability Level (test score)
i = 0
01
02
03
04
05
06
1
11
12
13
14
15
16
2
21
22
23
24
25
26
3
31
32
33
34
35
36
4
41
42
43
44
45
46
5
51
52
53
54
55
56
6
61
62
63
64
65
66
7
I. Single Cancellation Axiom (rows)
George Karabatsos, 4/3/2015
Each row:
Premise
Implication
• Key axiom for representing
person ability (test score) on an ordinal scale.
• All Item Response Theory Models, which are of the form
Pr(Yj = 1 | ) = Gj()
for non-decreasing Gj: ℝ [0,1], assume this axiom.
Examples of such IRT models:
1PL Rasch model: Pr(Yj = 1 | ) = exp( j) / [1+ exp( j)]
2PL: Pr(Yj = 1 | ) = exp(aj{ j}) / [1 + exp(aj{ j})]
3PL: Pr(Yj = 1 | ) = cj + (1 cj) / [1 + exp(aj{ j})]
MH Model: Pr(Yj = 1 | ) is non-decreasing in .
DM Model: Pr(Yj = 1 | ) is non-decreasing in , AND
IIO: Pr(Y1 = 1|) < Pr(Y2 = 1 | ) < < Pr(YJ = 1|) for all .
8
I. Single Cancellation Axiom (rows)
George Karabatsos, 4/3/2015
Test Items in easiness order j = 1 2 3 4 5 6
Ability Level (test score)
i = 0
01
02
03
04
05
06
1
11
12
13
14
15
16
2
21
22
23
24
25
26
3
31
32
33
34
35
36
4
41
42
43
44
45
46
5
51
52
53
54
55
56
6
61
62
63
64
65
66
9
I. Single Cancellation Axiom
George Karabatsos, 4/3/2015
Each
column:
Premise
Implication
Each row:
Premise
Implication
• Key axiom for representing
person ability (test score) and item easiness (difficulty)
on a common ordinal scale.
• Examples of IRT models that (fully) assume single cancellation:
1PL Rasch model: Pr(Yj = 1 | ) = exp( j) / [1+ exp( j)]
OPLM model:
Pr(Yj = 1 | ) = exp({ j}) / [1+ exp({ j})]
DM Model:
Pr(Yj = 1 | ) is non-decreasing in ,
and IIO:
Pr(Y1 = 1|) < Pr(Y2 = 1 | ) < < Pr(YJ = 1|) for all .
10
I. Single Cancellation Axiom
George Karabatsos, 4/3/2015
Premise
Implication
Test Items in easiness order j = 1 2 3 4 5 6
Ability Level (test score)
i = 0
01
02
03
04
05
06
1
11
12
13
14
15
16
2
21
22
23
24
25
26
3
31
32
33
34
35
36
4
41
42
43
44
45
46
5
51
52
53
54
55
56
6
61
62
63
64
65
66
11
I. Double Cancellation Axiom
George Karabatsos, 4/3/2015
Axiom
must hold
for all
3 3
submatrices
Test Items in easiness order j = 1 2 3 4 5 6
Ability Level (test score)
i = 0
01
02
03
04
05
06
1
11
12
13
14
15
16
2
21
22
23
24
25
26
3
31
32
33
34
35
36
4
41
42
43
44
45
46
5
51
52
53
54
55
56
6
61
62
63
64
65
66
12
I. Triple Cancellation Axiom
George Karabatsos, 4/3/2015
Premise
Implication
Axiom
must hold
for all
4 4
submatrices
• Key axioms for representing person ability (test score) and
item easiness (difficulty) on a common interval scale.
• All these axioms, together, are axioms for additive conjoint
measurement.
• Examples of IRT models that (fully) assume single cancellation:
1PL Rasch model (logistic):
Pr(Yj = 1 | ) = exp( j) / [1+ exp( j)]
Any 1PL model of the form:
Pr(Yj = 1 | ) = G( j), for non-decreasing G: ℝ [0,1] common to all test items.
• All previous discussions about measurement axioms and IRT also apply to polytomous IRT models.
13
I. Single, Double, Triple, and
all higher order cancellation axioms
George Karabatsos, 4/3/2015
• Even the probabilistic measurement axioms are deterministic.
They assert deterministic order relations among probabilities.
• Perline, Wright & Wainer (PWW; 1979, APM), to test the Rasch
model, analyzed data from a 10-item dichotomous-scored test
administered to 2500 released convicts (from Hoffman & Beck,
1974). The test inquires about the subject’s criminal history.
• PWW tested the conjoint measurement axioms on real data,
by counting the number of axiom violations.
For example, the number of rows violating single cancellation
and, the number of 3 3 submatrices violating double
cancellation.
This axiom testing approach does not distinguish between small
and large axiom violations. We illustrate this issue now.
14
How to Test Measurement Axioms?
George Karabatsos, 4/3/2015
15
True or
Random
Violation
of the
Single
Cancellation
Axiom ?
16
True or
Random
Violation
of the
Single
and
Double
Cancellation
Axioms ?
Apparent
single
cancellation
axiom
violations
in red
Apparent
double
cancellation
axiom
violations
in purple
• The number of axiom violations, as a statistic, has an intractable
sampling distribution, for the purposes of hypothesis testing.
• The false discovery rate approach to multiple testing
(Benjamini & Hochberg, 1995, JRSSB) is not easily
applicable because the different axioms such as single
cancellation and double cancellation are dependent
of on other.
17
How to Test Measurement Axioms?
George Karabatsos, 4/3/2015
• Data likelihood:
The Data:
n = (nij)(I+1)J , nij : # correct in test score group i for item j
N = (Nij)(I+1)J , Nij : # in test score group i who completed item j
MLE: p = (pij)(I+1) J = (nij / Nij)(I+1)J.
• Prior density, i.e., set of axioms:
• Example: single cancellation axiom (rows & columns),
A = { : ij < i+1,j for i = 0,1,…, I 1 & ij < i,j+1 for j =1,…, J 1}
(i: test score level; j indexes item in item easiness order) 18
II. Bayesian Model for Axiom Testing
be( | a,b): beta p.d.f.
Be( | a,b): beta c.d.f.
Be1(u | a,b): quantile.
1( A) = 1 if A.
1( A) = 0 if A.
Often in practice, a = b =1
(truncated uniform prior)
or a = b =½
(truncated reference prior).
George Karabatsos, 4/3/2015
Ln|N, i0
I
j1
JNij
n ij ij
n ij1 ij Nijn ij
i0
I
j1
J
be ij |aij,bij 1 A
i0
I
j1
J
be ij |aij,bij 1 Ad
• Posterior Density (Distribution):
19
II. Bayesian Model for Axiom Testing
George Karabatsos, 4/3/2015
|N,n,A Ln |N,
Ln |N, d
i0
I
j1
JNij
n ij ij
n ij1 ij Nijn ijbe ij |aij,bij 1 A
i0
I
j1
JNij
n ij ij
n ij1 ij Nijn ijbe ij |aij,bij 1 Ad
i0
I
j1
J
be ij |aij nij,bij N ij nij 1 A
i0
I
j1
J
be ij |aij,bij
1 A
• Posterior Density (Distribution): (c.d.f. ( | N, n, A) )
• Posterior cannot be numerically evaluated.
• MCMC full conditional posterior p.d.f.s (f.c.p.s):
π(θij | N, n, θ\ij) be(θij | aij + nij, bij + Nij nij)1(θ ∈ A), ∀ i, j
• Each MCMC sampling iteration: For every pair i, j in turn,
update/sample θij by drawing uij ~ U(0,1), and then taking:
(inverse c.d.f. sampling method; Devroye, 1986).
• As # of MCMC iterations S gets larger,
the MCMC chain {θ(s)}s=1,..,S converges to samples
from the posterior distribution (θ | N, n, A).
20
II. Bayesian Model for Axiom Testing
ij Be1 Beijmin |aij
, bij uij Beij
max |aij , bij
Beijmin |aij
, bij |aij
, bij
George Karabatsos, 4/3/2015
|N,n,A
i0
I
j1
JNij
n ij ij
n ij1 ij Nijn ijbe ij |aij,bij 1 A
i0
I
j1
JNij
n ij ij
n ij1 ij Nijn ijbe ij |aij,bij 1 Ad
• Possible ways to test axioms from model:
1. Check if pij = nij / Nij
is within 95% posterior interval of the
marginal posterior distribution (θij | N, n, A).
Decide violation of axiom(s) if pij
is located outside the 95% posterior interval.
2. Compute the posterior predictive p-value (Karabatsos Sheu 2004 APM):
with:
Decide violations of axioms if pvalueij < .05. (or smaller) 21
II. Bayesian Model for Axiom Testing
2pij;ij Nijpij Nijij
2
Nijij
nijrep
|Nij, ij binij |Nij, ij , with pijrep nij
rep/Nij
George Karabatsos, 4/3/2015
pvalueij 12 pij
rep; ij 2pij; ij pij
rep| |N,n,Adpij
repd
• Possible ways to test axioms from model (continued):
3. Consider the Deviance Information Criterion (DIC)
Consider DIC(A) of model under axiom (order) constraints,
and DIC(U) for unconstrained model (no order constraints).
Decide violations of axiom(s) if DIC(A) > DIC(U).
22
II. Bayesian Model for Axiom Testing
DIC D 2 D D
George Karabatsos, 4/3/2015
Deviance:
D 2i0
I
j1
J
nij log ij N ij nij log1 ij logN ij
nij
Deviance at posterior mean: D DE |N,n,A
Posterior mean of deviance: D Dd |N,n,A
D is goodness (badness) of fit term.
2 D D is model flexibility penalty,
given by 2 times the effective number of model parameters.
23
Apparent
single
cancellation
axiom
violations
in red
24
Test of single cancellation (over rows only)
results from
Karabatsos (2001, JAM)
No
significant
violation
of single
cancellation
over rows.
25
Test of single cancellation (over rows and columns)
results from Karabatsos (2001, JAM)
Significant
violation
of single
cancellation
axiom
26
True or
Random
Violation
of the
Single
and
Double
Cancellation
Axioms ?
Apparent
single
cancellation
axiom
violations
in red
Apparent
double
cancellation
axiom
violation
in purple
27
Test of single and double cancellation
(Karabatsos, 2001, JAM)
Significant
violation
of single
and double
cancellation
axiom
NAEP reading test data
28
George Karabatsos, 3/27/2015
NAEP data
100
examinees
6 items
results from
Karabatsos
& Sheu
(2004, APM)
Posterior
Predictive
Chi-square
test of
single
cancellation
(over rows).
Violations
indicated
by bold.
29
NAEP data
100
examinees
6 items
results from
Karabatsos
& Sheu
(2004, APM)
Posterior
Predictive
Chi-square
test of
single
cancellation
(over
columns).
Violations
indicated
by bold.
IV. Dealing With Axiom Violations • We have seen from the previous two empirical applications
that the measurement axioms can be violated,
even from data arising from carefully-constructed tests.
• One way to deal with the problem is by defining a more
flexible IRT model that can handle outliers.
• A flexible Bayesian Nonparametric outlier-robust IRT model.
• Will present and briefly illustrate the model through the analysis
of data arising from a teacher preparation survey from PIRLS.
• 244 respondents (teachers).
• Each rated (0-2) own level of teacher preparation on 10 items:
CERTIFICATE, LANGUAGE, LITERATURE, PEDAGOGY,
PSYCHOLOGY, REMEDIAL, THEORY, LANGDEV, SPED,
SECLANG.
Also included covariates AGE, FEMALE, Miss:FEMALE.
30
George Karabatsos, 4/3/2015
31
BNP-IRT model
Karabatsos (2015,
Handbook of Modern IRT) fD |X; p1
P
j1
J
fypj |xpi;
fypj |xpj; PYpj 1 |xpj; ypj1 PYpj 1 |xpj; 1ypj
PrY 1 |x; 1 F0 |x; 0
fy |x; dy
0
k
ny |k x,2 jx;,dy
kx;, k x
k 1 x
k,2 Nk | 0,
2U | 0,b
, N |0,2vdiag,JNJ N |0,
2 vINJ1
2 ,2 IG2 |a0 /2,a0 /2IG
2 |a/2,a/2.
#
#
#
#
#
#
#
#
Persons
(examinees)
indexed by
p = 1,…,P
Test items
indexed by
j = 1,…,J
32
33
Absolutely no item response outliers under the BNP-IRT model.
34
• The estimated posterior means of the person ability parameters
were found to be distributed with mean .00, s.d..46,
minimum .66, and maximum 3.68 for the 244 persons.
Valu
e
-5
0
5
10
Dependent variable = itemrespvs0
be
ta0
be
ta:C
ER
TIF
ICA
TE
(1)
be
ta:L
AN
GU
AG
E(1
)b
eta
:LIT
ER
AT
UR
E(1
)b
eta
:PE
DA
GO
GY
(1)
be
ta:P
SY
CH
OL
OG
Y(1
)b
eta
:RE
ME
DIA
L(1
)b
eta
:TH
EO
RY
(1)
be
ta:L
AN
GD
EV
(1)
be
ta:S
PE
D(1
)b
eta
:SE
CL
AN
G(1
)b
eta
:AG
E(1
)b
eta
:FE
MA
LE
(1)
be
ta:M
iss:F
EM
AL
E(1
)b
eta
:CE
RT
IFIC
AT
E(2
)b
eta
:LA
NG
UA
GE
(2)
be
ta:L
ITE
RA
TU
RE
(2)
be
ta:P
ED
AG
OG
Y(2
)b
eta
:PS
YC
HO
LO
GY
(2)
be
ta:R
EM
ED
IAL
(2)
be
ta:T
HE
OR
Y(2
)b
eta
:LA
NG
DE
V(2
)b
eta
:SP
ED
(2)
be
ta:S
EC
LA
NG
(2)
be
ta:A
GE
(2)
be
ta:F
EM
AL
E(2
)b
eta
:Mis
s:F
EM
AL
E(2
)sig
ma
^2
sig
ma
^2
_m
ub
eta
_w
0b
eta
_w
:CE
RT
IFIC
AT
E(1
)b
eta
_w
:LA
NG
UA
GE
(1)
be
ta_
w:L
ITE
RA
TU
RE
(1)
be
ta_
w:P
ED
AG
OG
Y(1
)b
eta
_w
:PS
YC
HO
LO
GY
(1)
be
ta_
w:R
EM
ED
IAL
(1)
be
ta_
w:T
HE
OR
Y(1
)b
eta
_w
:LA
NG
DE
V(1
)b
eta
_w
:SP
ED
(1)
be
ta_
w:S
EC
LA
NG
(1)
be
ta_
w:A
GE
(1)
be
ta_
w:F
EM
AL
E(1
)b
eta
_w
:Mis
s:F
EM
AL
E(1
)b
eta
_w
:CE
RT
IFIC
AT
E(2
)b
eta
_w
:LA
NG
UA
GE
(2)
be
ta_
w:L
ITE
RA
TU
RE
(2)
be
ta_
w:P
ED
AG
OG
Y(2
)b
eta
_w
:PS
YC
HO
LO
GY
(2)
be
ta_
w:R
EM
ED
IAL
(2)
be
ta_
w:T
HE
OR
Y(2
)b
eta
_w
:LA
NG
DE
V(2
)b
eta
_w
:SP
ED
(2)
be
ta_
w:S
EC
LA
NG
(2)
be
ta_
w:A
GE
(2)
be
ta_
w:F
EM
AL
E(2
)b
eta
_w
:Mis
s:F
EM
AL
E(2
)sig
ma
^2
_w
For BNP-IRT model,
boxplot of the marginal posterior
distributions of the item,
covariate, and prior parameters.
George Karabatsos, 4/3/2015
• The ability to measure persons and/or items on an ordinal or
interval scale depends on data satisfying a hierarchy of conjoint
measurement axioms, including single, double, triple cancelation,
and higher order cancellation conditions.
• We presented Bayesian model that can represent a set of
one or more axioms in terms of order constraints on binomial
parameters, with the constraints enforced by the prior distribution.
• This model provided a coherent approach to test the measurement
axioms on real data sets.
35
V. Conclusions
George Karabatsos, 4/3/2015
• Applications of the Bayesian axiom testing model
showed that the measurement axioms can be violated
from data arising even from carefully constructed tests.
• As a possible remedy to this issue,
we propose a more flexible, BNP-IRT model that
can provide estimates of person and item parameters
that are robust to any item response outliers in the data.
• In a sense the BNP-IRT model is not wrong for the data;
It is a highly flexible model which makes rather irrelevant
the practice of model-checking or axiom testing or model fit
analysis.
For related arguments, see Karabatsos & Walker 2009, BJMSP).
36
V. Conclusions
George Karabatsos, 4/3/2015
• The Bayesian axiom testing model of Karabatsos (2001),
was later used to
-- test decision theory axioms (e.g., Myung et al., 2005, JMP);
-- test measurement axioms (e.g., Kyngdon, 2011; Domingue
2012). The latter author suggested a minor modification to the
MH algorithm of Karabatsos (2001) to handle more orderings
under double cancellation.;
Like Karabatsos & Sheu (2004), this talk focused on a
Gibbs sampler which is usually preferable to a rejection
sampler like the MH algorithm, for MCMC practice.
etc.
• Karabatsos (2005, JMP) defined binomial parameter as the
probability of choice that satisfied an axiom. Then under a
conjugate beta prior for , we may directly calculate a
Bayes factor to test the axiom (H0) according to
H0: > c versus H1: < c for some large c, such as .95. 37
V. Conclusions
George Karabatsos, 4/3/2015
• Allow for random orderings for the cancellation axioms.
• Consider the joint posterior distribution:
(, , , A, | Y, N, n) = (, , A, | Y) ( | N, n, A,)
given Rasch model:
Posterior distribution:
• As before,
• A, is the random linear rank ordering that
the matrix ( P(Ypj = 1 | ,) )NJ induces on = (ij)(I+1)J .
• This ordering automatically satisfies all cancellation axioms. 38
Extensions of Axiom Testing Model (1)
, |Y ypjNJ
p1
N
j1
J expp jypj
1 expp jn, |0, INJ
p1
N
j1
J expp jypj
1 expp jdN, |0, INJ
George Karabatsos, 4/3/2015
PYpj 1 |p ,j expp j
1 expp j
|N,n,A, i0
I
j1
J
be ij |aij,bij
1 A,
• Then the joint posterior distribution: (, , , A, | Y, N, n)
can be estimated by using the usual MCMC methods.
• For each stage of the MCMC chain, {((s), (s), (s), A, (s))}s=1:S ,
the Gibbs sampler (inverse c.d.f.) method
would be used to provides a Gibbs sampling
update for (s), based on the updated ordering A, (s).
• Then the Bayesian axiom tests as before,
but now they are based marginalizing these tests
over the posterior distribution of A, .
39
Extensions of Axiom Testing Model (1)
George Karabatsos, 4/3/2015
• Extend the independent (truncated) Beta priors for the ijs
namely ~ ∏i∏j Be(ij | a, b) 1( A)
to a prior defined by a discrete mixture of beta distributions.
= (ij)(I+1)J ~iid ∏i∏j Be(ij | a, b)dG(a, b) 1( A),
G ~ DP(,G0)
where E[G(a, b)] = G0(a, b) := N2(log(a),log(b) | 0, V)
Var[G(a, b)] = G0(a, b) [1G0(a, b)] / ( + 1)
• Any smooth distribution defined on (0,1) can be approximated
arbitrarily-well by a suitable mixture of beta distributions.
Such a prior would define a more flexible Bayesian axiom testing
model, based on a richer class of prior distributions.
40
Extensions of Axiom Testing Model (2)
George Karabatsos, 4/3/2015
• Bayesian nonparametric inference of distribution function
under stochastic ordering: F1 < F2 < < FK
(Karabatsos & Walker, 2007, SPL).
o Considered Bernstein polynomial priors and
Polya tree priors for the Fs.
In each case, posterior inference based on order-constrained
beta posterior distributions (as in Karabatsos 2001).
• Bayesian nonparametric score equating model using a novel
dependent Bernstein-Dirichlet polynomial prior for the
test score distribution functions (FX, FY) used for equipercentile
equating (Karabatsos & Walker, 2009, Psychometrika).
• Bayesian inference for test theory without an answer key
(Karabatsos & Batchelder, 2003, Psychometrika).
• Comparison of 36 person fit statistics (Karabatsos 2003, AME).
41
Other Work / Collaborations
George Karabatsos, 4/3/2015
• Karabatsos, G., and Walker, S.G. (2012). A Bayesian
nonparametric causal model. J Statistical Planning & Inference.
o DP mixture of propensity score models for causal inference
in nonrandomized studies.
• Karabatsos, G., and Walker, S.G. (2012). Bayesian nonparametric
mixed random utility models. Computational Statistics & Data
Analysis, 56, 1714-1722.
o In terms of an IRT model, provides a DP infinite-mixture
of nominal response models, with person and item
parameters subject to the infinite-mixture.
• Fujimoto, K., and Karabatsos, G. (2014). Dependent Dirichlet
Process Rating Model (DDP-RM). Applied Psychological
Measurement, 38, 217-228.
o Model allows for clustering of ordinal category thresholds.
o Ken Fujimoto: former Ph.D. student. Now faculty at Loyola U. Chicago
42
Other Work / Collaborations
George Karabatsos, 4/3/2015
• Karabatsos, G., and Walker, S.G. (2012).
Adaptive-Modal Bayesian Nonparametric Regression (EJS).
o IRT version of this model, mentioned in this talk,
to appear in Handbook Of Item Response Theory (2015).
o Model extended to meta analysis:
Karabatsos, G., Walker, S.G., and Talbott, E. (2014). A
Bayesian nonparametric regression model for meta-analysis.
Research Synthesis Methods.
o Model extended for causal inference in non-randomized,
regression discontinuity designs:
(Karabatsos & Walker, 2015; (to appear in
Müller and R. Mitra (Eds.), Nonparametric Bayesian
Methods in Biostatistics and Bioinformatics).
43
Other Work / Collaborations
George Karabatsos, 4/3/2015