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```SICM Estimation Algorithm
To estimate the SICM model, a Metropolis-Hastings algorithm that uses Gibbs sampling
was implemented using Fortran (obtainable from the first author). The Metropolis–Hastings
algorithm defines the probability a candidate parameter proposed at iteration  ( ∗ ) is accepted
(i.e.,   =  ∗ ) over the previous value of the parameter at iteration  − 1 ( −1 ) as
(1, )
(A1)
where
=
(|∗ , ∗ , ∗ , ∗ ) (∗ , ∗ , ∗ , ∗ )Q( −1 | ∗ )
.
(| −1 , −1 , −1 , −1 ) (−1 , −1 , −1 , −1 )Q( ∗ | −1 )
(A2)
The term (| , , , ) is the probability of the data, given the model parameters, which
includes item parameters (λ), structural parameters (γ) , examinee attribute patterns (α), and
examinee abilities (θ). (, , , ) is the joint (prior) probability of the model parameters. The
term Q( ∗ | −1 ) is the candidate generating density, the density of the distribution from which
the candidate parameter ( ∗ ) is drawn given the previous value of the parameter( −1 ). When
the proposal distribution was not symmetric because of boundaries placed on the parameter
space (e.g., item parameters), a moving window proposal distribution was used (Henson &
Templin, 2003). A moving window proposal distribution draws  ∗ from a uniform distribution
with bounds UB and LB ((, )) where
=  ( −1 −

, ) ;  =  ( −1 − , ).
2
2
(A3)
The parameter  controls the width of the sampling interval. The parameters  and  constrain
the sampling intervals to lower and upper boundaries, respectively. The value of Q( ∗ | −1 ) is
calculated as the height of the density of the uniform distribution (, ):
Q(  | −1) =
1
.
−
(A4)
The density Q( −1 | ∗ ) is calculated as in Equation (A4) where  and  have values
=  ( ∗ −

, ) ;  =  ( ∗ − , ).
2
2
(A5)
The values of  that are accepted at each of the  stages,  1 ,  2 , … ,  T , comprise the
Markov chain. The first  entries of the Markov chain are discarded (burn-in period), where the
value of  is large enough for the chain to reach stationarity. Using a thinning interval of five,
stages  + 1 through  provide samples from the target posterior distribution to accurately
describe its shape and moments.
Using the Gibbs sampling, each parameter is updated individually, meaning the marginal
distribution ( ∗ ) of a parameter replaces the joint posterior distribution of all model parameters
(∗ , ∗ , ∗ , ∗ ) in Equation (A2). When updating the th parameter in a set of  parameters,
∗
=

−1
(Π=1,−1
,  =  , Π=+1,
)( ∗ )Q( −1 | ∗ )

(Π=1,−1
,
t−1
−1
, Π=+1,
)( −1 )Q( ∗ | −1 )
.
(A6)
=
where (·) is the conditional likelihood of the model (which differs for each parameter type,

specifications given in Table 1), given the first  − 1 parameters (Π=1,−1
) that were updated
−1
during the ℎ iteration and the  + 1 through ℎ item parameters (Π=+1,
) that have not
been updated in this iteration and retain their values from the ( − 1)ℎ iteration. For each
parameter type, Table 1 provides the specifications of (·), ( ∗ ) (the prior distribution or
marginal distribution of  ), and Q( ∗ | −1) that were used for estimation. When the candidate
generating density Q( ∗ | −1) was not symmetric, parameters , , and  that defined the
moving window function are also provided in Table 1.
References
Henson, R. A., & Templin, J. L. (2003). The moving window family of proposal distributions.
Educational Testing Service, External Diagnostic Research Group, Unpublished
Technical Report.
Table 1
Prior Distributions and Candidate Generating Densities for SICM Estimator

( ∗ )
(•)*
Q( ∗ | −1 )

Item Parameters
[ = ]
)
,
(−1000,1000) (−1
0, , .1)
[ = ]
)
,
(−1000,1000) (, )
.1
0
10,000
[ = ]
)
,
(−1000,1000) (, )
.1
0
10,000
[ = ]

(,),2 ∏=1(∏=1  |,  ) (−1000,1000) (, )
.1
,0
∏=1(∏=1  |
(),1
∏=1(∏=1  |
,
∏=1(∏=1  |
min( (),1 )+ 10,000
Structural Parameters
γ
∏=1( )
(−1000,1000)
( −1 , .1)
Examinee Parameters
++
[ = ]
)
,

∏=1 ∏=1( |

∏=1 ∏=1( |
[ = ]
)
,
∗
(−1
)
(0,1)
(−1 , .1)

−1
* (•) = (Π=1,−1
,  =  , Π=+1,
)
+ Denotes the smallest misconception main effect for option j on item i.
++ The parameter  (and thus this distribution) is a function of the parameters of the structural
model and serves as an (empirical) prior distribution for ∗ , mirroring common practice in factor
analytic, mixed effects, and item response models estimated with MCMC. Because ∗ is drawn
from the prior distribution, (∗ ) cancels with Q(∗ |−1 ) and Q(−1 |∗ ) cancels with
(−1 ).
```
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