The conditional likelihood of response \(x_{ij}\) to
item \(j\) from person \(i\) with item parameters
\(\xi_j\) and latent ability \(\theta_i\) is
$$L(x_i|\xi,\theta_i) = \prod_j \mathrm{Pr}(\mathrm{pick}=x_{ij} | \xi_j,\theta_i).$$
Items are assumed to be conditionally independent.
That is, the outcome of one item is assumed to not influence
another item after controlling for \(\xi\) and \(\theta_i\).
The unconditional likelihood is obtained by integrating over
the latent distribution \(\theta_i\),
$$L(x_i|\xi) = \int L(x_i|\xi, \theta_i) L(\theta_i) \mathrm{d}\theta_i.$$
With an assumption that examinees are independently and identically distributed,
we can sum the individual log likelihoods,
$$\mathcal{L}=\sum_i \log L(x_i | \xi).$$
Response models \(\mathrm{Pr}(\mathrm{pick}=x_{ij} |
\xi_j,\theta_i)\)
are not implemented in OpenMx, but are imported
from the RPF
package. You must pass a list of models obtained from the RPF
package in the `ItemSpec' argument. All item models must use the
same number of latent factors although some of these factor
loadings can be constrained to zero in the item parameter matrix.
The `item' matrix contains item parameters with one item per
column in the same order at ItemSpec.
The `qpoints' and `qwidth' argument control the fineness and
width, respectively, of the equal-interval quadrature grid. The
integer `qpoints' is the number of points per dimension. The
quadrature extends from negative qwidth to positive qwidth for
each dimension. Since the latent distribution defaults to standard
Normal, qwidth can be regarded as a value in Z-score units.
The optional `mean' and `cov' arguments permit modeling of the
latent distribution in multigroup models (in a single group, the
latent distribution must be fixed). A separate latent covariance
model is used in combination with mxExpectationBA81. The point
mass distribution contained in the quadrature is converted into a
multivariate Normal distribution by
mxDataDynamic. Typically mxExpectationNormal is used
to fit a multivariate Normal model to these data. Some intricate
programming is required. Examples are given in the manual.
mxExpectationBA81 uses a sample size of \(N\) for the covariance
matrix. This differs from mxExpectationNormal which uses a
sample size of \(N-1\).
The `verbose' argument enables diagnostics that are mainly of
interest to developers.
When a two-tier covariance matrix is recognized, this expectation
automatically enables analytic dimension reduction (Cai, 2010).
The optional `weightColumn' is superseded by the weight
argument in mxData. For data with many repeated
response patterns, model evaluation time can be
reduced. An easy way to transform your data into this form is to
use compressDataFrame. Non-integer weights are supported except for
EAPscores.
mxExpectationBA81 requires mxComputeEM. During a typical
optimization run, latent abilities are assumed for examinees
during the E-step. These examinee scores are implied by the
previous iteration's parameter vector. This can be overridden
using the `EstepItem' argument. This is mainly of use to
developers for checking item parameter derivatives.
Common univariate priors are available from
univariatePrior. The standard Normal
distribution of the quadrature acts like a prior distribution for
difficulty. It is not necessary to impose any additional Bayesian
prior on difficulty estimates (Baker & Kim, 2004, p. 196).
Many estimators are available for standard errors. Oakes is
recommended (see mxComputeEM). Also available are
Supplement EM (mxComputeEM), Richardson extrapolation
(mxComputeNumericDeriv), likelihood-based confidence
intervals (mxCI), and the covariance of the rowwise
gradients.