Scales can be said to be aligned if the item sufficient statistics
imply the same item parameter estimates, regardless of dimension. Scale
alignment is currently defined only for Rasch family models with
between-items multidimensionality (i.e., each scored item belongs to exactly
one dimension).
MODEL PARAMETERIZATIONS
The partial credit model is a general Rasch family model for polytomous item
responses. Within 'TAM', the partial credit model can be parameterized in two
ways. If a 'TAM' model is fit with the option irtmodel = "PCM", then the
following model is specified for an item with \(m + 1\) response categories:
$$\log(\frac{P(x | \theta_d)}{P(x-1 | \theta_d)}) = \alpha_d \theta_d - \xi_{i(d)x}$$
for response category \(x = 1,...,m\), and
$$P(x = 0 | \theta_d) = \frac{1}{\sum_{j=0}^{m}\exp \sum_{k=0}^j (\alpha_d \theta_d -
\xi_{i(d)k})}$$
for response category \(x=0\). \(\alpha_d\) is a dimension
steepness parameter, typically fixed to 1, \(\theta_d\) is a latent
variable on dimension \(d\), and \(\xi_i(d)x\) is a step parameter for item
step \(x\) on item \(i\) belonging to dimension \(d\).
If instead a TAM model is fit with the option irtmodel = "PCM2", the model is
specified as
$$\log(\frac{P(x | \theta_d)}{P(x-1 | \theta_d)}) = \alpha_d \theta_d - \delta_{i(d)} +
\tau_{i(d)x}.$$
MODEL TRANSFORMATIONS
Under Rasch family models, the latent trait metric can be linearly
transformed. For each dimension \(d\) the parameters on the transformed
metric (denoted by the \(\sim\) symbol) are found through the transformation
parameters \(r_d\) and \(s_d\) as described by the following equations:
$$\tilde{\theta}_d = r_{d} \theta_{d} + s_{d}$$
$$\tilde{\alpha}_d = \alpha_{d} / r_{d}$$
$$\tilde{\xi}_{i(d)x} = \xi_{i(d)x} + \alpha_d s_d / r_d$$
$$\tilde{\delta}_{i(d)} = \delta_{i(d)} + \alpha_d s_d / r_d$$
$$\tilde{\tau}_{i(d)x} = \tau_{i(d)x}$$
SUFFICIENT STATISTICS
Under Rasch family models, the item sufficient statistics are the number of
examinees that score in response category \(x\) or higher,
\(x = 1,...,m\). For the purpose of scale alignment, we consider sufficient
statistics to be the proportion of examinees that score in response category
\(x\) or higher. This definition allows for scale alignment in the
presence of missing data.
THURSTONE THRESHOLDS
Scales are aligned if the same sufficient statistics imply the
same item parameters, regardless of dimension. The success of scale alignment
is difficult to assess because the item sufficient statistics typically
differ across items and dimensions. Under the Rasch model for binary item
responses, the success of scale alignment can be assessed by looking at the
rank-order correlation (e.g., Kendall's tau) between item sufficient
statistics and item parameter estimates.
However, under the partial credit model, item sufficient statistics need not
be monotonically related to estimated item parameters. Under this model,
we can assess the quality of scale alignment by taking the rank-order
correlation between item sufficient statistics and Thurstone thresholds.
Thurstone thresholds are defined as the \(\theta\) value at which the
probability of responding in category \(x\) or higher equals .5. Thurstone
thresholds, in most cases, will be monotonically related to item sufficient
statistics (within dimensions). Note that the item difficulty estimates
under the Rasch model for binary items are also Thurstone thresholds.
ALIGNMENT METHODS
Two types of scale alignment methods have been developed.
The first class of methods, historically called delta-dimensional alignment
(DDA), requires fitting both a multidimenisonal model and a model in which
all items belong to a single dimension. With these two sets of parameter
estimates, the transformation parameters \(r_d\) and \(s_d\) are then
found so that, for each dimension, the means and standard deviation of
parameters from the transformed multidimensional models equal the means and
standard deviations of parameters from the unidimensional model. Under the
ordinary Rasch model, the estimated item difficulties can be used for
transformation (which is done if either method "DDA1" or "DDA2" is selected).
Under the partial credit model, either the \(\delta\) parameters or the
Thurstone thresholds from the two models may be used within the DDA (note
that DDA using item \(\xi\) parameters tends to be unsuccessful). Method
"DDA1" uses the item \(\delta\) parameters, and method "DDA2" uses the
Thurstone thresholds. If all items are binary, "DDA1" and "DDA2" are identical.
The second class of methods, called logistic regression alignment (LRA),
requires fitting a logistic regression between item sufficient statistics
and Thurstone thresholds for each dimension. The fitted logistic regression
coefficients can then be used to estimate \(r_d\) and \(s_d\) so that the
same logistic regression curve expresses the relationship between sufficient
statistics and Thurstone thresholds for all dimensions.
For either the DDA or LRA method, a reference dimension (by default, the
first dimension) is specified such that \(r_d = 1\) and \(s_d = 0\) for
the reference dimension.