PARSCALE

IRT Models Supported in PARSCALE

GRM in PARSCALE is, in the logistic form, given by

$$\begin{array}{l}{P}_{jk}\left(\theta \right)=\frac{\exp \left(D{a}_j\left(\theta -b_j+d_{jk}\right)\right)}{1+\exp \left(D{a}_j\left(\theta -b_j+d_{jk}\right)\right)}\\ -\frac{\exp \left(D{a}_j\left(\theta -b_j+d_{jk+1}\right)\right)}{1+\exp \left(D{a}_j\left(\theta -b_j+d_{jk+1}\right)\right)},\end{array}$$

where D is the scaling coefficient, which is either 1 (for logistic metric) or 1.7 (for the normal metric); aj and bj are a common slope parameter (i.e., item discrimination parameter) and a threshold parameter (i.e., item difficulty parameter) for item j, respectively; and djk is the score-category parameter for category k. Unlike PARSCALE, in Samejima’s original GRM work in 1969 and 1972, –bj + djk was simply expressed as –bjk. PARSCALE also supports the normal-ogive version of GRM, which is

$$P_{jk}\left(\theta \right)=\frac{1}{\sqrt{2\pi }}{\displaystyle {\int }_{a_j\left(\theta -b_j+d_{jk}+1\right)}^{a_j\left(\theta -b_j+d_{jk}\right)}\exp \left(\frac{-t^2}{2}\right)}dt.$$

Another IRT model for polytomously scored response data supported by PARSCALE is the Masters partial credit model and its generalized version, GPCM, by Muraki. The GPCM is given by

$$p_{jk}\left(\theta \right)=\frac{\exp \left({\displaystyle {\sum }_{v=0}^{k}D{a}_j}\left(\theta -b_{jv}\right)\right)}{{\displaystyle {\sum }_{w=0}^{mj}\exp \left({\displaystyle {\sum }_{v=0}^{w}D{a}_j\left(\theta -b_{jv}\right)}\right)}},$$

[Page 1208]where k = 1, 2, … , mj.

For dichotomously scored response data, PARSCALE supports the three-parameter logistic model, in which the lower asymptote of item characteristic function is addressed by c-parameter in

$$P_j\left(\theta \right)=c_j+\left(1-c_j\right)\frac{1}{1+\exp \left(-D{a}_j\left(\theta -b_j\right)\right)}.$$

Estimation Methods and Processes

PARSCALE divides the model estimation process into four “phases.” Phase 0 is for processing the syntax input and other data input files and calibration settings. Phase 1 is for summarizing the response data matrix (reporting frequencies of response categories for each item) and for calculating the mean and standard deviation of response for each item as well as the Pearson product-moment correlation and polyserial correlation coefficients, which are used for computing the initial values for a- and b-item parameters for each item.

Phase 2 is for calibrating item parameters. PARSCALE uses the marginal maximum-likelihood estimation (MMLE) method, which is an instance of the expectation–maximization (EM) algorithm. In the expectation (E) step of the EM algorithm with MMLE at the beginning, given the initial item parameter values from Phase 1, the expected posterior distribution is computed based on marginalized θ distribution, which is from an integral of probabilities of different response patterns across quadrature nodes with preset weights. In PARSCALE, the integral is approximated using the Gauss–Hermite quadrature method. In the maximization (M) step of Phase 2, using the expected values from the E step, item parameter values that maximize the log-likelihood function are searched using the Newton–Gauss (i.e., Fisher scoring) iterative procedure. By default, PARSCALE assumes a log-normal prior distribution for a parameters (i.e., slope parameter), a normal prior distribution for b parameters (i.e., threshold parameter), and a β prior distribution for c parameters (i.e., lower asymptote parameter). Once the M step search finds the item parameter estimates, the E and M steps are repeated until either the number of iterations reaches a maximum level or the convergence criterion is met. PARSCALE allows users extensive controls over the MMLE setting, including the number and weight of quadrature points, the maximum number of EM iterations, the maximum number of Newton–Gauss iterations, and conversion criteria for the EM and Newton–Gauss procedures.

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