Samejima (1973) proposed an IRT model for continuous item scores as a limiting form of the graded response model. Even though the Continuous Response Model (CRM) is as old as the well-known and popular binary and polytomous IRT models, it is not commonly used in practice. This may be due to the lack of accessible computer software to estimate the parameters for the CRM. Another reason might be that the continuous outcome is not a type of response format commonly observed in the field of education and psychology. There are few published studies that used the CRM (Ferrando, 2002; Wang & Zeng, 1998). In the field of education, the model may have useful applications for estimating a single reading ability from a set of reading passages in the Curriculum Based Measurement context. Also, this type of continuous response format may be more frequently observed in the future as the use of computerized testing increases. For instance, the examinees or raters may check anywhere on the line between extremely positive and extremely negative in a computerized testing environment rather than responding to a likert type item.
Wang & Zeng (1998) proposed a re-parameterized version of the CRM. In this re-parameterization, the probability of an examinee i with a spesific θ obtaining a score of x or higher on a particular item j with a continuous measurement scale ranging from 0 to k and with the parameters a, b, and α is defined as the following:
$$
P(X_{ij} \geq x|\theta_{i},a_{j},b_{j},\alpha_{j})=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{v}{e^{\frac{-t^2}{2}}dt}
$$
\(
v=a_{j}(\theta_{i}-b_{j}-\frac{1}{\alpha_{j}}ln\frac{x_{ij}}{k_{j}-x_{ij}})
\)
where a is a discrimination parameter, b is a difficulty parameter, and α represents a scaling parameter that defines some scale transformation linking the original observed score scale to the θ scale (Wang & Zeng, 1998). k is the maximum possible score for the item. a and b in this model have practical meaning and are interpreted same as in the binary and polytomous IRT models. α is a scaling parameter and does not have a practical meaning.
In the model fitting process, the observed X scores are first transformed to a random variable Z by using the following equation:
$$
Z_{ij}=ln(\frac{X_{ij}}{k_{j}-X_{ij}})
$$
Then, the conditional probability density function of the random variable Z is equal to:
$$
{f}(z_{ij}|\theta_{i},a_{j},b_{j},\alpha_{j})=\frac{a_{j}}{\sqrt{2\pi}\alpha_{j}}exp^{-\frac{[a_{j}(\theta_{i}-b_{j}-\frac{z_{ij}}{\alpha_{j}})]^2}{2}}
$$
The conditional pdf of Z is a normal density function with a mean of α(θ-b) and a variance of α^2/ a^2.
Wang & Zeng (1998) proposed an algorithm to estimate the CRM parameters via marginal maximum likelihood and Expectation-Maximization (EM) algorithm. In the Expectation step, the expected log-likelihood function is obtained based on the integration over the posterior θ distribution by using the Gaussian quadrature points. In the Maximization step, the parameters are estimated by solving the first and second derivatives of the expected log-likelihood function with rescpect to a, b, and α parameters via Newton-Raphson procedure. A sequence of E-step and M-step repeats until the difference between the two consecutive loglikelihoods is smaller than a convergence criteria. This procedure is available through type="Wang&Zeng" argument. If type is equal to "Wang&Zeng", then user can specify BFGS argument as either TRUE or FALSE. If BFGS argument is TRUE, then the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is used to approximate Hessian. If BFGS argument is FALSE, then the Hessian is directly computed.
Shojima (2005) simplified the EM algorithm proposed by Wang & Zeng(1998). He derived the closed formulas for computing the loglikelihood in the E-step and estimating the parameters in the M-step (please see the reference paper below for equations). He showed that the equations of the first derivatives in the M-step can be solved algebraically by assuming flat (non-informative) priors for the item parameters. This procedure is available through type="Shojima" argument.