EstCRMitem {EstCRM}R Documentation

Estimating Item Parameters for the Continuous Response Model


Estimate item parameters for the Continuous Response Model (Samejima,1973)


EstCRMitem(data, max.item, min.item, max.EMCycle = 500, converge = 0.01,



a data frame with N rows and m columns, with N denoting the number of subjects and m denoting the number of items.


a vector of length m indicating the maximum possible score for each item.


a vector of length m indicating the minimum possible score for each item.


a number of maximum EM Cycles used in the iteration. Default 500.


a criteria value indicating the difference between loglikelihoods of two consecutive EM cycles to stop the iteration. Default .01


type of optimization. Takes two values, either "Shojima" or "Wang&Zeng". Default is the non-iterative EM developed by Shojima(2005). See details.


a valid argument when type is equal to "Wang&Zeng". If TRUE, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is used to update Hessian.


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}


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:


Then, the conditional probability density function of the random variable Z is equal to:


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.


EstCRMitem() returns an object of class "CRM". An object of class "CRM" is a list containing the following components:


original data


descriptive statistics for the original data and the transformed z scores


estimated item parameters in the last EM Cycle


a list that reports the information for each EM cycle


The difference of loglikelihoods between the last two EM cycles


* The ID variable ,if included, has to be excluded from the data before the analysis. If the format of the data is not a "data frame", the function does not work. Please use "" to change the format of the input data.

* A previous published simulation study (Shojima, 2005) was replicated as an example to check the performance of EstCRMitem().Please see the examples of simCRM in this package.

*The example below is a reproduction of the results from another published study (Ferrando, 2002). Dr. Ferrando kindly provided the dataset used in the published study. The dataset was previously analyzed by using EM2 program and the estimated CRM item parameters were reported in Table 2 in the paper. The estimates from EstCRMitem() is comparable to the item parameter estimates reported in Table 2.


Cengiz Zopluoglu


Ferrando, P.J.(2002). Theoretical and Empirical Comparison between Two Models for Continuous Item Responses. Multivariate Behavioral Research, 37(4), 521-542.

Samejima, F.(1973). Homogeneous Case of the Continuous Response Model. Psychometrika, 38(2), 203-219.

Shojima, K.(2005). A Noniterative Item Parameter Solution in Each EM Cycyle of the Continuous Response Model. Educational Technology Research, 28, 11-22.

Wang, T. & Zeng, L.(1998). Item Parameter Estimation for a Continuous Response Model Using an EM Algorithm. Applied Psychological Measurement, 22(4), 333-343.

See Also

EstCRMperson for estimating person parameters, fitCRM for computing item-fit residual statistics and drawing empirical 3D item category response curves, plotCRM for drawing theoretical 3D item category response curves, simCRM for generating data under CRM.


##load the dataset EPIA

##Check the class. "data.frame" is required.

##Define the vectors "max.item" and "min.item". The maximum possible
##score was 112 and the minimum possible score was 0 for all items

max.item <- c(112,112,112,112,112)
min.item <- c(0,0,0,0,0)

##The maximum number of EM Cycle and the convergence criteria can be


##Estimate the item parameters
CRM <- EstCRMitem(EPIA, max.item, min.item, max.EMCycle, converge)

##Other details


##load the dataset SelfEff


##Check the class. "data.frame" is required.


##Define the vectors "max.item" and "min.item". The maximum possible
##score was 11 and the minimum possible score was 0 for all items

max.item <- c(11,11,11,11,11,11,11,11,11,11)
min.item <- c(0,0,0,0,0,0,0,0,0,0)

##Estimate the item parameters
CRM2 <- EstCRMitem(SelfEff, max.item, min.item, max.EMCycle=200, converge=.01)

##Other details


[Package EstCRM version 1.4 Index]