fitCRM {EstCRM}R Documentation

Compute item fit residual statistics for the Continuous Response Model


Compute item fit residual statistics for the Continuous Response Model as described in Ferrando (2002)


fitCRM(data, ipar, est.thetas, max.item,group)



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


a matrix with m rows and three columns, with m denoting the number of items. The first column is the a parameters, the second column is the b parameters, and the third column is the alpha parameters


object of class "CRMtheta" obtained by using EstCRMperson()


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


an integer, number of ability groups to compute item fit residual statistics. Default 20.


The function computes the item fit residual statistics as decribed in Ferrando (2002). The steps in the procedure are as the following:

1- Re-scaled θ estimates are obtained.

2- θ estimates are sorted and assigned to k intervals on the θ continuum.

3- The mean item score is computed in each interval for each of the items.

4- The expected item score and the conditional variance in each interval are obtained with the item parameter estimates and taking the median theta estimate for the interval.

5- An approximate standardized residual for item m at ability interval k is obtained as:

z_{mk}= \frac{\bar{X}_{mk} - E(X_{m}|\theta_{k})}{\sqrt{\frac{\sigma^2(X_{m}|\theta_{k})}{N_{k}}}}



a data frame with k rows and m+1 columns with k denoting the number of ability intervals and m denoting the number of items. The first column is the ability interval. Other elements are the standardized residuals of item m in ability interval k.


a list of length m with m denoting the number of items. Each element is a 3D plot representing the item category response curve based on the empirical probabilities. See examples below.


Cengiz Zopluoglu


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

See Also

EstCRMperson for estimating person parameters, EstCRMitem for estimating item parameters plotCRM for drawing theoretical 3D item category response curves, simCRM for generating data under CRM.


##load the dataset EPIA


##Due to the run time issues for examples during the package building
##I had to reduce the run time. So, I run the fit analysis for a subset
##of the whole data, the first 100 examinees. You can ignore the
##following line and just run the analysis for the whole dataset.
##Normally, it is not a good idea to run the analysis for a 100

EPIA <- EPIA[1:100,]  #Please ignore this line

##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)

##Estimate item parameters

CRM <- EstCRMitem(EPIA, max.item, min.item, max.EMCycle = 500, converge = 0.01)
par <- CRM$param

##Estimate the person parameters

CRMthetas <- EstCRMperson(EPIA,par,min.item,max.item)

##Compute the item fit residual statistics and empirical item category
##response curves

fit <- fitCRM(EPIA, par, CRMthetas, max.item,group=10)

##Item-fit residual statistics


##Empirical item category response curves
fit$emp.irf[[1]]   #Item 1

[Package EstCRM version 1.4 Index]