fitCRM {EstCRM} R Documentation

## Compute item fit residual statistics for the Continuous Response Model

### Description

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

### Usage

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


### Arguments

 data a data frame with N rows and m columns, with N denoting the number of subjects and m denoting the number of items. ipar 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 est.thetas object of class "CRMtheta" obtained by using EstCRMperson() max.item a vector of length m indicating the maximum possible score for each item. group an integer, number of ability groups to compute item fit residual statistics. Default 20.

### Details

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}}}} 

### Value

 fit.stat 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. emp.irf 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

### References

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

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.

### Examples



data(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
##subjects

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$param ##Estimate the person parameters CRMthetas <- EstCRMperson(EPIA,par,min.item,max.item) ##Compute the item fit residual statistics and empirical item category ##response curves fit <- fitCRM(EPIA, par, CRMthetas, max.item,group=10) ##Item-fit residual statistics fit$fit.stat

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



[Package EstCRM version 1.4 Index]