EstCRM (version 1.6)

fitCRM: 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)

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.

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.

Author

Cengiz Zopluoglu

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:

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

References

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.

Examples

  if (FALSE) {

    ##load the dataset EPIA
    
    data(EPIA)
    
    ##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
  }