TAM (version 1.6-0)

tam.modelfit: Model Fit Statistics in TAM

Description

This function computes several model fit statistics. It includes the Q3 statistic (Yen, 1984) and an adjusted variant of it (see Details). Effect sizes of model fit (MADaQ3, $MADRESIDCOV$, $SRMR$) are also available. The function IRT.modelfit is a wrapper to tam.modelfit, but allows convenient model comparisons by using the IRT.compareModels (CDM) function. The tam.modelfit function can also be used for fitted models outside the TAM package by applying tam.modelfit.IRT or tam.modelfit.args.

Usage

tam.modelfit(tamobj, progress = TRUE)

## S3 method for class 'tam.modelfit':
summary(object,...)

## S3 method for class 'tam.mml':
IRT.modelfit(object, \dots)
## S3 method for class 'tam.mml.3pl':
IRT.modelfit(object, \dots)
## S3 method for class 'tamaan':
IRT.modelfit(object, \dots)

## S3 method for class 'IRT.modelfit.tam.mml':
summary(object, \dots)
## S3 method for class 'IRT.modelfit.tam.mml.3pl':
summary(object, \dots)
## S3 method for class 'IRT.modelfit.tamaan':
summary(object, \dots)

tam.modelfit.IRT( object , progress=TRUE )

tam.modelfit.args( resp , probs , theta , post , progress=TRUE )

Arguments

tamobj

Object of class tam

progress

An optional logical indicating whether progress should be displayed

object

Object of class tam.modelfit (for summary) or objects for which IRT.data, IRT.irfprob and IRT.posterior have been defined (for tam.modelfit.IRT).

resp

Dataset with item responses

probs

Array with item response functions evaluated at theta

theta

Matrix with used $\bold{\theta}$ grid

post

Individual posterior distribution

...

Further arguments to be passed

Value

• A list with following entries
• stat.MADaQ3
  Global fit statistic MADaQ3 and global model test with p value obtained by Holm adjustment
• chi2.stat
  Data frame with chi square tests of condional independence for every item pair (Chen & Thissen, 1997)
• fitstat
  Model fit statistics $100 \cdot MADRESIDCOV$, $SRMR$ and $SRMSR$
• modelfit.test
  Test statistic of global fit based on multiple testing correction of $\chi^2$ statistics
• stat.itempair
  Q3 and adjusted Q3 statistic for all item pairs
• residuals
  Residuals
• Q3.matr
  Matrix of $Q_3$ statistics
• aQ3.matr
  Matrix of adjusted $Q_3$ statistics

Details

For each item $i$ and each person $n$, residuals $e_{ni}=X_{ni}-E(X_{ni})$ are computed. The expected value $E(X_{ni})$ is obtained by integrating the individual posterior distribution. The Q3 statistic of item pairs $i$ and $j$ is defined as the correlation $Q3_{ij} = Cor( e_{ni} , e_{nj} )$. It is known that under local independence, the expected value is slightly smaller than zero. Therefore, an adjusted Q3 statistic (aQ3; $aQ3_{ij}$) is computed by subtracting the average of all Q3 statistics from Q3. To control for multiple testing, a p value adjustment by the method of Holm (p.holm) is employed (see Chen, de la Torre & Zhang, 2013). An effect size of model fit (MADaQ3) is defined as the average of absolute values of $aQ3$ statistics. The SRMSR (standardized root mean square root of squared residuals, Maydeu-Olivaras, 2013) is based on comparing residual correlations of item pairs $$SRMSR = \sqrt{ \frac{1}{ J(J-1)/2 } \sum_{i < j} ( r_{ij} - \hat{r}_{ij} )^2 }$$ Additionally, the SRMR is computed as $$SRMR = \frac{1}{ J(J-1)/2 } \sum_{i < j} | r_{ij} - \hat{r}_{ij} |$$ The $MADRESIDCOV$ statistic (McDonald & Mok, 1995) is based on comparing residual covariances of item pairs $$MADRESIDCOV = \frac{1}{ J(J-1)/2 } \sum_{i < j} | c_{ij} - \hat{c}_{ij} |$$ This statistic is just multiplied by 100 in the output of this function.

References

Chen, J., de la Torre, J., & Zhang, Z. (2013). Relative and absolute fit evaluation in cognitive diagnosis modeling. Journal of Educational Measurement, 50, 123-140. Chen, W., & Thissen, D. (1997). Local dependence indexes for item pairs using item response theory. Journal of Educational and Behavioral Statistics, 22, 265-289. Maydeu-Olivares, A. (2013). Goodness-of-fit assessment of item response theory models (with discussion). Measurement: Interdisciplinary Research and Perspectives, 11, 71-137. McDonald, R. P., & Mok, M. M.-C. (1995). Goodness of fit in item response models. Multivariate Behavioral Research, 30, 23-40. Yen, W. M. (1984). Effects of local item dependence on the fit and equating performance of the three-parameter logistic model. Applied Psychological Measurement, 8, 125-145.

Examples

#############################################################################
# EXAMPLE 1: data.cqc01
#############################################################################

data(data.cqc01)
dat <- data.cqc01

#*****************************************************
#*** Model 1: Rasch model
mod1 <- tam.mml( dat )
# assess model fit
res1 <- tam.modelfit( tamobj = mod1 )
summary(res1)
# display item pairs with five largest adjusted Q3 statistics
res1$stat.itempair[1:5,c("item1","item2","aQ3","p","p.holm")]

# IRT.modelfit
fmod1 <- IRT.modelfit(mod1)
summary(fmod1)

#*****************************************************
#*** Model 2: 2PL model
mod2 <- tam.mml.2pl( dat )
# IRT.modelfit
fmod2 <- IRT.modelfit(mod2)
summary(fmod2)

# model comparison
IRT.compareModels(fmod1 , fmod2 )

#############################################################################
# SIMULATED EXAMPLE 2: Rasch model
#############################################################################

set.seed(8766)
N <- 1000    # number of persons
I <- 20      # number of items
# simulate responses
library(sirt)
dat <- sirt::sim.raschtype( rnorm(N) , b=seq(-1.5,1.5,len=I) )
#*** estimation
mod1 <- tam.mml( dat )
summary(dat)
#*** model fit
res1 <- tam.modelfit( tamobj = mod1)
summary(res1)

#############################################################################
# EXAMPLE 3: Model fit data.gpcm | Partial credit model
#############################################################################

data(data.gpcm)
dat <- data.gpcm

# estimate partial credit model
mod1 <- tam.mml( dat)
summary(mod1)

# assess model fit
tmod1 <- tam.modelfit( mod1 )
summary(tmod1)