Description

The function tam.modelfit 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 CDM::IRT.compareModels 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.

The function tam.Q3 computes the Q_3 statistic based on weighted likelihood estimates (see tam.wle).

Usage

Arguments

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} ). The residuals in tam.modelfit are calculated by integrating values of the individual posterior distribution. Residuals in tam.Q3 are calculated by using weighted likelihood estimates (WLEs) from tam.wle.

It is known that under local independence, the expected value of Q_3 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. An equivalent statistic based on the Q_3 statistic is similar to the standardized generalized dimensionality discrepancy measure (SGDDM; Levy, Xu, Yel & Svetina, 2015).

The SRMSR (standardized root mean square root of squared residuals, Maydeu-Olivaras, 2013) is based on comparing residual correlations of item pairs

SRMSR=√{ \frac{1}{ J(J-1)/2 } ∑_{i < j} ( r_{ij} - \hat{r}_{ij} )^2 }

Additionally, the SRMR is computed as

SRMR=\frac{1}{ J(J-1)/2 } ∑_{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 } ∑_{i < j} | c_{ij} - \hat{c}_{ij} |

This statistic is just multiplied by 100 in the output of this function.

Value

A list with following entries

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. doi: 10.1111/j.1745-3984.2012.00185.x

Chen, W., & Thissen, D. (1997). Local dependence indexes for item pairs using item response theory. Journal of Educational and Behavioral Statistics, 22, 265-289.

Levy, R., Xu, Y., Yel, N., & Svetina, D. (2015). A standardized generalized dimensionality discrepancy measure and a standardized model-based covariance for dimensionality assessment for multidimensional models. Journal of Educational Measurement, 52(2), 144–158. doi: 10.1111/jedm.12070

Maydeu-Olivares, A. (2013). Goodness-of-fit assessment of item response theory models (with discussion). Measurement: Interdisciplinary Research and Perspectives, 11, 71-137. doi: 10.1080/15366367.2013.831680

McDonald, R. P., & Mok, M. M.-C. (1995). Goodness of fit in item response models. Multivariate Behavioral Research, 30, 23-40. doi: 10.1207/s15327906mbr3001_2

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. doi: 10.1177/014662168400800201

Examples