Calculate the \(Q_3\) statistic to evaluate the assumption of independent items.
Arguments
- x
A measrdcm object.
- ...
Unused.
- crit_value
The critical value threshold for flagging the residual correlation of a given item pair. The default is 0.2, as described by Chen and Thissen (1997).
- summary
A summary statistic to be returned. Must be one of
"q3max"or"q3star"(see Details). IfNULL(the default), no summary statistic is return, and all residual correlations are returned.
Value
If summary = NULL, a tibble with the residual correlation and
flags for all item pairs. Otherwise, a numeric value representing the
requested summary statistic.
Details
Psychometric models assume that items are independent of each other, conditional on the latent trait. The \(Q_3\) statistic (Yen, 1984) is used to evaluate this assumption. For each observed item response, we calculate the residual between the model predicted score and the observed score and then estimate correlations between the residuals across items. Each residual correlation is a \(Q_3\) statistic.
Often, a critical values is used to flag a residual correlation above a given threshold (e.g., Chen & Thissen, 1997). Alternatively, we may use a summary statistic such as the maximum \(Q_3\) statistic (\(Q_{3,max}\); Christensen et al., 2017), defined as
$$Q_{3,max} = \text{max}_{i>j}\left|Q_{3,ij}\right|$$
Or the mean-adjusted maximum \(Q_3\) statistic (\(Q_{3,*}\); Marais, 2013), defined as
$$ \overline{Q}_3 = \begin{pmatrix} I\\\ 2\end{pmatrix}^{-1} \displaystyle\sum_{i>j}Q_{3,ij} \\ Q_{3,*} = Q_{3,max} - \overline{Q}_3 $$
References
Chen, W.-H., & Thissen, D. (1997). Local dependence indexes for item pairs using item response theory. Journal of Educational and Behavioral Statistics, 22(3), 265-389. doi:10.3102/10769986022003265
Christensen, K. B., Makransky, G., & Horton, M. (2017). Critical values for Yen's Q3: Identification of local dependence in the Rasch model using residual correlations. Applied Psychological Measurement, 41(3), 178-194. doi:10.1177/0146621616677520
Marais, I. (2013). Local dependence. In K. B. Christensen, S. Kreiner, & M. Mesbah (Eds.), Rasch models in health (pp. 111-130). Wiley.
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(2), 125-145. doi:10.1177/014662168400800201
Examples
# example code
model_spec <- dcm_specify(qmatrix = dcmdata::mdm_qmatrix,
identifier = "item")
model <- dcm_estimate(dcm_spec = model_spec, data = dcmdata::mdm_data,
identifier = "respondent", method = "optim",
seed = 63277)
yens_q3(model)
#> # A tibble: 6 × 4
#> item_1 item_2 resid_corr flag
#> <chr> <chr> <dbl> <lgl>
#> 1 mdm1 mdm2 -0.132 FALSE
#> 2 mdm1 mdm3 -0.0910 FALSE
#> 3 mdm1 mdm4 -0.123 FALSE
#> 4 mdm2 mdm3 -0.110 FALSE
#> 5 mdm2 mdm4 -0.249 TRUE
#> 6 mdm3 mdm4 -0.0988 FALSE