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Abstract
The linear logistic test model (LLTM) is a well-recognized psychometric model for examining the components of difficulty in cognitive tests and validating construct theories. The plausibility of the construct model, summarized in a matrix of weights, known as the Q-matrix or weight matrix, is tested by (1) comparing the fit of LLTM with the fit of the Rasch model (RM) using the likelihood ratio (LR) test and (2) by examining the correlation between the Rasch model item parameters and LLTM reconstructed item parameters. The problem with the LR test is that it is almost always significant and, consequently, LLTM is rejected. The drawback of examining the correlation coefficient is that there is no cut-off value or lower bound for the magnitude of the correlation coefficient. In this article we suggest a simulation method to set a minimum benchmark for the correlation between item parameters from the Rasch model and those reconstructed by the LLTM. If the cognitive model is valid then the correlation coefficient between the RM-based item parameters and the LLTM-reconstructed item parameters derived from the theoretical weight matrix should be greater than those derived from the simulated matrices.
Highlights
The linear logistic test model (LLTM; Fischer, 1973) is an extension of the Rasch model (RM, Rasch, 1960/1980) which imposes some linear constraints on the item parameters
Based on the logic of parallel analysis (PA), we suggest that if the cognitive model is valid the correlation coefficient between the Rasch model item parameters and the LLTM-reconstructed item parameters derived from the theoretical weight matrix should be greater than the correlation coefficients derived from random simulated weight matrixes with the same number of items and cognitive operations
The correlation between the RM item parameters and the item parameters reconstructed by the LLTM was r = 0.8506 for the listening test and r = 0.7208 for the reading test
Summary
The linear logistic test model (LLTM; Fischer, 1973) is an extension of the Rasch model (RM, Rasch, 1960/1980) which imposes some linear constraints on the item parameters. Where qij is the given weight of the basic parameter j on item i, ηj is the estimated difficulty of the basic parameter j, and c is a normalization constant. The motivation behind this extension of the Rasch model is to investigate and parameterize the cognitive operations and mental processes. Q-Matrix Validation for LLTM that are involved in solving the items (basic parameters) (Fischer, 1995). Under the LLTM, theoretically, the difficulty parameters of the processes hypothesized to be involved in solving the items ηj add up and constitute the Rasch model item difficulty parameters βi.
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