Summed Score Likelihood-Based Indices for Testing Latent Variable Distribution Fit in Item Response Theory

Area of Science:

• Statistics
• Psychometrics
• Item Response Theory

Background:

• Standard item response theory (IRT) assumes normally distributed latent variables.
• Violating this assumption can bias item parameter estimates.
• Assessing latent variable distribution fit is crucial for accurate IRT analysis.

Purpose of the Study:

• To develop and evaluate novel Satorra-Bentler type moment adjustments for likelihood-based statistics.
• To test the calibration and power of these adjusted statistics for detecting latent variable nonnormality.
• To compare the performance of the proposed indices against the traditional M2 statistic.

Main Methods:

• Development of Satorra-Bentler type moment adjustments for summed score likelihood-based statistics.
• A simulation study under various conditions to assess calibration and power.
• Comparison with the M2 goodness-of-fit statistic.

Main Results:

• The proposed adjusted statistics exhibit tail-area probabilities well-approximated by central chi-squared distributions under the null hypothesis.
• The adjusted statistics demonstrate good power for detecting latent variable distributional violations.
• These indices are not sensitive to other misspecifications like multidimensionality.
• The adjusted statistics show considerably higher power against nonnormality compared to the M2 statistic.

Conclusions:

• The developed moment-adjusted statistics provide a powerful and reliable tool for testing latent variable normality in IRT.
• These methods enhance the robustness of IRT models by accurately identifying distributional assumption violations.
• The findings suggest improved accuracy in item parameter estimation when latent variable distribution is properly assessed.