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Assessing Dimensionality in Non-Positive Definite Tetrachoric Correlation Matrices: Does Matrix Smoothing Help?

Justin D Kracht1, Niels G Waller1

  • 1University of Minnesota.

Multivariate Behavioral Research
|December 30, 2020
PubMed
Summary

Matrix smoothing modestly improves parallel analysis for binary data tetrachoric correlation matrices, aiding dimensionality recovery in factor analysis. Further research is needed to explore practical significance.

Keywords:
Matrix smoothingbinary itemsdimensionality recoverynon-positive definiteparallel analysis

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Area of Science:

  • Psychometrics
  • Statistical Modeling

Background:

  • Assessing dimensionality in factor analysis is crucial for model accuracy.
  • Non-positive definite (NPD) tetrachoric correlation matrices pose challenges for standard analyses.
  • Parallel analysis is a common method for determining the number of factors.

Purpose of the Study:

  • To investigate the impact of matrix smoothing on dimensionality recovery using parallel analysis for NPD tetrachoric correlation matrices.
  • To extend previous research by examining a wider range of factors, model error, and realistic item parameters.
  • To evaluate the practical utility of smoothing algorithms in improving parallel analysis for binary data.

Main Methods:

  • Two simulation studies were conducted, replicating and extending prior work.
  • Non-positive definite (NPD) tetrachoric correlation matrices were generated and smoothed using three algorithms.
  • Parallel analysis was applied to both original and smoothed matrices to assess dimensionality.
  • Factor analysis was applied to an empirical dataset (Adjective Checklist data) to demonstrate the effects.

Main Results:

  • Matrix smoothing provided modest improvements in the performance of parallel analysis with binary data.
  • The benefits of smoothing were often small in practical terms.
  • The study explored the influence of varying numbers of factors and levels of model error.

Conclusions:

  • Matrix smoothing can enhance parallel analysis for NPD tetrachoric correlation matrices with binary data, though improvements are often marginal.
  • The findings suggest that while smoothing is beneficial, its practical impact on dimensionality assessment may be limited.
  • The study highlights the importance of considering matrix properties and smoothing techniques in factor analysis research.