Robustness of the ilr Approach in Likert Scales W-Shaped Data Analysis

Accurate psychometric profiling and the selection of appropriate therapeutic interventions are essential components of any psychotherapeutic treatment. Developing a precise psychological profile not only benefits the patient but also saves time and reduces costs. Likert scales produce compositional data, as each level of agreement with an item assertion corresponds to a corresponding level of disagreement. By utilizing an isometric log-ratio (ilr) transformation, the bivariate information can be converted to a real-valued interval scale, resulting in unbiased statistical outcomes that enhance the statistical power of the Pearson correlation significance test, provided that the central limit theorem (CLT) holds true for statistics. However, in practical applications, the CLT depends on factors such as the number of components (i.e., items) and the distributional shape of the (ilr transformed) data generating process (DGP). Through simulations, we demonstrate that the ilr approach remains effective even when the CLT assumptions are violated. This indicates that the ilr method is robust in scenarios with few components and extremely shaped DGPs, thereby boosting the statistical power of correlation tests. Using the ilr approach sample sizes can be reduced while maintaining the statistical power and saving sampling costs. W-shaped DGPs can occur if individuals are indifferent or favor extreme values, e.g., concerning the acceptance or rejection of life-prolonging measures in hospital or the death penalty. The findings of this study extend previous research, highlighting the versatility and reliability of the ilr approach in analyzing psychometric data. The results support the medical treatments of patients as well as the improvement of health information systems. This has implications for psychometric health economics (e.g., grant funding and QALY index values), patient well-being, grant allocation decisions, economic planning, and overall profitability.