Robustness of the Compositional Data Approach in Bipolar Psychometric Likert Scales Big Skewed Data Analysis

Managerial decision making, the enhancement of recommender systems and the prediction of human behaviour depend on individual attitudes and preferences. To gather these, bipolar psychometric questionnaires are used among other methods. Recently, the compositional data structure known as the Simplex was identified in bipolar Likert scale data. This concept posits that any level of agreement with an item assertion implies a level of disagreement, resulting in bivariate compositional information. Through the use of an isometric log-ratio (ilr) transformation, bivariate data can be converted to a real-valued interval scale. It is well-established that the ilr approach enhances the statistical power of correlation tests and two-sample t-tests based on Student's t-distribution, provided that the Central Limit Theorem (CLT) holds true. However, in practical applications, adherence to the CLT depends on factors such as the number of items and the distribution shape of the data generating process (DGP). Psychometric data may consist of only a few item responses, and DGPs can exhibit skewness, leading to violations of the CLT. Through simulation studies, we demonstrate that the ilr approach performs effectively even when the CLT assumptions regarding skewness and small item numbers are violated, thereby increasing the statistical power of correlation tests. This research extends previous findings and underscores the versatility of the ilr approach as a dependable tool in psychometric big data analysis.