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

Managerial decision making, the enhancement of recommender systems, the prediction of human behaviour and the selection of adequate psychotherapeutic measures all depend on individual attitudes, preferences and psychometric profiles. 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. Applying standard statistical procedure (e.g., computation of mean, standard deviation, linear regression etc.) to Simplex data yields biased results. Through the use of an isometric log-ratio (ilr) transformation, bivariate data can be converted to a real-valued interval scale where means and sums of item responses remain unbiased. Correlations form the basis of many psychometric big data analyses including, but not limited to, linear regression models (including structural equation models, partial least squares path modeling and moderator analysis), development and validation of bipolar Likert scales and econometric decision making. It is well-established that the ilr approach enhances the statistical power of correlation 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 bimodality, leading to violations of the CLT. Through simulation studies, we demonstrate that the ilr approach performs effectively even when the CLT assumptions regarding bimodality and small item numbers are violated, thereby increasing the statistical power of correlation tests. This research extends previous findings.