Breakdown of the compositional data approach in psychometric Likert scale big data analysis: about the loss of statistical power of two-sample t-tests applied to heavy-tailed big data

Bipolar psychometric scale data play a crucial role in psychological healthcare and health economics, such as in psychotherapeutic profiling and setting standards. Creating an accurate psychological profile not only benefits the patient but also saves time and costs. The quality of psychotherapeutic measures directly impacts grant funding decisions, influencing managerial choices. Moreover, the accuracy of consumer data analyses affects costs, profits, and the long-term sustainability of decisions. Considering psychometric bipolar scale data as compositional data can enhance the statistical power of well-known paired and unpaired two-sample t-tests, supporting managerial decision-making and the development or implementation of health interventions. This increase in statistical power is observed when the central limit theorem (CLT) holds true in statistics. Through stochastic simulation, this study explores the impact of violating the CLT on statistical power of the unpaired t-test under heavy-tailed data generating processes (DGPs) with finite variance. The findings reveal a reduction in statistical power based on specific parameters like the psychometric limit of quantification, the number of items in a questionnaire, the response scale used, and the dispersion of the DGP.