Sampling size determination under budget constraints using two-stage procedure: Examining impact of PERMA well-being intervention in nature among school students

Recent advancements in designing optimal sampling frameworks, particularly under cost or budget constraints, have gained significant attention. Several studies aim to maximize statistical power for hypothesis tests within these constraints. However, existing methods lack the flexibility to accommodate interventions while maximizing power for four specific tests under budgetary limitations. To address this gap, we propose a two-stage procedure that estimates the optimal sample sizes for two groups and simultaneously tests four hypotheses with maximum power under budget constraints. Through extensive simulation analyses, we demonstrate the performance of the proposed procedure. To illustrate the application of our procedure, we implemented a PERMA (Positive Emotions, Engagement, Relationships, Meaning, and Accomplishment) intervention experiment. This experiment involved two groups, viz. experimental group and control group. This was conducted in a natural setting to enhance the well-being of school students in India. The experimental group participated in a five-day PERMA intervention, while the control group did not receive any intervention. Participants in both groups completed pre- and post-tests using the WHO-5 well-being measure. Given the limited budget for this experiment, optimizing sample size to achieve maximum statistical power was critical. This study provides a practical framework for optimizing sample sizes in budget-constrained experiments, particularly in psychological research.