A Mathematical Index for Quantifying and Reducing Student Depression Risk Through Personalised Behavioural Guidance

Student depression represents one of the most pressing mental health challenges in higher education, with prevalence rates exceeding 30% globally and significant consequences for academic outcomes and long term wellbeing. Existing screening tools either rely on opaque machine learning classifiers that cannot explain their outputs or employ static clinical checklists that provide no actionable lifestyle guidance. This paper proposes the Psychological Risk Index (PRI), a mathematically explicit and fully interpretable framework for quantifying depression risk in students using behavioural and lifestyle data. The PRI introduces two innovations. First, it separates risk and resilience into two mathematically distinct indices the Raw Risk Index (RRI) and the Resilience Index (RI) combined through a multiplicative dampening equation that reflects the psychological reality that resilience actively buffers rather than simply subtracts from risk. Second, it extends the index through partial derivative analysis to producea personalised intervention engine that identifies the single behavioural change producing the greatest risk reduction for each individual student. Empirical weights are derived from logistic regression applied to 27,898 students, and all parameters are optimised through exhaustive cross-validation across 1,600 combinations, improving AUC from 0.894 to 0.904. A nonlinear quadratic formulation captures the empirically observed sleep optimum at 5.5 hours. Subgroup analysis confirms consistent performance across gender (Male AUC = 0.904, Female AUC = 0.905) and all degree types (range 0.899 to 0.939). Population-level analysis reveals that 57.5% of students achieve greatest risk reduction through dietary improvement, providinguniversities with a concrete, evidence-based target for mental health intervention programmes. The PRI demonstrates that interpretable mathematical frameworks can match black-box model performance within 1% AUC while delivering the transparency and personalised actionability that clinical and educational settings require.