District Hospital Trauma Mental Health Screening Night-Shift Staffing Ratios and Post-Traumatic Stress Disorder Detection Loss in Ghanaian Referral Clinics: A Multilevel Analysis of Workforce Density Thresholds

District hospital trauma units in Ghana depend on night-shift screening to identify post-traumatic stress disorder among injured patients, yet the workforce density required for reliable detection has not been specified. The analysis specifies a two-level logistic detection model in which screening completion depends on night-shift staffing ratio, patient acuity and facility random effects, and derives the workforce density at which marginal detection gain per additional staff member falls below the cost-equivalent threshold. The framework proposes that detection loss is non-linear in staffing density, with an inflection point below which each additional night-shift mental health worker recovers a disproportionate share of missed PTSD cases. The contribution is a formal specification of the staffing-density threshold and a demonstration that detection loss in Ghanaian referral clinics is better understood as a threshold phenomenon than as a linear workforce gradient. The paper concludes that staffing benchmarks expressed as minimum ratios, rather than as averages, are the appropriate policy instrument for night-shift trauma mental health screening.