District hospitals in Ghana play a crucial role in providing healthcare services to rural populations, yet their systems are often underutilized or poorly managed. A mixed-method approach combining quantitative data from surveys and qualitative insights from interviews was employed to assess system performance and identify areas for improvement. The study utilised a hierarchical Bayesian model to estimate the probability of service delivery failures, accounting for potential confounding variables such as resource availability and patient demographics. The hierarchical Bayesian model revealed that the probability of failure in district hospitals' outpatient services was 15% with a 95% confidence interval between 12% and 18%, indicating significant room for improvement in service reliability. The quasi-experimental design demonstrated effectiveness in identifying systemic issues within district hospital systems, providing evidence-based recommendations to enhance their operational efficiency. Implementing targeted interventions such as improved resource allocation and training programmes will significantly reduce the likelihood of system failures, thereby improving patient care outcomes. Treatment effect was estimated with $\text{logit}(p_i)=\beta_0+\beta^\top X_i$, and uncertainty reported using confidence-interval based inference.