Logistic Regression Model with Jumps for Depression

Abstract

This study develops a Bayesian logistic regression model incorporating treatment-like jumps to predict depression transitions among police officers. Six symptoms (insomnia, guilt, suicidal ideation, psychomotor retardation, anxiety and fatigue), each scored from 0 to 3, are linked to latent normal or depressed states through a logit function. Treatment is modelled as a jump that reduces one symptom score, with scores floored at zero, yielding a finite-mixture recovery probability. Data were simulated in R 4.4.5 for 100,000 officers, without using clinical records. Officers were classified as depressed if any symptom score exceeded zero, and recovery probabilities were generated for depressed officers using the treatment-adjusted model. Model parameters were estimated using Metropolis-within-Gibbs Markov chain Monte Carlo (MCMC), with 30,000 iterations and a 10,000-iteration burn-in. The estimates recovered the simulation's true parameter values, with convergence occurring within approximately 1,000–1,600 iterations. However, symptom coefficient estimates under treatment jumps of sizes one, two and three had posterior standard deviations comparable to their means, indicating substantial uncertainty and no clearly distinguishable effects from zero. Modelled transition probabilities ranged from 0.5001 to 0.5037 across the treatment scenarios. Probabilities decreased for officers with symptom scores above one but remained unchanged for symptom-free officers. Within the simulation, the jump model produced lower depression-transition probabilities than the treatment model, which in turn produced lower probabilities than the conventional logistic regression model. The findings demonstrate the model's potential for incorporating symptom-specific treatment effects while highlighting the need for validation using clinical data.