Fig. 1

Observed and bias-adjusted odds ratios for firearm suicide using probabilistic quantitative selection bias analysis. Note: Odds ratios (ORs) are plotted on the log scale. Observed ORs are from separate models adjusted for sex, age, death year, marital status, educational attainment, urbanicity. The final bias-adjusted OR is adjusted for systematic selection bias and random error via the formula: \(\text{exp}\left(\text{ln}({\widehat{\text{OR}}}_{\text{adj}})+N\left(\text{0,1}\right)\times \text{SE}(\text{ln}({\widehat{\text{OR}}}_{\text{observed}})\right)\) where \({\widehat{\text{OR}}}_{\text{adj}}= {\widehat{\text{OR}}}_{\text{observed}}\times {\text{OR}}_{\text{select}}\) where \({\widehat{\text{OR}}}_{\text{observed}}\) is the observed OR (adjusted for sex, age, death year, marital status, educational attainment, urbanicity) and \({\text{OR}}_{\text{select}}=\frac{{S}_{\text{case},0}{S}_{\text{control},1}}{{S}_{\text{case},1}{S}_{\text{control},0}}\) where S is a selection proportion and exposed and unexposed are indexed by 1 and 0, respectively. We assumed all cases were selected, thus \({\text{OR}}_{\text{select}}\) simplified to \(\frac{{S}_{\text{control},1}}{{S}_{\text{control},0}}\). In the absence of known selection probabilities for the control group, we used the comparisons of exposure rates among MVC decedents and the general California population in eTable 1, Additional file 1 as a benchmark for \({\text{OR}}_{\text{select}}\). For example, the \({\text{OR}}_{\text{select}}\) for suicidal ideation/attempt was 8.00/4.67 = 1.71. We used a Monte Carlo simulation with 50,000 iterations, specifying a triangular distribution with mode equal to \({\text{OR}}_{\text{select}}\) and lower and upper limits ranging from 20% lower to 20% higher than the mode. The median of this distribution is the final bias-adjusted OR and the 2.5th and 97.5th percentiles constitute the 95% simulation interval