Prior research (Kyung-Sook et al., 2018), and based on statistical interactions we found in preliminary analyses, indicated that gender modifies the relationship between the main exposures of interest and suicidality. Consequently, we carried out the analyses stratified by gender. Cox proportional hazards regression was used to investigate deaths by suicide. The proportional hazards assumption was tested for using Schoenfeld residuals. For hospital admissions due to self-harm, however, the proportional hazards assumption was not met for loneliness, so data were reanalysed using a Royston Parmar model (Royston and Lambert, 2011), with Akaike information criterion from preliminary analyses indicating that two knots should be used to model the baseline hazard, and single time varying parameter, for loneliness.

For each participant the start date for the follow-up period used in analyses was the date of their first attendance at a UK Biobank centre at baseline, which ranged from March 2006 to October 2010. Participants were censored upon death, and for the death by suicide analyses, the last date (February 2018 for England and Wales and until June 2017 for Scotland) that mortality records were available, and for the analyses of self-harm, the last date (March 2015) that hospital records were available.

Six different models are presented for both deaths by suicide and hospital admissions for self-harm. The first three models are presented for each of living arrangements, loneliness, and emotional support separately. Models 1 are univarable regression models only including each of the main independent variables. Models 2 adjust for all sociodemographic variables, and Models 3 additionally adjust for the health variables. Models 4 adds loneliness and Models 5 adds emotional support to Models 3. In Models 6, all variables were included.

We accounted for missing data using multiple imputation by chained equations, generating twenty imputed data sets. Imputation models were stratified by gender and included age, living arrangements, loneliness, emotional support, all variables used in the models, the Nelson-Aalen estimate of cumulative hazard, survival status (Cleves et al., 2016), and additional variables to improve model fit, including household income, participation in social groups, contact with friends and family, parental depression, limiting longstanding illness and self-rated health. These variables were not included in the main models because they either had comparatively high rates of missing data which limited their utility in preliminary complete case analysis, or, in the case of health variables, might mediate the relationship between our exposures of interest and outcomes. Our models were fitted to each imputed data set and combined in accordance with Rubin's rules. All analyses were carried out using Stata 16.0.

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