7.9.1 COX REGRESSION AND INTERACTION ANALYSES
In sub-study I, the associations between ACEs and hospital-presenting self- harm in adolescence and young adulthood were modelled with Cox
regression, adjusting for region of residence, parental education and childhood income. The main analyses were conducted separately for
maternal and paternal characteristics. In these models, standard errors were clustered by mother’s or father’s identification number in respective models.
Interaction analyses between low childhood income (lowest childhood income quartile) and the adverse experiences were conducted by adding the binary income indicator, an indicator of an adverse experience and the interaction term of these two variables into the model. These were repeated for all the different ACEs. In addition, relative excess risk due to interaction (RERI) was calculated to measure the presence of an additive interaction effect. Besides interaction between income and ACEs, interactions between any maternal and any paternal adversity and between adverse experiences and offspring sex were also tested.
The common way to measure interaction with effect estimates expressed as a ratio (odds ratios, relative risks, hazard ratios) is to use the multiplicative scale, and test whether the effect of two exposures combined is larger than the product of the effects of these two exposures when considered separately (Li & Chambless, 2007; VanderWeele & Knol, 2014). However, two
exposures may also have an interaction effect on an additive scale, which refers to the extent to which the joint exposure of two exposures exceed the sum of the effects of these two assessed separately (VanderWeele & Knol, 2014). In linear regression, the additive scale is the scale used to assess interaction by statistical software, but interaction effects from models
designed for dichotomous outcomes are usually reported at the multiplicative scale. RERI is a measure which can be calculated from the estimated odds or hazard ratios to obtain an estimate of the direction and statistical
significance of the interaction effect on the additive scale from these models (Li & Chambless, 2007; VanderWeele & Knol, 2014). RERI does not include the size of the effect, but it nevertheless has public health significance: an interaction effect on the additive scale informs that the exposure has an effect on a larger number of individuals in one population subgroup than in
another, which can be useful information for intervention designs (VanderWeele & Knol, 2014).
7.9.2 PARAMETRIC G-FORMULA
Sub-study II was a causal mediation analysis on the direct and indirect associations between childhood income and hospital-presenting self-harm in late adolescence and young adulthood. For this purpose, the parametric g- formula based on logistic regressions was used. The g-formula is rooted in the framework of counterfactual mediation analysis and simulation of potential outcomes (Keil et al., 2014; Westreich et al., 2015) and is a highly useful tool in epidemiological mediation analysis (Bijlsma et al., 2017; Lin et al., 2017; Mikkonen et al., 2020; Wang & Arah, 2015). The traditional mediation analysis framework by Baron and Kenny (1986) cannot be reliably used to study mediation with binary outcomes due to the infamous issues with non-collapsibility when comparing nested models (Vansteelandt &
Keiding, 2011). The g-formula is a generalisation of direct standardisation and thus circumvents this problem of scaling by standardising covariate values across covariate strata (Keil et al., 2014). The parametric version of the g-formula uses parametric methods to calculate the conditional probabilities needed in standardisation (Hernán & Robins, 2020). In essence, the parametric g-formula produces marginal effect estimates, from which causal inference is possible, given all the assumptions of
counterfactual causal inference hold (including no unobserved confounding), and there are no gross misspecifications in the model or variable
measurement (Hernán & Robins, 2020; Keil et al., 2014). However, in
Data and Methods
the causal nature of the estimates remains debatable (Hernán & Robins, 2020).
To estimate the effects of low childhood income on self-harm in sub-study II, the g-formula was implemented by designing a hypothetical intervention of raising those in the lowest childhood income quintile into the second lowest, effectively giving all the children the average childhood income of children in the second lowest quintile. The actual procedure started sampling with replacement from the observational data and by fitting logistic regression models to the sampled data for the outcome and all the mediators examined.
The models for mediators included childhood income and all the childhood confounders and covariates described above. In the model for the outcome, the mediator variables were also included. The coefficients from these models were then used to produce a new simulated dataset without the hypothetical intervention (natural course scenario, NC) and then another simulated dataset with the hypothetical intervention present (counterfactual scenario, CF). Then, to assess mediation, a third simulated dataset was produced in which the intervention was again implemented but mediator values were derived from the NC data. After these three simulations, the average values of the outcome were saved for each scenario. This was repeated 100 times to reduce Monte Carlo (MC) error. These MC stabilised estimates were then used to calculate the total effect (TE), total direct effect (TDE) and natural indirect effect (NIE) (Wang & Arah, 2015). The whole process was repeated 250 times to obtain 95% bootstrap confidence intervals.
As effect estimates, both relative and absolute effect sizes were used. The TE was calculated as the difference between the average of the estimates obtained after MC correction in the NC and CF scenarios, while relative TE was the ratio between these averages, subtracted from 1. TDE was calculated as the difference between the mediation scenario and the NC, while NIE was just the difference between TE and TDE. These effects were calculated after each bootstrap iteration and the average of the 250 bootstrap effect
calculations used as the final estimate. The magnitude of mediation was illustrated as percentage mediated, which was calculated after all the bootstrap iterations as the ratio of the average NIE estimate to the average TE estimate. This differed from other effect calculations, which were done within each bootstrap iteration. The different calculation approach was used due to unstable estimates of percentage mediated when calculating within bootstraps, which was related to small TDEs obtained. Finally, absolute and relative average treatment effects for the treated (ATT) were also calculated, which are the TE and relative TE among those subject to the hypothetical intervention. These effects were additionally calculated in population subgroups defined by ACEs and family stability to examine effect moderation.
In addition, to indirectly assess the relative importance of the individual mediators, the simulation of the mediation scenarios was repeated allowing the intervention to have an effect on each of the mediators one by one while holding the other mediators at NC values. This procedure adds the indirect effect through a particular mediator into the direct effect and thus the smaller the percentage mediated (NIE/TE), the higher importance of a particular mediator. These examinations were also repeated in population subgroups defined by ACEs and family stability.
7.9.3 TRAJECTORIES OF TREATMENT AND MULTINOMIAL MODELS In sub-studies III and IV, instead of using the event of hospital-presenting self-harm as a variable in the exposure–outcome regression framework, a rolling panel of observations of equal duration was created around the date of the event, spanning two years before and after the event. In sub-study III, the panel was based on calendar months and the month of self-harm was excluded from the main trajectory analysis. In sub-study IV, the panel was calculated based on 90-day periods around the date of self-harm. In the statistical models, the indicator for time relative to event was used as the main exposure variable. The approach allowed for observing temporal changes in the prevalence of an outcome, relative to the event of interest. In sub-study III, General Estimation Equations (GEE) were used to consider clustering between observations when modelling the trajectory of psychiatric treatment, and in sub-study IV, logistic regression with clustered standard errors was chosen for this purpose. The models were controlled for the covariates presented above. The results from these studies were presented as predicted probabilities, which were calculated using marginal
standardisation. For each class of exposure variable, marginal
standardisation produces the predicted probabilities which would have been observed if every observed individual was constrained to belong in the respective class, standardised to confounder distributions in the total population. Marginal standardisation can also be viewed as a special case of the g-formula discussed above (Muller & MacLehose, 2014), which is a generalisation of direct standardisation (Keil et al., 2014).
Although the estimates derived from these models are not strictly causal given the omission of an individual-level fixed effects estimator, the approach is attractive for these types of studies, where the interest is in differences between population subgroups. Population-averaged models allow for examining differences defined by time-constant (or fairly constant) variables, which are not estimable with, e.g., individual fixed-effects methods. In sub- study III, socioeconomic differences in psychiatric treatment use were assessed based on parental educational level, and the trajectories were
Data and Methods
categorical time indicator, which allowed for separate levels of psychiatric treatment at each time point between these groups, in effect producing group-specific treatment trajectories adjusted for observed confounders. In sub-study IV, an overall population-averaged trajectory was estimated, and sociodemographic differences by parental employment, education and living arrangements with the child were assessed with interaction analyses. Sub- study IV also examined differences between mothers and fathers by stratifying all analyses by parent.
The downside of the trajectory approach is that it does not take into account whether individuals with an outcome at each time point are the same.
Therefore, in sub-study III cumulative psychiatric treatment at different time points (one month, one year and two years before and after self-harm) were calculated, and hierarchically categorised into inpatient treatment,
specialised outpatient treatment, medication only and no treatment.
Socioeconomic differences in the group membership of these categorical outcomes were assessed using multinomial logistic models and by predicting probabilities of group membership by parental education at each examined cumulative time point. In addition, cumulative observed averages throughout the follow-up were calculated. In sub-study IV, these cumulative measures were not included. Instead, for additional insights into the treatment trajectories, similar trajectories to those around self-harm were also estimated around offspring traffic accidents and accidental poisonings, as well as around a random date for a general population reference.