Participants were drawn from the stony brook temperament study (SBTS), an ongoing longitudinal study in the USA , involving children from the community, recruited through commercially obtained mailing lists, with no significant medical conditions or developmental disabilities, and living with at least one English speaking biological parent. Participants were primarily European–American (87%), came from two-parent homes (94%) and had a middle-class background .
All procedures contributing to this work comply with the ethical standards of the relevant national and institutional committees on human experimentation and with the Helsinki Declaration of 1975, as revised in 2008. Institutional Review Board approval for the current study was obtained from Stony Brook University (study name: Observations of Active and Inactive Children, protocol number: 88933–35). In the current study, we considered a subset of children from the Stony Brook temperament study for whom data on both parenting and observed ES were available at age 3, and who were included in our previous analysis on the interaction between early parenting and children’s ES in the prediction of behavioural problems at ages 3 and 6 . For the purpose of this study, we included additional data at ages 9 and 12. At age 3, data were available for 292 children (see ), at age 9, due to attrition over time, data were available for 214 children, and at age 12 the sample included 196 children (43% female). Importantly, the sample with complete data (n = 196) was comparable to the original sample across all study variables (N = 292, see supplementary material file, Sect. 1).
When children were 3 years old, mothers provided information on parenting, and children’s ES was observed in a laboratory context. At 9 years, children provided information on ruminative coping strategies and at both 9 and 12 years on depressive symptoms.
Environmental sensitivity. Children’s sensitivity was investigated at age 3 with the highly sensitive child-rating system (HSC-RS) . The HSC-RS consists of a set of 10 rating scales that code global behaviours associated with sensitivity observed in the context of the Lab-TAB procedure . Each scale ranges from 1 to 7, with higher scores reflecting higher levels of ES. Psychometric proprieties of the HSC-RS were satisfactory .
Parenting. Parenting style was assessed when children were aged 3 using the parenting styles and dimensions questionnaire (PSDQ) . Parents reported on three parenting styles: permissive (five items capturing an indulgent caring attitude with difficulties in setting rules), authoritarian (12 items capturing low emotional support and hostility), and authoritative parenting (15 items capturing emotional support and rule setting reasoning). Internal consistency based on the 414 mothers who completed the questionnaire when children were 3 years old was good with Cronbach’s α = 0.74, 95% CI (0.70–0.77) for permissive parenting, α = 0.74, 95% CI (0.70–0.77) for authoritarian parenting, and α = 0.82, 95% CI (0.80–0.84) for authoritative parenting.
Rumination. At age 9, children completed the ruminative response subscale from the child response styles questionnaire (CRSQ-Rumination) . Children were instructed to select the option that indicates how they usually respond to feeling sad on a four-point scale. Internal consistency based on data from the 425 children who completed the questionnaire in the entire SBTS sample was good with α = 0.84, 95% CI (0.82–0.86).
Depressive symptoms. At 9 and 12 years, children reported levels of depressive symptoms using the children’s depression inventory (CDI) . Children were instructed to select the response for each item that best describes how they were thinking and feeling during the past week. Items are scored on a third-point scale, with higher scores reflecting greater depressive symptoms. Internal consistency, based on the entire SBTS sample, was acceptable with Cronbach’s α = 0.74, 95% CI (0.71–0.77), at age 9 (N = 481) and α = 0.82, 95% CI (0.79–0.84) at age 12 (N = 357).
Analyses were performed using the statistical software R , including blavaan  using STAN for implementing Markov chains Monte Carlo (MCMC) sampling [40, 41] and ggplot2  packages. First, we computed bivariate correlations among study variables using Pearson’s r. Afterwards, we compared and explored a series of multivariate regression models adopting a Bayesian approach for estimating parameters. The specific models that we compared were: model 0, the null model, i.e. a model assuming that there is no correlation among study variables, model 1, representing the association of parenting at age 3 with depression at 9 and 12, mediated through rumination at age 9, model 2 which was similar to model 1 but included the additive effect of ES on rumination at age 9, and model 3 which added the interaction term between parenting and sensitivity on rumination to test whether parenting predicted depression through the mediating role of rumination, conditional on ES levels (see Fig. 1). Models 0–3 were repeated separately for each of the three parenting styles. Depression at ages 9 and 12 was simultaneously included in all tested models. The following comparative indices were used to compare models: leave-one-out cross-validation information criterion (Loo IC) , with lower values reflecting a better fit of the model to data, the log Bayes factor, specific to the comparison with the null model, with higher values providing a stronger support to the model with respect to the null one, and the model weight criterion , with higher values reflecting a stronger support for the model. Loo IC and the log Bayes factor were used to compare each model against the null one, and the model weight criterion to compare each model against the previously tested model. In addition, for each endogenous variable, the variance explained by predictors was explored using R2 associated 90% highest posterior density intervals (HPDI) [45, 46] on the best model selected. HPDI values provide a direct representation of the most credible values of estimated parameters after accounting for prior beliefs. If HPDIs do not contain the zero, or only a small proportion of values are close to zero, then an effect can be reasonably supported.
Prior distributions. We defined informative prior distributions to incorporate our expectations (defined by prior mean value) and associated uncertainty (defined by prior standard deviation) into the analysis of the new data. We used standardized prior distributions, priors were defined based on results from the same sample when participants were aged 3 and 6 years . We expected the pattern of findings that we previously identified in relation to internalizing behavioural problems to be stable at ages 9 and 12, that is, to identify an association between permissive parenting and internalizing symptoms specific for children scoring high in ES. At the same time, we assumed a moderate degree of uncertainty to consider the possibility of different findings. More specifically, pertaining to the impact of parenting at age 3 on rumination at age 9 (path a, Fig. 1), we considered data from the same sample  showing that parenting had a small impact on behavioural and social outcomes at age 6, whilst also considering literature reporting that parenting styles matter for children’s adjustment . Hence, we hypothesized a relatively small but noticeable impact, with a moderate degree of uncertainty, operationalized as a normal distribution with a mean of M = 0.1 and a standard deviation of SD = 0.1, formally written as normal (0.1, 0.1), with a positive direction of effects of permissive and authoritarian parenting, and a negative direction for authoritative parenting. Similarly, pertaining to parenting predicting depression at ages 9 (path c1) and 12 (path c2), we expected a small but noticeable relation, equal to normal (0.1, 0.1) and normal (0.05, 0.1) for ages 9 and 12, respectively. Regarding predicting the impact of rumination at age 9 on depression at age 9 (path b1), considering the literature showing that the two are strongly associated [27,28,29], we hypothesized a relatively strong association with normal (0.50, 0.10). For rumination at age 9 on depression at age 12 (path b2, Fig. 1), we expected a smaller effect size compared to that observed for the two variables at age 9, with normal (0.35, 0.10). For the association between ES at age 3 and rumination at age 9 (path w), we adopted a sceptical prior with normal (0, 0.2), as we did not expect sensitivity at age 3 to predict rumination at age 9 irrespective of environmental quality, but we still allowed the possibility of an association between the two. For the interaction term between parenting and ES (path k, Fig. 1), based on previous empirical literature [14, 22], we assumed that permissive parenting would be a specific risk factor for highly sensitive children, hence we operationalized the interaction between permissive parenting and ES with normal (0.3, 0.1), and we assumed no significant interaction between ES and authoritative and authoritarian parenting [normal (0, 1)]. Given that residual variances are positive by definition, we used the default prior distribution for residual variances in the blavaan package for , namely a gamma distribution formally written as gamma (1, 0.5).
Computational details. Posterior distributions for each parameter were estimated using four Markov chains Monte Carlo (MCMC), each running at least for 4000 replicates. MCMC convergence was assessed by calculating the potential scale reduction statistic, PSRF. This statistic measures the ratio of the average variance of samples within each chain to the variance of the pooled samples across chains.
Interpretation of posterior distributions. Once the best model was identified, we considered standardized posterior distributions of model parameters to interpret effects. Each posterior was summarized by its mean value and associated 90% highest posterior density intervals, as described above. In addition to this, following Kruschke and Liddell , we evaluated effects considering the region of practical equivalence (ROPE), which defines values that are equivalent to the null effect. The lower the percentage of overlap between the ROPE and HPDI, the stronger is the support provided for the investigated effect. To summarize effects, we used the inverse of the overlap computed with I = 1− (HPDI ∩ ROPE)/HPDI), with I varying from 0.0 to 1, so that higher values corresponded to stronger effects. The ROPE was set from −0.1 to + 0.1 for all model parameters representing direct effects . For indirect effects, as these are function of investigated parameters (a×b1 and a × b2 for indirect effects, and a × b1 + k × b1 and a × b2 + k × b2 for conditional indirect effects), ROPEs were (−0.01, 0.01) and (−0.02, 0.02), respectively, for indirect and conditional indirect effects.
Model predictions and interaction effects. Finally, we illustrated interaction effects and findings for extreme groups [i.e. highly sensitive children (scoring in the top 30%) and low-sensitive children (scoring in the bottom 30%)] .
To ensure that findings were not biased by missing data, we repeated all analyses in an imputed data set, applying a Bayesian estimation method described in the supplementary material (Sect. 3). Given that the imputed results were very similar to non-imputed data, we decided to only report results based on the available data (we provide results from the imputation and the associated sensitivity analysis in supplementary materials).