Participants
The elementary school students and their parents from Beijing (in northern China), Shandong province (in eastern China), and Yunnan province (in south-western China) were enrolled to participant in a large-scale longitudinal research project. The basic demographic information, ODD symptoms and other family factors of both children and parents were recorded and assessed in the consecutive years of 2013–2014 [4]. In total, there were 307 boys (62.1%) and 187 girls (37.8%) involved in this study. The children's ages ranged from 6 to 13 years old (M = 9.37, SD = 1.59), and parents' ages ranged from 25 to 64 years old (\({M}_{mother}\)= 37.03, SD = 3.98; \({M}_{father}\) = 39.23, SD = 4.91).
Ethical considerations
Prior to conducting the study, the Institutional Review Board of Beijing Normal University in China approved the research protocol, including the consent procedure. Invitation letters and informed consent were sent to the parents and the children with a complete set of consent forms were involved in the further investigation.
Measures
ODD symptoms The children’s parents assessed the ODD symptoms on an 8-item scale derived from the DSM-IVR [39] for the diagnosis of ODD both in the first and second years. The severity of ODD symptoms was evaluated by adding the scores of items, with a higher total score indicating more ODD symptoms. The Cronbach’s \(\alpha\) of the scale was 0.85 in this study.
PCR The Chinese version of the Child-Parent Relationship Scale (CPRS) was used to assess PCR [40] (22 items; e.g., I share an affectionate, warm relationship with my child). The higher the total summed score, the better and stronger the parent–child relationship. In this study, the Cronbach’s \(\alpha\) of the scale was 0.82.
Monthly income A 5-point scale for monthly income was used to evaluate the entire family economic level. The higher score suggested higher monthly income in a family.
Family adaptability/cohesion The Family Adaptability and Cohesion Evaluation Scale (FACES-II) was used to evaluate the cohesion and adaptability in a family [41] (30 items; e.g., At home, we do things together). The higher score of FACES suggested stronger adaptability and cohesion in the family. The Cronbach’s \(\alpha\) for FACES-II was 0.84 in the study.
Marital relationship The Dyadic Adjustment Scale (DAS) was used to measure the perception of the relationship with an intimate partner [42] (32 items; e.g., Do you confide in your mate). The higher score of DAS indicated the higher quality of the marriage. The Cronbach’s \(\alpha\) for the DAS was 0.89 in this research.
Parenting style The Authoritative Parenting Index (API) consisted of two subscales: responsiveness and demandingness [43]. The responsiveness subscale (API-r) of the Authoritative Parenting Index (API) [43] was reported by the parent and reflected the support of parents for their children (7 items; e.g., I listen patiently to my child). The higher the score, the higher the degree of support from parents. The Cronbach's \(\alpha\) for this subscale was 0.82. The demandingness subscale (API-d) of API was reflected the control of parents over their children (9 items; e.g., I always tell my child what to do). The higher the score, the higher the degree of control from parents. In the research, the Cronbach’s \(\alpha\) for this subscale was 0.81.
Parent emotion regulation The self-reported Difficulties in Emotion Regulation Scale (DERS) was used to assess the ability of emotion regulation of parents [44] (36 items; e.g., When I am upset, I become angry with myself for feeling that way). The higher score of DERS indicated the greater difficulty in emotion regulation of the parents. The Cronbach’s \(\alpha\) was 0.84 in this study.
Child emotion regulation The Emotion Regulation Checklist (ERC) was used to assess children’s positive and negative emotion-related behaviors reported by parents [45] (24 items; e.g., Is easily frustrated). The higher score of ERC indicated poorer emotion regulation in children. In this study, the Cronbach’s \(\alpha\) was 0.82.
CF modelling
This experimental procedure was implemented on R studio based on GRF package. GRF is a plug-in package for forest-based statistical estimation and inference, which includes the functions of causal forest and regression forest. For details on the package, please refer to its technical reference (https://github.com/grf-labs/grf).
Model definition
Independent variable \({X}\) We defined the \(X\) values according to the difference of CPRS scores between the two consecutive years. As shown in Formulas 1 and 2, samples with the difference exceeding 5 (10% change) were defined as the improved group, while those with the difference within the range of -5 to + 5 were in the control group. The CPRS1 and CPRS2 in Formulas 1 and 2 represented the CPRS scores in the first and second years. Therefore, in total there were 423 qualified samples selected for causal forest modeling, with 155 children (37%) in the improved group and 268 children (63%) in the control group.
Dependent variable \({Y}\) The outcome ODD, the ODD assessment score in the second year was defined as dependent variable. As shown in Table 1, the average of the ODD scores was lower than 4, a common criterion when defining ODD. So both ODD and risk ODD children were involved in this study.
Covariates \({C}{s}\) The other measurements in the first year were used as \(Cs\) to assess their impacts on causality. According to the multilevel family model [10], the monthly income and FACES reflected factors from the entire family level, the measurements of DAS, API-r and API-d reflected the factors from dyadic level, and the DERS and ERC were from the individual level. Furthermore, the baseline ODD, the ODD score in the first year, was also involved as one of the \(Cs\) in the CF model.
After defining the variables in the CF model, the EM (Expectation-Maximum) method in SPSS 26.0 was used to fill in the missing values.
Modal construction
First, the regression forest was used to estimate the influence of \(Cs\) on \(Y\)and \(X\) respectively, which was to estimate the influences of the related family \(Cs\) on the causality of PCR and ODD symptoms. Specifically, we put the previously defined \(Cs\) and\(Y\) into the regression forest model to estimate the \(\widehat{Y}\) and the residual error between \(Y\) and \(\widehat{Y}\). In the same way, we further put the \(Cs\) and \(X\)into the regression forest model to estimate the \(\widehat{X}\) and the residual error between \(X\) and \(\widehat{X}\). Then all the parameters (\(X, \widehat{X}, Y, \widehat{Y}, Cs\)) were put into the causal forest to adjust the influences of \(Cs\) on the estimation of causal effect.
Second, we adjusted the number of trees in the forest to 300 to minimize the variance of the causal effect. After that, the causal effect for each sample could be estimated for further analysis at the group level.
Model evaluation
First, the reliability of the estimated causal effect and the heterogeneity were evaluated. A test with the best linear predictor (BLP) was performed to assess the goodness fitting of the forest. Furthermore, it could evaluate whether the heterogeneity of the causal effect had been well calibrated [46]. With the BLP test, the predicted dependent variable could be estimated as Formulas 3 and 4.
According to Formula 3 for the control group (\(X=0\)), the output result was the baseline value \({b}_{0}\left(C\right)\). While according to Formula 4 for the improved group (\(X=1\)), the output result included two more components.The average causal effect \(\tau \left(C\right)\)reflected the overall difference between the improved and control groups. In addition, the heterogeneity (\(\varDelta \tau =\tau -\stackrel{-}{\tau }\)) reflected the fluctuation of the causal effect of different individuals. Basically, a coefficient of 1 for \({\beta }_{1}\) suggested that the mean forest prediction was reasonable, whereas a coefficient of 1 for \({\beta }_{2}\) additionally suggested that the heterogeneity estimated from the forest were well calibrated.
Second, based on the model evaluation, the heterogeneity of the causal effect could be further analyzed in two ways generally. The first one was to compare the different causal effects between subgroups based on single \(Cs\). Here we divided the group into two subgroups according to the median value of each\(C\) and then compared the differences between the high and low subgroups. The second way was to analyze the heterogeneity based on a representative tree selected from the CF model, which could analyze the heterogeneous causal effects in several subgroups by considering the important \(Cs\) together [36].