Bayesian network modelling to identify on-ramps to childhood obesity

Data for the analyses came from ‘Growing Up in Australia: The Longitudinal Study of Australian Children’ (LSAC) [9], Australia’s nationally representative children’s longitudinal study, focusing on social, economic, physical, and cultural impacts on health, learning, social and cognitive development. The study tracks two cohorts of children, referred to as the birth (B) cohort (5107 infants from 0 to 1 years old) and the kindergarten (K) cohort (4983 children from ages 4 to 5 years). Data were collected over seven biennial visits (“Waves”) from 2004 to 2016.

A selection of ~25 variables (Table 1) was chosen from the questionnaires for inclusion in Bayesian network models, informed by the existing literature on childhood obesity; e.g. the literature indicates that parental body mass index (BMI), socio-economic status, birthweight score and screen time are causally associated with childhood BMI.

We analysed 12 of the cross-sectional datasets (waves 2–7 in the B cohort and waves 1–6 in the K cohort). For each wave and cohort, a Bayesian network (BN) [6] was used to model the factors surrounding childhood BMI. At each time point (wave) the cross-sectional dataset was used to construct the distribution of possible network structures, allowing for inference on the causal pathways to childhood BMI at that time point. By comparing cross-sectional networks, we could then follow the evolution of these causal pathways over time.

To investigate the causal factors of childhood BMI in different genders, we further split each data set into boys and girls and made inferences on the corresponding Bayesian networks separately.

When aiming to infer causality, graph structures are sought which do not contain any cycles/loops (such loops lead to self-causality, which is hard to interpret). These structures are called directed acyclic graphs (DAGs). Figure 1a illustrates a hypothetical DAG containing four variables: socio-economic status, BMI of the primary caregiver (BMI1), BMI of the second parent (BMI2), and BMI of the child (BMI). The interpretation of this DAG is as follows: First, socio-economic status is antecedent to parents’ BMI, i.e. socio-economic status is causal to the parents’ BMI and not the other way around. Second, both caregivers’ BMIs are causal to the child’s BMI. Third, conditional on the caregivers’ BMIs, a child’s BMI is independent of socio-economic status, i.e. socio-economic status has no impact on child BMI, given the parents’ BMI.

A BN is a graphical representation of the equations in a structural equation model (SEM). In a Bayesian paradigm, one starts with a prior belief about the subject of interest (here, the DAG structure) based on existing knowledge. Then, on observing data, this prior belief is updated via what is known as a ‘likelihood function’ to arrive at a revised (‘posterior’) belief. In the context of BNs, the subject of interest has two components: first, the parameters of a particular DAG configuration, which we denote generically by θ_G, including quantities such as the strength of the connection between two factors; and second, the DAG itself, denoted by G. We wish to infer both θ_G and G, which is done via the joint posterior distribution P(θ_G, G∣ data) = P(θ_G∣ G, data)P(G∣ data). We first make inference regarding the structure G, by attaching probabilities to structures, P(G∣ data) and then, given a structure, infer the parameters needed to prescribe that structure P(θ_G∣ G, data). In the first step, P(G∣ data) is computed by integrating over all the possible values of parameters. This is different from traditional SEM which either assumes G is known or selects a single G,\(\hat{G}\) say, using a model selection technique and then makes inference only about \({\theta}_{\hat{G}}\) [11, 12]. However, structure learning is arguably more fundamental to causal inference than parameter estimation, since the parameters can only be estimated once the structure is known.

The review by McLachlan and colleagues [7] refers to three approaches for estimating a BN structure: data-driven, expert knowledge-driven, and hybrid approaches. These approaches are all Bayesian, which correspond to varying prior beliefs. The solely data-driven approach is analogous to a prior belief which assumes that each possible DAG is equally likely. The expert approach is analogous to a prior belief which assumes that the expert-constructed network is the true network, with probability 1. The hybrid approach, as used here, allows the strength of prior beliefs to vary both within and across structures; hence, information from different sources can be incorporated in a logically consistent manner, allowing the relative contributions of information from experts and from data to be measured. Importantly, hybrid approaches provide an ideal platform for formalising the collaboration between subject domain experts and specialist data experts: both groups are essential for success.

Although Bayesian networks have the potential to implement causal inference using observational data, they are not without drawbacks. First, the number of possible DAGs grows super-exponentially with respect to the number of variables, and it is computationally infeasible to compute the likelihood for each possible DAG once there are more than only a moderate number (~10) of variables. Second, for linear Gaussian Bayesian networks, the structure learning algorithms can only learn up to a DAG’s equivalence class, in which all the DAGs are equally likely [6]. The equivalence class is represented by a completed partially directed acyclic graph (CPDAG) [6]. CPDAGs contain undirected links which could be in either direction. Figure 1b shows the CPDAG of the DAG in Fig. 1a. In Fig. 1b, the undirected link between socio-economic status and BMI1 indicates we cannot distinguish the causal directions. For computational reasons, almost all the existing algorithms to estimate network structures assume that continuous variables cannot be ‘parents’ of discrete variables [10]. In our data, there are both discrete and continuous variables. The algorithm we used to conduct structure learning is Partition Markov chain Monte Carlo (PMCMC) [7] and the code is available at the Comprehensive R Archive Network (https://​cran.​r-project.​org/​web/​packages/​BiDAG/​index.​html). All the analyses in this paper were undertaken in R 4.0.4 (https://​www.​R-project.​org/​). PMCMC reduces the abovementioned computational challenges by collapsing the DAG space into partition space. We have adopted a strategy which considers every variable to be a Gaussian random variable to tackle the challenge caused by the existence of a mixture of continuous and discrete random variables in the data [13]. The details can be found in Additional file 1 [section of “The strategy in Partition MCMC to handle hybrid Bayesian networks”].

By applying PMCMC to the LSAC data, we obtained posterior samples of DAG structures at each time point for each wave and cohort of the LSAC data. Following the changes in DAG structures across waves allowed us to observe how causal patterns change as children age.

We also calculated the posterior probability of each DAG (top left corner), which describes the probability of each DAG given the data. These probabilities are expressed as a proportion of the sum of the posterior probability densities corresponding to the top 100 graphs. The larger the value, the more probable is the graph. Mathematically, the probability is defined as \(\frac{d_i}{\sum_{t=1}^{100}{d}_t}\), where d_i is the likelihood of the ith graph; i.e. a value of 70% indicates that when considering the subset of the top 100 graph structures, that graph has a posterior probability of 0.70 if each graph is equally likely a priori.

The CPDAG derived from the most probable DAG for B cohort waves 5 (age 8–9) is shown in Fig. 2. It clearly shows that socio-economic status played a central role in the obesity networks we studied. For every wave in the B cohort, socio-economic status sits in the central position of the CPDAG structure. This implies that socio-economic status drives almost everything else in the network structure. The same conclusion applies to other waves. For Fig. 2, the effect size of socio-economic status on a child’s BMI z-score is about −0.062. In other words, a unit change in socio-economic status can lead to a decrease of 0.062 in a child’s BMI z-score on average. In LSAC, socio-economic status was derived from family income, parents’ education and parents’ occupational status (Gibbings and colleagues [15]); however, our results indicate that socio-economic status represents an important influence on child BMI over and above any of its constituents alone. In addition, more than 99% of the posterior samples of DAG structures contain a pathway from socio-economic status or parental high school level to child BMI. The detail of the percentages is found in Table 3.

DAG structures from every wave show the importance of both parents finishing high school (P1E for mother, P2E for father). These two variables are correlated with socio-economic status, and the relationships are present in every DAG. The importance of this relationship is especially apparent in the network for K cohort wave 1 (Fig. 3), for which no specific socio-economic status variable was available. Figure 3 shows that in the absence of a specific socio-economic status variable, the parental high school level becomes the central node of the network.

For children up to the age of 6 years (Fig. 4), a child’s BMI is on the periphery of the DAG and is connected to the other variables only via the BMI of the child’s carers (BMI1 and BMI2) and the child’s birth weight z-score. After the age of 6 years, the drivers of childhood obesity become more complex. There is a formation of another sub-graph around child-specific variables, such as conduct disorder, emotional problems, sleep quality and quantity and electronic games, although there is considerable uncertainty associated with the direction and strength of these relationships at different waves.

Figure 2 shows that after age 8 years, free time activity (e.g. dancing and sports) becomes an important driver of obesity, and this, in turn, is driven by socio-economic status and the extent of electronic gaming by the child. In Fig. 2, the effect size of free time activity for a child’s BMI z-score is −0.118, which is significant. In other words, a unit change in free time activity can lead to a decrease of 0.118 in a child’s BMI z-score on average. Figure 4 also indicates that gender begins to impact a child’s BMI from age 6 (B cohort wave 4). However, gender does not directly influence a child’s BMI; rather, it passes its influence through other paths, e.g. SEX → electronic gaming→ free time activity → child BMI, which is shown in Fig. 2. To further investigate the impact of gender, we applied PMCMC to boys and girls separately. The CPDAG derived from the most likely DAG of B cohort wave 5 is presented in Fig. 5 for boys (Fig. 5a) and girls (Fig. 5b), respectively. For boys, the causal pathway electronic gaming → free time activity → child BMI emerges. However, for girls, sleep → free time activity → child BMI is the main pathway regarding how free time activity impacts child BMI. It would appear that boys and girls have different upstream factors influencing free time activity.

To illustrate the difference between BN and multiple regression, we conducted analyses using both techniques on a dataset containing variables: child BMI, parents’ BMI, socio-economic status, and parental high school level. Child BMI was the dependent variable in multiple regression analysis, and we compared its results to that of BN. The most probable DAG obtained by PMCMC showed the complete set of direct and indirect causal pathways from each of the variables to the child’s BMI. However, multiple regression only revealed the direct paths between parental BMIs and children’s BMI, with the other indirect relationships not detected. More details of this comparison can be found in the Supplementary Material.

Table 4 shows that in B cohort wave 5, several links are so strong that they appear in almost all the posterior samples, such as BMI1/BMI2 → BMI, GW→ BWZ, BWZ → BMI and SE→BMI1. These links are well supported by the literature. We have created similar tables for other waves of data in both B and K cohorts. The details can be found in Supplementary Material. The tables imply that the above links are also the most common links for other datasets. It can also be seen that socio-economic status is a driving node in all the networks. It confirms the central role of socio-economic status.

The LSAC data were collected in Australia which is a developed country. Thus, the children in this data set may only be representative of wealthy countries. It does not necessarily cover the characteristics of children from low- and middle-income countries.

Our study used Bayesian networks to model the variables surrounding childhood obesity. Whereas BNs are powerful, they are not without their drawbacks. They are computationally expensive, due to the super-exponential growth of the number of possible graph structures. For example, a system with 20 factors has an order of 2¹⁹⁰ possible graph structures, which is greater than the number of atoms in the universe. Therefore, an exhaustive search is impossible and some constraints on the number of possible graph structures need to be imposed.

All the presented causal pathways are only valid for the LSAC data. There is the possibility that some confounders were not measured in these data and misleading causal links may have resulted. For example, there could be further ‘upstream’ variables influencing both socio-economic status and parental high school levels which might explain the apparent undirected link between those two variables. However, under the current dataset, socio-economic status and parental high school levels are co-dependent.