A Review of the Techniques Used to Control Confounding Bias and How Spatiotemporal Variation Can Be Controlled in Environmental Impact Studies

Abstract

Inferring causality has long been a challenging task in environmental impact studies and monitoring programs, mostly because of the problem of confounding bias, i.e. the difficulty of separating impact from natural variation. Traditional approaches for dealing with confounding, despite improvements in study design and statistical analysis, are inadequate. Using aquatic biota as a case study, this review explains the limitations of traditional methods used to separate the impact of human-made pollution from natural variation in the environment. Advantages and disadvantages of the traditional and novel techniques are enumerated. Bayesian networks (BNs) and structural equation modelling (SEM) as causal modelling techniques are introduced as approaches to improve environmental impact monitoring.

Appendix

1.1 Bayesian Network and a Theory of Inferred Causation

Bayesian networks (BNs) are graphical models for reasoning under uncertainty that describe the relationships among variables, using probabilistic expressions (Barton et al. 2012). A directed acyclic graph (DAG) is a causal structure of a set of variables, in which the nodes represent variables and arcs represent direct connections between the variables. These direct connections are often causal connections (Pearl 2000; Korb and Nicholson 2010).

A simple yet typical BN is illustrated in Fig. 4. It shows relationships among season (X₁), catching a cold (X₂) or suffering from hay fever (X₃), sneezing (X₄) and reaching for a tissue (X₅). The lack of a direct link between X₁ and X₅ shows that the impact of seasonal changes on the need for tissues is mediated by other variables. To describe the relationship among variables in Fig. 4, once we know the season, it is revealed that the sneeze is because of a cold or hay fever (assuming that a cold is more common in winter and hay fever happens more in spring). In modelling language, by knowing the value of X₁ (season) and conditioning on it, the middle variables, X₂ (cold) and X₃ (hay fever), are independent. On the other hand, the case where two causes have a common effect acts the opposite way, e.g. X₂ (cold) and X₃ (hay fever), will be dependent if the person is sneezing and reaches for a tissue in that by rejecting one of these explanations, the probability of the other increases (Pearl 2000).

In nature, there are arbitrary functional relationships between each effect and its cause(s) imposed by stable causal mechanisms. If these relationships are perturbed by arbitrary disturbances introduced by nature, then it results in hidden or unmeasurable conditions that can be summarised in some probability function. Scientists attempt to identify these mechanisms, organised in the form of acyclic directed graphs (Pearl and Verma 1991). Directed graphs are an intuitive way of expressing causal knowledge (Steyvers et al. 2003), in which each variable is modelled as a function of its direct causes (parents). By definition, a directed acyclic graph can form the causal structure of a set of variables, with each node representing a variable of V, and each link relating to a direct functional relationship among the corresponding variables. The causal structure serves as a blueprint for a causal model which, in turn, specifies how each variable is influenced by its immediate causes (known as Markovian parents in a DAG) (Pearl 2000).

It is important to know how strongly the variables are related, and this can be obtained by utilising the quantitative component of the BN (Aguilera et al. 2011). Taking into account a BN as a carrier of conditional independence relationships, it holds that joint probability distribution over all the variables is equal to the product of the conditional distributions attached to each node, as follows:

$$ p\left({x}_1,\dots .,{x}_n\right)={\prod}_{i=1}^np\left({x}_i| pa\left({x}_i\right)\right) $$

(11)

where pa(x_i) is a set of predecessors of x_i.

Suppose we have a set of variables as in V = {X₁,  … , X_n}, and p(v) is the joint probability distribution on these variables. Then, a set of variables PA_j is said to be Markovian parents of X_j if PA_j is a minimal set of predecessors of X_j that renders X_j independent of all its other predecessors¹ (Pearl 2000). In simple terms, the Markov condition is the state of any variable being probabilistically independent of its non-descendants given the state of its parents (Steyvers et al. 2003). The Markov condition guides us to decide when a set of parents is complete in terms of including all the relevant immediate causes of a variable. Once a causal model is formed, it defines a joint probability distribution of the variables in the system in which each variable must be independent of its ancestors given its parents (Pearl and Verma 1991). At this stage, the scientist has a chance to inspect a subset of the observed variables and the probability distribution over those observed variables, but the underlying causal model and causal structure are still not revealed (Pearl 2000).

As the causal structure of a set of variables is unknown, there is no unique model that would fit a given distribution of those variables. There are numerous models that each might fit data with different sets of hidden variables, and each might connect the observed variables through different causal relationships. The solution to the problem of having no unique model is to follow standard norms of scientific induction and find a simpler, less-elaborate model that is equally consistent with the data. The model which is selected using this process is referred to as ‘minimal’ (Pearl and Verma 1991). From this, inferred causation can be defined as follows: a variable X is said to have a causal influence on a variable Y if a directed path from X to Y exists in every minimal latent structure consistent with the data (Pearl 2000).

In situations where all of the variables in a causal model are observable, conditional independencies are sufficient to infer causal relationships. However, this relationship cannot be attributed to the unobservable variables known as latent or hidden variables (Pearl and Verma 1991). The Markov condition permits us to exclude some of the causes out of the set of parents (to be summarised by probabilities), but not if they are also the cause of other variables in the model (Pearl 2000). For example, disturbances, dictated by nature, that perturb the causal relationships and influence several families of parent–child in the model, will be latent variables (Pearl and Verma 1991). Under the assumption of model-minimality and stable distribution, causal relationships can be uncovered. This method does not claim to identify stable physical mechanisms in nature, but it can identify the mechanisms that may be used for causal inference, plausibly from non-experimental data (Pearl 2000).

1.2 The Relationship Between BNs and SEM and a General Framework for Causal Modelling

Similar to the BN approach, SEM is a causal modelling approach for inferring causal relationships from observational data (Gupta and Kim 2008). SEM is a family of causal modelling (Pearl 2000) and is defined as the process of developing and evaluating structural equation models (Grace et al. 2012). This method provides a framework for learning about causal processes by combining the cause–effect information and statistical data to determine the quantitative relationship among studied variables (Gupta and Kim 2008). Structural equation models can be shown with related mathematical equations, e.g. regression equations in that regression models illustrate the influence of variables on another variable. This influence can be shown using an arrow pointing to the variable of interest from the variable that is influencing it (Byrne 2010). Generally, a structural causal model consists of a set of equations, and can be written as:

$$ {x}_i={f}_i\left(p{a}_i,{u}_i\right)\kern0.5em i=1,\dots, n, $$

(12)

where pa_i is the set of variables that directly determine the value of X_i, and U_i is representative of errors. Structural equation models are schematically depicted in causal diagrams. Given a causal model in Eq. 12, if we draw an arrow to each variable from its immediate causes, the resulting graph will be a causal diagram. If the causal diagram is acyclic and the errors are jointly independent, the model is a Markovian model. Markovian models connect causation and probabilities via Markov condition (Spirtes et al. 1993; Pearl 2000). Two assumptions need to be met for justification of causal Markov condition. First, every variable that is a cause of two or more variables should be included in the model. Second, ‘common cause principle’ should be met which states that if two variables are dependent, either one is the cause of the other or there is a third variable that they share as a common cause (Reichenbach 1956). Based on the causal Markov condition, a child–parent relationship can be specified as a deterministic function instead of the usual conditional probability used in BNs. This leads to the same distribution that characterises BNs (Pearl 2000). In other words, for every BN with a distribution P (as in Eq. 11), there is a structural equation model (as in Eq. 12) that produces a distribution identical to P (Druzdzel and Simon 1993).

SEM models not only have the graphical feature and the logic of causality that accompanies BNs, but also they have a number of advantages over BNs (Gupta and Kim 2008). Pearl (2000) states some reasons for preferring functional models. First, functional models are more general than stochastic models. Using functional models, one can go beyond predicting the effect of interventions to analysing counterfactuals. For example, functional models can answer questions such as ‘what would be the concentration of algae if we could stop discharging effluent to the streams’, whereas SEM can answer questions like ‘what would have been the algal concentration if discharging had not been occurring’ (Paul et al. 2016). Another advantage of functional representation is that standard statistical methods and software can now be used for the building and testing of SEM models (Paul and Anderson 2013).

Graph theoretical SEM is an approach for translating a causal diagram into a structural causal model and is the most common method of testing causal models. This process is shown in Fig. 4 (Paul et al. 2016). The process of structural causal modelling starts with drawing a causal diagram. The next part of the process involves translating the causal diagram into a set of structural equation and d-separation statements. d-Separation is used as a translation tool for translating the language of causality to the language of probability distribution (Shipley 2000). The statistical models and independence relationships underpinning SEM can be fitted and tested using the structural equation and d-separation statements (Pearl 1998). The third step of the process is to fit and check the statistical models for each of the structural equations, and to test all the independencies entailed in the causal structure (Pearl 2000; Shipley 2000). In the process of building a causal diagram, based on the results from model testing, the causal diagram might need to be adjusted in an iterative process. As SEM is strictly confirmatory, the hypothesised model might be accepted or rejected. Based on the results of the hypothesised model and how perfect it fits the data, the model might be modified and re-estimated (Byrne 2010). Finally, the model is used to predict effects of interventions and to inform management and policy (Pearl 2000).