An incremental approach to causal inference in the behavioral sciences

Abstract

Causal inference plays a central role in behavioral science. Historically, behavioral science methodologies have typically sought to infer a single causal relation. Each of the major approaches to causal inference in the behavioral sciences follows this pattern. Nonetheless, such approaches sometimes differ in the causal relation that they infer. Incremental causal inference offers an alternative to this conceptualization of causal inference that divides the inference into a series of incremental steps. Different steps infer different causal relations. Incremental causal inference is consistent with both causal pluralism and anti-pluralism. However, anti-pluralism places greater constraints the possible topology of sequential inferences. Arguments against causal inference include questioning consistency with causation as an explanatory principle, charging undue complexity, and questioning the need for it. Arguments in favor of incremental inference include better explanation of diverse causal inferences in behavioral science, tailored causal inference, and more detailed and explicit description of causal inference. Incremental causal inference offers a viable and potentially fruitful alternative to approaches limited to a single causal relation.

Appendix

1.1 Campbellian notation

In its most recent form (Shadish et al. 2002), $R$ indicates random group asignment whereas NR indicates non-random group assignment. Non-random groups are also separated by dashed lines. $X$ indicates a treatment and $X\!\!\!\!\!-$ indicates cessation of ongoing treatment intervention, whereas $O$ indicates an observation. Numeric subscripts identify time points at which these take place whereas letter subscripts indicate measures used. Braces group observations that take place at the same time point.

1.2 Potential outcomes approach

As formalized by Holland (1986b), $R=<U,\,K,\,Y,\,S>$ where $R$ is Rubin’s model, $U$ is the set of experimental units commonly referred to as cases or observations, $K$ is the set of treatments to which members of $U$ are assigned, $Y$ is a function from paired values of $U$ and $K$ to values of the observed response variable, and $S$ is a function from members of $U$ to members of $K$ representing the treatment to which the given member was indeed assigned. The causal effect of $K$ on $Y$ is then defined as $Y(u,\,t)=Y(u,\,c)$ for $K=\{t,\,c\}$ which readily generalizes to any two treatments. The Stable Unit Treatment Value Assumption (SUTVA) states that $Y(u,\,k)$ is stable which entails (a) that there are no unrepresented treatment variables and (b) that for all $u \quad \ne u',\,K(u')$ does not affect $Y(u,\,k)$ (Rubin 2010).

1.3 Bayes net approach

The three central conditions formalized by Spirtes et al. (2000) take the following form. Let $P$ denote the probability distribution over the set of vertices, $V$, represented in graph $G$. Let $W$ denote any subset of $V$. Let $A(v)$ denote the ancestors of $v$, the causes of $v$ represented the graph and let $D(v)$ denote the descendants of $v$, the effects of $v$ represented in the graph. The Causal Markov Condition states that for every $W$ in $V,\,W$ remains statistically independent of all remaining variables in $V$ excluding $A(W)$ and $D(W)$. The Causal Minimality Condition states that there is no subgraph of $G$ that satisfies the Casual Markov Condition with respect to $P$. The Faithfulness Condition states that every conditional independence relation in $P$ is implied by the application of the Causal Markov Condition to $G$.

Pearl’s (2009) five properties characterizing causation take the following form. Let $W,\,X$, and $Y$ denote sets of variables. Let $W_{x}(u)$ denote the value of $W$ that results from manipulating $X$ (in the model) to take the value $x$. Composition states that if $W_{x}(u) = \hbox {w}$ then $Y_{xw}(u)=Y_{x}(u)$. Effectiveness states that $X_{xw}(u)=x$. Reversibility states that if ($Y_{xw}(u)=y$ and $W_{xy}(u)=w$) then $Y_{x}(u)=y$. Existence states that there exists an $x \varepsilon X$ such that $X_{y}(u)=x$. Uniqueness states that for univariate $X$, if ($X_{y}(u)=x$ and $X_{y}(u)=x$’) then $x=x$’. For detailed discussion, see Pearl (2009 chapter 7 and Markus 2011).

1.4 Granger causation

Granger (2001, chapter 2) gives the following definition of Granger causation. Let $t$ index discrete non-overlapping times. Let $\Omega _{t}$ denote the values of all variables up to and including time $t$. Axiom A states that the past can cause the future but not vice versa. Axiom B states that $\Omega _{t}$ contains no redundant information as would be the case if one variable were a determinate function of one or more others. Let $A$ denote some subset of values of a putative effect variable $Y_{t+1}$. $X_{t}$ causes $Y_{t+1}$ if $\hbox {P}(Y_{t+1 }\varepsilon A \vert \Omega _{t})\ne \,\hbox {P}(Y_{t+1}\varepsilon A {\vert } \Omega _{t}-X_{t})$, where P(.) denotes probability and $\Omega _{t}-X_{t}$ denotes the complement of $X_{t}$ within $\Omega _{t}$. Granger’s later definition of non-causation suggests that one can safely interpret the above definition as a biconditional rather than just a conditional as stated.