Causal Inference - General. The counterfactual framework mentioned at the end of the previous section has been used as a basic theory of causation in philosophy since the 1700s [Xxxx, 1748], but the most well-known analysis is Xxxxx’x in 1973, updated and revised in 1999 [Menzies, 2008]. Al- though the philosophical theory and analysis differs from how statisticians think of causal effect analysis, we do rely on the philosophical notion of counterfactuals as “unactualised possibilities” [Menzies, 2001]. More formally, Pearl describes the case with binary true/false variables x and y and defines a counterfactual as “a probability statement about the truth of y, had x been true, when it is known that y had been false when x was false [Xxxxxxx, 2002].” Although the specific mathematical model to estimate causal effects (Xxxxx’x causal model) was not established until the 1970s [Xxxxx, 1974], the counterfactual language was being used as early as 1918 by Xxxxxx If we say, ‘This boy has grown tall because he has been well fed,’ we are not merely tracing out the cause and effect in an individual instance; we are sug- gesting that he might quite probably have been worse fed, and that in this case he would have been shorter. The formalization of counterfactuals in statistical analyses was introduced by Xxxxxx [1990] in the context of agricultural experiments. Causal inference is a challenging topic in statistics, where we have a much longer history of declaring associations between variables with confidence, and tend to translate inference into causal relationships with great trepidation. The first critical difference (of many) between a traditional analysis resulting in association and an analysis resulting in a causal link is the imposition of order on a set of variables. If x is a cause of y, then x should occur chronologically prior to y. Additionally, if x is a cause of y, it becomes conceptually problematic if y could also coherently be considered a cause of x [Holland, 1986]. Holland emphasizes the importance of “measuring the effects of causes” [1986] as the place within concepts of causation that statistics has the most to offer. In setting up a model for causal inference (referred to as ‘Xxxxx’x model’ in most literature), Holland begins with a population U of individual units u ‘. . . on which causes or treatments may act.’ [1986] Holland further defines an indicator variable S, with the value S(u) for a specific unit u within the population equal to t for treatment and c for cont...
