The model posits a population of N
cases, each of which does or does not exhibit the presence of some outcome, Y. With probability prob_X
, each case also exhibits the presence or absence of some potential cause, X. The outcome Y can be realized through four distinct causal relations, distributed through the population of cases according to process_proportions
. First, the presence of X might cause Y. Second, the absence of X might cause Y. Third, Y might be present irrespective of X. Fourth, Y might be absent irrespective of X.
Our inquiry is a "cause of effects" question. We wish to know whether a specific case was one in which the presence (absence) of X caused the presence (absence) of Y.
Our data strategy consists of selecting one case at random in which both X and Y are present. As part of the data strategy we seek two pieces of evidence in favor of or against the hypothesized causal relationship, H, in which X causes Y.
The first (second) piece of evidence is observed with probability p_E1_H
(p_E2_H
) when H is true, and with probability p_E1_not_H
(p_E2_not_H
) when H is false.
Conditional on H being true (false), the correlation between the two pieces of evidence is given by cor_E1E2_H
(cor_E1E2_not_H
).
The researcher uses Bayes<U+2019> rule to update about the probability that X caused Y given the evidence. In other words, they form a posterior inference, Pr(H|E). We specify four answer strategies for forming this inference. The first simply ignores the evidence and is equivalent to stating a prior belief without doing any causal process tracing. The second conditions inferences only on the first piece of evidence, and the third only on the second piece of evidence. The fourth strategy conditions posterior inferences on both pieces of evidence simultaneously.
We specify as diagnosands for this design the bias, RMSE, mean(estimand), mean(estimate) and sd(estimate).