Builds a design in which two pieces of evidence are sought and used to update about whether X caused Y using Bayes' rule.

## Usage

```
process_tracing_designer(
N = 100,
prob_X = 0.5,
process_proportions = c(0.25, 0.25, 0.25, 0.25),
prior_H = 0.5,
p_E1_H = 0.8,
p_E1_not_H = 0.2,
p_E2_H = 0.3,
p_E2_not_H = 0,
cor_E1E2_H = 0,
cor_E1E2_not_H = 0,
label_E1 = "Straw in the Wind",
label_E2 = "Smoking Gun",
args_to_fix = NULL
)
```

## Arguments

- N
An integer. Size of population of cases from which a single case is selected.

- prob_X
A number in [0,1]. Probability that X = 1 for a given case (equal throughout population of cases).

- process_proportions
A vector of numbers in [0,1] that sums to 1. Simplex denoting the proportion of cases in the population in which, respectively: 1) X causes Y; 2) Y occurs regardless of X; 3) X causes the absence of Y; 4) Y is absent regardless of X.

- prior_H
A number in [0,1]. Prior probability that X causes Y in a given case in which X and Y are both present.

- p_E1_H
A number in [0,1]. Probability of observing first piece of evidence given hypothesis that X caused Y is true.

- p_E1_not_H
A number in [0,1]. Probability of observing first piece of evidence given hypothesis that X caused Y is not true.

- p_E2_H
A number in [0,1]. Probability of observing second piece of evidence given hypothesis that X caused Y is true.

- p_E2_not_H
A number in [0,1]. Probability of observing second piece of evidence given hypothesis that X caused Y is not true.

- cor_E1E2_H
A number in [-1,1]. Correlation between first and second pieces of evidence given hypothesis that X caused Y is true.

- cor_E1E2_not_H
A number in [-1,1]. Correlation between first and second pieces of evidence given hypothesis that X caused Y is not true.

- label_E1
A string. Label for the first piece of evidence (e.g., "Straw in the Wind").

- label_E2
A string. Label for the second piece of evidence (e.g., "Smoking Gun").

- args_to_fix
A character vector. Names of arguments to be args_to_fix in design.

## Details

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’ 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).

## Examples

```
# Generate a process-tracing design using default arguments:
pt_1 <- process_tracing_designer()
draw_estimands(pt_1)
#> inquiry estimand
#> 1 did_X_cause_Y TRUE
draw_estimates(pt_1)
#> estimator posterior_H result inquiry
#> 1 No tests (Prior) 0.5000000 TRUE did_X_cause_Y
#> 2 Straw in the Wind 0.2000000 FALSE did_X_cause_Y
#> 3 Smoking Gun 0.4117647 FALSE did_X_cause_Y
#> 4 Straw in the Wind and Smoking Gun 0.1489362 00 did_X_cause_Y
draw_data(pt_1)
#> ID causal_process X Y S test_results E1 E2
#> 1 043 Y_regardless TRUE TRUE 1 10 TRUE FALSE
if (FALSE) {
diagnose_design(pt_1, sims = 1000)
}
# A design in which the smoking gun and straw-in-the-wind are correlated
pt_2 <- process_tracing_designer(cor_E1E2_H = .32)
if (FALSE) {
diagnose_design(pt_2, sims = 1000)
}
# A design with two doubly-decisive tests pointing in opposite directions
pt_3 <- process_tracing_designer(p_E1_H = .80,p_E1_not_H = .05,
label_E1 = "Doubly-Decisive: H",
p_E2_H = .05,p_E2_not_H = .80,
label_E2 = "Doubly-Decisive: Not H")
draw_estimates(pt_3)
#> estimator posterior_H result
#> 1 No tests (Prior) 0.50000000 TRUE
#> 2 Doubly-Decisive: H 0.17391304 FALSE
#> 3 Doubly-Decisive: Not H 0.05882353 TRUE
#> 4 Doubly-Decisive: H and Doubly-Decisive: Not H 0.01298701 01
#> inquiry
#> 1 did_X_cause_Y
#> 2 did_X_cause_Y
#> 3 did_X_cause_Y
#> 4 did_X_cause_Y
if (FALSE) {
diagnose_design(pt_3, sims = 1000)
}
```