Code from examples in manuscript

library(causaloptim)
#> Loading required package: igraph
#> 
#> Attaching package: 'igraph'
#> The following objects are masked from 'package:stats':
#> 
#>     decompose, spectrum
#> The following object is masked from 'package:base':
#> 
#>     union
library(igraph)

confounded exposure and outcome

b <- graph_from_literal(X -+ Y, Ur -+ X, Ur -+ Y)
V(b)$leftside <- c(0,0,0)
V(b)$latent <- c(0,0,1)
E(b)$rlconnect <- E(b)$edge.monotone <- c(0, 0, 0)

obj <- analyze_graph(b, constraints = NULL, effectt = "p{Y(X = 1) = 1} - p{Y(X = 0) = 1}")
optimize_effect_2(obj)
#> lower bound =  
#> MAX {
#>   -p10_ - p01_
#> }
#> ----------------------------------------
#> upper bound =  
#> MIN {
#>   1 - p10_ - p01_
#> }

multiple instruments

Not run, this takes a very long time to compute.

b <- graph_from_literal(Z1 -+ X, Z2 -+ X, Z2 -+ Z1, Ul -+ Z1, Ul -+ Z2,
                        X -+ Y, Ur -+ X, Ur -+ Y)
V(b)$leftside <- c(1, 0, 1, 1, 0, 0)
V(b)$latent <- c(0, 0, 0, 1, 0, 1)
E(b)$rlconnect <- c(0, 0, 0, 0, 0, 0, 0, 0)
E(b)$edge.monotone <- c(0, 0, 0, 0, 0, 0, 0, 0)

obj <- analyze_graph(b, constraints = NULL, effectt = "p{Y(X = 1) = 1} - p{Y(X = 0) = 1}")

bounds.multi <- optimize_effect_2(obj)

b2 <- graph_from_literal(Z1 -+ X, Ul -+ Z1,
                         X -+ Y, Ur -+ X, Ur -+ Y)
V(b2)$leftside <- c(1, 0, 1, 0, 0)
V(b2)$latent <- c(0, 0, 1, 0, 1)
E(b2)$rlconnect <- c(0, 0,  0, 0, 0)
E(b2)$edge.monotone <- c(0, 0, 0, 0, 0)


## single instrument
obj2 <- analyze_graph(b2, constraints = NULL, effectt = "p{Y(X = 1) = 1} - p{Y(X = 0) = 1}")
bounds.sing <- optimize_effect_2(obj2)


joint <- function(df, alpha, pUr, pUl) {

  Z1 <- df$Z1
  Z2 <- df$Z2
  X <- df$X
  Y <- df$Y

  pUr * pUl * (((pnorm(alpha[1] + alpha[2] * 1)) ^ Z1 * (1 - pnorm(alpha[1] + alpha[2] * 1)) ^ (1 - Z1)) *
                 ((pnorm(alpha[3] + alpha[4] * 1 + alpha[5] * Z1)) ^ Z2 *
                    (1 - pnorm(alpha[3] + alpha[4] * 1 + alpha[5] * Z1)) ^ (1 - Z2)) *
                 ((pnorm(alpha[6] + alpha[7] * Z1 + alpha[8] * Z2 + alpha[9] * 1)) ^ X *
                    (1 - pnorm(alpha[6] + alpha[7] * Z1 + alpha[8] * Z2 + alpha[9] * 1)) ^ (1 - X)) *
                 (pnorm(alpha[10] + alpha[11] * X + alpha[12] * 1)) ^ Y *
                 (1 - pnorm(alpha[10] + alpha[11] * X + alpha[12] * 1)) ^ (1 - Y)) +
    pUr * (1 - pUl) * (((pnorm(alpha[1] + alpha[2] * 0)) ^ Z1 * (1 - pnorm(alpha[1] + alpha[2] * 0)) ^ (1 - Z1)) *
                         ((pnorm(alpha[3] + alpha[4] * 0 + alpha[5] * Z1)) ^ Z2 *
                            (1 - pnorm(alpha[3] + alpha[4] * 0 + alpha[5] * Z1)) ^ (1 - Z2)) *
                         ((pnorm(alpha[6] + alpha[7] * Z1 + alpha[8] * Z2 + alpha[9] * 1)) ^ X *
                            (1 - pnorm(alpha[6] + alpha[7] * Z1 + alpha[8] * Z2 + alpha[9] * 1)) ^ (1 - X)) *
                         (pnorm(alpha[10] + alpha[11] * X + alpha[12] * 1)) ^ Y *
                         (1 - pnorm(alpha[10] + alpha[11] * X + alpha[12] * 1)) ^ (1 - Y)) +
    (1 - pUr) * pUl * (((pnorm(alpha[1] + alpha[2] * 1)) ^ Z1 * (1 - pnorm(alpha[1] + alpha[2] * 1)) ^ (1 - Z1)) *
                         ((pnorm(alpha[3] + alpha[4] * 1 + alpha[5] * Z1)) ^ Z2 *
                            (1 - pnorm(alpha[3] + alpha[4] * 1 + alpha[5] * Z1)) ^ (1 - Z2)) *
                         ((pnorm(alpha[6] + alpha[7] * Z1 + alpha[8] * Z2 + alpha[9] * 0)) ^ X *
                            (1 - pnorm(alpha[6] + alpha[7] * Z1 + alpha[8] * Z2 + alpha[9] * 0)) ^ (1 - X)) *
                         (pnorm(alpha[10] + alpha[11] * X + alpha[12] * 0)) ^ Y *
                         (1 - pnorm(alpha[10] + alpha[11] * X + alpha[12] * 0)) ^ (1 - Y)) +
    (1 - pUr) * (1 - pUl) * (((pnorm(alpha[1] + alpha[2] * 0)) ^ Z1 * (1 - pnorm(alpha[1] + alpha[2] * 0)) ^ (1 - Z1)) *
                               ((pnorm(alpha[3] + alpha[4] * 0 + alpha[5] * Z1)) ^ Z2 *
                                  (1 - pnorm(alpha[3] + alpha[4] * 0 + alpha[5] * Z1)) ^ (1 - Z2)) *
                               ((pnorm(alpha[6] + alpha[7] * Z1 + alpha[8] * Z2 + alpha[9] * 0)) ^ X *
                                  (1 - pnorm(alpha[6] + alpha[7] * Z1 + alpha[8] * Z2 + alpha[9] * 0)) ^ (1 - X)) *
                               (pnorm(alpha[10] + alpha[11] * X + alpha[12] * 0)) ^ Y *
                               (1 - pnorm(alpha[10] + alpha[11] * X + alpha[12] * 0)) ^ (1 - Y))


}


## get conditional probabilities
## key = XY_Z1Z2

get_cond_probs <- function(p.vals) {

  z1z2.joint <- unique(p.vals[, c("Z1", "Z2")])
  for(j in 1:nrow(z1z2.joint)) {
    z1z2.joint$Prob.condz1z2[j] <- sum(subset(p.vals, Z1 == z1z2.joint[j, "Z1"] & Z2 == z1z2.joint[j, "Z2"])$Prob)

  }

  p.vals.2 <- merge(p.vals, z1z2.joint, by = c("Z1", "Z2"), sort = FALSE)

  p.vals.2$Prob.cond.fin <- ifelse(p.vals.2$Prob ==0, 0.0, p.vals.2$Prob / p.vals.2$Prob.condz1z2)
  res <- as.list(p.vals.2$Prob.cond.fin)
  names(res) <- with(p.vals.2, paste0("p", X, Y, "_", Z1, Z2))

  ## conditional on Z1 only

  xyz1.joint <- unique(p.vals[, c("Z1", "X", "Y")])
  for(j in 1:nrow(xyz1.joint)) {

    xyz1.joint$Prob.xyz1[j] <- sum(subset(p.vals, Z1 == xyz1.joint$Z1[j] &
                                            X == xyz1.joint$X[j] & Y == xyz1.joint$Y[j])$Prob)

  }

  z1.marg0 <- sum(subset(xyz1.joint, Z1 == 0)$Prob.xyz1)
  z1.marg1 <-   sum(subset(xyz1.joint, Z1 == 1)$Prob.xyz1)

  xyz1.joint$Prob.z1[xyz1.joint$Z1 == 0] <- z1.marg0
  xyz1.joint$Prob.z1[xyz1.joint$Z1 == 1] <- z1.marg1

  xyz1.joint$Prob.cond <- with(xyz1.joint, Prob.xyz1 / Prob.z1)
  res2 <- as.list(xyz1.joint$Prob.cond)
  names(res2) <- with(xyz1.joint, paste0("p", X, Y, "_", Z1))

  list(multi = res, sing = res2)

}



## simulate and compare the two
nsim <- 50000
f.multi <- interpret_bounds(bounds.multi$bounds, obj$parameters)
f.single <- interpret_bounds(bounds.sing$bounds, obj2$parameters)
result <- matrix(NA, ncol = 6, nrow = nsim)


for (i in 1:nsim) {

  alpha <- rnorm(12, sd = 2)
  pUr <- runif(1)
  pUl <- runif(1)

  p.vals.joint <- obj$p.vals
  p.vals.joint$Prob <- joint(p.vals.joint, alpha, pUr, pUl)
  if(any(p.vals.joint$Prob == 0)) next

  condprobs <- get_cond_probs(p.vals.joint)

  bees <- do.call(f.multi, condprobs$multi)
  bees.sing <- do.call(f.single, condprobs$sing)

  result[i, ] <- unlist(c(sort(bees), abs(bees[2] - bees[1]), sort(bees.sing), abs(bees.sing[2]- bees.sing[1])))

}
colnames(result) <- c("bound.lower",
                      "bound.upper", "width.multi",
                      "bound.lower.single", "bound.upper.single", "width.single")
bounds.comparison <- as.data.frame(result)

plot(width.multi ~ width.single, data = bounds.comparison, pch = 20, cex = .3,
     xlim = c(0, 1), ylim = c(0, 1), xlab= "Single IV", ylab = "Two IV",
     main = "Comparison of width of bounds intervals")
abline(0, 1, lty = 3)

measurement error in the outcome

b <- graph_from_literal(Ul -+ X -+ Y -+ Y2, Ur -+ Y, Ur -+ Y2)
V(b)$leftside <- c(1, 1, 0, 0, 0)
V(b)$latent <- c(1, 0, 1, 0, 1)
E(b)$rlconnect <- c(0, 0, 0, 0, 0)
E(b)$edge.monotone <- c(0, 0, 0, 0, 0)

obj <- analyze_graph(b, constraints = "Y2(Y = 1) >= Y2(Y = 0)",
                     effectt = "p{Y(X = 1) = 1} - p{Y(X = 0) = 1}")

optimize_effect_2(obj)
#> lower bound =  
#> MAX {
#>   -1,
#>   -1 + 2p0_0 - 2p0_1
#> }
#> ----------------------------------------
#> upper bound =  
#> MIN {
#>   1,
#>   1 + 2p0_0 - 2p0_1
#> }