# Code from examples in manuscript

    library(causaloptim)
    #> Loading required package: igraph
    #> Warning: package 'igraph' was built under R version 4.1.1
    #>
    #> Attaching package: 'igraph'
    #> The following objects are masked from 'package:stats':
    #>
    #>     decompose, spectrum
    #> The following object is masked from 'package:base':
    #>
    #>     union

## 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)
    V(b)$nvals <- c(3,2,2)
    E(b)$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 {
    #>   -1 + p00_ + p11_
    #> }
    #> ----------------------------------------
    #> upper bound =
    #> MIN {
    #>   1 - p10_ - p01_
    #> }

    obj2 <- analyze_graph(b, constraints = NULL, effectt = "p{Y(X = 2) = 1} - p{Y(X = 0) = 1}")
    optimize_effect_2(obj2)
    #> lower bound =
    #> MAX {
    #>   -p10_ - p20_ - p01_ - p11_
    #> }
    #> ----------------------------------------
    #> upper bound =
    #> MIN {
    #>   1 - p20_ - p01_
    #> }

    obj3 <- analyze_graph(b, constraints = NULL, effectt = "p{Y(X = 2) = 1} - p{Y(X = 1) = 1}")
    optimize_effect_2(obj3)
    #> lower bound =
    #> MAX {
    #>   -p00_ - p20_ - p01_ - p11_
    #> }
    #> ----------------------------------------
    #> upper bound =
    #> MIN {
    #>   1 - p20_ - p11_
    #> }

## multiple instruments

Not run, this takes a few minutes 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)
    V(b)$nvals <- c(2, 2, 2, 2, 2, 2)
    E(b)$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)
    V(b2)$nvals <- c(2, 2, 2, 2, 2)
    E(b2)$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)

    b3 <- graph_from_literal(Z3 -+ X, Ul -+ Z3,
                             X -+ Y, Ur -+ X, Ur -+ Y)
    V(b3)$leftside <- c(1, 0, 1, 0, 0)
    V(b3)$latent <- c(0, 0, 1, 0, 1)
    V(b3)$nvals <- c(4, 2, 2, 2, 2)
    E(b3)$rlconnect <- c(0, 0,  0, 0, 0)
    E(b3)$edge.monotone <- c(0, 0, 0, 0, 0)

    ## single instrument
    obj3 <- analyze_graph(b3, constraints = NULL, effectt = "p{Y(X = 1) = 1} - p{Y(X = 0) = 1}")
    bounds.quad <- optimize_effect_2(obj3)

    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))

      ## conditioning on Z3

      z3.joint <- unique(p.vals[, c("Z3"), drop = FALSE])
      for(j in 1:nrow(z3.joint)) {
        z3.joint$Prob.condz3[j] <- sum(subset(p.vals, Z3 == z3.joint[j, "Z3"])$Prob)
      }

      p.vals.3 <- merge(p.vals, z3.joint, by = c("Z3"), sort = FALSE)

      p.vals.3$Prob.cond.fin <- ifelse(p.vals.3$Prob ==0, 0.0, p.vals.3$Prob / p.vals.3$Prob.condz3)
      res3 <- as.list(p.vals.3$Prob.cond.fin)
      names(res3) <- with(p.vals.3, paste0("p", X, Y, "_", Z3))

      list(multi = res,
           sing = res2,
           quad = res3)

    }

    ## 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)
    f.quad <- interpret_bounds(bounds.quad$bounds, obj3$parameters)

    result <- matrix(NA, ncol = 9, nrow = nsim)

    set.seed(211129)
    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)

      p.vals.joint$Z3 <- with(p.vals.joint, ifelse(Z1 == 0 & Z2 == 0, 0,
                                                   ifelse(Z1 == 0 & Z2 == 1, 1,
                                                          ifelse(Z1 == 1 & Z2 == 0, 2,
                                                                 3))))

      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)
      bees.quad <- do.call(f.quad, condprobs$quad)

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

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

    #pdf("figsim.pdf", width = 8, height = 4.25, family = "serif")
    par(mfrow = c(1,2))
    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 binary IV/Single 4-level IV",
         main = "Width of bounds intervals")
    abline(0, 1, lty = 3)

    plot(bound.lower.quad ~ bound.lower, data = bounds.comparison[1:100,], pch = 20, cex = 1,
         xlim = c(-1, 1), ylim = c(-1, 1), xlab = "Two binary IV", ylab = "Single 4-level IV",
         main = "Bounds values")
    points(bound.upper.quad ~ bound.upper, data = bounds.comparison[1:100,], pch = 1, cex = 1)

    legend("bottomright", pch = c(1, 20), legend = c("upper", "lower"))
    #dev.off()

    summary(bounds.comparison) # contains 467 NA's to avoid division by 0
    # Verify that a single quad-level instrument yield the same bounds as two linked binary ones.
    all(round(x = bounds.comparison$bound.lower, digits = 12) ==
            round(x = bounds.comparison$bound.lower.quad, digits = 12) &&
            round(x = bounds.comparison$bound.upper, digits = 12) ==
            round(x = bounds.comparison$bound.upper.quad, digits = 12),
        na.rm = TRUE)

## 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)
    V(b)$nvals <- c(2, 2, 2, 2, 2)
    E(b)$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
    #> }