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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':
#>
#> unionb <- initialize_graph(graph_from_literal(X -+ Y, Ur -+ X, Ur -+ Y))
V(b)$nvals <- c(3,2,2)
obj <- analyze_graph(b, constraints = NULL, effectt = "p{Y(X = 1) = 1} - p{Y(X = 0) = 1}")
optimize_effect_2(obj)
#> lower bound =
#> MAX {
#> -1 + p00_ + p11_
#> }
#> ----------------------------------------
#> upper bound =
#> MIN {
#> 1 - p10_ - p01_
#> }
#>
#> A function to compute the bounds for a given set of probabilities is available in the x$bounds_function element.
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_
#> }
#>
#> A function to compute the bounds for a given set of probabilities is available in the x$bounds_function element.
obj3 <- analyze_graph(b, constraints = NULL, effectt = "p{Y(X = 2) = 1} - p{Y(X = 1) = 1}")
optimize_effect_2(obj3)
#> lower bound =
#> MAX {
#> -p00_ - p20_ - p01_ - p11_
#> }
#> ----------------------------------------
#> upper bound =
#> MIN {
#> 1 - p20_ - p11_
#> }
#>
#> A function to compute the bounds for a given set of probabilities is available in the x$bounds_function element.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$bound.upper.quad, digits = 12),
na.rm = TRUE)b <- graph_from_literal(Ul -+ X -+ Y -+ Y2, Ur -+ Y, Ur -+ Y2)
V(b)$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
#> }
#>
#> A function to compute the bounds for a given set of probabilities is available in the x$bounds_function element.These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.