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solve_closed <- function(prob, delta, lhs) {
# Closed-form solution for theta, given
# 1 - lhs = prob * inv_logit(theta) + (1 - prob) * inv_logit(theta + delta)
# as presented in Rosenbaum & Rubin
a <- lhs * exp(delta)
b <- (lhs - prob) * exp(delta) + lhs - 1 + prob
c <- lhs - 1
w <- (-b + sqrt(b ^ 2 - 4 * a * c)) / (2 * a)
theta <- log(w)
return(theta)
}
solve <- Vectorize(solve_closed)
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