# Bounds for counterfactual outcome probabilities in instrumental variables scenarios `ivbounds` computes non-parametric bounds for counterfactual outcome probabilities in instrumental variables scenarios. Let $Y$, $X$, and $Z$ be the outcome, exposure, and instrument, respectively. $Y$ and $X$ must be binary, whereas $Z$ can be either binary or ternary. Ternary instruments are common in, for instance, Mendelian randomization. Let $p(Y_x=1)$ be the counterfactual probability of the outcome, had all subjects been exposed to level $x$. `ivbounds` computes bounds for the counterfactuals probabilities $p(Y_1=1)$ and $p(Y_0=1)$. Below, we define $p_{yx.z}=p(Y=y,X=x|Z=x)$. ```r ivbounds(data, Z, X, Y, monotonicity=FALSE, weights) ``` ## Arguments - `data`: either a data frame containing the variables in the model, or a named vector `(p00.0,...,p11.1)` when $Z$ is binary, or a named vector `(p00.0,...,p11.2)` when $Z$ is ternary. - `Z`: a string containing the name of the instrument $Z$ in `data` if `data` is a data frame. In this case $Z$ has to be coded as (0,1) when binary, and coded as (0,1,2) when ternary. `Z` is not specified if `data` is a vector of probabilities. - `X`: a string containing the name of the exposure $X$ in `data` if `data` is a data frame. In this case $X$ has to be coded as (0,1). `X` is not specified if `data` is a vector of probabilities. - `Y`: a string containing the name of the outcome $Y$ in `data` if `data` is a data frame. In this case $Y$ has to be coded as (0,1). `Y` is not specified if `data` is a vector of probabilities. - `monotonicity`: logical. It is sometimes realistic to make the monotonicity assumption $z \geq z' \Rightarrow X_z \geq X_{z'}$. Should the bounds be computed under this assumption? - `weights`: an optional vector of `prior weights' to be used in the fitting process. Should be NULL or a numeric vector. Only applicable if `data` is a data frame. ## Details `ivbounds` uses linear programming techniques to bound the counterfactual probabilities $p(Y_1=1)$ and $p(Y_0=1)$. Bounds for a causal effect, defined as a contrast between these, are obtained by plugging in the bounds for $p(Y_1=1)$ and $p(Y_0=1)$ into the contrast. For instance, bounds for the causal risk difference $p(Y_1=1)-p(Y_0=1)$ are obtained as $[min\{p(Y_1=1)\}-max\{p(Y_0=1)\},max\{p(Y_1=1)\}-min\{p(Y_0=1)\}]$. In addition to the bounds, `ivbounds` evaluates the IV inequality $$ \max\limits_{x}\sum_{y}\max\limits_{z}p_{yx.z}\leq 1. $$ ## Returns An object of class `"ivbounds"` is a list containing - **call**: the matched call. - **p0**: a named vector with elements `"min"` and `"max"`, containing the evaluated lower and upper bounds for $p(Y_0=1)$, respectively. - **p1**: a named vector with elements `"min"` and `"max"`, containing the evaluated lower and upper bounds for $p(Y_1=1)$, respectively. - **p0.symbolic**: a named vector with elements `"min"` and `"max"`, containing the lower and upper bounds for $p(Y_0=1)$, respectively, on a symbolic form (i.e. as strings). - **p1.symbolic**: a named vector with elements `"min"` and `"max"`, containing the lower and upper bounds for $p(Y_1=1)$, respectively, on a symbolic form (i.e. as strings). - **IVinequality**: logical. Does the IV inequality hold? - **conditions**: a character vector containing the violated condiations, if `IVinequality=FALSE`. ## References Balke, A. and Pearl, J. (1997). **Bounds on treatment effects from studies with imperfect compliance**. **Journal of the American Statistical Association** 92 (439), 1171-1176. Sjolander A., Martinussen T. (2019). Instrumental variable estimation with the R package ivtools. **Epidemiologic Methods** 8 (1), 1-20. ## Author(s) Arvid Sjolander. ## Examples ```r ##Vitamin A example from Balke and Pearl (1997). n000 <- 74 n001 <- 34 n010 <- 0 n011 <- 12 n100 <- 11514 n101 <- 2385 n110 <- 0 n111 <- 9663 n0 <- n000+n010+n100+n110 n1 <- n001+n011+n101+n111 #with data frame... data <- data.frame(Y=c(0,0,0,0,1,1,1,1), X=c(0,0,1,1,0,0,1,1), Z=c(0,1,0,1,0,1,0,1)) n <- c(n000, n001, n010, n011, n100, n101, n110, n111) b <- ivbounds(data=data, Z="Z", X="X", Y="Y", weights=n) summary(b) #...or with vector of probabilities p <- n/rep(c(n0, n1), 4) names(p) <- c("p00.0", "p00.1", "p01.0", "p01.1", "p10.0", "p10.1", "p11.0", "p11.1") b <- ivbounds(data=p) summary(b) ```

ivbounds function