ivbounds: Bounds for counterfactual outcome probabilities in...
In ivtools: Instrumental Variables

Description Usage Arguments Details Value Author(s) References Examples

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).

1	ivbounds(data, Z, X, Y, monotonicity=FALSE, weights)

`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 ≥q z' \Rightarrow X_z ≥q 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.

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}∑_{y}\max\limits_{z}p_{yx.z}≤q 1.

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`.

Arvid Sjolander.

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.

##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)