README.md

In gabrielfgm/rbounds: rbounds: Bounds Estimators for Partially Identified Parameters

rbounds: Estimating Bounds on Conditional Expectation Functions

The goal of rbounds is to estimate bounds on the location of parameters in models in which the value of the parameter can only be partially identified.

The package was developed to estimate the conditional mean of a binary outcome with non-random missing outcomes. At the moment the package contains one function which estimates bounds on a conditional mean using non-parametric kernel regression estimators from the np package. In the future I hope to extend the package to cover treatment effect estimation, and to work with some additional assumptions such as monotone regression and monotone treatment response.

Example of bounding the conditional mean

Imagine that we are interested in estimating E(y\|x) where y ∈ {0, 1}. Unfortunately, we are missing some observations of y. Whether an observation is missing is denoted by a dummy variable z, where when z = 1 y is observed and otherwise the value of y is not observed. We have some covariates x whose value we observe for all observations.

By the law of total probability it follows that

E(y\|x) = E(y\|x, z = 1)P(z = 1\|x) + E(y\|x, z = 0)P(z = 0\|x).

Each of the pieces of this equation can be estimated from the available data with the exception of E(y\|x, z = 0), which is the conditional expectation of the outcome for the cases in which the outcome value is unobserved.

However, since 0 ≤ y ≤ 1, the unobserved function can take on the value of at most 1 – if all the unobserved outcomes were 1, or at least 0. Thus the true conditional mean must fall within the bounds:

E(y\|x, z = 1)P(z = 1\|x) ≤ E(y\|x) ≤ E(y\|x, z = 1)P(z = 1\|x) + (1 − P(z = 1\|x)).

More generically, if we denote the minimum value the conditional outcome can take on as γ0 and the maximum by γ1, then the conditional mean can be bounded by

E(y\|x, z = 1)P(z = 1\|x) + γ0P(z = 0\|x) ≤ E(y\|x) ≤ E(y\|x, z = 1)P(z = 1\|x) + γ1P(z = 0\|x).

The conditional expectations in these expressions can be estimated parametrically or non-parametrically, and confidence intervals can be derived for the locations of the endpoints.

Example

Using simulated data we present a simple example of the pidoutcomes() function which we use to estimate partially identified conditional conditional means.

## We generate some fake data
library(rbounds)
set.seed(42)

N <- 100
x <- rnorm(N)
z <- rbinom(N, 1, pnorm(x))
y <- rbinom(N, 1, pnorm(x))

df <- data.frame(y, x, z)

res <- pidoutcomes(y~x, z, df)
res
#> Av. Lower CI Av. Lower Bound Av. Upper Bound Av. Upper CI
#>   0.19763        0.30612         0.82252        0.8856

# We can plot it with a generic plotting method
plot(res)

gabrielfgm/rbounds documentation built on May 8, 2019, 8:09 a.m.