In bsaul/sandwichShop: Extends sandwich Package

In causal inference and other settings, estimating equations are often used. Target parameters may depend on one or more models. For example, inverse probability weighted estimators often fit parametric models for the probability of treatment. Doubly robust estimators use models for both the outcome and treatment.

This package provides tools to "stack" estimating equations from multiple models and derive asymptotically valid variance estimates via M-estimation [@stefanski2012].

General Notation

|Notation | Description| |-----------------|-----------------------| | $m$ | the number of statistically independent units (e.g. clusters) indexed by $i$ | | $n_i$ | where $m > 1$, the number of (possibly) correlated units within cluster $i$ | |$O_i$ | data for group $i$, e.g. $O_i = {Y_i, A_i, L_i}$ could be the outcome, treatment, and covariate vectors (or matrices) for cluster $i$ | | $\psi(O_i, \theta_l)$ | the set of estimating equations corresponding the set of parameters $\theta_l$ from the $l$th model. E.g, for doubly robust estimation, $\theta_1$ could be the parameters in a logistic regression treatment model; $\theta_2$ may be parameters in a linear model for the outcome; and $\theta_3 = \mu$ may be the target parameter | | $\theta = {\theta_1, \dots, \theta_L, \mu}$ | the set of all parameters in estimating equations. In most (all?) cases, $\theta_l \notin \theta_{l'}$ for all $l$. | |$\psi(O_i, \theta)$ | estimating equations for all parameters in $\theta$ |

Under regularity conditions, the solution, $\hat{\theta}$ to: [ \sum_{i = 1}^m \psi(O_i, \theta) = 0 ]

is a consistent estimator for $\theta$, and $\hat{U}^{-1} \hat{V} (\hat{U}^{-1})^T$ is the estimated variance-covariance for $\theta$ where:

[ \hat{V} = \frac{1}{m} \sum_i \psi(O_i, \hat{\theta}) \psi(O_i, \hat{\theta})^T ]

and

[ \hat{U} = \frac{1}{m} \sum_i \psi'(O_i, \hat{\theta}) ]

In [@stefanski2012], $V = B$ (aka the "meat" matrix) and $U^{-1} = A^{-1}$ (aka the "bread" matrix).

Math to Code

Let $\psi_{\mu} = \psi(O_i, \mu)$ be the estimating equations of the target parameter; $\psi_{\gamma} = \psi(O_i, \gamma)$ be the estimating equations of the outcome model; and $\psi_{\alpha} = \psi(O_i, \lambda)$ estimating equations of the target parameter. Dropping the $i$ notation for convenience.

For the DR estimator: [ \psi(O_i, \mu) = g(O_i, \alpha, \gamma) - \mu ]

when, for example, $\mu$ is overall mean,

[ g(O_i, \lambda, \gamma) = \frac{1}{n_i} \sum_{j = 1}^{n_i}\left{ \frac{ Y_{ij} - h_{ij} (Y_i, L_i, A_i; \gamma) } { f(A_i, L_i, \lambda )} \pi(A_i; \alpha) + \sum_{a_i} h_{ij} (Y_i, L_i, A_i; \gamma) \pi(a_i; \alpha) \right} ]

We then need to create 4 functions.

|function| inputs | output | pseudocode example | |--------|--------|--------|-------| | h | outcome model | scalar | function(gamma, model1, ...) { inv_link(model.matrix(model1) %*% gamma) } | | f | treatment model | scalar | function(lambda, model2, ...) { inv_link(model.matrix(model2) %*% lambda) } | | g | | scalar | function(theta, lambda), model1, model2) { mean(outccome) * h(theta[gamma]) / f(theta[lambda])} | | psi | outcome model & treatment model | vector or matrix if d/dtheta psi | function(theta, model1, model2) {g(theta[1:p]) - theta[p] } |

Doubly Robust Example

We consider the case of clustered data.

library(sandwich)
library(sandwichShop)
library(inferference)
library(lme4)
library(dplyr)

exampledt <- vaccinesim %>% filter(group <= 50)


# Create list of arguments for treatment and outcome model

model_args = list(
  model_treatment = list(
    formula = A ~ X1 + (1|group),
    method  = glmer,
    options = list(family = binomial)
  ) ,
  model_outcome = list(
    formula = y ~ X1 + A + (1|group),
    method  = glmer,
    options = list(family = binomial)
  )
#   ,
#   causal_effect = list(
#     method = ipw_func,
#     user   = TRUE
#   )
)

models <- make_models(model_args = model_args, data = exampledt)

estfun_stacker(models) 
# methods(estfun)

bsaul/sandwichShop documentation built on May 13, 2019, 8:13 a.m.