Estimating equations for the g-formula (g-computation)

Returns a stacked set of estimating equations for the g-formula. When X0 = NULL, estimates a single causal mean under the plan encoded by X1. When X0 is provided, estimates the average causal effect (difference between two plans).

Usage

ee_gformula(theta, y, X, X1, X0 = NULL, force_continuous = FALSE)

Arguments

theta

Numeric vector. If X0 = NULL, length is 1 + p (causal mean + regression coefficients). If X0 is provided, length is 3 + p (ACE, mean under X1, mean under X0, regression coefficients).

y

Numeric vector of n observed outcomes.

X

Numeric n-by-p design matrix (observed data).

X1

Numeric n-by-p design matrix under action plan 1.

X0

Optional n-by-p design matrix under action plan 0. Default NULL.

force_continuous

Logical. Force linear regression even when y is binary? Default FALSE.

Value

A matrix of estimating equation contributions. When X0 = NULL the first row is named causal_mean; when X0 is provided the first three rows are named ACE, E[Y^1], and E[Y^0], where 1 and 0 index the two plans. The outcome model rows are named X_1 through X_p for the columns of X.

Examples

# A binary treatment, two confounders, and a continuous outcome whose true
# average causal effect is 1.5.
set.seed(42)
n <- 1000
W1 <- rnorm(n)
W2 <- rbinom(n, 1, 0.4)
A <- rbinom(n, 1, inverse_logit(-0.5 + 0.5 * W1 + 0.3 * W2))
Y <- 2 + 1.5 * A + W1 - 0.5 * W2 + rnorm(n)

X <- cbind(1, A, W1, W2) # Observed design matrix
X1 <- cbind(1, 1, W1, W2) # Everyone treated
X0 <- cbind(1, 0, W1, W2) # Everyone untreated

psi <- function(theta) ee_gformula(theta, y = Y, X = X, X1 = X1, X0 = X0)

# theta holds the average causal effect, the mean under treatment, and the
# mean under no treatment, followed by the four outcome model coefficients.
m <- m_estimate(stacked_equations = psi, init = rep(0, 7))
coef(m)[1:3]
#>      ACE   E[Y^1]   E[Y^0] 
#> 1.578994 3.309943 1.730949