Estimating equations for weighted sensitivity analysis of the mean

Estimates the mean of an outcome with missing data using a weighted sensitivity analysis approach. Handles MCAR, MAR, and MNAR mechanisms by specifying a user-defined sensitivity function q_eval and a monotone increasing distribution function H_function.

Usage

ee_mean_sensitivity_analysis(theta, y, delta, X, q_eval, H_function)

Arguments

theta: Numeric vector of length 1 + b, where b is the number of columns in X. The first element is the corrected mean; the remainder are regression coefficients.
y: Numeric vector of n outcome values. Missing values should be indicated via the delta parameter.
delta: Numeric vector of n indicators: 1 if y is observed, 0 if missing. Must not contain NA.
X: Numeric n-by-b design matrix for the missingness model. Should include an intercept column. Must not contain NA.
q_eval: Numeric vector of n evaluated sensitivity function values, i.e. q(Y, alpha).
H_function: A function mapping real values to [0, 1] that is monotone increasing (e.g., inverse_logit()).

Value

A (1+b)-by-n matrix of estimating equation contributions, with the first row named corrected_mean and the missingness model rows named X_1 through X_b for the columns of X.

Examples

# An outcome observed for only part of the sample, with missingness driven by
# the measured covariate W.
set.seed(42)
n <- 500
W <- rbinom(n, 1, 0.5)
Y_full <- 200 - 35 * W + rnorm(n, sd = 5)
delta <- rbinom(n, 1, inverse_logit(2 + W))

# Missing outcomes never enter the estimating equation, so any placeholder
# value works; a zero keeps the arithmetic finite.
Y <- ifelse(delta == 1, Y_full, 0)
X <- cbind(1, W) # Missingness model design matrix

# A sensitivity function of zero everywhere assumes the outcome is missing at
# random given W. Nonzero values encode departures from that assumption.
psi <- function(theta) {
  ee_mean_sensitivity_analysis(
    theta,
    y = Y,
    delta = delta,
    X = X,
    q_eval = rep(0, n),
    H_function = inverse_logit
  )
}

# theta holds the corrected mean, started near the observed outcomes,
# followed by the two missingness model coefficients.
m <- m_estimate(stacked_equations = psi, init = c(180, 0, 0))
coef(m)
#> corrected_mean            X_1            X_2 
#>     183.174625       1.698074       1.061031