Estimating equation for regression calibration

Corrects for measurement error in a binary predictor using external validation data. Scales the naive coefficient by the calibration factor.

Usage

ee_regression_calibration(theta, beta, a, a_star, r, X = NULL, weights = NULL)

Arguments

theta

Numeric vector: the corrected coefficient first, then the calibration coefficients for the design cbind(a_star, X) (the a_star coefficient first). Length 2 + ncol(X) when X is supplied, or length 3 when X = NULL (an intercept-only calibration model).

beta

Numeric scalar. External estimate of the coefficient for the mismeasured predictor on the outcome.

a

Numeric vector of gold-standard action measurements (validation sample only, where r = 0).

a_star

Numeric vector of mismeasured action values.

r

Numeric indicator: 0 for validation, 1 for main study.

X

Optional design matrix for calibration model. Default NULL uses intercept only.

weights

Optional numeric vector of n weights. Default NULL.

Value

A length(theta)-by-n matrix (2 + ncol(X) rows, or 3 rows when X = NULL). The first row is named corrected_beta and the second a_star. The remaining rows are named X_1 through X_p for the columns of X, or intercept when X = NULL.

Examples

# A binary exposure is measured with error in the main study. The external
# validation study regresses the gold-standard exposure on the mismeasured
# one, and that calibration slope rescales the naive outcome coefficient.
set.seed(789)
n <- 500
a_true <- rbinom(n, 1, 0.5)
a_star <- ifelse(a_true == 1, rbinom(n, 1, 0.85), rbinom(n, 1, 0.1))
r <- c(rep(0, 200), rep(1, 300))

# The gold standard is observed only in the validation sample, so the main
# study positions carry a 0 placeholder. It never reaches an estimate: the
# calibration model is multiplied by (1 - r).
a <- ifelse(r == 0, a_true, 0)

# `beta` is the naive coefficient for the mismeasured exposure, supplied here
# as a fixed external value. Stack an outcome model and pass its coefficient
# instead to propagate the uncertainty in that estimate as well.
psi <- function(theta) {
  ee_regression_calibration(theta, beta = 0.8, a = a, a_star = a_star, r = r)
}

m <- m_estimate(stacked_equations = psi, init = c(1, 0.1, 0.5))

# Corrected coefficient, then the calibration slope and intercept
coef(m)
#> corrected_beta         a_star      intercept 
#>      1.0853202      0.7371097      0.1764706