Density ratio between the target and the pooled population

Estimates \(c_i = P(D_i = 1 | W_i) / \hat\pi\) with \(\hat\pi = n^{-1}\sum_i D_i\) by a logistic regression of the population indicator D on the features W (typically surrogates S and, optionally, covariates). Because \(P(D=1|W)/\pi = f_{target}(W)/f_{pooled}(W)\), \(c_i\) is the weight that aligns the pooled sample with the target population and is used by the doubly robust auxiliary estimator (aux = "dr" in heart_assoc()).

Usage

heart_density_ratio(
  D,
  W,
  quadratic = FALSE,
  ridge = 1e-08,
  maxit = 50L,
  tol = 1e-08
)

Arguments

D

0/1 population indicator (1 = target).

W

Features (matrix or data.frame); factors are expanded by stats::model.matrix().

quadratic

If TRUE, squared terms of the numeric columns are added.

ridge

Initial ridge penalty.

maxit, tol

Maximum IRLS iterations and convergence tolerance.

Value

Numeric vector c_i with attributes coef (fitted logistic coefficients), e_hat (fitted P(D = 1 | W)) and pi_hat.

Details

The logistic model is fitted by iteratively reweighted least squares with a small ridge penalty that is increased automatically if the weighted Gram matrix is numerically singular (e.g. under quasi-separation).

Examples

sim <- heart_simulate(n_tar = 500, n_src = 2000, J = 10, seed = 1)
c_i <- heart_density_ratio(sim$D, cbind(S = sim$S, sim$X))
tapply(c_i, sim$D, mean)
#>         0         1 
#> 0.9774112 1.0903551