PGC-2 mediation estimator: two-stage proximal mediation with path-specific bridges

Extends the proximal-inference approach to the two-linked-DAG mediation setting. Uses path-specific negative controls — W1 capturing the X->M confounder conf_XM and W2 capturing the M->Y confounder conf_MY — to purge confounding at both stages via bridge functions.

Usage

fit_pgc_mediation2(y, X, M, g, W1, W2, gm = NULL, covars = NULL, min_f = 10)

Arguments

y: Numeric outcome vector (length n).
X: Numeric exposure vector (length n).
M: Numeric mediator vector (length n).
g: Numeric instrument for X (length n).
W1: Numeric negative-control matrix (n x q) or vector for the X->M path (captures conf_XM). If a matrix, the bridge uses all q columns.
W2: Numeric negative-control matrix (n x q) or vector for the M->Y path (captures conf_MY). If a matrix, the bridge uses all q columns.
gm: Optional numeric mediator instrument vector (length n). When NULL (default), Stage 2 uses pure NC identification. When supplied, Gm helps isolate conf_MY before bridging — robust to Gm-U correlation.
covars: Optional data frame of additional covariates (n rows).
min_f: Minimum acceptable partial F-statistic for the excluded instrument G1. Default 10.

Value

Named list: NDE, NDE_se, NDE_p, NIE, NIE_se, NIE_p. Returns all-NA if the first-stage partial F for G1 is below min_f.

Details

When gm is NULL (no mediator instrument), Stage 2 residualises M on the cleaned exposure X_hat to isolate the conf_MY-driven component, then bridges on W2. This is pure negative-control identification at both stages — no mediator instrument is required.

When gm is supplied (mediator instrument present but its exogeneity may be in doubt), Stage 2 uses Gm to help isolate conf_MY's effect on M before bridging on W2. The bridge W_hat_M does the confounding removal, so the estimator is robust to Gm being correlated with conf_MY — the residual bias from Gm-U correlation is smaller than IV2SLS2's.

Strategy (three stages):

Bridge for X (purge conf_XM from X): X_resid = residuals(X ~ G1 + C); W_hat_X = bridge(X_resid ~ W1) (fitted values, proxy for conf_XM); X_hat = fitted(X ~ G1 + W_hat_X + C). Weak-IV check: partial F for G1 >= min_f.
Bridge for M (purge conf_MY from M): M_resid = residuals(M ~ X_hat + C) (gm = NULL), or M_resid = residuals(M ~ Gm + C) (gm supplied); W_hat_M = bridge(M_resid ~ W2) (fitted values, proxy for conf_MY); M_hat = fitted(M ~ X_hat + W_hat_M + C) (gm = NULL), or M_hat = fitted(M ~ X_hat + Gm + W_hat_M + C) (gm supplied); alpha_M = coefficient on X_hat.
Outcome: Y ~ X_hat + M_hat + W_hat_X + W_hat_M + C; NDE = coefficient on X_hat, beta_M = coefficient on M_hat.

NIE = alpha_M * beta_M (delta-method SE).

The key advantage over fit_iv2sls_mediation2: PGC-2 does not require instrument exogeneity. When the mediator instrument Gm is correlated with the confounder conf_MY (rho_G2 > 0), IV2SLS2 is biased but PGC-2's bridge absorbs conf_MY regardless of the instrument violation. The tipping-point simulation maps where PGC-2 bias crosses below IV2SLS2 bias.

References

Miao, W., Geng, Z., & Tchetgen Tchetgen, E. (2018). Identifying causal effects with proxy variables of an unmeasured confounder. Biometrika, 105(4), 987-993.

Examples

set.seed(1)
dat <- generate_toy_data(n = 500, mo_confounding = 0.8,
rho_G2 = 0.3, lambda_XM = c(1, 0), lambda_MY = c(0, 1),
omega_1 = 0.7, omega_2 = 0.7, seed = 1)
# Without mediator instrument (pure NC identification)
fit_pgc_mediation2(dat$Y[, 1], dat$X, dat$M, dat$G1, dat$W1, dat$W2)
#> $NDE
#> [1] -0.5101239
#> 
#> $NDE_se
#> [1] 0.02827378
#> 
#> $NDE_p
#> [1] 2.55428e-56
#> 
#> $NIE
#> [1] 0.739548
#> 
#> $NIE_se
#> [1] 0.02236263
#> 
#> $NIE_p
#> [1] 7.839686e-240
#> 
#> $alpha_M
#> [1] 0.5002583
#> 
#> $alpha_se
#> [1] 0.0116492
#> 
#> $beta_M
#> [1] 1.478332
#> 
#> $beta_M_se
#> [1] 0.02851672
#> 
# With (possibly imperfect) mediator instrument
fit_pgc_mediation2(dat$Y[, 1], dat$X, dat$M, dat$G1, dat$W1, dat$W2,
gm = dat$Gm)
#> $NDE
#> [1] -0.5101239
#> 
#> $NDE_se
#> [1] 0.02827378
#> 
#> $NDE_p
#> [1] 2.55428e-56
#> 
#> $NIE
#> [1] 0.739548
#> 
#> $NIE_se
#> [1] 0.02236263
#> 
#> $NIE_p
#> [1] 7.839686e-240
#> 
#> $alpha_M
#> [1] 0.5002583
#> 
#> $alpha_se
#> [1] 0.0116492
#> 
#> $beta_M
#> [1] 1.478332
#> 
#> $beta_M_se
#> [1] 0.02851672
#>