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Estimating equations for IPW marginal structural model

Source: R/ee-causal.R
ee_ipw_msm.Rd

Estimates the parameters of a marginal structural model using inverse probability weighting with a logistic propensity score model.

Usage

ee_ipw_msm(
  theta,
  y,
  A,
  W,
  V,
  distribution,
  link,
  hyperparameter = NULL,
  truncate = NULL,
  weights = NULL
)

Arguments

theta

Numeric vector of length c + b, where c is the number of MSM parameters and b is the number of propensity score model parameters.

y

Numeric vector of n observed outcomes.

A

Numeric vector of n binary treatment indicators (0/1).

W

Numeric n-by-b design matrix for the propensity score model.

V

Numeric n-by-c design matrix for the marginal structural model.

distribution

Character string for the GLM distribution of the outcome model. Every name ee_glm() takes except the three that estimate a nuisance parameter, which the theta partition here reserves no slot for: "normal" (or "gaussian"), "binomial" (or "bernoulli", or "bin"), "poisson", "inverse_normal" (or "inverse_gaussian"), and "tweedie". See hyperparameter for what naming one of the three does.

Character string for the GLM link function. See ee_glm() for options.

hyperparameter

Optional numeric hyperparameter passed straight through to the marginal structural model's ee_glm() call. Used by the tweedie outcome distribution, where it fixes the variance power v(mu) = mu^hyperparameter. Default NULL. Note that the theta partition reserves exactly ncol(V) slots for the MSM and no slot for an estimated nuisance parameter, so outcome families that carry one ("gamma", "negative_binomial", "nb") cannot be used as the MSM outcome model. Naming one is refused by name, before the outcome model is formed and whatever hyperparameter is set to. Python Delicatessen cannot fit those families either, where the attempt fails as a shape error.

truncate

Optional length-2 numeric vector c(lower, upper) to clip estimated propensity scores. Bounds must be in ascending order (lower <= upper). Default NULL.

weights

Optional numeric vector of n weights. Default NULL.

Value

A (c+b)-by-n matrix of estimating equation contributions. The marginal structural model rows are named MSM alpha_0 through MSM alpha_(c-1), matching the zero-based subscripts the literature gives those parameters. The propensity score rows are named W_1 through W_b for the columns of W.

Examples

# A confounded binary treatment whose true effect on the outcome is -2.
set.seed(42)
n <- 500
W <- rbinom(n, 1, 0.5)
A <- rbinom(n, 1, 0.25 + 0.5 * W)
Y <- 5 + 2 * W - 2 * A + rnorm(n)

W_ps <- cbind(1, W) # Propensity score design matrix
V <- cbind(1, A) # Marginal structural model design matrix

psi <- function(theta) {
  ee_ipw_msm(
    theta,
    y = Y,
    A = A,
    W = W_ps,
    V = V,
    distribution = "normal",
    link = "identity"
  )
}

# theta holds the two marginal structural model coefficients, whose slope is
# the causal effect, followed by the two propensity score coefficients.
m <- m_estimate(stacked_equations = psi, init = rep(0, 4))
coef(m)
#> MSM alpha_0 MSM alpha_1         W_1         W_2 
#>    5.953823   -2.066997   -1.024504    2.234017