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Sandwich variance estimator for likelihood-based estimator

var_beta_mle.Rd

Computes a sandwich variance estimator for the parameter vector \(\theta\) obtained from estimate_beta_likelihood_optimx. The function allows for a general outcome model y ~ AW + Z1 + ... + Zp and a possibly different AFT model for X | Z specified by model_xz.

Usage

var_beta_mle(data_yXZ, theta, model, model_xz = NULL, aw_var = "AW")

Arguments

data_yXZ

A data frame containing at least the outcome y, the auxiliary covariate A, the observed covariate W, the event indicator D, the exposure covariate AW (or another name specified by aw_var), and any additional covariates appearing in model and optionally model_xz.

theta

Numeric vector of parameter estimates, typically taken from estimate_beta_mle(...)[["beta_est"]]. Must be ordered as \((\beta, \psi, \gamma_x, \text{shape}_x)\).

model

A formula specifying the outcome regression model, e.g. y ~ AW + Z1 + Z2.

model_xz

Optional formula specifying the AFT model for X | Z. If NULL (default), the model for X | Z is taken to be Weibull with log-mean linear in (1, Z1, ..., Zp), where Z1, ..., Zp are all covariates on the right-hand side of model except aw_var. If provided, model_xz can be either a right-hand-side formula (e.g. ~ Z1 + Z2) or a full Surv formula (e.g. Surv(W, D) ~ Z1 + Z2); in both cases, only the RHS is used here to construct the design for X | Z.

aw_var

Character string giving the name of the exposure covariate that is defined as A - X (default is "AW"). This variable must appear on the right-hand side of model and as a column in data_yXZ.

Value

A list with components

beta_est

Estimated regression coefficients \(\beta\).

psi_est

Estimated residual standard deviation \(\psi\).

se_beta

Sandwich standard errors for \(\beta\).

se_psi

Sandwich standard error for \(\psi\).

sandwich_var

Full sandwich variance matrix for \(\theta\).

Details

The parameter vector \(\theta\) is assumed to be of the form \(\theta = (\beta, \psi, \gamma_x, \text{shape}_x)\), where:

  • \(\beta\) are the regression coefficients in the outcome model,

  • \(\psi\) is the residual standard deviation on the original scale,

  • \(\gamma_x\) indexes the (log-)mean of \(X | Z\),

  • \(\text{shape}_x\) is the Weibull shape parameter for \(X | Z\).