SIMEX

SIMEX, short for Simulation-Extrapolation, is a statistical method used to correct bias caused by measurement error in covariates within regression models. It was introduced by Cook and Stefanski in 1994 and has since been applied in various fields, including epidemiology, genetics, and econometrics. The core idea is to study how estimates change as additional noise is imposed on the observed covariates, then extrapolate back to a scenario with no measurement error.

The method starts with a classical additive measurement error model where an observed predictor W equals the true covariate X plus an error U, W = X + U, with U independent of X and of known or estimable variance. If replicate measurements or external data are available, the measurement error variance can be estimated. The procedure then proceeds in stages. First, for a sequence of nonnegative values λ, synthetic noise is added to W to form W_λ = W + sqrt(λ) e, where e is drawn from the same distribution as U. For each W_λ, the regression model is fitted to obtain an estimate θ̂(λ) of the parameter of interest. Next, θ̂(λ) is modeled as a function of λ (often linear or quadratic) and extrapolated to λ = −1, which corresponds to removing all measurement error. Bootstrapping or other resampling methods may be used to quantify uncertainty.

SIMEX has extensions to generalized linear models, survival analysis, and multivariate measurement error, as well as Bayesian variants. Its performance depends on the correct specification of the measurement error distribution and variance, the choice of extrapolation model, and computational resources.