# Implementation Aspects in Simulation Extrapolation-Based Stocking-Lord Linking
## 3 Proposed Computational Shortcuts in SIMEX-Based Linking
### A Glimpse of the SIMEX Technique  
This section outlines the core principle of the SIMEX method introduced by [PERSON]. [22; 25; 26]. The SIMEX is a general approach for providing bias correction due to measurement errors in variables for any multivariate statistical method. Consider a dataset composed of two parts: a multivariate random variable \(\mathbf{Z}\) measured without error, and a univariate random variable \(W\), which contains measurement errors. Assume that independent realizations \(\mathbf{z}_{i}\) and \(w_{i}\) of \(\mathbf{Z}\) and \(W\) are available for cases \(i=1,\ldots,I\) in a sample of \(I\) cases. Furthermore, assume the classical measurement error model [27]  
\[w_{i}=w_{i}^{*}+e_{i}, \tag{8}\]where \(w_{i}^{*}\) represents the measured value of \(W\) without error, and \(e_{i}\) denotes the variable referring to the measurement error. In the following, we suppose that the measurement error variance \(\mathsf{Var}(e_{i})=\tau^{2}\) is either known or can be consistently estimated from the data. Additionally, we make the assumption that the measurement errors \(e_{i}\) follow a normal distribution with a mean of zero. A statistical procedure yields a parameter estimate \(\widehat{\delta}=f(\mathbf{Z},W)\) for a parameter vector \(\delta\), where \(f\) is a known function of the dataset represented by the random variables \(\mathbf{Z}\) and \(W\). The known function \(f\) can represent any parametric multivariate statistical method. The bias of the estimated parameter \(\widehat{\delta}\) is affected by the measurement error variance \(\tau^{2}\). The SIMEX method seeks to eliminate or at least mitigate this bias in \(\widehat{\delta}\) caused by measurement errors through a simulation-based technique. The core idea is to intentionally introduce additional measurement errors into the data and then compute the parameter estimate \(\widehat{\delta}\) based on these altered datasets that include extra measurement errors. The values \(w_{i}\) of variable \(W\) are replaced with  
\[\widehat{w}_{i}=w_{i}+\widetilde{e}_{i}=w_{i}^{*}+e_{i}+\widetilde{e}_{i}, \tag{9}\]  
where \(\widetilde{e}_{i}\) represents a random draw from a normal distribution with a mean of zero and a variance of \(\lambda\tau^{2}\) with \(\lambda\geq 0\). As a result, the adjusted values \(\widehat{w}_{i}\) (i.e., the pseudo-values) contain measurement error \(e_{i}+\widetilde{e}_{i}\), leading to a variance of \((1+\lambda)\tau^{2}\). The SIMEX method provides estimates \(\widehat{\delta}\) as a function of \(\lambda\) across a grid of \(\lambda\) values. The literature suggests using the following grid of \(\lambda\) values: 0.5, 1.0, 1.5, and 2.0 (see [27; 28]). To mitigate Monte Carlo errors due to simulating extra measurement errors in \(\widetilde{e}_{i}\) (see (9)), the estimation is carried out through repeated simulations of the dataset for each fixed \(\lambda\) value. As a result, the parameter estimate is expressed as a function of \(\lambda\), leading to a parameter curve \(\widehat{\delta}(\lambda)\). This first stage of the procedure pertains to the simulation step in SIMEX, while the second stage involves an extrapolation. In that stage, a regression function \(\widehat{\delta}(\lambda)\) is estimated, and the predicted value at \(\lambda=-1\) yields a bias-reduced estimate for \(\delta\) that has been extrapolated to a scenario with no measurement error variance (i.e., \(\tau^{2}=0\)). Any functional form for \(\widehat{\delta}(\lambda)\) could be assumed. However, a quadratic extrapolating function [27] is found to be effective in applications and frequently recommended in the literature using the specification  
\[\widehat{\delta}(\lambda)\approx\mathbf{\alpha}_{0}+\mathbf{\alpha}_{1}\lambda+\mathbf{ \alpha}_{2}\lambda^{2}. \tag{10}\]  
The final parameter estimate of \(\delta\) in SIMEX is given by \(\widehat{\delta}(-1)=\mathbf{\alpha}_{0}-\mathbf{\alpha}_{1}+\mathbf{\alpha}_{2}\).