# Monte Carlo Simulation

The history of Monte Carlo simulations can be traced back to Metropolis and Ulam [61]. It is widely used to assess the performance of statistical estimation methods in the statistical literature. This method obtains numeric results by using repeated random sampling, which is not only useful for practical identifiability analysis, but also helpful for statistical inference. In general, a Monte Carlo simulation procedure can be outlined as follows:

- 1. Determine the nominal parameter values 0
_{0}for simulation studies, which can be obtained by fitting the model to the experimental data if available. Otherwise it can be obtained from the literature. - 2. Use the nominal parameter values to numerically solve the ODE model and obtain the solution
**x**(t) at the experimental design time points. - 3. Generate N sets of simulated data from the model (11.12) with a given measurement error level.
- 4. Fit the ODE model to each of the N simulated data sets and obtain parameter estimates 0/, i = 1, 2, ..., N.
- 5. Calculate the average relative estimation errors (ARE) for 0 as

The ARE can be used to assess whether each of the parameter estimates is acceptable or not. If the ARE of a parameter estimate is unacceptably high, we can claim that this parameter is not practically or statistically identifiable. Some parameters may not be sensitive to measurement errors and can always be well estimated, but some other parameters may be quite sensitive to measurement errors, and their AREs are large even with a small measurement error. In practice, there is no clear-cut rule of ARE to claim an "unidentifiable" parameter. Thus, the practical identifiability relies on the underlying problem and the judgment of investigators. Also notice that various statistical estimation approaches can be employed to obtain the parameter estimates, and the ARE may depend on the estimation methods.