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# Individual parameters estimation

### Purpose

Although the population parameter estimation algorithm gives a rough estimation of the individual parameters, it can be estimated two more precise estimators: the conditional mode and the conditional mean.

If the option ”Estimate the conditional modes” is selected, the individual parameters are estimated by maximizing the conditional probabilities $p(\psi_i|y_i;\hat{\theta})$, i.e.

$$\hat{\psi}_i^{mode} = \underset{\psi_i}{\textrm{arg max }}p(\psi_i|y_i;\hat{\theta})$$

It corresponds to the “optimal” value for the fit. If the option “Estimate the cond. means and s.d.” is selected, the conditional means and standard deviations are estimated by MCMC. The conditional distribution $p(\psi_i|y_i;\hat{\theta})$ of the vector of individual parameters $\psi_i$ can be estimated for each individual $i$ using the Metropolis-Hastings (MH) algorithm. For each $i$, this algorithm generates a sequence $(\psi_i^{(k)}, i\leq k \leq K)$ which converges in distribution to the conditional distribution $p(\psi_i|y_i;\hat{\theta})$ and can be used for estimating any summary statistic of it (mean, standard deviation, quantiles, etc.).

The MH algorithm therefore allows us to define an initial estimator of the individual parameter $\psi_i$ that approximates the conditional mean:

$\hat{\psi}_i^{mean} = \frac{1}{K}\sum_{k=1}^{K}\psi^{k}$

For each parameter, the mean of these quantities over all the subjects is displayed together with a $r_{mcmc}$ interval. Two files indiv_parameters.txt and indiv_eta.txt are created with the estimated individual parameters and random effects in table format. Also, if there were defined priors on some fixed effects, and it was selected prior distribution method for some of them, a new file called simulatedPopParams.txt is created simulations using their posterior distribution laws.

### Advanced settings for individual parameter estimation:

MCMC: $m_1$, $m_2$, $m_3$ and $m_4$ are the numbers of transitions of the four different kernels used in the MCMC algorithm. The default value are $m_1=2$, $m_2=0$, $m_3=2$, and $m_4 = 2$. $L_{mcmc}$ and $r_{mcmc}$ define the stopping rule of the MCMC algorithm. The number of iterations of MCMC increases with $L_{mcmc}$ and decreases with $r_{mcmc}$. It is accessible in Menu/Settings/Indiv. parameters.

### Best practices : When do we look at the conditional mode and when do we look at the conditional mean?

The choice of using the conditional mean $\psi^{mean}$ or conditional mode $\psi^{mode}$ is arbitrary. By default, Monolix uses the conditional mode for computing predictions, taking the philosophy that the “most likely” values of the individual parameters are the most suited for computing the “most likely” predictions.