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Conditional distribution

Purpose

Although the population parameter estimation algorithm can give a rough estimation of the individual parameters, it can be estimated by a more precise estimator using the conditional distribution.

Conditional distribution

If the option “Condition distribution” 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 using the Metropolis-Hastings (MH) algorithm. For each individual 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.). Thus, we draws from the condition

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 an interval. When the mean and the standard error does not vary anymore, we stop the draws.

Display and outputs

During the evaluation of the conditional distribution, the following plot is proposed.

We see the conditional expectation and standard error for each parameters. We continue to draw from the conditional distribution until these value are stable enough, i.e. they do not vary more that 5 % during the last 50 draws. The bars above and below represent the 5% above and the 5% lower respectively. When all the 50 points are in the interval, the bars are green, red otherwise.

Settings

To change the settings for the convergence of the conditional mean, you can click on the settings of the conditional distribution task.

and the following settings window appears

In the interface, a summary of the individual parameters is proposed (min, max and some quartiles) as shown in the figure below.

In terms of output, a folder called IndividualParameters is created in the result folder where the following files are created

  • individualParameters.txt. All the individual parameters for each subject-occasion from SAEM, the conditional mean and/or the conditional mode (with extensions _SAEM, _mean and _mode respectively) are written in that file.
  • individualRandomEffects.txt. All the random effects for each subject-occasion from SAEM, the conditional mean and/or the conditional mode (with extensions _SAEM, _mean and _mode respectively) are written in that file.
  • simulatedParameters.txt. All the simulated individual parameters are written in that file. A column rep is added to allow the simulation of several individual parameters for the same subject-occasion.
  • simulatedRandomEffects.txt. All the simulated individual random effects are written in that file. A column rep is added to allow the simulation of several individual parameters for the same subject-occasion.

Notice that the covariates are also added to each files for each subject-occasion.

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.