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Standard error using the Fisher Information Matrix

Purpose

The variance of the maximum likelihood estimate (MLE) \hat{\theta}, and thus confidence intervals, can be derived from the observed Fisher information matrix (FIM), itself derived from the observed likelihood (i.e., the pdf of observations y):

$$ I_{y}(\hat{\theta})\triangleq -\frac{\partial^2}{\partial\theta^2}\log({\cal L}_y(\hat{\theta})) $$

Fisher Information Matrix calculation methods

There are two different algorithms: by linearization or by stochastic approximation. When “linearization” is used, the structural model is linearized, and the full statistical model is approximated by a Gaussian model. When “stochastic approximation” is used, the exact model is used, and the Fisher information matrix (F.I.M) is approximated stochastically.

Display

The final estimations are displayed together with the population parameters:

  • the estimated fixed effects, their standard-errors, the absolute and relative p-values obtained from the Wald test (only for the coefficients of the covariates),
  • the estimated variances (or standard deviations) and their standard-errors,
  • the estimated residual error parameters and their standard-errors,

Remarks

  • If the two methods are used to computed the standard errors, both are displayed in the Estimated Population Parameters table.
  • To help the user in the interpretation, we provide a graphical information using colors for both the p-value and the RSE
    • For the p-value: between .01 and .05, between .001 and .01, and less than .001.
    • For the RSE: between 50% and 100%, between 100% and 200%, and more than 200%.

In addition, in the Results frame, we propose the full correlation matrix as in the figure below
where we compute

  • the RSE for each parameter,
  • the correlation matrix of the fixed effect estimates and the the variance components estimates,
  • the smallest and largest eigenvalues,

Remarks

  • Again, if the two methods are used to computed the standard errors, both are displayed in the Results/Standard Errors frame.
  • To help the user in the interpretation, we provide a graphical information using colors for both the correlation and the RSE
    • For the correlation: if the absolute between .5 and .8, if the absolute between .8 and .9, and if the absolute is higher than .9.
    • For the RSE: between 50% and 100%, between 100% and 200%, and more than 200%.
    • If you look at a specific value with the mouse, both parameters are highlighted to know easily which parameter you are looking at as on the figure below.

Outputs

In terms of outputs, a folder called FisherInformation is created in the result folder where

  • the jacobian is stored in a file jacobian.txt
  • the covariance estimates are stored in a file covarianceEstimatesSA.txt (or covarianceEstimatesLin.txt if the linearization is used)
  • the correlation estimates are stored in a file correlationEstimatesSA.txt (or correlationEstimatesLin.txt if the linearization is used)

the table of parameter name and value is defined.

Best practices: when to use “linearization” and when to use “stochastic approximation”

Firstly, it is only possible to use the linearization algorithm for the continuous data. In that case, this method is generally much faster than stochastic approximation and also gives good estimates of the FIM. The FIM by model linearization will generally be able to identify the main features of the model. More precise– and time-consuming – estimation procedures such as stochastic approximation and importance sampling will have very limited impact in terms of decisions for these most obvious features. Precise results are required for the final runs where it becomes more important to rigorously defend decisions made to choose the final model and provide precise estimates and diagnostic plots.