Model estimation by maximum likelihood

John M. Drake

Model calibration

Model calibration requires tuning the parameters to “best” represent reality

Evaluation of this representation is by

  • goodness-of-fit
  • generalizability

Least squares estimation

Model calibration is performed by adjusting the parameters to minimize some objective function (aka loss function)

For instance, least squares estimation minimizes the sum of squared errors

SSE=ni=1(yif(xi))2

where yi is the ith observation, xi are the inputs to the model, and f() is the model.

Least squares estimation

The “adjustment” of parameters is typically performed by numerical optimization

Model comparison: Likelihood ratio test

Likelihood ratio test for nested models

The likelihood ratio test statistic is given by

\lambda_{LR} = -2 \left[ \mathcal l(\theta_0) - \mathcal l(\theta_1) \right]

where \theta_0 is a null hypothesis and \theta_1 is an alternative hypothesis.

Wilks theorem establishes that if the null hypothesis is true, \lambda_{LR} is asymptotically distributed with a \chi^2 distribution. This can be used to perform custom hypothesis tests when we encode null and alternative hypotheses in the \theta_0 and \theta_1 vectors.

Model comparison: Akaike's Information Criterion

Akaike's Information Criterion (AIC) can be used for non-nested models.

The AIC of a model is given by

AIC = 2k-2\log(\mathcal L (\theta))

where k is the number of parameters of the model.

AIC differences greater than 2 are generally considered to be statistically significantly different.

Acknowledgements

Presentations and exercises draw significantly from materials developed with Pej Rohani, Ben Bolker, Matt Ferrari, Aaron King, and Dave Smith used during the 2009-2011 Ecology and Evolution of Infectious Diseases workshops and the 2009-2019 Summer Institutes in Statistics and Modeling of Infectious Diseases.

Licensed under the Creative Commons attribution-noncommercial license, http://creativecommons.org/licenses/bync/3.0/. Please share and remix noncommercially, mentioning its origin.