Besides the various measures of Minimum Description Length (e.g., normalized maximum likelihood, Fisher Information approximation), there are two other methods worth to mention:
- Parametric Bootstrap. It's a lot easier to implement than the demanding MDL measures. A nice paper is by Wagenmaker and colleagues:
Wagenmakers, E.-J., Ratcliff, R., Gomez, P., & Iverson, G. J. (2004). Assessing model mimicry using the parametric bootstrap. Journal of Mathematical Psychology, 48, 28-50.
The abstract:
We present a general sampling procedure to quantify model mimicry, defined as the ability of a model to account for data generated by a competing model. This sampling procedure, called the parametric bootstrap cross-fitting method (PBCM; cf. Williams (J. R. Statist. Soc. B 32 (1970) 350; Biometrics 26 (1970) 23)), generates distributions of differences in goodness-of-fit expected under each of the competing models. In the data informed version of the PBCM, the generating models have specific parameter values obtained by fitting the experimental data under consideration. The data informed difference distributions can be compared to the observed difference in goodness-of-fit to allow a quantification of model adequacy. In the data uninformed version of the PBCM, the generating models have a relatively broad range of parameter values based on prior knowledge. Application of both the data informed and the data uninformed PBCM is illustrated with several examples.
Update: Assessing model mimicry in plain English. You take one of the two competing models and randomly pick a set of parameters for that model (either data informed or not). Then, you produce data from this model with the picked set of parameters. Next, you let both models fit the produced data and check which of the two candidate models gives the better fit. If both models are equally flexible or complex, the model from which you produced the data should give a better fit. However, if the other model is more complex, it could give a better fit, although the data was produced from the other model. You repeat this several times with both models (i.e., let both models produce data and look which of the two fits better). The model that "overfits" the data produced by the other model is the more complex one.
- Cross-Validation: It is also quite easy to implement. See the answers to this question. However, note that the issue with it is that the choice among the sample-cutting rule (leave-one-out, K-fold, etc) is an unprincipled one.
