# Akaike Information Criterium (*AIC*) in model selection

Data analysis often requires selection over several possible
models, that could
fit the data.
With noisy data, a more complex model gives better fit to the data (smaller
sum-of-squares, *SS*) than less
complex model. If only *SS* would be used to select the model that best fits the data,
we would conclude that a very complex model that fits every noise peak is the best.
Therefore, the model complexity needs to be taken into account in model selection.
Akaike Information Criterium is a commonly used method for model comparison, and conclusions from
other model selection methods are usually the same.

In the special case of sum-of-squares optimization, the basic *AIC* formula is expressed as:

, where *n* is the number of observations (for example PET frames), *k* is the
number of estimated parameters in the model (excluding fixed parameters), and *SS* is
the sum-of-squares, *Σe _{i}^{2}* (where

*e*are the estimated residuals). Although the

_{i}*AIC*formula appears to be very simple, its derivation is well founded on information theory, and the penalty term

*2×k*is not just an arbitrary value (Burnham and Anderson, 1998).

Information criterion is not a null hypothesis test: do not use terms like "(not) significant" or "rejected" in reporting results.

Based on information criteria, you must not test whether one model is "significantly" better than another model.

When sample size *n* is small compared to the number of parameters (*n/k* < 40,
that is, almost always in PET data analysis) the use of a second-order corrected *AIC*
(*AICc*) is recommended
(Burnham and Anderson, 1998):

*SS* is in the square of the units of the measured data. Therefore, the *AIC* is
on a relative scale, and it is critical to compute and present the *AIC* differences
(*ΔAIC*), instead of *AIC* or *AICc* values, over candidate models
(Burnham and Anderson, 1998;
Motulsky and Christopoulos, 2004).
Define A to be a simpler model and B to be a more complicated model
(*k _{A}<k_{B}*). The difference in

*AIC*is:

, and the difference in *AICc* is:

Equations (3) and (4) can be used only after both models A and B are
fitted to the data. A more practical approach may
be to calculate the *AIC*s separately for each model fit, and later calculate the difference
simply as:

, since, based on the properties of the logarithm, Eq (1) can also be written as

and then

*ΔAIC* or *ΔAICc* should calculated related to the smallest
*AIC* or *AICc*, so that the best model will have *ΔAIC* = 0
(Burnham and Anderson, 2004).
Although original *AIC* values may be very large compared to
the differences, that does not mean that the difference would not be important;
**only** the differences in *AIC* are interpretable as to the strength of
evidence (Burnham and Anderson, 2004).
The transformation *exp(-ΔAIC/2)* provides the likelihood of the model
(Akaike, 1981;
Burnham and Anderson, 2004).

Models can only be compared using information criteria when they have been fitted to exactly the same set of data with the same weights.

## MSC

Model selection criterion (*MSC*) is a reciprocal modification of Akaike information
criterion, used in *Scientist* software (MicroMath, Sant Louis, Missouri, USA).
*MSC* is independent on the magnitude (scaling) of the data.
Larger *MSC* means better fit, and when comparing models the most appropriate model for
the data is that with the largest *MSC*.

, where *w _{i}* are the weights for each data sample (

*i*),

*C*are the predicted (simulated, fitted) values,

_{SIM}*C*are the measured values, and C

_{PET}_{PET}is the mean of measured values.

## Comparison of *AIC* to other tests

*AIC* has been reported to find the "true" model more reliably than for example F-test
(Glatting et al, 2007;
Kletting et al, 2009a).
Compared to F-test, *AIC* has the advantage of being suited both for nested and
non-nested models.
Whether F-test tends to choose more complex or simple models than *AIC* depends on
the selected α value.
Glatting and Kletting conclude that *AIC* is effective and efficient approach.

Golla et al (2017) compared five model
selection criteria (*AIC*, *AICc*, *MSC*,
Schwartz Criterion, and F-test) on data from six PET tracers, and noted that all methods resulted in
similar conclusions.

Bayesian information criterion (*BIC*) is another sometimes used method.
*BIC* is not related to information theory, despite its name, and use of *AICc* over
*BIC* is recommended by
Burnham and Anderson (2004).

## See also:

- Fitting compartmental models
- Fitting PTACs
- Fitting plasma parent fractions
- Model validation
- Comparing PET TACs

## Literature

Akaike H. Likelihood of a model and information criteria.
*J Econometrics* 1981; 16(1): 3-14.
doi: 10.1016/0304-4076(81)90071-3.

Alves IL, García DV, Parente A, Doorduin J, Dierckx R, Marques da Silva AM, Koole M, Willemsen A,
Boellaard R. Pharmacokinetic modeling of [^{11}C]flumazenil kinetics in the rat brain.
*EJNMMI Res.* 2017; 7:17.
doi: 10.1186/s13550-017-0265-4.

Bonate PL: *Pharmacokinetic-Pharmacodynamic Modeling and Simulation*, 2nd ed., Springer,
2011. doi: 10.1007/978-1-4419-9485-1.

Burnham KP and Anderson DR. *Model Selection and Inference: A Practical
Information-Theoretical Approach*. Springer, 1998. ISBN 978-1-4757-2917-7.
doi: 10.1007/978-1-4757-2917-7.

Burnham KP, Anderson DR. Multimodel inference: understanding AIC and BIC in model selection.
*Sociol Methods Res.* 2004; 33(2): 261-304.
doi: 10.1177/0049124104268644.

Forster: Key Concepts in Model Selection: Performance and Generalizability.
*J Math Psychol.* 44, 205-231, 2000.
doi: 10.1006/jmps.1999.1284.

Glatting G, Kletting P, Reske SN, Hohl K, Ring C: Choosing the optimal fit function: Comparison
of the Akaike information criterion and the F-test. *Med Phys.* 34(11): 4285-92, 2007.
doi: 10.1118/1.2794176.

Golla SSV, Adriaanse SM, Yaqub M, Windhorst AD, Lammertsma AA, van Berckel BNM, Boellaard R.
Model selection criteria for dynamic brain PET studies. *EJNMMI Phys.* 2017; 4: 30.
doi: 10.1186/s40658-017-0197-0.

Kletting P, Glatting G: Model selection for time-activity curves: The corrected Akaike
information criterion and the F-test *Z Med Phys.* 19: 200-206, 2009a.
doi: 10.1016/j.zemedi.2009.05.003.

Kletting P, Kull T, Reske SN, Glatting G: Comparing time activity curves using the Akaike
information criterion. *Phys Med Biol.* 54: N501-N507, 2009b.
doi: 10.1088/0031-9155/54/21/N01

Turkheimer, Hinz, Cunningham: On the undecidability among kinetic models: from model selection
to model averaging. *J Cereb Blood Flow Metab.*, 23:490-498, 2003. doi: 10.1097/01.WCB.0000050065.57184.BB.

Zhou Y, Aston JAD, Johansen AM. Bayesian model comparison for compartmental models with
applications in positron emission tomography. *J Appl Statistics* 2013; 40(5): 993-1016.
doi: 10.1080/02664763.2013.772569.

Tags: Modeling, Compartmental model, Fitting, Validation, Statistics

Updated at: 2021-02-23

Created at: 2010-09-07

Written by: Harri Merisaari, Jambor I, Lars Jødal, Vesa Oikonen