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. 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.

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, Σei2 (where ei are the estimated residuals). Although the 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).

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 (kA<kB). 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 AICs 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).

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.

Comparison to F-test and BIC

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. Another method, Bayesian information criterion (BIC), is not related to information theory, despite its name, and use of AICc over BIC is recommended by 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 wi are the weights for each data sample (i), CSIM are the predicted (simulated, fitted) values, CPET are the measured values, and CPET is the mean of measured values.

References

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Updated at: 2018-12-12
Created at: 2010-09-07
Written by: Harri Merisaari, Jambor I, Lars Jødal, Vesa Oikonen