Signature
AICc = 2 * k - 2 * lnL + 2 * k * (k + 1) / (n - k - 1)
| Inputs | Definition | Unit |
|---|---|---|
k | Number of parameters estimated when the model is fitted | count |
lnL | Natural logarithm of the maximised likelihood of the model | none |
n | Sample size of the data to which the model was fitted | count; must exceed k plus 1 |
AICc | Akaike information criterion with the second-order small-sample correction | none; read only as a difference from other models fitted to the same data |
|---|
Function
Penalised-likelihood ranking of candidate survival models
Maps the maximised log-likelihood and the number of estimated parameters of each model fitted to the same data to an information criterion, a fit statistic with a penalty for each extra parameter. Lower values rank higher, and only differences between models fitted to the same observations carry meaning. In health technology assessment the candidates are usually parametric survival curves fitted to trial time-to-event data before extrapolation. The criteria measure fit within follow-up only, so they inform but do not settle the choice of curve.
Try this function
Implementations
Excel
AICc in one cell with the small-sample correction
With named cells LogLik, Params and SampleSize, the formula returns AICc.
=2*Params-2*LogLik+2*Params*(Params+1)/(SampleSize-Params-1)
Assumptions
Sample size exceeds the parameter count plus one
The correction is defined only when n is greater than k plus 1. All candidates are compared on AICc, not a mixture of AIC and AICc.
Worked examples
AICc of the generalised gamma curve with 300 patients
With 300 patients the ratio of sample size to parameters is 100 for the generalised gamma, and the correction adds only about 0.08 to its AIC of 1211.6. The ranking in the six-curve example is unchanged.
k = 3; lnL = -602.8; n = 300; AICc = 1211.68
AICc of a three-parameter curve in a hypothetical arm of 30 patients
In a hypothetical arm of 30 patients, a three-parameter model with a maximised log-likelihood of -100 has an AIC of 206 and an AICc of about 206.92, because the correction rises to about 0.92.
k = 3; lnL = -100; n = 30; AICc = 206.92
Common errors
Using AIC in a small arm where AICc is needed
Burnham and Anderson call the use of AIC where AICc is needed a pervasive mistake. In an arm of 30 patients the correction is about 0.14 for a one-parameter model and about 0.92 for a three-parameter model, so plain AIC understates the extra penalty on the three-parameter model by about 0.78 units.
Sources
Second-order small-sample correction to AIC
Burnham KP, Anderson DR. Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods & Research. 2004;33(2):261-304. Page 270, which gives the AICc formula and recommends it unless the ratio of sample size to parameters exceeds about 40.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0