Akaike weight of a candidate model

Converts AIC differences into weights that sum to 1 across the R candidate models. The weight w_i equals exp(-delta_i/2) divided by S, where S is the sum of exp(-delta_r/2) over all R candidates. It is read as the weight of evidence that the model is the best approximating model in the set, given the data. Adding or removing a candidate changes every weight.

Signature

w = exp(-delta / 2) / S
Inputs
InputsDefinitionUnit
deltaAIC difference of the model from the best-scoring model in the setAIC units
SSum of exp(-delta_r/2) over all R candidate models, including the best-scoring model, which contributes 1none
Output
wAkaike weight of the candidate modelproportion from 0 to 1; the weights of all candidates sum to 1

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.

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Implementations

  • Excel

    Akaike weight from a column of AIC differences

    With all AIC differences in a range named DeltaRange and the current model's difference in a cell named Delta, SUMPRODUCT forms the sum S without an array entry.

    =EXP(-Delta/2)/SUMPRODUCT(EXP(-DeltaRange/2))

Assumptions

  • Weights conditional on the full candidate set

    The weights depend on which models are in the set. The ratio of two weights does not, because it equals exp of half the difference between their AIC differences.

Worked examples

  • Akaike weight of the log-normal curve

    The six exponentials of minus half the AIC differences sum to about 2.2561 in the article's example (2.25605 unrounded). The log-normal, with a difference of 0, has a weight of 1 divided by 2.25605, about 0.4433, shown as 0.44 in the article's table.

    delta = 0; S = 2.25605; w = 0.4433

  • Akaike weight of the Weibull curve

    The Weibull, 3.8 units behind, has a relative likelihood of about 0.1496 and a weight of about 0.066, shown as 0.07 in the article's table.

    delta = 3.8; S = 2.25605; w = 0.0663

  • Akaike weight of the generalised gamma curve

    The generalised gamma, 1.2 units behind, has a relative likelihood of about 0.5488 and a weight of about 0.24.

    delta = 1.2; S = 2.25605; w = 0.2433

Common errors

  • Dropping the factor of one half from the weight

    Using exp(-delta) instead of exp(-delta/2) exaggerates the lead of the best model. In the six-curve example it raises the log-normal weight from about 0.44 to about 0.64 and cuts the Weibull weight from about 0.066 to about 0.014.

  • Reading the largest Akaike weight as plausibility of the extrapolation

    An Akaike weight of 0.44 for the log-normal describes its fit to the observed trial data relative to the other candidates. It says nothing about whether the curve's hazard beyond follow-up is clinically plausible, which has to be assessed against external evidence.

Sources

  • Definition and interpretation of Akaike weights

    Burnham KP, Anderson DR. Multimodel inference: understanding AIC and BIC in model selection. Sociological Methods & Research. 2004;33(2):261-304. Pages 271 to 272, which define the likelihood of a model given the data, the Akaike weights and their dependence on the full model set.

    View source →

Canonical Identity

Persistent URI
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