untitled

There are two parameters, because we estimated the mean and residual variation in r:

p 4 =^2

AIC 4 =−30.692

The model with the lowest AIC score is taken as the best. Smaller scores imply a

better fit to the data. We can evaluate the merit of each model relative to the best

model, using the difference between their AIC scores (∆AICi=AIC score of model

i minus the AIC score of the best model). If ∆AIC <4, then the difference in explana-

tory power is considered trivial by statistical experts so the models provide similar

explanation (Burnham and Anderson 1998).

On the basis of this simple evaluation, one would conclude that the threshold theta

logistic model is the best (most parsimonious) descriptor of the observed changes in

Serengeti wildebeest abundance recorded over the past 40 years, followed by the Ricker

logistic model (∆AIC =7.2). The Gompertz logistic model (∆AIC =11.4) and geo-

metric growth model (∆AIC =18.9) are even less consistent with the observed data.

We can more formally assess the likelihood of each of the competing models by

calculating their Akaike weights. The Akaike weight for a given model iis calculated

by dividing the likelihood of that model (exp[−∆AICi/2]) by the summed likelihoods

of all of the competing models (∑exp[−∆AICj/2]):

The Akaike weight for the theta logistic model is 0.99, whereas the Akaike weights

of all the other models are less than 0.01. This implies that there is a 99% prob-

ability that the theta logistic model is the best (i.e. the most parsimonious) model

in the set considered. This is not to say that the theta logistic model is correct – only

that it offers the most parsimonious (efficient) prediction of the data, balancing the

need for accurate prediction with a tolerably small number of parameters. Akaike

weights offer a practical means of assessing how seriously to take alternative pre-

dictions derived from each of the models.

The simplest biological interpretation of this pattern is that wildebeest require some

resource, such as food of suitable quality, whose availability is strongly related to

wildebeest abundance. This interpretation is further strengthened by the observation

that wildebeest survival rates measured locally are positively related to the amount

of green grass available per individual animal, which in turn varies with monthly

rainfall (Mduma et al. 1999).

To simplify our example, we evaluated only four alternative models. One can read-

ily imagine other plausible models that might be even more useful in predicting changes

in wildebeest abundance over time, such as age-dependent models or models incor-

porating interactions with predators. These new models should also be ranked in terms

wi i

j

j

exp( )

exp( )

=

−

−

=

∑

∆

∆

AIC

AIC

1

4

AIC 4 log ( )

=− +

−−

⎛

⎝⎜

⎞

⎠⎟

22

1

44

4

e p

n

np

Λ

MODEL EVALUATION AND ADAPTIVE MANAGEMENT 263

15.4.5Evaluation of
the models