untitled

The profile curve shown in Fig. 15.3 is useful in demonstrating how the negative

log-likelihood would change with different parameter values. We see that the value

of Kthat minimizes the negative log-likelihood is the value identified using least-

squares regression. The minimum log-likelihood is the maximum likelihood, so we

refer to the “best-fit” value of Kas a maximum likelihood estimate. Just as we can

use likelihood to evaluate the most probable parameter values, in analogous fashion

we can use likelihood to assess the plausibility of alternative models to explain the

observed data.

We now want to decide which of numerous mathematical expressions best represent

the data we have at hand. A famous physicist named Ludwig Boltzmann provided the

theoretical basis for making such decisions. This concept was reformalized 75 years

later by Kullback and Liebler, using the concept of “information.” Information can

be loosely defined as the “distance” between the true causal relationship and an approx-

imating model. Interestingly, we do not really need to know the “true” relationship

in order to evaluate the predictive ability of the alternative model. That is a good

thing, because in ecology we will never know the real relationship or even the real

parameters for any of the candidate explanatory models. Nonetheless, we can still

evaluate the plausibility of each of the alternative models, by using the likelihood

(Λ) and the number of parameters (p) in the estimating function.

The most general approach that can be applied to non-nested models is Akaike’s

information criterion (AIC) or one of its variants (Burnham and Anderson 1998).

AIC is derived from Kullback–Liebler information (Akaike 1973), when applied to

experimental or field data whose parameters must be estimated and for whom the

form of mathematical relationships is inexact. AIC values become smaller with

increasing likelihood, but larger with each increase in the number of parameters. One

would ordinarily expect that the likelihood of models should go up when a larger

LK n

rr

N

i K

i

i

n

( ) log( ) log( )

(^) max
=+⎡
⎣⎢

⎤

⎦⎥

+

− −

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣

⎢

⎤

⎦

⎥

=

−

σ ∑ σ

1

2

2

1

2

2

2

0

1

π

258 Chapter 15

900 1000 1100 1200 1300 1400

K

L (

K

)

–26

–24

–22

Fig. 15.3Negative log- –20
likelihood for a range of
values of Kin the
Ricker logistic model
applied to the
wildebeest population in
a Serengeti ecosystem.

15.4 Evaluating the likelihood of alternative models using AIC