untitled

sample size that we will call n). For each of these n estimates of r, we can calculate

the deviation, residi, around the regression line:

The standard deviation of these residuals (σ=0.055) can then be calculated by squar-

ing each residual, summing together the squared residuals, and dividing by the total

number of replicates.

In our example (Fig. 15.2), the residuals seem to be reasonably well depicted by

a normal distribution, although the sample size is too small to be totally sure. On

the presumption that the residuals do follow a normal distribution with mean of zero

and σ=0.055, we assess how well each model explains the existing data through

use of the concept of likelihood (Λ). For example, we could use this approach to

evaluate the likelihood that the carrying capacity of Serengeti wildebeest is any par-

ticular value:

Likelihood is proportional to the probability that a given model is correct, given a

particular set of data. Likelihood is calculated from the probability function defining

the residual variability that affects each estimate (in this case the normal distribu-

tion) and the value predicted by the model. Hence, we can use the equation defining

the normal distribution, which we obtain from any statistics textbook, with an

expected value for each observation, derived from the Ricker logistic model (see Sec-

tion 15.4.1).

Because likelihoods are often very small or very large numbers, it is customary to

evaluate their negative natural log-transformed values (termed the negative log-like-

lihood). This transforms the function from a “dome” to a “valley” shape, in which

the most likely parameter is the value that is at the bottom of the valley (Fig. 15.3):

−− −

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⎟

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⎢

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rr

N

i K

i

max^1

2

2

σ^2

Λ = exp

=

−

∏

1

(^02)
1
i σ π
n
residiirr i

N

K

=−max −

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1

MODEL EVALUATION AND ADAPTIVE MANAGEMENT 257

5 4 3 2 1 0

• 0.1 0 0.1

Histogram

Normal distribution

Fig. 15.2Frequency
distribution of residuals
around the Ricker
logistic regression of
rversus N(histogram)
and a normal
probability curve with
the same mean and
variance, plotted against
these data.