Ecological Archives A016-001-A4

N. Thompson Hobbs and Ray Hilborn. 2006. Alternatives to statistical hypothesis testing in ecology: a guide to self teaching. Ecological Applications 16:5–19.

Appendix D. Using more than one source of data in calculating likelihoods.

Many models that use information on age-structure follow an index of numbers at age for a cohort.  The rate at which the index decays with age provides information about the natural mortality (m) and the fishing mortality (u).  In its simplest form, this method is known as catch-curve-analysis (Ricker 1958, Hilborn and Walters 1992).  The basic equation for the dynamics of the cohort is:

(D.1)

 

where is the predicted number at age a, is the observed number at age a, m is the annual natural mortality rate, and u is the annual fishing mortality rate. If we assume that the errors between predicted and observed are normally distributed, then the likelihood of a set of observed observations is

(D.2)

 

where is the standard deviation of the process error that leads to differences between observed and predicted.  The total likelihood is the product of the individual likelihoods.

The table below shows some hypothetical data giving estimated numbers (or an index of numbers) at different ages.


Age

N


5

1030

6

880

7

640

8

490

9

401

10

315


This method assumes that the fishing and natural mortality rates are constant over all ages. For demonstration, we also assume we know to be 50. It is clear that from these data alone, we can only estimate the total mortality rate and cannot distinguish between m and u

Figure D1 shows the likelihood profile for u.  The data suggest a total mortality rate of about 0.2, so all values of fishing mortality are equally likely, if u = 0, then m = 0.2; if u = 0.1, then m = 0.1; and if   = 0.2, then m  = 0.  Values above u = 0.2 are increasingly unlikely.

The most common practice in fisheries models has been to fix the natural mortality rate at a "best guess" which then makes the data much more informative about u as shown in the likelihood profile if we fix m at 0.1 (Fig. D2).

The data are now telling us that if we know m = 0.1 then we have a pretty good idea of what u is.  However, this is quite unrealistic in that we would never know the natural mortality rate and at best, we would have an idea of what it is, perhaps from analysis of age structure of the population before fishing began, perhaps from a tagging study. 

If we can summarize the information we have on natural mortality rate as a likelihood (again choosing normal for convenience),

(D.3)

 

where  is the most likely value of u given the pre-existing data and  is the standard deviation of the estimated u from pre-existing data. We now have two likelihoods; the likelihood of the observed age data, and the likelihood of m from previous data.  These can be combined simply by multiplying them together:

(D.4)

 

Assuming  = 0.1 and  = 0.05 we find the likelihood profile shown in Fig. D3.

As expected, our understanding of the value of u is less certain than if we pretend we know m without error, but much more certain than if we have no information about m.


 
   FIG. D1. Likelihood profile for the fishing mortality rate.

 

 

 
   FIG. D2. Likelihood profile for the fishing mortality rate assuming that natural mortality = 0.1.

 

 

 
   FIG. D3. Likelihood profile for fishing mortality assuming that the average fishing mortality rate is 0.1.

 

LITERATURE CITED

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Burnham, K. P., and D. R. Anderson. 2002. Model Selection and Multi-Model Inference:  A Practical Information-Theoretic Approach. Springer-Verlag, New York, New York, USA.

Deangelis, D. L., R. A. Goldstein, and R. V. O'Neill. 1975. Model for trophic interaction. Ecology 56:881–892.

Hilborn, R., and M. Mangel. 1997. The Ecological Detective: Confronting Models with Data. PrincetonUniversity Press, Princeton, New Jersey, USA.

Hilborn, R., and C. J. Walters. 1992. Quantitative fisheries stock assessment: choice, dynamics and uncertainty. Chapman and Hall, New York, New York, USA.

Holling, C. S. 1959. Some characteristics of simple types of predation and parasitism. Canadian Entomologist 91:385–398.

Ricker, W. E. 1958. Handbook of computations for biological statistics of fish populations. Bulletin of the Fisheries Research Board of Canada 119:300.

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Vucetich, J. A., R. O. Peterson, and C. L. Schaefer. 2002. The effect of prey and predator densities on wolf predation. Ecology 83:3003–3013.



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