Table 3.7: Confidence Levels and data classification of the 90 S-R populations for a Beverton-Holt stock-recruitment model; the{-1, 0,+1} coding based on the ˆ

η1 distribution indicates the presence of: strong evidence for reliably identifying η1 < 0, inconclusive evidence where the sign of η1 can not be identified, and

strong evidence for identifying η1 > 0, respectively.

Coding of ˆη1distribution Confidence Level (%) −1 0 +1 60 30 43 17 70 26 50 14 80 22 61 7 90 15 72 3 95 11 78 1 99 7 82 1

3.3

Results

Statistical analysis based on the frequentist paradigm shows, for a Beverton-Holt model, the existence of seven populations having their approximate 99% con- fidence interval for η1 lying entirely in the negative region (Table 3.7). Those

seven populations are from six different fish species in six locations, indicating that this classification result is not peculiar to a particular species or location. Standard confidence levels were increased gradually to reflect the sensitivity of the classification labels of the 90 S-R populations to the choice of cut-offs. For low confidence levels I observed many populations classified with label -1, but this classification declined as I increased the confidence level.

Next, I compared results obtained from the frequentist approach to those ob- tained from Bayesian methods. In the Bayesian framework the credible interval is obtained from the marginal posterior distribution using the equal-tailed cred- ible interval. Figure 3.6 illustrates a comparison between the frequentist and Bayesian inference (for different priors for η1) applied to a single population,

namelyDFO-QUE-COD3Pn4RS-1964-2007. Note that in the frequentist method I am using a parametric bootstrap replication of ˆη1; however, in the Bayesian setting I

am estimating the marginal posterior distribution of η1given a particular popula-

tion and a prior. I am in general interested in whether these methods produce the same result or not; the bootstrap estimates (red plot), the posterior distribution with respect to π1(dotted-dashed green plot) and the posterior with respect to π2

their approximate 95% confidence interval and approximate 95% credible inter- vals were more likely to agree, despite the difference of their shapes. I should also inform the reader that in this figure I used half of the posterior samples (i.e. 5,000 samples) from both posteriors so as to avoid overlap of p(η1|R, S, π1) and p(η1|R, S, π2) in plots. I generalised this comparison —by analysing the output

Figure 3.6: Density plots of: 1,000 parametric bootstrap replications of ˆη1 (solid

red plot); marginal posterior distribution of η1 with respect to π1 (dotted-dashed

green plot); marginal posterior distribution of η1 with respect to π2 (dashed blue

plot). The analysis is applied to the DFO-QUE-COD3Pn4RS-1964-2007population.

−6 −4 −2 0 0.0 0.1 0.2 0.3 0.4 η1 Density distribution of η^ *1 p(η1 | R,S,π1) p(η1 | R,S,π2)

of frequentist and Bayesian methods—for all 90 S-R populations, as illustrated in Figure 3.7. The MLE used for bootstrap simulations is represented by an asterisk (*); the black error bars represent the approximate 95% confidence intervals and the square shape denotes the mode of simulated MLEs distribution. The red error bars represent the approximate 95% credible intervals with respect to π1,

and the blue ones represent the approximate 95% credible intervals with respect to π2. I observed a large approximate 95% confidence interval for the following

population numbers: 9, 10, 13, 22, 25, 50, 55, 62 and 63, caused essentially by the small sample sizes: 12, 28, 29, 17, 20, 12, 8, 10, and 9 data points respec- tively. Moreover, I found for some other populations (20, 50, 53 and 63) different marginal posteriors with respect to the choice of the prior; however, for the re- maining populations I found robust posterior inference with respect to the choice

3.3. Results 101

Figure 3.7: Comparison between the frequentist and Bayesian method to inference for a Beverton-Holt model. The black error bars show an approximate 95% BCa confidence interval where the asterisk symbol represents the MLE of η1 and

the square symbol represents the mode of simulated MLEs with bootstrapping. The red error bars and the blue error bars show the approximate 95% credible interval with respect to π1 and π2 respectively. The vertical axis represents the η1

parameter and the horizontal axis represents the sequential population number; ranging from 1 to 30, 31 to 60, and 61 to 90 respectively.

0 5 10 15 20 25 30 −10 −5 0 5 10 Dataset no η1 η^1 distribution of η^ *1 p(η1 | R,S,π1) p(η1 | R,S,π2) 30 35 40 45 50 55 60 −10 0 1 0 2 0 3 0 Dataset no η1 η^1 distribution of η^ *1 p(η1 | R,S,π1) p(η1 | R,S,π2) 60 65 70 75 80 85 90 −20 −10 0 1 0 2 0 3 0 Dataset no η1 η^1 distribution of η^ *1 p(η1 | R,S,π1) p(η1 | R,S,π2)

Table 3.8: Comparison between frequentist and Bayesian methods (with π1and π2

priors) for a Beverton-Holt stock-recruitment model for evaluating the reliability of η1 in survival across the 90 S-R fish populations.

Coding of η1 distribution

frequentist Bayesian π1 Bayesian π2

Confidence Level (%) −1 0 +1 −1 0 +1 −1 0 +1 60 30 43 17 32 43 15 31 47 12 70 26 50 14 29 47 14 27 51 12 80 22 61 7 21 62 7 20 65 5 90 15 72 3 12 73 5 10 77 3 95 11 78 1 11 75 4 10 77 3 99 7 82 1 9 79 2 7 82 1

Table 3.9: Confidence Levels and data classification of the 90 S-R populations using model selection; the{-1, 0,+1} coding based on the ˆη1 distribution indicate

the presence of: strong evidence for reliably identifying η1 < 0, inconclusive

evidence where the sign of η1 can not be identified, and a strong evidence for

identifying η1 > 0, respectively. Label Confidence Level (%) −1 0 +1 60 31 48 11 70 28 53 9 80 23 62 5 90 16 70 4 95 7 82 1 99 3 87 0

of the prior (i.e. π1 or π2). Table 3.8 illustrates a comparison of the estimation

error for η1 assessed by the frequentist and Bayesian approaches when applied to

the 90 S-R populations. I observed that both frequentist and Bayesian methods classified approximately the same number of populations, labelled with −1. To adjust my results, I used the fitted models (derived from model selection) and tested whether I could reliably estimate the sign of η1 with different confidence

levels, as described in Table 3.9. For the case where the confidence level is 95%, I found seven populations labelled with−1, 82 populations labelled with 0 and one population labelled with +1. The entire classification list for the 95% confidence level is illustrated in Appendix A (Table A.1).

Finally, I applied the edge effect analysis to populations classified with label -1 and to populations longer than 55 data points (Table A.1). The former revealed an agreement in the classification of six of the seven populations so that I can

In document Statistical Modelling of Marine Fish Populations and Communities (Page 122-126)