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

*η*1 distribution indicates the presence of: strong evidence for reliably identifying
*η*1 *< 0, inconclusive evidence where the sign of η*1 can not be identiﬁed, and

*strong evidence for identifying η*1 *> 0, respectively.*

Coding of ˆ*η*1distribution
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 η*1 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 η*1) applied to a single population,

namelyDFO-QUE-COD3Pn4RS-1964-2007. Note that in the frequentist method I am
using a parametric bootstrap replication of ˆ*η*1; however, in the Bayesian setting I

*am estimating the marginal posterior distribution of η*1given 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 π*1*(dotted-dashed green plot) and the posterior with respect to π*2

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(η*1* |R, S, π*1) and

*p(η*1

*2) in plots. I generalised this comparison —by analysing the output*

**|R, S, π**Figure 3.6: Density plots of: 1,000 parametric bootstrap replications of ˆ*η*1 (solid

*red plot); marginal posterior distribution of η*1 *with respect to π*1 (dotted-dashed

*green plot); marginal posterior distribution of η*1 *with respect to π*2 (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 π*1,

and the blue ones represent the approximate 95% credible intervals with respect
*to π*2. 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 η*1 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 π*1 *and π*2 *respectively. The vertical axis represents the η*1

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 π*1*and π*2

priors) for a Beverton-Holt stock-recruitment model for evaluating the reliability
*of η*1 in survival across the 90 S-R ﬁsh populations.

*Coding of η*1 distribution

frequentist *Bayesian π*1 *Bayesian π*2

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 ˆη*1 distribution indicate

*the presence of: strong evidence for reliably identifying η*1 *< 0, inconclusive*

*evidence where the sign of η*1 can not be identiﬁed, and a strong evidence for

*identifying η*1 *> 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. π*1 *or π*2). Table 3.8 illustrates a comparison of the estimation

*error for η*1 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 η*1 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