60 
Fishery Bulletin 1 12(1) 
not include negative pd estimates. On the basis of 
ABIC, they were comparable with those BIG sta- 
tistics for their counterparts without MWET and, 
therefore, should be considered for making infer- 
ences (Table 2). In short, except M2N, M2rUN, and 
M2BN, | ABIC | e [4.13,5.70]: models with greater 
BIC were substantially less well supported despite 
among-model similarity of residual patterns and 
magnitude (Fig. 2). (Should the actual deviance 
for the parameter a be dropped from the BIC for 
M2, M2rU, and M2B, then | ABIC | «2. This result 
would indicate lack of differences between models 
with and without MWET (Ml, MlrU, and M1B), 
and both types of models should be considered for 
making inferences. However, because a would still 
have no explanatory power [i.e., the BIC for M2, 
M2rU, and M2B would still be slightly larger than 
the BIC for Ml, MlrU, and M1B], the parsimonious 
models [without MWET] would still be preferred). 
On the other hand, the LBF statistic consistently 
rejected models with the prior r or ro~U(0.01,1.5), 
models incorporating MWET, or both models; 
even upon some evidence against models without 
MWET, this evidence was weak (Table 2). 
Therefore, comparisons of models indicated that 
the complexity of (base and sensitivity) M2 brought 
about by the introduction of MWET was not war- 
ranted by the data. Furthermore, the models with 
the prior r or ro~C/(0.01,1.5) were discredited on 
the grounds that the values for their depletion 
component were negative and because of the dis- 
agreement between BIC and LBF statistics. Prefer- 
ence was given to results from the LBF statistic for 
2 reasons. The BF is among the formal solutions 
to the model-choice problem (Plummer, 2008). Al- 
though Equation 7b is computationally unstable, 
it is consistent as the simulation size S increases 
and, in practice, often gives results that are accu- 
rate enough for interpretation on the logarithmic 
scale (Kass and Raftery, 1995). The whole process 
of model comparisons with BIC and BF (LBF) sta- 
tistics therefore selected Ml and M1B only. Unless 
otherwise indicated, the following results related 
to Ml and M1B. 
Extent of climate forcing 
The plots of discrepancy checks (not shown) and 
the Bayesian P-values (0.53-0.55) indicated that 
the fitted linear models were adequate for the rela- 
tionships between 1) the process errors and MWET, 
an increase in D and D associated with the depletion 2) surplus production and MWET, and 3) instanta- 
component, with D > D leading to negative pv values. neous surplus production and MWET (Fig. 5). These 
This result was symptomatic of an ill-specified (inap- relationships were positive because positive values had 
propriate?) prior, ill-fitted models, or was symptomatic most of the mass under the posterior, bell-shaped dis- 
of both and, hence, of unreliable inferences from MlrU tributions of their slopes. This result was reflected in 
and M2rU runs, although the corresponding overall p D positive credible means and medians of those slopes 
estimates were positive. and by large posterior probabilities of increase 
The BIC statistics for M2N, M2rUN, and M2BN did (P*>0.88). 
1 .4 
1 .2 
1 .0 
0.8 
0.6 
0.4 
0.2 
0.0 
I 
1.2 
1.0 
0.8 ■ 
B 
■M2rU 
■ M2rUN 
1970 
1980 
1990 
2000 
2010 
O 0.6 
CD 
•!2 0.4 ■ 
0.2 ■ 
0.0 
-M2B 
-M2BN 
J 'I ! *. ; ! I ! ; \ : 
» , % / if »i \! \; 
t / \ • ' U II v • 
1970 
1980 
1990 
2000 
2010 
Figure 4 
Time series of the predicted posterior medians of the intrin- 
sic rate of population increase (r) for the Atlantic Croak- 
er stock off the U.S. Atlantic coast derived from Bayesian 
state-space biomass dynamic models incorporating the 
minimum winter estuarine temperature (M2), 1972-2008. 
Results relate to (A) M2 base runs, (B) M2 sensitivity runs 
with the prior (7(0.01,0.15) for the scaling parameter /-q 
(M 2rU), and (C) M2 sensitivity runs with the southeastern 
(North Carolina-east Florida) shrimp trawl fishery bycatch 
(M2B). M2N, M2rUN, and M2BN are M2, M2rU, and M2B in 
which the prior for the coefficient controlling MWET effects 
(a) was centered on zero with a precision of 0.02. The 95% 
central intervals of r were not plotted for clarity. 
