56 
Fishery Bulletin 1 12(1) 
The Bayesian approach to fitting the linear regres- 
sions and assessing their adequacy for the temporal 
trends in standardized residuals was adapted after 
Kery (2010). The adequacy in question was based on 
the posterior predictive checks as reflected in Bayes- 
ian P-values and plots of the sum-of-squares for trends 
in replicated (“perfect”) standardized residuals against 
the sum-of-squares for trends in actual standardized 
residuals. When a model is adequate for the actual 
data, about half of the points lie above a 1:1 line on 
the plot. Equivalently, the Bayesian P-value is “close” 
to 0.5, and values “near” zero or one indicate doubtful 
fit of the model. Unfortunately, the range of Bayesian 
P-values for a good fit is unclear (Kery and Schaub, 
2012). By analogy to Ono et al.’s (2012) similar statis- 
tic, a Bayesian P-value of 0.45-0.55 was assumed close 
to 0.5. 
The types of association between various standard- 
ized residuals and year were identified on the basis of 
1) the signs of the posterior means and medians of the 
trend slopes, 2) the location of zero in the posterior 
distributions of slopes (i.e., whether the 95% BCI of 
these slopes covered zero), and 3) the computation of 
the probability of decline, P*. This probability should 
be “close” to 0.5 (i.e., zero centered at the 95% BCI) for 
the lack of trend; its larger value (typically approach- 
ing one) indicated a negative trend and vice-versa for a 
smaller value approaching zero. The previous 3 proce- 
dures were jointly used to draw pragmatic conclusions 
because it was unclear what value of P* indicated that 
a trend was not strong enough to be considered posi- 
tive or negative. 
The deviance information criterion (DIC) and the 
Bayes factor (BF) were used to compare various BSSB- 
DMs. Although DIC can be problematic in MCMC sim- 
ulations, it is the most popular method of a Bayesian 
model fit and selection that is routinely implemented 
in the WinBUGS software. Typically, DIC selects among 
models by trading off goodness of fit and model com- 
plexity (Spiegelhalter et al., 2002; Wilberg and Bence, 
2008) when competing models are fitted to the same 
data sets. It is given by 
DIC = 2D - D = D + P Q, 
D(0) = -21ogL(0)-21og[P(O | ©)], (6) 
PD = D-D, 
where D = deviance (measure of goodness of fit); 
D = the posterior mean deviance; 
D = the deviance of posterior means of the el- 
ements in 0; and 
Pd = the “effective number of parameters.” 
The statistic pd is unstable to estimate, is not an inte- 
ger, does not necessarily correspond with the number 
of parameters and, although it should be positive, can 
even be negative. The latter problem usually arises 
separately or jointly from ill-specifying priors or an ill- 
fitting model (data-prior conflict), and is symptomatic 
of suspicious inferences or of non-normal posteriors of 
the parameters on which priors have been placed (Spie- 
gelhalter et al., 2002). 
The model with the smaller DIC is better supported 
by the data. In practice, comparisons of models are per- 
formed by using the difference in DIC (ADIC) among 
the competing models. As a rule of thumb, ADIC>10 in- 
dicates models with no support for the model with the 
higher DIC; if 3<ADIC<7, the model with the higher 
DIC has considerably less support; and ADIC<2 indi- 
cates lack of substantial differences between models 
compared. All models with ADIC<2 units from the low- 
est DIC model should receive consideration in making 
inferences (Spiegelhalter et al., 2002). 
The BF comparing how well any two models M x (as- 
sociated with the null hypothesis) and M y (correspond- 
ing to the alternative hypothesis) fitted the biomass 
indices was 
BF. 
P(°l M y) 
” r(o|M,)' 
(7a) 
where P(0 | M) = the marginal likelihood for M e {M x , 
My) and was approximated as (Newton and Raftery, 
1994; Kass and Raftery, 1995): 
n-l 
P(O|M) = |i|p(O|0 s r 1 
^ S=1 
where S = the number of simulations and 
P(O|0 s ) = e“ a5Z)(0s) . 
(7b) 
The model that predicted the biomass indices better 
was considered to have more evidence supporting them 
and, hence, was preferred. Model preference relied on 
the guidelines of Kass and Raftery (1995) inferred from 
the natural log of BF (LBF), LBF yx = 21og (BF yx ). Here, 
BF yx <1 o LBF yx < 0 supported M x ; evidence for M x 
was considered negligible if 1 < BF yx <3 o 0 < LBF^ 
< 2; and BF yx > 3 <*> LBF yx > 2 supported M y . 
The competing models included the same types of 
fishery removals. However, they differed in whether 
they included MWET, in the type of priors used, or in 
whether they included the SESTF bycatch. 
Environmental anomalies 
Assessing MWET effects on the Atlantic Croaker popu- 
lation off the U.S. east coast relied upon 3 approaches. 
First, in Equation 4 for Ml, MlrU, and M1B, any po- 
tential environmental effects were implicitly lumped in 
the posterior process errors of these models, ft Gt = 
log(P t ) - log(S t ) ° £t = log(&t> - log(&t))- These errors 
were expected to be theoretically positively correlated 
with MWET because MWET is considered to be the 
dominant environmental factor affecting the popula- 
tion dynamics of the species. The relationship between 
the posterior process errors and MWET was checked 
by regressing the credible medians of St+i, for year t+ 1, 
