Campbell et al.: Release mortality in the fishery targeting Lutjanus campechanus 
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rates were recalculated for discrete depth groups from 
the original data by aggregating sites by depth. 
Ideally, the frequency at which fish were vented 
would be calculated; however, some studies reported 
that venting occurred irregularly and at the choice of 
participants. If a study reported at least some amount 
of venting, then it was categorized as a venting treat- 
ment; otherwise, it was considered a nonventing treat- 
ment. Caging studies that reported a venting treat- 
ment were maintained as reported, but it should be 
noted that those experiments included recompression 
of fish (i.e., their air bladders) by submergence back to 
depth in cages regardless of whether a fish had been 
vented. Because few studies reported hook size, it was 
not included. Finally, the intent of this meta-analysis 
was to evaluate release mortality under normal fishing 
conditions; therefore, estimates from the hyperbaric- 
chamber study (Burns et al. 6 ) were not included in our 
study. 
The meta-analytical model used in our study is a 
special case of a weighted general linear model as de- 
tailed in the metafor package (Viechtbauer, 2010), a 
meta-analysis package for R software. The analysis 
was performed on effect size (es) rather than on raw 
proportions, where es was the logit-transformed propor- 
tion and was calculated with the following equation: 
es = log 
( 1 ) 
where jq = the total number of individuals that experi- 
enced mortality; and 
Hi = the total sample size. 
The estimate and the corresponding sampling vari- 
ance were calculated by using the escalc function in 
the metafor package (Viechtbauer, 2010) in R software, 
vers. 2.15.1 (R Core Team, 2012). 
We fitted estimates of effect size in a mixed-effects 
model to evaluate the effects of depth, fishing sector, 
timing of the mortality estimate, venting treatment, 
season, and hook type (Viechtbauer, 2010). For the 
categorical variables, the absence of group member- 
ship (i.e., setting that value to 0) by default defines 
the opposite group; therefore, there is no need to have 
all variables included. For instance, identifying esti- 
mates associated with commercial data as 1 automati- 
cally defines values set equal to 0 as being associated 
with recreational estimates. The full estimated model 
is shown below: 
Prb ( mortality ) ~ depth + sector + timing + venting 
+ season + hook type + rate + timing*venting, 
where depth of capture in meters is modeled as a con- 
tinuous variable and all other variables are modeled 
as categorical. Sectors were defined as commercial or 
recreational. Timing was defined as either immediate 
mortality or delayed mortality, referred to hereafter 
simply as immediate or delayed. Venting treatments 
included venting and nonventing. Season variables in- 
cluded spring, summer, fall, or winter. Hook types were 
tested as circle or as J- and mixed hooks combined be- 
cause we were interested in the effect of circle hook 
regulations. The rate variable represents each indi- 
vidual estimate and was modeled as a random effect 
(i.e. estimated mortality rate). Therefore, the model 
treated multiple estimates coming from a single study 
as unique estimates from the available population. 
Treatment of multiple estimates from the same study 
as unique estimates occurred when a study was con- 
ducted over different seasons or over a range of depths. 
Finally, because we wanted to test whether the vent- 
ing treatment was confounded with the study type and 
timing of the estimate (immediate), we also included 
an interaction term (timing* v enting) . 
Several additional model runs were performed to 
evaluate sensitivity of the model to various issues. The 
commercial data set was represented by a single study 
and, although it was a fairly extensive study that pro- 
duced many estimates, it may not be representative of 
all commercial fisheries for red snapper. Therefore, we 
made model runs that excluded the data from Nieland 
et al. (2007). 
Heterogeneity (x 2 ) was estimated by using restricted 
maximum-likelihood. Coefficients for p, (io,--.,Pp then 
were estimated with weighted least squares in which 
each estimate of effect size was weighted by the inverse 
of its variance. Wald-type tests and confidence intervals 
were calculated for p, |3o>--->Pp> assuming normality. On 
the basis of the fitted model, we calculated predicted 
values and residuals. Cochran’s Q-test was used to as- 
sess the amount of heterogeneity among studies (i.e., a 
null hypothesis of x 2 =0). Model predictions were calcu- 
lated with the predict function in the metafor package. 
The predict function allows for the input of a range 
of values (e.g., depths) over which to calculate model 
predictions and also allows for the adjustment of coef- 
ficient weights so that individual treatment effects can 
be isolated (e.g., venting and season). Predicted values 
and associated upper and lower bounds were then con- 
verted back to proportions by taking the inverse of the 
logit-transformed effect-size data with the following 
equation: 
Proportion - — — T , (2) 
(l + exp es ) 
Average model predictions were evaluated by giving 
equal weighting to the coefficients within fishing sec- 
tor, timing of mortality, venting, season, and hook type 
and by inputting a depth range of 10-100 m. Model 
predictions for the various venting and season treat- 
ments were then calculated through adjustment of the 
weighting scheme submitted to the predict function. 
For instance, to evaluate the effect of 100% venting, all 
of the weight for the venting treatments was put onto 
the treatment with 100% venting, and model predic- 
tions were recalculated. 
