Barlow and Berkson Evaluating methods for estimating rare events with zero-heavy data 
357 
Table 4 
Number of simulations representing interactions of sea turtles with the U.S. Atlantic pelagic longline fishery in which the simu- 
lated amount of bycatch fell outside the 95% confidence interval (Cl). We ran 1000 simulations for each of the 5 spatial scenarios 
and 2 spatiotemporal scales of delta-lognormal estimation. Underestimation occurs when the total simulated amount of bycatch 
falls above the Cl, and overestimation occurs when the total simulated amount of bycatch falls below the Cl. The co-occurrence 
clumping scenario and sets-only clumping scenario were considered the most realistic spatial scenarios. 
Spatiotemporal scale for estimation 
Spatial scenario 
Stratum 
level 
All sets pooled 
Underestimate 
Overestimate 
Underestimate 
Overestimate 
Co-occurrence clumping (Turtles dump , Sets clump . turtles ) 
14 
0 
61 
8 
Independent clumping C Turtles damp , Sets dump . sets ) 
2 
55 
2 
77 
Sets-only clumping ( Turtles un - foTm , Sefs clump _ sets ) 
1 
1 
15 
10 
Turtles-only clumping (Turtles c|ump , Sets um{or J 
0 
1 
0 
15 
Fully uniform (Turtles uniform , Sets uniform ) 
3 
0 
60 
2 
CIs from each of 1000 simulations, it was expected 
that the total simulated bycatch would fall below the 
Cl in 25 simulations and be above the Cl in 25 simula- 
tions. The stratum-level CIs for the more realistic spa- 
tial scenarios had far fewer than 25 estimates above 
and 25 estimates below; therefore the stratum-level 
CIs were too conservative (Table 4). Alternatively, the 
CIs from all sets pooled performed well in the sets- 
only clumping scenario (Turtle s uniform , Sefs c]ump . sets ) 
but, in the co-occurrence clumping scenario ( Turtles - 
clump’ S ^ s ciump-turties)’ the y contained values that were 
less than the true amount of bycatch more often than 
expected (Table 4). 
Discussion 
Performance of the estimation methods 
The delta-lognormal method with stratum-level esti- 
mates was the most suitable method in the most realistic 
spatial scenarios, the co-occurrence clumping scenario 
{Turtles clump , Sefs clump . turtles ) and sets-only clumping sce- 
nario (Turtles unitorm , Sets dump _ se J. This result was seen 
because observed sets were representative of unobserved 
sets, sample sizes of observed bycatch were sufficient for 
estimating bycatch within strata, and model assump- 
tions were not violated. 
Observed fishing sets were representative in the co- 
occurrence clumping scenario (Turtles clump , Sefs clump . tur . 
Ues ) because all sets fished where sea turtles were pres- 
ent. Likewise, observed sets were representative in the 
sets-only clumping scenario (Turtles uniform , Sefs clump _ sets ) 
because each set had the same probability of encoun- 
tering a turtle when turtles had a uniformly random 
distribution. Further, because these 2 spatial scenarios 
had enough observed bycatch within strata to make 
stratum-level estimates, strata did not have to be pooled 
to achieve larger sample sizes. Therefore, differences 
between strata could be captured and potential biases 
associated with pooling were avoided. 
On the other hand, the GLMs could be used only to 
estimate bycatch for all sets pooled because of conver- 
gence problems related to the small amount of observed 
bycatch in strata. Moreover, the relationship between 
environmental and fishing conditions and the amount 
of bycatch was probably not well established in these 
models because bycatch was rare and observer coverage 
was low. The use of poorly fitted models could explain 
why the GLM estimates had lower precision than the 
delta-lognormal estimates. 
The GLMs were as accurate as the delta-lognormal 
methods in the fully uniform scenario ( Turtles „ 
S ^ s uniform> because this spatial scenario was the only 
one that did not violate the GLM-P assumption that 
counts are independent and randomly distributed in 
space (McCracken 2004, Sileshi 2006). Violations of 
GLM-P assumptions introduced biases in the other spa- 
tial scenarios. Additionally, it is likely that the GLM- 
NB did not perform better than the GLM-P because 
overdispersion was not a problem (White and Bennetts, 
1996; Sileshi, 2006). 
In the 2 scenarios where sea turtles were clumped but 
sets did not mimic their clumping pattern, a low level 
of bycatch was seen. Under the independent clumping 
scenario (Turtles clump , Se*s clump . set8 ) and turtles-only 
clumping scenario {Turtles clump , Sets uni{orm ), some sets 
were not expected to encounter any turtles, whereas 
other sets were expected to encounter many turtles, but 
the overall frequency of encountering turtles was low. 
The lowest mean observed bycatch rate occurred in the 
turtles-only clumping scenario (Turtles clumft , Sets um(orm ) 
with 0.004 turtles/set. This low observed bycatch rate 
is likely related to the delta-lognormal method having 
the most bias in this spatial scenario as well. The delta- 
lognormal method of estimating stratum-level bycatch 
