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Fishery Bulletin 1 10(1 ) 
ciated with the phase of the spawning season in which 
samples occurred. That is, small larvae were more 
likely to be available for capture when sampling was 
conducted near the peak-spawning season. The CPFV 
index from the previous year was included as a general 
measure of stock size. It was used to account for the 
fact that larvae may have been less likely to occur in 
otherwise suitable habitat in some years simply because 
the population was smaller. We note that abundance 
estimates from recent stock assessments (Crone et al., 
2009) could not be used directly in the model because 
no estimates were made before 1962. The correlation 
between the CPFV index and abundance from the stock 
assessment during 1962-2008 was z'=0.81. The value 
for the previous year, rather than the current one, was 
used so that the measure was relatively independent of 
Pacific mackerel movement during the spawning season. 
The probability of capturing one or more larvae was 
modeled with a semiparametric logistic model with the 
“gam” function (i.e., generalized additive model) in the 
“mgcv” package (Wood, 2006) for R. The form of the 
model was 
v 1 - y J k 
where /3 0 = the intercept, 
S lt (-) = the smoothing function, and 
x h - the value of the Mh covariate. 
The response variable, y, was presence or absence of 
larvae. The smoothing function was either a restricted 
cubic spline with shrinkage (the “cs” curve in mgcv; 
cf.. Wood, 2006) or a parameter estimate if a term was 
entered as a simple linear predictor. 
Several constraints were added to develop models that 
were parsimonious enough to prevent over-fitting yet 
flexible enough to be biologically realistic for a species’ 
expected response along an environmental gradient 
(e.g., monotonic, unimodal, or skewed unimodal pat- 
terns). First, we limited the number of knots in the cu- 
bic splines to three. Thus, only curves that were skewed 
and unimodal or simpler were considered. The second 
constraint was that we increased the penalty per degree 
of freedom fit to each term by setting the “gamma” op- 
tion in the “gam” function to 1.4 to minimize potential 
over-fitting (Wood, 2006). 
We performed model selection using the shrinkage 
features in the “gam” procedure rather than fitting a 
large set of potential candidate models (i.e., subsets of 
environmental variables fitted with different amounts of 
flexibility for each term). The “select” option was set to 
true for all models. This procedure allowed coefficients 
with little or no predictive ability to be shrunk to zero, 
effectively dropping them from the model. The stock- 
size variable was entered as a linear term rather than 
a spline in the logistic models, because a monotonically 
increasing response was the only biologically sensible 
response to increasing stock size. The plankton-vol- 
ume variable was allowed to be monotonic or simpler, 
rather than constrained to a linear term, because very 
high plankton volumes could indicate that invertebrate 
predators on eggs and larvae were present, which could 
negatively affect the suitability of the habitat. 
A second model was fitted by using the same proce- 
dures listed above, except that temperature and day 
of year were entered as tensor product (Wood, 2006) 
interactions with latitude. This competing model was 
considered because some Pacific mackerel exhibited 
peak spawning in August near Punta Eugenia, Mexico, 
rather than in April as most Pacific mackerel did in 
the SCB (Lo et ah, 2010). This procedure resulted in a 
small second mode for histograms of temperature and 
day of year where Pacific mackerel were captured. The 
rationale for the use of this model was that the broader 
survey area likely contained a mixture of Pacific mack- 
erel that were likely to spawn near the SCB at cooler 
temperatures in the spring and Pacific mackerel likely 
to spawn at warmer temperatures in the summer near 
Punta Eugenia. The interaction terms were fitted by 
allowing five knots for temperature or day of year and 
latitude, thereby allowing for a more flexible prediction 
surface with two peaks (e.g., a peak in April at high 
latitudes in the SCB and a second peak in August at 
lower latitudes near Punta Eugenia). This model was 
compared to the original model using Akaike’s informa- 
tion criterion (AIC; Akaike, 1974). The model with the 
lowest AIC of the two was selected as the final model 
for interpretation. 
We initially fitted a model to predict densities of lar- 
vae for samples where at least one larva was captured. 
The intention was to calculate expected densities as 
the product of the two models with a two-stage or A- 
generalized-linear model (Stefansson, 1996; Welsh et 
ah, 1996). However, variability in the models was so 
great that the approach provided little or no additional 
information, and the approach was abandoned. 
Results 
Distributions of larval Pacific mackerel varied greatly 
among years, but large clusters of larvae frequently 
were captured near Punta Eugenia in Mexican waters 
and nearshore in the southern California Bight (Fig. 2). 
Corrected densities varied by several orders of magni- 
tude within and among years. The greatest numbers of 
Pacific mackerel larvae were captured in the early 1980s 
and fewest from 1999 through 2008. During years when 
both U.S. and Mexican waters were sampled (1951-84), 
greater larval densities generally occurred in Mexican 
waters until 1975, but larger catches occurred in the 
SCB in 1978, 1981, and 1984. Within Mexican waters, 
densities were typically greater near Punta Eugenia in 
the southern portion of the sampled region than they 
were near the U.S. -Mexican border. 
The logistic model that included interactions for tem- 
perature and day was selected in preference to the 
model with no interactions based on AIC values of 3876 
versus 3348. The difference of 528 units of AIC in- 
