Lovell and Carter: Sampling weights in regression models of recreational fishing-site choice 
245 
djj = 1 if angler i chooses site j and 0 otherwise. 
We will refer to Equation 4 as the unweighted estima- 
tor because this specification is based on the assump- 
tion that the anglers are selected to be interviewed by 
simple random sampling. Manski and Lerman (1977) 
suggested a way to weight the conditional logit estima- 
tor by using information on the estimated population 
proportion of anglers observed at each of the j sites. 
The log likelihood of the weighted exogenous sampling 
maximum likelihood (WESML) estimator is 
is) 
where Qj and H j are the population and sample propor- 
tions at site j. 
Note that the variance matrix of the estimated param- 
eters must also be corrected to reflect the nonrandom 
nature of the data (Lerman and Manski, 1981). 
The estimated parameters of the conditional logit 
model can be used to determine an angler’s willing- 
ness to pay (WTP) for changes in site characteristics or 
for the loss or addition of a site (Haab and McConnell, 
2002). The most common measure used in the NMFS 
recreational site-choice demand models with the APA- 
IS data is the value of a one-unit change in harvest 
rate. If we define one element of q, say q\, as the har- 
vest rate, then the average angler WTP for a one-unit 
change in this attribute is given by 
WTP( qi ) = (6) 
Pc 
We calculated confidence intervals for this nonlinear 
combination of parameters using the method of Krin- 
sky and Robb (1986) with 1000 random draws from the 
multivariate normal distribution defined by the esti- 
mated mean parameter vector [3 and its related esti- 
mated covariance matrix. The mean marginal WTP by 
anglers is given by the mean of the simulated WTP 
vector calculated by evaluating expression 5 with the 
1000 parameter draws. Similarly, the upper and lower 
95% confidence intervals are given by the 5 th and 95 th 
quantile of the simulated WTP vector. The unweighted 
conditional logit and the WESML versions of the model 
and WTP confidence intervals were estimated with the 
software NLogit 2 , vers. 5 (Econometric Software, Pla- 
inview, NY). 
The simulated WTP vectors can also be used to for- 
mally test whether the mean WTP estimates generated 
by the weighted versions of the model are significant- 
ly different from the mean WTP estimates produced 
with the unweighted version of the model. We use the 
method of convolutions suggested by Poe et al. (2005) 
to test the null hypothesis that the estimates of mean 
unweighted WTP and mean weighted WTP are equal 
2 Mention of trade names or commercial companies is for iden- 
tification purposes only and does not imply endorsement by 
the National Marine Fisheries Service, NOAA. 
against the alternative that the unweighted estimate 
is larger than the weighted estimate. The calculations 
with the method of convolutions were performed with 
the mded package for R, vers. 3.0.0 (R Core Team, 
2013). 
Sources and description of data 
Data with which to estimate the conditional logit de- 
mand models came from a sample of private-boat 
anglers who were interviewed in 2009 as part of the 
APAIS in the Gulf of Mexico. The APAIS is a strati- 
fied, multistage, cluster-sample survey of angler trips 
designed to obtain estimates of mean catch per angler 
fishing day by species type and angler effort for each 
state, wave (2-month period), fishing mode (private 
boat, for-hire boat, and shore), and area fished (inland, 
state waters, and federal waters). The sample frame is 
based on the number of days at coastal fishing sites 
(site-days) that are accessible to the general public. 
Before 2011, simple random sampling was assumed 
during estimation procedures for estimates of mean 
catch and total effort, although the data were collected 
through the use of a complex sample design. The APA- 
IS sampling weights were designed to incorporate the 
complex sample design and nonrandom nature of the 
sample (Breidt et al. 1 ). Available as part of the APAIS 
data sets, these weights were calibrated by the NMFS 
Marine Recreational Information Program so that they 
summed to total estimated angler effort by year, wave, 
state subregion, fishing mode, and area fished. This cal- 
ibration was based on estimates of total angler effort 
from data from the APAIS and CHTS, both of which 
are part of the NMFS Marine Recreational Information 
Program. The CHTS is a random-digit-dial telephone 
survey of households stratified by coastal counties to 
obtain the number of private boat and shore trips made 
by each member of a household in a 2-month period. 
Summation of the weightings within or across strata 
can then provide an estimate of total fishing effort by 
those same strata or combinations of strata (Foster 3 ). 
We selected APAIS trips by anglers who launched 
from sites along the coasts of western Florida, Ala- 
bama, Mississippi, and Louisiana and who reported 
targeting (by hook and line) red snapper or any of the 
following species of groupers: rock hind (Epinephelus 
adscensionis), speckled hind (E. drummondhayi ), red 
hind (E. guttatus ), red grouper (E. morio), yellowedge 
grouper (Hypoi'thodus. flauolimbatus), misty grouper 
(H. mystacinus), Warsaw grouper ( H . nigritus), snowy 
grouper ( H . niveatus), black grouper ( Mycteroperca bo- 
naci ), yellowmouth grouper (M. inter stitialis), gag (M. 
microlepis), scamp (M. phenax ), tiger grouper (M. ti- 
gris), and yellowfin grouper (M. venenosa). Anglers who 
indicated they targeted any type of grouper also were 
3 Foster, J. 2013. Personal commun. Office of Science and 
Technology, Natl. Mar. Fish. Serv., NOAA, Silver Spring, MD 
20910. 
