Using balanced sampling in creel surveys
Section 3. A creel survey for striped bass in the Gaspé Peninsula
The Gaspé Peninsula is on the Canadian East Coast in the
Province of Québec. In 2015 a creel survey for striped bass was conducted in
this peninsula as recreational striped bass fishing had just been reintroduced
after a long moratorium.
The study area, presented in Figure 3.1, is
scattered over more than 250 kms, on the Gaspé Peninsula coast. The survey
is carried out by a single wildlife agent; it is not possible for him to visit
two distant sites on the same day. For that reason, neighboring sites are
grouped into three sectors as shown in Figure 3.1. We consider the survey
for the 33 holidays. The survey variable is the fishing effort, in number of
hours of fishing. As some sites attract more fishermen than others, the number
of visits to site
of sector
has to be proportional to its importance
as given in Table 3.1. In addition, for
the purpose of the survey, a day is divided into three periods (AM, PM, EV),
where EV stands for evening, and six subperiods (AM1, AM2, PM1, PM2, and EV1,
EV2). For instance AM1 goes from 8:00 to 10:00 while AM2 is from 10:00 to
12:00. A working day contains two periods and four subperiods. For instance if
the agent works AM and PM, then he has a free evening. Thus during a working
day he is able to visit four sites, two per working period.
The survey population on a day consists of 54
quadruplets,
4 of which are sampled. To denote population
units the following indices are useful:
-
represents
the days;
-
stands for
the sectors in Figure 3.1;
-
denotes a
period within a day;
-
represents
the subperiods within a period;
-
represents
the sites, see Figure 3.1, within a sector.
The goal is to estimate the fishing effort for combination of subperiod (6 levels) and site (9 levels). We want to plan a survey with a predetermined sample size for the 54 cells of the cross-classified table. The basic selection probabilities are
where replacing
or
by
means that a summation is taken on the
corresponding index. Observe that the sum of
over the indices
is equal to 4, the number of units visited by
the wildlife technician on a single day.
At a first glance, the sample could possibly be drawn in
a single stage using selection probabilities (3.1) by balancing on the 54 site
by subperiod indicator variables. This is not feasible because of operational
constraints. The first one is that on a single day the technician visits sites
from the same sector to limit the traveling between sites. The second
constraint is that on a working day the technician is off duty for the two
subperiods of the same period. In order to meet these operational constraints
we propose, in the next section, a design having three levels of sampling where
sectors are selected at level 1, periods are selected at level 2 and
sites are selected at level 3.
![Figure 3.1 of article 54954 issue 2018002](fig/fig3_1-eng.jpg)
Description for Figure 3.1
Geographical map showing the nine sites to be surveyed for striped bass. The map is divided in three sectors: East, Centre and West. There are three surveyed sites in each sector. In the East sector, there are Boom Défense, E. St-Jean and Barachois. In the Centre sector, there are Ste-T. de Gaspé, Chandler and Malbaie. In the West sector, there are Bonaventure, P. Henderson and C. Carleton.
Table 3.1
Average and expected number of visits to each site
Table summary
This table displays the results of Average and expected number of visits to each site. The information is grouped by Sector (appearing as row headers), Site and (équation) (appearing as column headers).
Sector |
Site |
|
|
|
|
East
|
Boom Défense
|
2 |
20.308 |
20.286 |
0.850 |
E. St-Jean
|
1 |
10.154 |
10.153 |
0.621 |
Barachois
|
2 |
20.308 |
20.296 |
0.881 |
Centre
|
Ste-T. de Gaspé
|
1 |
10.154 |
10.176 |
0.865 |
Malbaie
|
1 |
10.154 |
10.155 |
0.880 |
Chandler
|
1 |
10.154 |
10.162 |
0.881 |
West
|
Bonaventure
|
2 |
20.308 |
20.311 |
1.004 |
P. Henderson
|
1 |
10.154 |
10.153 |
0.681 |
C. Carleton
|
2 |
20.308 |
20.309 |
1.016 |
3.1 A balanced multi-stage design for creel survey
This
section describes the three stages of the survey that ensures that the
operational constraints presented in the previous section are met. It also
gives, for each stage, the balancing variables.
The
first stage is stratified by day; for each day a single sector is drawn with
selection probabilities
At level two, for each sector selected at
level 1, two periods are selected out of 3 using simple random sampling (i.e.,
with selection probabilities 2/3). At level three, a sector*period is
stratified by subperiod and one site is selected for each subperiod, the
selection probabilities are
In summary the selection probabilities at the
three levels are
As expected the product
is equal to (3.1), the target selection
probability.
The
goal is still to get a sample with predetermined sample sizes for the 54 site
by subperiod combinations. Thus balanced sampling needs to be implemented at
each stage. At level 1 we need to balance on the indicator variables for
the three sectors while at level 2 balancing on the 9 indicator variables
for the sector by period combinations is needed. Balancing at level 3 is
slightly more complicated as it involves several strata.
At
level 2,
sector*periods have been selected. Each one is
stratified by subperiod so we are facing 132 strata at level 3 and one
site is selected from each one. Balancing is needed with respect to the 54 site
by subperiod indicator functions. This is a complex problem and the balancing
constraints (2.3) involve the inverse of a large variance covariance matrix.
Thus to implement a rejective algorithm in this context one would need an
alternative to criterion (2.3) for accepting a sample. For now we discuss the
implementation of balanced sampling for this design with the cube method.
Comparisons between the cube method and rejective sampling in the context of a
simplified creel survey are presented in Section 4.
Among
the 132 third stage strata, the number of strata for one subperiod, say AM2, in
sector
is an integer close to
that depends on the stage 2 sample. This
integer plays the role of
in equation (2.2) for balancing the sites of
sector
at stage 3 while, for the
site, the probability in (2.2) is
The stage 3 calibration equations for the 54
site by subperiod indicator functions can be described in a similar way.
Clearly, it is not possible to meet exactly the 54 balancing equations and the
cube method will give a sample that is approximately balanced.
The
approximation occurs at the landing phase of the algorithm where balancing
constraints are dropped in order to complete the selection of the sample, as
introduced in Deville and Tillé (2004).
As the stage 3 sample is highly stratified, we use the implementation of the
landing phase in the function balancedstratification2
developed in Hasler and Tillé (2014),
with a small correction that prevents it from stopping when the sample is
already balanced at the start of the landing phase. In the matrix of balancing
constraints, the site constraints were given more importance than those which
make visits to each site equally distributed among subperiods at level 3.
They were the last ones to be dropped at the landing phase of the cube method.
To
investigate how a failure to meet all balancing equations impacted the sample
design, we generated
random replications of the balanced sample.
The number of visits
to site
was noted. Table 3.1 compares the average
of
over the Monte Carlo replications,
to its
expectation,
For all practical purposes, the two are equal
and a failure to meet some balancing equations has no impact on the site
selection probabilities. Table 3.1 also reports the standard deviations
Most
of the standard deviations are less than 1 in Table 3.1. Thus the absolute
differences between target and realized sample sizes are less than or equal to
2 for most Monte Carlo samples.
Table 3.2
gives the expected number of visits in the 6 subperiods; they are all equal to
22, up to two decimal points, with standard deviations less than 0.2. Thus the
period and subperiod constraints are met. Table 3.3 gives a realized
sample for the first five days of the creel survey. It shows a harmonious
permutation of sectors at level 1, periods at level 2, and sites at
level 3 through the days because of the way in which the sample design was
constructed. Given a balanced sample produced by the cube algorithm, an
arbitrary permutation of the days gives an alternative balanced sample. Indeed
the sampling design is invariant to a relabeling of the days. For instance,
with the sample of Table 3.3 the technician has to travel from the western
to the eastern sector between days 4 and 5. To avoid this long trip one could
interchange days 1 and 5: the first two days would then be spent in the eastern
sector and between days 4 and 5 the technician would travel from the western to
the central sector. The alternative and the original samples have the same
estimated totals for the calibration variables.
Table 3.2
Average and expected number of visits at each subperiod
Table summary
This table displays the results of Average and expected number of visits at each subperiod. The information is grouped by Period (appearing as row headers), Subperiod and (équation) (appearing as column headers).
Period |
Subperiod |
|
|
|
Morning
|
8h00-10h00
|
22 |
22.000 |
0.000 |
10h00-12h00
|
22 |
22.000 |
0.000 |
Afternoon
|
12h00-15h00
|
22 |
21.999 |
0.184 |
15h00-18h00
|
22 |
21.999 |
0.184 |
Evening
|
18h00-20h30
|
22 |
22.001 |
0.184 |
20h30-23h00
|
22 |
22.001 |
0.184 |
Table 3.3
Units selected in a balanced sample for the first five days
Table summary
This table displays the results of Units selected in a balanced sample for the first five days. The information is grouped by H (appearing as row headers), Sector, Period, Subperiod and Site (appearing as column headers).
H |
Sector |
Period |
Subperiod |
Site |
1 |
Centre
|
Afternoon
|
12h-15h
|
Chandler
|
15h-18h
|
Malbaie
|
Evening
|
18h-20h30
|
Chandler
|
20h30-23h
|
Ste-T. de Gaspé
|
2 |
East
|
Morning
|
8h-10h
|
E. St-Jean
|
10h-12h
|
Boom Défense
|
Evening
|
18h-20h30
|
Barachois
|
20h30-23h
|
E. St-Jean
|
3 |
Centre
|
Morning
|
8h-10h
|
Malbaie
|
10h-12h
|
Ste-T. de Gaspé
|
Afternoon
|
12h-15h
|
Malbaie
|
15h-18h
|
Chandler
|
4 |
West
|
Morning
|
8h-10h
|
P. Henderson
|
10h-12h
|
Bonaventure
|
Afternoon
|
12h-15h
|
C. Carleton
|
15h-18h
|
C. Carleton
|
5 |
East
|
Afternoon
|
12h-15h
|
Boom Défense
|
15h-18h
|
Barachois
|
Evening
|
18h-20h30
|
Boom Défense
|
20h30-23h
|
Barachois
|
3.2 Estimation of the fishing effort and of its
variance
Once
the survey is completed, the sample is a set of site
subperiod
with sampling weights equal to the inverse of
the selection probabilities given in (3.1). As the balancing equations for the
54 cells of the site by subperiod cross-classified table are not met exactly,
we propose, following Deville and
Tillé (2004), calibrating the survey weights on the total,
of the indicator variables for these 54 cells.
All the sampled units in cell
have the same weight, namely
where
defined in (3.1), does not depend on
The calibrated weight for a sampled unit in
cell
is
where
is the sample size for cell
it is the number of days for which site
of sector
has been visited during subperiod
of period
In general
is a random variable. When the samples are
perfectly balanced, (2.2) implies that
the calibrated and basic weights are then
equal. Now if
represents the fishing effort for population
unit
the fishing effort in cell
is
Its calibrated estimator is
where
is the average fishing effort for the
units sampled for that cell of the cross
classified table. An estimator for the total fishing effort is obtained by
summing the cells’ estimated totals.
The
evaluation of a design based variance estimator for the calibrated estimator of
the total fishing effort is complex. A simple variance estimator for the
estimated total for a single cell of the cross-classified table is available.
The sample of days selected for cell
is a Bernoulli sample with selection
probabilities
neglecting the balancing constraints. Thus by
conditioning on the sample size,
is
times the sample mean of a simple random
sample. It is a design-unbiased estimator whose variance can be estimated using
the formula for the variance of an estimated total in a simple random sampling
design. We claim that these results are still valid when the balancing
constraints are taken into account since the balanced sample design is
invariant to a relabelling of the days. The estimated fishing efforts for the
54 cells of the cross-classified table are however dependent and it seems
difficult to come up with a conditionally unbiased design based variance
estimator for their total. A model based estimator seems to be only approach
available for this total.
For
the survey actually conducted in 2015, the methods used to estimate fishing
effort and total catch are among those proposed in Pollock et al. (1994). It was a roving survey and the
fishing effort at a sampled site was calculated as the average number of
anglers on the site during the subperiod times the length, in hours, of the
subperiod. Fishing efforts were estimated using calibrated weights; additional
results are available in (Daigle,
Crépeau, Bujold and Legault, 2015).
ISSN : 1492-0921
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