SWAK ss Part B 022307

SUPPORTING STATEMENT
REGIONAL ECONOMIC DATA COLLECTION PROGRAM
FOR SOUTHWEST ALASKA
OMB CONTROL NO.: 0648-xxxx

B. COLLECTIONS OF INFORMATION EMPLOYING STATISTICAL METHODS
1. Describe (including a numerical estimate) the potential respondent universe and any
sampling or other respondent selection method to be used. Data on the number of entities
(e.g. establishments, State and local governmental units, households, or persons) in the
universe and the corresponding sample are to be provided in tabular form. The tabulation
must also include expected response rates for the collection as a whole. If the collection has
been conducted before, provide the actual response rate achieved.
For the vessel surveys, the overall population consists of all fishing vessels landing raw fish at a
port in Southwest, Alaska during 2005. For that year, there were 2,117 vessels. This population
consists of three vessel classes – small, medium, and large vessel classes. The population sizes
are 1,479, 421, and 217, respectively for small, medium, and large vessels classes. An unequal
probability sampling (UPS) procedure is used to determine the sample sizes needed for the
analysis for each vessel class, which is described in Item #2 below and in Attachment D. The
population sizes of local businesses and fish processors are 172 and 41, respectively.
The expected response rates for the vessel surveys are based on consideration of the following
factors. First, compared with a previous data collection project conducted for Southeast Alaska
(Hartman 2002), which achieved an overall response rate of about 30%, the number of questions
in the present project is much smaller and the quantity of information being asked is much
smaller. Second, in the present study, questions about sensitive information such as vessel cost
and expenditures are omitted. The previous Southeast study included these sensitive questions,
which significantly contributed to the low response rate. Third, input from select members of the
respondent populations helped guide survey design and question wording. Fourth, follow-up
telephone calls will also increase the response rate. Based on these factors, it is expected that,
overall, the response rate for mail survey of fishermen for the present project will be about 55%
which is much higher than in the Southeast study. For telephone interviews with local
businesses (including fish processors), a response rate of 65% is expected. For a more detailed
description of the methods we used, and will use, to increase the response rate, see Item #3
below.
Vessel Class

Population
size

Small vessel
Medium vessel
Large vessel
Local businesses
including fish
processors

1,479
421
217
213

Mail or
phone
interview
sample size
491
225
164
213

1

Expected
number of
respondents

Expected
response rate

270
124
90
139

55%
55%
55%
65%

2. Describe the procedures for the collection, including: the statistical methodology for
stratification and sample selection; the estimation procedure; the degree of accuracy
needed for the purpose described in the justification; any unusual problems requiring
specialized sampling procedures; and any use of periodic (less frequent than annual) data
collection cycles to reduce burden.
Since the majority of gross revenue within each harvesting sector comes from a small number of
vessels, a simple random sampling (SRS) of vessels would only include a small portion of the
total ex-vessel value, and therefore, would be misleading. As a result, for the present project an
unequal probability sampling (UPS) method without replacement is used that accounts for this
unequal harvest in each target population. The objective of implementing the sampling task is to
estimate the employment and labor income information for each of three disaggregated
harvesting sectors using as an auxiliary variable the ex-vessel revenues provided by Commercial
Fisheries Entry Commission (CFEC) earnings data. Since each sector will be used as a separate
economic sector in IMPLAN model, we face three separate problems for three different sectors
in sampling. For each sector, we use a UPS without replacement method to identify sampling
units. Details on our sampling methodology are described in Attachment D.
3. Describe the methods used to maximize response rates and to deal with non-response.
The accuracy and reliability of the information collected must be shown to be adequate for
the intended uses. For collections based on sampling, a special justification must be
provided if they will not yield "reliable" data that can be generalized to the universe
studied.
(a) Maximizing Response Rates
Previous applications of voluntary commercial fishing surveys in Alaska (e.g., Hartman 2002)
were hampered by low response rates that principally resulted from the use of long and
complicated survey instruments. Commercial fishermen are frequently asked, and often
required, to participate in surveys from numerous organizations including NOAA, Alaska
Department of Fish and Game (ADF&G), and universities. As a result, commercial fishermen
are less likely to complete voluntary surveys that are lengthy, poorly-designed, and do not clearly
involve issues that are important to them. In this data collection, significant efforts were made to
ensure the survey instruments were short in length, contained well-designed questions, and
clearly conveyed the importance of the data collection to issues that are important to commercial
fishermen.
The mail surveys are short (6 to 7 questions depending upon the survey version, all of which
span eight pages) and avoid many sensitive questions compared with many previously-fielded
commercial fishing surveys. The set of questions was limited to only those that are essential for
achieving the objectives of the project as outlined in Part A, Item #1 above. There is only a
fraction of the number of questions asked compared with the Southeast Alaska commercial
fishing survey discussed earlier, which achieved an overall response rate of about 30%. In the
mail surveys, numerous questions on vessel expenditures that are often included in surveys of
commercial fishermen are omitted here to avoid the added complexity and likely sensitivity of
asking for this type of information from respondents. 1
1

Vessel expenditures will be estimated using (1) the sales data collected from telephone interviews with local
2

The telephone scripts for use in interviews with local businesses and fish processors were
developed with similar goals in mind. Specifically, each phone script was constructed to include
only the most essential questions to ensure the telephone interviews were short in length to
minimize the time burden on respondents.
Pretesting activities that included a small focus group and several interviews with fishermen and
fish processors (totaling less than 10 individuals) were used to evaluate the content and
presentation of the survey materials, as well as to ensure input by the fishing community.
Feedback from these pretesting activities aided in non-trivial ways to the development of the
survey questions. For instance, considerable effort was made to ensure that the survey
instrument reflected considerations for the record-keeping systems kept by fishermen and used
common terms and wording used by fishermen. Participants in pretesting activities also
indicated that previous voluntary surveys often did not provide adequate assurances that the
information being requested would be handled confidentially, which often deterred them from
responding. To ensure respondents that the data they share will be kept confidential, a detailed
confidentiality statement is presented on the first page of the mail survey and mentioned upfront
in the telephone interviews. A similar statement is made in the cover letter accompanying the
mail survey.
Another reason believed to have caused low response rates in previous survey efforts is the
disinterest among respondents toward the survey purpose. Surveys that collect information that
will clearly benefit or interest respondents are more likely to be completed. The importance and
benefits of this data collection project to the respondents (fishermen, local businesses, and fish
processors) will be emphasized in the advance letter, cover letter, mail survey, and telephone
interviews. In these letters and phone interviews, the investigators clearly state that with the help
of the respondents, the important role of the respondents’ fishing and business activities in the
regional economy can be better identified and that the information they provide will be used to
enhance the fishery management practices of NOAA Fisheries, and, thereby, to increase the
long-run economic benefits to the fishermen and local businesses. Making a clear link between
the survey, their participation, and the fishery and regional economy is expected to help increase
the response rate relative to previous studies.
In addition to the above steps taken to maximize response rates, the survey instruments (mail and
telephone) were subjected to significant review by several researchers with expertise on Alaska
fisheries and economic surveys to ensure the quality of the materials.
In addition to high-quality survey instruments, the set of survey protocols to be followed in
implementation was designed to maximize response rates. For the mail survey, a modified
Dillman (2000) approach will be employed that includes four survey contacts as follows (All the
letters, postcard reminder, and follow-up phone scripts for these four contacts are attached in
Attachment C):
•
•

An advance letter notifying the respondents a few days before they receive the survey
questionnaire. This will be the first contact with the respondent.
An initial mailing sent a few days after the advance letter. Each mailing will contain

businesses and fish processors and (2) a cost engineering approach.
3

•
•

a cover letter, personalized questionnaire, and a pre-addressed stamped return
envelope.
A postcard follow-up reminder mailed 5-7 days following the initial mailing.
A follow-up phone call to encourage response and identify individuals that have
misplaced or need another copy of the survey. If the respondent agrees, the mail
survey will be completed over the phone. 2 Up to three attempts will be made to
contact each respondent for the telephone interview. Individuals needing an additional
copy of the survey will be sent one with another cover letter and return envelope.

A strict Dillman approach is not warranted, given negative input from commercial fishermen
about repeated contacts beyond the phone contact.
The result of the efforts described above are compact and high-quality survey instruments that
contain questions vessel owners, local businesses, and fish processors can answer with
minimal effort. As a result, the expected response rate for the mail survey of fishermen is
expected to exceed previous survey efforts and achieve a response rate of approximately 55%.
This response rate is much higher than that in the longer and more complicated Southeast
Alaska study (30% response rate). For the telephone interviews with local businesses
(including fish processors), a response rate of 65% is assumed based on previous experience. 3
(b) Non-response
To better understand the differences between them, comparisons will be drawn between
respondents and non-respondents with respect to several observable characteristics: (1)
geographical area of landed fish, (2) ex-vessel value, and (3) species that vessels catch. This
information is available from government data for each vessel. If significant and systematic
differences between the two groups are discovered, the population parameter estimates of
interest may be adjusted by using weights formed from these variables.
4. Describe any tests of procedures or methods to be undertaken. Tests are encouraged as
effective means to refine collections, but if ten or more test respondents are involved OMB
must give prior approval.
There are no plans to conduct a pilot survey or other tests involving more than ten respondents.
5. Provide the name and telephone number of individuals consulted on the statistical
aspects of the design, and the name of the agency unit, contractor(s), grantee(s), or other
person(s) who will actually collect and/or analyze the information for the agency.
John Slanta (Census Bureau, PH 301-763-4773) and Dr. Dan Lew (NMFS, PH 206-526-4252)
assisted in the development and review of sampling procedures for this project.
2

In this case, the ex-vessel values (by species) of the vessel will be provided to the vessel owners so that they will
not have to access their records, which should greatly simplify the question and allow them to calculate the crew and
skipper payments easily. In doing this, we will make sure that the person we will be interviewing on the phone is
the true owner of the vessel. This is because we do not want to breach the confidentiality by providing the sensitive
information to the wrong person. As is seen in the mail survey questions (Attachment A), however, this ex-vessel
information will not be given to the respondent in the mail survey.
3
See Section A #12, Footnote 6.
4

Several NMFS economists with experience in economic survey design and implementation
reviewed the survey materials and survey protocols, including Dr. Dan Lew, Dr. Ron Felthoven,
and Dr. Brian Garber-Yonts.
Professor Hans Geier (University of Alaska, Fairbanks) is the contractor who will conduct the
data collection project, revise the IMPLAN data, and participate in developing regional
economic models.
Dr. Chang Seung (Alaska Fisheries Science Center) will conduct the statistical analysis of the
information collected, and develop regional economic models with Professor Geier.

5

ATTACHMENT A. SAMPLING PROCEDURES FOR HARVESTING SECTORS1
The overall project objective is to estimate the employment and labor income information for
each of three disaggregated harvesting sectors using data to be collected via a mail survey.
Using ex-vessel revenue information, an unequal probability sampling (UPS) procedure will be
employed to determine the sampling plan for each of the three harvesting sectors. The procedure
is described below.
In the literature, there exist many methods for conducting UPS without replacement (see, for
example, Brewer and Hanif 1983; Sarndal 1992). One critical weakness with most of these
methods is that the variance estimation is very difficult because the structure of the 2nd order
inclusion probabilities (πij)2 is complicated. One method that overcomes this problem is Poisson
sampling. However, one problem with Poisson sampling is that the sample size is a random
variable, which increases the variability of the estimates produced. An alternative method that is
similar to Poisson sampling but overcomes the weakness of the Poisson sampling is Pareto
sampling (Rosen 1997)3 which yields a fixed sample size.
In this project, there are two tasks that we need to do for estimating the population parameters
using UPS without replacement. First, the optimal sample size needs to be determined. Second,
once the optimal sample size is determined, the population parameters and confidence intervals
need to be estimated. For the first task, we will use the variance of Horvitz-Thompson (HT)
estimator from Poisson sampling in Part I below.4 For the second task, we will use the Pareto
sampling method described in Part II below (Slanta 2006). In determining the optimal sample
size in Part I, we will use information on an auxiliary variable (ex-vessel revenue). To estimate
the population parameters in Part II, we use actual response sample information on the variables
of interest (employment and labor income).
Part I: Estimating Sample Size
Step 1: Estimation of Optimal Sample Size (n*)
(A) Obtaining Initial Probabilities
To obtain the initial values of the inclusion probabilities (πi) for unit i in the population, we
multiply the auxiliary value of unit i (Xi, i.e., the ex-vessel value of vessel i in the population) by
a proportionality constant (t)5:

π = tX
i

(1)

i

where πi
Xi

: probability of vessel i being included in the survey sample
: value of the auxiliary variable (ex-vessel value of vessel i in the
population)

1

Here, t is given by
N

t=

∑X

i

i

(2)

N

∑X

V +

2
i

i

where N
V

: population size
: desired variance (of HT estimator of the population total); Poisson
variance. Here, V is given as:
2
⎛ εX ⎞
⎜
⎟
V =
⎜z
⎟
⎝ 1−(α / 2) ⎠
where ε is the error allowed by the investigator [e.g., if ε is 0.1, then 10% error of
true population total ( X =

N

∑X
i =1

i

) is allowed]; and z is percentile of the standard

normal distribution. Therefore, choosing a desired variance V is equivalent to
N
(1 − π i ) X i2
setting the values of ε and z. The value of V calculated using V = ∑
i =1

πi

(Poisson variance; Brewer and Hanif 1983, page 82) with πi’s being the final
values of N inclusion probabilities obtained from Step 1, will be equal to the
desired variance given at the beginning of Step 1.
Some of the resulting πi’s could be larger than one. The number of certainty units (i.e., the
number of units for which πi >1) is denoted C1. If πi > 1, then we force this inclusion probability
to equal one (πi = 1).
(B) Iterations and Determination of Optimal Sample Size
We recalculate t using the noncertainty units (i.e., the units for which πi <1) obtained in (A)
above, i.e.,
M1

t=

∑X

i

i

V +

(2’)

M1

∑X

2
i

i

where M1

: number of noncertainty units from (A), where M1 = N – C1.

Using equation (1) above, we calculate the inclusion probabilities for the noncertainty units by
multiplying the t value [from equation (2’)] by the ex-vessel values of the noncertainty units. If
the resulting πi’s are larger than one, we force them to equal one. The resulting numbers of
certainty and noncertainty units are denoted C2 ( = C1 + additional number of certainty units) and
M2 ( = M1 – additional number of certainty units), respectively, where C2 + M2 = N. Next, for
M2 units of noncertainty, we calculate the t and πi’s again. This is an iterative process. We
2

continue this process until the noncertainty population stabilizes (i.e., until there is no additional
certainty unit).
If the noncertainty population stabilizes after kth iteration, there will be Ck units of certainty units
and Mk units of noncertainty units and Ck+ Mk = N. Summing over the probabilities for all these
certainty and noncertainty units, we obtain the optimal sample size (n*) as:
n* =

N

∑π

(3)

i

i

At this stage the optimal sample size may not be an integer number. In this stage, we also
compute the optimal sample size under simple random sampling (SRS)6, nsrs, and compare it
with n*.
Step 2: Determining Number of Mailout Surveys
(A) Adjustment of Probabilities
Once the optimal sample size (n*) is determined in Step 1, we divide the sample size (n*) by the
expected response rate (obtained from previous studies) to determine the number of surveys that
need to be mailed out to achieve n*. The number thus derived is denoted na (this number may
not still be an integer value). We next adjust the inclusion probabilities for the Mk noncertainty
units obtained in Step 1 above as:

⎤
⎡
⎢ π ⎥
π i = (na − C k ) ⎢ M k i ⎥
⎥
⎢
⎢ ∑π i ⎥
⎦
⎣ i

(4)

If the resulting probabilities are larger than one (πi > 1), we make them certainties (πi = 1). The
resulting numbers of certainty and noncertainty units are denoted Ck+1 and Mk+1, respectively.
Next, we adjust the probabilities of the new set of noncertainty units (Mk+1) in a similar way
using equation (4’) below:
⎤
⎡
⎢ π ⎥
π i = (na − C k +1 ) ⎢ M k +1i ⎥
(4’)
⎥
⎢
⎢ ∑π i ⎥
⎦
⎣ i
We continue this process until the noncertainty population stabilizes. The resulting numbers of
certainty and noncertainty units are Cq and Mq, respectively.
(B) Apply Minimum Probability Rule
At this point, we impose a minimum probability rule. UPS can have excessively large weights
(= 1/πi) and if they report a large value, then the population estimate and its variance would be
very large. In order to avoid this problem, we can impose a minimum value of the inclusion
3

probabilities. If m is the minimum imposed probability, then we do the following:
If πi < m, then set πi = m for each i, where i = 1, ..., N.
The value for m here is determined arbitrarily. The only cost involved in using this rule is a
small increase in sample size.7
(C) Finding an Integer Value for Sample Size
Next, we add up all the resulting inclusion probabilities. The resulting sum is denoted nb ( > na),
which may not be an integer value. Next, we adjust again the probabilities for noncertainty units
including the units for which the minimum probabilities were imposed as:
⎤
⎡
⎢
π ⎥
π i = ( nc − C q ) ⎢ M q i ⎥
⎥
⎢
⎢ ∑π i ⎥
⎦
⎣ i

(5)

where nc is the smallest integer value larger than nb (e.g., if nb = 15.3, then nc = 16). Finally, we
add up the resulting (certainty and noncertainty) probabilities. The sum of all these probabilities
is the final survey sample size (i.e., the number of surveys to be sent out to), and is denoted nm (=
nc).

Part II: Estimation of Population Parameters and Confidence Intervals

Step 3: Implementation of Pareto Sampling
After the mailout sample size (nm) for each sector is determined in Step 2, the mailout sample is
selected from each sector’s population using Pareto sampling. The probability of each unit
(vessel) being in the sample in a given sector is proportional to the unit’s (vessel’s) ex-vessel
revenue. Because the majority of gross revenue within each sector comes from a small number
of vessels, a random sample of vessels would only include a small portion of the total ex-vessel
values.
According to Brewer and Hanif (1983), there are fifty different approaches that are used for
UPS. Most of these approaches suffer from the weakness that it is very hard to estimate the
variance. Poisson sampling overcomes this problem, and is relatively easy to implement.
However, the limitation of Poisson sampling is that the sample size is a random variable.
Therefore, in this project, we will use Pareto sampling (Rosen 1997 and Saavedra 1995) which
overcomes the limitation of Poisson sampling. The mailout sample size will be nm as determined
in Step 2 (C) above. We will use the inclusion probabilities obtained from Equation (5) above in
implementing Pareto sampling.
The procedure of this sampling method (Block and Crowe 2001) is briefly described here:
1. Determine the probability of selection (πi) for each unit i as in Equation (5) above.
4

2. Generate a Uniform (0,1) random variable Ui for each unit i
3. Calculate Qi = Ui (1 – πi ) / [πi (1 - Ui )]
4. Sort units in ascending order by Qi, and select nm smallest ones in sample.
From the above, it is clear that we will have a fixed sample size with Pareto sampling.
Step 4: Mailing out Surveys and Obtaining Actual Response Sample
Next, we will send out the surveys to the nm units (vessel owners). Actual response sample will
be obtained and the size of the actual response sample is denoted r.
Step 5: Estimation of Population Parameters (Population Total)
Using the information in the actual response sample, we calculate population parameters for
variables of interest (employment and labor income in our project), not for ex-vessel revenue,
using HT estimator (Horvitz and Thompson 1952). We are interested in estimating the
population totals (not population means) of the variables of interest. The HT estimator is given
as:
r

YˆHT = ∑ wi y i

(6)

i =1

where r
wi
yi

: number of respondents
: weight for ith unit ( = 1/πi ). Note that the weights are calculated here
using the information on the auxiliary variable, not that on the variables
of interest
response
sample data of ith unit (employment or labor income)
:

However, the HT estimator needs to be adjusted for non-response. The estimator is adjusted in
the following way.
⎛ N
⎜ ∑Xj
⎜ j =1
Yˆ = ⎜ r
⎜⎜ ∑ wi X i
⎝ i =1

⎞
⎟
⎟ˆ
⎟ YHT
⎟⎟
⎠

where N
Xi

: population size
: auxiliary variable of ith unit (respondents only)

(7)

Usually, we apply this adjustment to the certainties separately from the noncertainties, and then
add the two together to get a final estimate. If there are no respondents within any of the two
groups of certainty units and noncertainty units, then we collapse the two groups before applying
the adjustment. Specifically, the final estimate of population total is given by:
5

⎛ N1
⎜ ∑Xj
⎜ j =1
Yˆ = ⎜ r1
⎜ ∑ wi X i
⎜
⎝ i =1

⎞
⎛ N2
⎟ r
⎜ ∑Xj
⎟ 1
⎜ j =1
⎟ ∑ wi y i + ⎜ r2
⎟ i =1
⎜ ∑ wi X i
⎟
⎜
⎠
⎝ i =1

⎞
⎟ r
⎟ 2
⎟ ∑ wi y i
⎟ i =1
⎟
⎠

(8)

where N1
: number of certainty units in the population
N2
: number of noncertainty units in the population
: number of respondents from certainty units
r1
: number of respondents from noncertainty units, and
r2
N1 + N2 = N and r1 + r2 = r.
Step 6: Estimation of Variance for YˆHT and Yˆ

Here we will calculate the variances of the population estimates for the variables of interest. The
variance estimate for Pareto sampling is given in Rosen (1997, Equation (4-11), p. 173) as:
⎧
⎪
⎪
n
m
Var (YˆHT ) =
⎨
nm − 1 ⎪
⎪
⎩

⎡ nm
⎛y
⎢∑ (1 − π i )⎜⎜ i
⎢⎣ i =1
⎝πi

⎞
⎟⎟
⎠

2

⎡ nm ⎛ 1 − π i
⎢∑ y i ⎜⎜
⎣ i =1 ⎝ π i

⎤
⎥ −
⎥⎦

nm

∑ (1 − π
i =1

i

)

⎞⎤
⎟⎟⎥
⎠⎦

2

⎫
⎪
⎪
⎬
⎪
⎪
⎭

(9)

Since we have adjusted for nonresponse, we need to incorporate the variability due to
nonresponse into the variance. If we assume that the response mechanism is fixed 8, then we
have a ratio estimator and its variance can be found in Hansen, Hurwitz, and Madow (1953, page
514). This variance is a Taylor expansion, and is given as:
⎛ σˆ 2 ( A) σˆ 2 (B ) 2 COV ( A, B ) ⎞
⎟⎟
+
−
Var Yˆ = Yˆ 2 ⎜⎜
2
2
AB
A
B
⎠
⎝

()

(10)

where
r

A = ∑ wi y i
i =1
r

B = ∑ wi X i
i =1

2
⎧
⎡ r
⎤ ⎫
⎪
⎢∑ (1 − π i )(wi yi )⎥ ⎪
nm ⎪⎡ r
2⎤
⎣ i =1
⎦ ⎪
2
σˆ ( A) =
⎨⎢∑ (1 − π i )(wi yi ) ⎥ −
⎬
nm
nm − 1 ⎪⎣ i = 1
⎦
(1 − π i ) ⎪⎪
∑
⎪
i =1
⎩
⎭

6

2
⎧
⎡ r
⎤ ⎫
⎪
⎢∑ (1 − π i )(wi X i )⎥ ⎪
nm ⎪⎡ r
2⎤
⎣ i =1
⎦ ⎪
2
σˆ (B ) =
⎬
⎨⎢∑ (1 − π i )(wi X i ) ⎥ −
nm
nm − 1 ⎪⎣ i = 1
⎦
⎪
(1 − π i )
∑
⎪
⎪
i =1
⎭
⎩

⎧
⎡ r
⎤⎡ r
⎤⎫
(
)(
)
(
)(
)
−
−
1
π
1
π
w
y
w
X
⎪ r
⎢∑
i
i
i ⎥⎪
i
i i ⎥ ⎢∑
⎤ ⎣ i =1
nm ⎪⎡
i =1
⎦
⎣
⎦⎪ .
2
COV ( A, B ) =
⎨⎢∑ (1 − π i )wi y i X i ⎥ −
⎬
nm
nm − 1 ⎪⎣ i =1
⎦
⎪
(1 − π i )
∑
⎪
⎪
i =1
⎩
⎭

Step 7: Calculation of Confidence Intervals
Confidence intervals are calculated using response sample statistics obtained in steps 5 and 6.
We only choose one sample, but if there were many independent samples chosen then we would
expect on average that approximately 100(1-α) % of the confidence intervals constructed in the
following manner will contain the truth.
⎛⎜ Yˆ − z
Var (Yˆ ) , Yˆ + z
Var (Yˆ ) ⎞⎟
α /2
α /2
⎠
⎝

where Yˆ

(11)

: Estimated population total for employment or labor income.

Note that it is possible to use t-statistics if the sample size is small.

7

Footnotes

1. In the process of developing this document, several experts in UPS sampling assisted me
by providing helpful comments and inputs. The experts include John Slanta (U.S. Census
Bureau), Bengt Rosen (Uppsala University), Pedro Saavedra (ORC Macro), Holmberg
Anders (Statistics Sweden), Paolo Righi (ISTAT, Italy), and Bob Fay (U.S. Census). In
particular, I would like to thank John Slanta very much for his time and effort in
providing valuable inputs and advice. His suggestions and comments contributed
significantly to the development of the sampling procedures in this document. Many
thanks go to Dan Lew (NMFS) for his rigorous review and valuable suggestions which
contributed in a significant way to the improvement of this document. I also benefited
from discussions of UPS with Norma Sands at NWFSC and from the Excel file that she
developed.
2. 2nd order inclusion probability (πij) is defined as the joint probability of including in
sample the ith and jth population units.
3. Saavedra (1995) independently developed the same sampling methodology as Rosen
(1997), which he called Odds Ratio Sequential Poisson Sampling (ORSPS).
4. Although we do not use Poisson sampling itself, we do use the Poisson variance of HT
estimator of the population total.
5. Equation (1) is derived as follows.
X
HT estimator, Xˆ HT = ∑ i , has variance,
i

2

πi

2

X
X
V ( Xˆ HT ) = ∑ i (1 − π i ) = ∑ i − ∑ X i2 (Brewer and Hanif 1983, page 82) (A)
N

i =1

πi

N

i =1

πi

N

i =1

For an expected sample size n,
⎞
⎛
⎟
⎜
Xi ⎟
⎜
πi = n N
⎟
⎜
⎜ ∑ Xi ⎟
⎠
⎝ i =1
Substituting (B) into (A) and solving for n,

(B)

2

N
⎞
⎛ ˆ
⎞
⎛ N
⎜V ( X HT ) + ∑ X i2 ⎟
n = ⎜⎜ ∑ X i ⎟⎟
⎟
⎜
i =1
⎠
⎝
⎝ i =1 ⎠
Substituting (C) into (B),

(C)

N
⎡
⎤
Xi
∑
⎢
⎥
i =1
⎥ X i , i = 1, 2, ... , N,
πi =⎢
N
⎢ ˆ
2 ⎥
⎢V ( X HT ) + ∑ X i ⎥
i =1
⎣
⎦

8

(D)

where V ( Xˆ HT ) is the desired variance.

6. The optimal sample size under SRS is determined using the following standard formula:

n srs ≥

z 2 N (CV p ) 2
z 2 (CV p ) 2 + ( N − 1) ε 2

where nsrs
CVp

(Levy and Lemeshow, formula (3.14) on page 74)

: optimal sample size under SRS
: coefficient of variation of the population parameter. Since the
information on the population parameters (i.e., employment and
labor income) is not available, we use ex-vessel revenue, for
which the population information is available from CFEC.
Therefore, CVp is defined as standard deviation of the ex-vessel
revenue in the population divided by the mean.

7. This minimum probability rule is used, for example, in the Manufacturing and
Construction Division of the Census Bureau. To date, there has not been any research on
the minimum probability in the sampling literature. It is an arbitrary value and in
applications has sometimes varied between strata in the same survey. Some researchers
determine the minimum probability such that the resulting weight, which is the reciprocal
of the minimum probability, is less than or equal to the population size. Generally
speaking, this minimum probability rule has little effect on the sample size.
8. Fixed response mechanism means that a unit included in a sample is always a respondent
or non-respondent no matter what sample the unit is included in. In other words, the
probability of the unit being a respondent is either one or zero but nothing in-between.

9

References

Block, C. and Crowe, S. (2001). Pareto-πps Sampling. Unpublished Document. Statistics
Canada.
Brewer, K. and Hanif, M. (1983). Sampling with Unequal Probabilities. Springer Verlag, New
York.
Hansen, Hurwitz, and Madow (1953).
Methods and Applications.

Sampling Survey Methods and Theory. Volume 1.

Horvitz, D. and Thompson, D. (1952). A Generalization of Sampling without replacement from
a Finite Universe. Journal of American Statistical Association Vol. 47, pp. 663-685.
Levy, P. and Lemeshow, S. (1999). Sampling of Populations – Methods and Applications.
Third Edition. Wiley and Sons.
Rosén, B. (1997). On Sampling with Probability Proportional to Size. Journal of Statistical
Planning and Inference, 62, 159-191.
Särndal, C.-E., Swensson, B. & Wretman, J. (1992). Model Assisted Survey Sampling. Springer
Verlag, New York.
Saavedra, P. 1995. Fixed Sample Size PPS Approximations with a Permanent Random
Number. Joint Statistical Meetings, American Statistical Association, Orlando, Florida.
Slanta, J. (2006). Personal Communication.

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