United States 

Department of

Agriculture

National

Agricultural

Statistics

Service

Research and

Development Division

Washington DC 20250

RDD Research Report

Number RDD-11-02

November 2011

Annual Land Utilization 

Survey (ALUS): Design 

and Methodology

Hailin Sang

Pam Arroway

Kenneth K. Lopiano

Denise A. Abreu

Andrea C. Lamas

Linda J. Young

This report was prepared for limited distribution to the research community outside the United 

States Department of Agriculture. The views expressed herein are not necessarily those of the 

National Agricultural Statistics Service or of the United States Department of Agriculture. 
2

EXECUTIVE SUMMARY

The National Agricultural Statistics Service (NASS) obtains annual estimates of farm numbers 

from its annual June Area Survey (JAS). Every five years, the annual numbers of farms 

estimates are compared to the one obtained from the quinquennial census of agriculture 

(conducted during years ending in 2 and 7). The JAS annual numbers have been declining 

steadily between censuses, especially between the 2002 and 2007 Censuses. Furthermore, in 

2002 and 2007, the JAS farm numbers indications1were considerably lower than farm numbers 

from the census and, in 2007, the difference could not be attributed to sampling error alone. 

Additionally, results from a 2007 qualitative study revealed that agricultural operations were 

being incorrectly classified as non-agricultural during the screening procedures of the JAS. 

In an attempt to get a better understanding of the misclassification leading to this undercount in 

the JAS number of farms indications, a post-JAS intensive screening called the Farm Numbers 

Research Project (FNRP) was undertaken in the fall of 2009 (Abreu, McCarthy and Colburn, 

2010). The design of the JAS includes rotating in new segments each year, while rotating out the 

oldest segments. Segments stay in the JAS sample for five years. Each year’s sample is 

comprised of segments from each of five rotations. Thus, the 2009 JAS contained segments that 

were rotated into the sample in 2009, 2008, 2007, 2006 and 2005. The sampling design of the 

FNRP targeted the 20-percent of JAS segments that were newly rotated in 2009. It targeted three 

types of tracts in these segments: 1) non-agricultural tracts; 2) agricultural tracts that refused to 

participate in the JAS and were estimated; and 3) agricultural tracts that were inaccessible in 

June and were estimated. The FNRP results suggest misclassification may be reduced through 

improvements in the screening questionnaire and by enhanced enumerator training guidelines. 

To fully address this concern, NASS and the National Institute of Statistical Sciences (NISS) 

formed a research team to review methodology associated with the design of the JAS and to 

suggest improvements. This team was one of the three academic-government teams of the twoyear collaborative research program between NASS and NISS called Cross-Sector Research in 

Residence Program. This research report documents the design and methodology of the Annual 

Land Utilization Survey (ALUS), which is a follow-on survey to the JAS. ALUS builds upon

the experience gained in the FNRP. From a methodological perspective, the two primary 

differences between the ALUS and the FNRP are the following: (1) All estimated and nonagricultural JAS tracts have a probability for inclusion in ALUS, but only the 20-percent newly 

rotated in estimated and non-agricultural JAS tracts were included in the FNRP; and (2) All of 

the FNRP-eligible tracts were included in the FNRP sample, but only a sample of the ALUSeligible tracts will be included in the ALUS. Because a sample of all estimated and nonagricultural JAS tracts will be collected, ALUS represents the second phase of a two-phase 

sample with the first phase being the traditional JAS. 

As in the JAS, ALUS will be a stratified sample of segments, using JAS strata and sampling 

across rotations. Segments that are eligible for inclusion in ALUS must have at least one tract 

that was pre-screened as non-agricultural (regardless of potential) or that was estimated in the 

JAS (as either farm or non-farm). The sample allocation of ALUS segments to each state-stratum 

 

1

Indications refer to point estimates derived from the survey results and could differ from official published 

estimates.
3

combination considers two factors: the proportion of the ALUS population in the stratum and the 

proportion of the FNRP adjustment from non-agricultural tracts in the stratum. For a selected 

segment, all tracts satisfying one of these criteria will be re-evaluated using a modified combined 

JAS-FNRP questionnaire. In the 2009 and 2010 JAS, over 90-percent of all segments would 

have been considered eligible for ALUS. The collection of eligible segments in a particular year 

will be called the ALUS population. The 2010 ALUS-eligible population is used to demonstrate 

the sampling design.

As stated earlier, the combination of JAS and ALUS can be considered as a two-phase sampleJAS being the first phase and ALUS the second. The two-phase JAS-ALUS stratified design, 

can be applied to any estimate produced by the JAS. Given that each phase makes use of a 

probability sampling design with known inclusion probabilities, standard results can be used to 

construct a design-based estimator (Sarndal and Swensson, 1987). However, non-response is also 

expected to occur in ALUS. In this report, methodology for a three-phase sampling design is 

developed by extending the methodology of Sarndal and Swensson (1987). A general sampling 

design is assumed in each phase. The estimator and its estimated variance are unbiased. Here we 

use this methodology for the two-phase JAS-ALUS, with the third phase being non-response.

This methodology can be applied not only to estimates of the number of farms but to all 

variables collected in the ALUS. Thus, although the primary impetus for this work is to improve 

the estimates for the number of farms, it can also be used to improve estimates of other important 

variables. 
4

RECOMMENDATIONS

1) A follow-on survey to the JAS, called ALUS, should be conducted so that the farm number 

indications (as well as any included commodities) from the JAS can be adjusted for 

misclassification.

2) The ALUS questionnaire should be a modified version of the JAS-FNRP questionnaire. 

While essentially a shortened version of the JAS questionnaire, the FNRP questionnaire was also

designed to target misclassification of farms and capture data on the type of farms that were 

believed to have been misclassified via multiple modes of data collection (face-to-face, phone, or 

mail). Additional JAS variables added to this questionnaire would allow the ability to adjust 

these variables as well. 

3) The methodology for a two-phase study with nonresponse (considered as a third phase), as 

developed in this report, should be used for the analysis of the JAS-ALUS.

Note: The recommendations above were discussed at the January 2011 Senior Executive Team

meeting where the decision was made to not conduct the ALUS in 2011 due to budget 

constraints.
5

Annual Land Utilization Survey (ALUS): Design and Methodology

Hailin Sang2, Pam Arroway3, Kenneth K. Lopiano4, 

Denise A. Abreu5, Andrea C. Lamas5, Linda J. Young4

Abstract

Each year, the National Agricultural Statistics Service (NASS) publishes an estimate of the 

number of farms in the United States based on the June Area Survey (JAS). Independent studies 

showed that the JAS number of farm indications have significant undercount due to 

misclassification. To adjust for this undercount, a follow-on survey to the JAS called the Annual 

Land Utilization Survey (ALUS) has been proposed. ALUS is designed and developed based on 

the Farm Numbers Research Project (FNRP). NASS conducted the FNRP in the fall of 2009

(Abreu, McCarthy and Colburn, 2010). ALUS samples from all JAS segments containing any 

estimated or non-agricultural JAS tracts. For a selected segment, all estimated and nonagricultural JAS tracts will be re-evaluated. The collection of eligible segments in a particular 

year will be called the ALUS population. The sample allocation of ALUS segments to each statestratum combination considers two factors: the proportion of the ALUS population in the stratum 

and the proportion of the FNRP adjustment from non-agricultural tracts in the stratum. ALUS 

can be treated as a second phase to the JAS. The two-phase stratified design, JAS-ALUS, can be 

applied to any estimate produced by the JAS. However, ALUS has non-response. In this paper,

methodology for a three-phase sampling design is developed by extending the two-phase 

sampling design methodology proposed by Sarndal and Swensson (1987). A general sampling 

design is allowed in each phase; that is, the inclusion probabilities in each phase are arbitrary. 

The estimator is unbiased, and an unbiased estimator for the variance is provided. Here, this 

method is applied to the two-phase JAS-ALUS with the third phase being response/nonresponse. 

KEYWORDS: three-phase sampling design estimation, unbiased estimator, variance 

estimation, non-response

 

2 National Institute of Statistical Sciences, 19 T.W. Alexander Drive, Research Triangle Park, NC 27709

3

Department of Statistics, North Carolina State University, Raleigh, NC 27695

4

Department of Statistics, University of Florida, Gainesville, FL 32611

5

National Agricultural Statistics Service, USDA, 3251 Old Lee Hwy, Fairfax VA 22030
6

1. INTRODUCTION

The National Agricultural Statistics Service (NASS) uses its annual June Area Survey (JAS) as 

the vehicle to generate annual estimates of the number of farms. Every five years, the annual

numbers of farms estimates are compared to the one obtained from the quinquennial census of 

agriculture (conducted during years ending in 2 and 7). The annual numbers have been declining 

steadily between censuses, especially between the 2002 and 2007 Censuses. Furthermore, in 

2002 and 2007, the JAS farm numbers indications were considerably lower than farm numbers 

from the census and, in 2007, the difference could not be attributed to sampling error alone. 

Additionally, results from a 2007 qualitative study revealed that agricultural operations were 

being incorrectly classified as non-agricultural during the screening procedures of the JAS.

In an attempt to get a better understanding of the misclassification leading to this undercount in 

the JAS number of farms indication, the NISS/NASS farm numbers research team6proposes a 

yearly follow-on survey to the JAS called the Annual Land Utilization Survey (ALUS). The 

purpose of ALUS is to provide information about misclassification of farms and non-farms, 

focusing on tracts that are a) determined to be non-agricultural in June or b) are estimated in 

June. ALUS results could be used to directly augment the JAS indications of farm numbers. In 

addition, data collection will include several other variables, allowing indications of other 

commodities to be adjusted using ALUS.

ALUS is modeled on the 2009 Farm Numbers Research Project (FNRP). FNRP was a one-time 

follow-on survey to the JAS segments (Abreu, McCarthy and Colburn, 2010). The design of the 

JAS includes rotating in new segments each year. Segments stay in the JAS sample for five 

years. Each year’s sample is comprised of segments from each of five rotations. Thus, the 2009 

JAS contained segments that were rotated into the sample in 2009, 2008, 2007, 2006 and 2005. 

The sampling design of the FNRP targeted the 20-percent of JAS segments that were newly 

rotated in for 2009 (“2009 segments”). All tracts in 2009 segments that were non-agricultural or 

estimated in JAS were selected for FNRP. 

Current NASS procedures define a tract as a unique land operating arrangement. However, for 

densely populated tracts, it is possible that multiple operations (places of interest) may be 

present. For a selected tract, all places of interest are considered subtracts. Subtracts are 

subsampled if there are 8 or more per tract. The FNRP sample consisted of 10,204 tracts, which 

resulted in a total of 17,191 subtracts. 

The FNRP recommended changes to the screening procedures to improve the quality of 

information obtained in the JAS, based on analysis of the misclassification of tracts as 

farms/non-farms. The results of those recommendations may first be seen in the 2010 JAS, but 

misclassification will certainly persist. An annual follow-up like ALUS will allow researchers to 

 

6 NASS had a two year collaborative research program with the National Institute of Statistical Sciences (NISS) 

called the Cross-Sector Research in Residence Program. This program was composed of three academicgovernment teams focusing on important NASS research issues. One of the teams was entrusted to work on 

potential improvements to the methodology and design of the June Area Survey.
7

monitor misclassification rates for farms/non-farms, as well as measurement error for additional

variables. 

There are two primary differences between the ALUS and the FNRP: (1) FNRP only included

the 20-percent newly rotated in estimated and non-agricultural JAS tracts while ALUS will have 

a positive probability of inclusion of all estimated and non-agricultural JAS tracts and (2) all 

FNRP-eligible tracts were included in the FNRP sample, but only a sample of the ALUS-eligible 

tracts will be included in the ALUS. Therefore, ALUS represents the second phase of a twophase sample with the first phase being the traditional JAS. 

Although non-response was not allowed in FNRP, the second-phase ALUS sample will have

non-response. A design-based approach to estimation is to use methodology for a three-phase 

sampling design, with the third phase addressing the non-response in the second phase. The 

three-phase design-based estimator developed here is general in the sense that the inclusion 

probabilities in each phase are arbitrary. The estimator is shown to be unbiased. The variance of 

the estimator is derived and an unbiased estimator of it is developed. At the end, this

methodology is applied to JAS-ALUS, with the third phase design addressing non-response in 

ALUS. We also provide an estimator and its variance estimation in the JAS terminologies. This 

methodology can be applied not only to estimates of the number of farms but to all variables 

collected in the ALUS.

2. THE DESIGN OF ALUS

2.1 Major Findings From FNRP

An analysis of the impact of JAS screening procedures used in FNRP was completed by Abreu, 

McCarthy and Colburn (2010). A major finding of this work is that, assuming misclassification 

rates are the same for all rotations, the JAS indication of number of farms would increase by 

approximately 580,000 farms using FNRP data. We will refer to this as the “FNRP adjustment” 

to the JAS indication (see Table 1). The bulk of these farms were “found” in tracts that had been 

identified as non-agricultural with no potential in the JAS. On the order of 45-percent of tracts 

are pre-screened into this category in a typical JAS. In FNRP, 6-percent of the sampled subtracts 

selected from this category were determined to be farms, resulting in 500,000 of the FNRP 

adjustment. Another 75,000 of the FNRP adjustment came from tracts pre-screened as nonagricultural with either potential for agriculture or unknown potential. The remaining FNRP 

adjustment came from tracts that had to be estimated in JAS. Although most tracts (92-percent) 

that had been estimated as farms in JAS were confirmed as such in FNRP, approximately 30-

percent of those that had been estimated as non-farms were identified as farms in FNRP. The net 

FNRP adjustment from estimated tracts was about 5,000 farms.
8

Table 1: FNRP Results by type of tract

Type of Tract

FNRP 

Sample Size 

(subtracts)

Number of 

FNRP Farms

Net Expanded 

Number of 

Farms

Estimated as farm 1,591 1,466 (7,822)

Estimated as non-farm 121 37 13,032

Non-agricultural with potential 487 95 38,346

Non-agricultural with unknown 

potential 364 56 37,479

Non-agricultural with no potential 14,628 905 500,338

FNRP Total 17,191 2,559 581,373

Rates of “conversion” (subtracts that were identified as non-farms in JAS and as farms in FNRP) 

varied by state and strata. Nationally, tracts that had agricultural potential or unknown potential 

in the JAS had conversion rates of about 20-percent and 15-percent, respectively. However, 

within a state, sample sizes were typically less than 10 tracts per stratum, making estimates of 

conversion rates unreliable at the state level. The conversion rate for tracts that were identified 

as having no agricultural potential in JAS was 6-percent overall. About 95-percent of the strata 

had conversion rates of less than 17-percent for tracts with no-potential. However, tracts with no

potential from strata in the 40’s (low rates of cultivation) contributed over half of the FNRP 

adjustment.

FNRP results are used as guidelines for the ALUS design, but ALUS will be able to detect 

different types of trends as well. Due to the experience the enumerators gained in conducting 

FNRP, the changes in JAS protocols made following FNRP and, the fact that FNRP included 

only 2009 segments, results from ALUS may be quite different. 

2.2 Overall Sampling Design

As in the JAS, ALUS will be a stratified sample of segments, using JAS strata and sampling 

across rotations. Segments that are eligible for inclusion in ALUS must have at least one tract 

that was pre-screened as non-agricultural (regardless of potential) or that was estimated in JAS 

(as either a farm or non-farm). For a selected segment, all tracts satisfying one of these criteria 

will be re-evaluated using the FNRP questionnaire (see Appendix A). In the 2009 and 2010 JAS, 

over 90-percent of all segments would have been considered eligible for ALUS. The collection 

of eligible segments in a particular year will be called the ALUS population. 

The sample allocation of segments to each state-stratum combination considers two factors: the 

proportion of the ALUS population in the stratum and the proportion of the FNRP adjustment 

from non-agricultural tracts in the stratum. The latter simultaneously accounts for the number of 

converted non-agricultural tracts and the expansion factors associated with them, allowing states 

and strata that contributed most to the FNRP adjustment to be targeted. Tracts that were 

estimated as farms or non-farms in JAS contributed little to the FNRP adjustment, so this 

information is not included in choosing allocations to the strata in the ALUS sample. 
9

Table 2: Guidelines for ALUS allocation scheme

Strata

Proportion of 

FNRP 

adjustment 

from nonagricultural 

tracts

Proportion of 

ALUS-eligible 

segments in 

2009 JAS

Proportion of 

ALUS-eligible 

segments in 

2010 JAS

Suggested 

Proportion 

of ALUS 

sample

10s 16% 53% 52% 27%

20s 34% 26% 27% 30%

30s <1% 3% 3% 3%

40s 50% 17% 17% 40%

50s <1% <1% <1% 0%

Total 576,000 farms 10,168 segments 10,121 segments

In the JAS, the sampling scheme favors cultivated areas. For ALUS, the sampling will lean more 

heavily on moderately and less cultivated strata where the largest portion of the FNRP 

adjustment originates. For example, strata 10s (10, 11, ...) are highly cultivated areas. The exact 

stratum definition varies from state to state, but this may be more than 50-percent cultivated 

land. In the JAS, over half of the selected segments are from these strata. However, 10s made 

up only 16-percent of the FNRP adjustment arising from non-agricultural tracts so only about 27-

percent of the ALUS sample will come from these strata. The sample will be evenly distributed 

over the five rotations, with approximately 20-percent of the ALUS sample selected from each. 

This will allow modeling of the effect of the number of years a segment has been in the survey 

on misclassification rates.

Within each stratum of the ALUS population, segments will be selected with probability 

proportionate to size (pps) sampling where the size measure of a segment is defined as 

. * number of tracts pre-screened number of tracts number of tracts size= 0 1 as non-agricultural estimated as non-farm estimated as farm

.

The rationale for the size measure comes from FNRP, but does not depend heavily on the 

specific results of that study. As noted previously, non-agricultural tracts/subtracts made up the 

vast majority of the FNRP adjustment. Estimated tracts had less impact on the FNRP adjustment. 

These tracts do not affect allocation (sample size) in each stratum, but are used in helping to 

select segments once allocations are determined. In a typical JAS, few tracts are estimated as 

non-farms (around 400 in 2009), but one-third of these estimated tracts converted to farms in 

FNRP. Thus, estimated tracts will be over-sampled relative to their contribution to the FNRP 

adjustment. Because most tracts (92-percent) estimated as farms in the JAS were confirmed as 

farms in FNRP, ALUS will not target segments that have only JAS tracts from estimated farms. 

This is reflected in the multiplier of 0.1 on the number of tracts estimated as farms in the size 

measure. If a segment is selected, all ALUS-eligible tracts within that segment will be in the

sample, including those estimated as farms.
10

Within selected tracts, sampling rates of subtracts will be the same as FNRP. That is, if the tract 

contains 7 dwellings or less, then all are sampled. If the tract contains 8-20 dwellings, half are 

sampled. If there are more than 20 dwellings, one-sixth are sampled.

2.3 Example Allocations

Both the standard error and the cost of the proposed ALUS were investigated using FNRP data. 

This required development of specific example allocations for each state and stratum. In practice, 

ALUS allocations will need to be determined each year after the JAS data are collected because a 

segment’s weight for the probability proportionate to size (pps) sampling will depend on its JAS 

classification. 

For different national sample sizes, a proposed stratified allocation of segments was developed 

using strata that are combinations of state and JAS strata. National sample sizes ranging from 

500 to 5000 segments were considered. Note that the approximate size of the FNRP sample was 

about 2200 segments. The numbers of segments allocated to each stratum (across the nation) are 

summarized in Table 3.

Table 3: Example (approximate) allocation of segments to strata 

Sample size 

(segments)

JAS stratum

10s 20s 30s 40s 50s

5000 1350 1500 150 2000 0

4500 1215 1350 135 1800 0

4000 1080 1200 120 1600 0

3500 845 1050 105 1400 0

3000 810 900 90 1200 0

2500 675 750 75 1000 0

2000 540 600 60 800 0

1000 270 300 30 400 0

500 135 150 15 200 0

The design attempts to maintain a minimum number of segments (between 1 and 4 depending on 

the total sample size) in each state and stratum combination. The JAS 2010 data were used to 

approximate the number of segments in the ALUS population for a particular stratum in a typical 

JAS. If the stratum allocation was larger than the ALUS population in the 2010 JAS for that 

stratum, then the allocation was reduced. The example allocations considered are presented in 

Appendix B. In practice, these allocations would need to be adjusted at least slightly each year 

based on the JAS data and resulting ALUS population sizes. In addition, any stratum that had a 

sample size of zero in FNRP was not included in the example allocations. This was done for 

estimation purposes and is not recommended for actual ALUS allocations.

Table 4 summarizes the anticipated results and costs of each sample size. The anticipated 

national-level coefficient of variation (CV) and standard error on the number of farms 

adjustment are calculated following the method used for FNRP. That is, the appropriate formulae 

for follow-on surveys were used (Kott, 1990).
11

 

Table 4: Summary of proposed ALUS allocations

Approximate 

Sample Size 

(segments)

Anticipated 

CV of ALUS 

adjustment

Anticipated 

Standard 

Error of 

ALUS 

adjustment

Anticipated 

Cost to 

States

5000 6.5 37,000 $897,000

4500 6.8 39,000 $806,000

4000 7.1 41,000 $725,000

3500 7.7 44,000 $647,000

3000 8.2 47,000 $545,000

2500 9.0 52,000 $459,000

FNRP: 2200 10.9 63,000 $412,000

2000 10.0 58,000 $373,000

1000 13.8 80,000 $181,000

500 18.8 108,000 $100,000

The JAS 2008 cost data are available for each state on a per segment basis. These costs are based 

on an enumerator visiting every tract within a selected segment. For ALUS, we assume that 56-

percent of tracts in a selected segment will be ALUS-eligible. The value of 56-percent is derived 

from calibrating the cost of a FNRP size sample (2200 segments) to match the actual cost of 

FNRP ($412,000). Anticipated costs are summarized in Table 4. Cost data are only available at 

the state level, not at the stratum level. The anticipated cost assumes that segments in all strata 

have the same cost. Although this calculation is quite rough, more sophisticated methods would 

probably not result in marked improvement of the cost approximations. Appendix C contains 

anticipated costs for each state. Note that these estimates only include approximate costs to the 

states. In FNRP, real estate parcel data, which cost an additional $92,000 for a one-year license, 

were used to improve the quality of the names and addresses for non-agricultural tracts. This cost 

is not reflected in the estimates presented in Appendix C.

2.4 Using ALUS Results to Adjust JAS Indications

The FNRP questionnaire (Appendix A) was designed to target misclassification of farms and

capture data on the type of farms that were believed to have been misclassified via multiple 

modes of data collection (face-to-face, phone, or mail). It was essentially a shortened version of 

the JAS questionnaire. NASS should consider re-designing this questionnaire for ALUS to 

collect data on as many JAS variables as possible for farms newly identified in ALUS. This will 

allow ALUS results to be used to adjust more than just farm number indications. In particular, 

use of the full JAS questionnaire, the Agricultural Coverage Evaluation Survey (ACES)7

questionnaire, or an extended version of the FNRP questionnaire should be considered. 

 

7 ACES is conducted during census years and is designed to improve coverage through targeted sampling of small 

and minority owned farms. 
12

The combination of JAS and ALUS can be considered a two-phase sample. JAS is the first phase 

of the sample; then a sub-sample of JAS segments are selected for ALUS. Provided that each 

phase makes use of a probability sampling design for which the inclusion probabilities are 

known, standard results can be used to construct a design-based estimator (Sarndal and 

Swensson, 1987). It is expected that misclassification and non-response will still occur in ALUS. 

However, this follow-up survey will provide valuable information for adjusting estimates and 

should reduce the amount of non-response.

This methodology can be applied not only to estimates of number of farms but to all variables 

collected in the ALUS. Thus, although the primary impetus for this work is to improve estimates 

of the number of farms, it can improve estimates of other important variables. In particular, 

farms that are “missed” in JAS will not have values for many JAS variables. Those that are 

newly identified in ALUS will have accompanying data that can be used to adjust any variables 

common to both ALUS and JAS. For this reason, the FNRP questionnaire should be reviewed 

within NASS to determine whether other information should be gathered during ALUS. 

3 ALUS METHODOLOGY

Each JAS tract may be classified into one of seven categories (see Table 5). If an enumerator 

completed an interview during the JAS, then the agricultural tract is in category 6 (farm) or 7 

(non-farm). It is assumed that all completed interviews result in an accurate classification of farm 

status, so no error is associated with these two categories. Further, because these tracts are 

associated with completed JAS interviews, they are not ALUS-eligible. The remaining categories 

(1-5) represent ALUS-eligible tracts. A tract for which the JAS farm status was estimated is in 

category 1 (farm) or 2 (non-farm). A tract determined to be non-agricultural during JAS prescreening is in category 3 (non-agricultural tract with potential), 4 (non-agricultural tract without 

potential), or 5 (non-agricultural tract with unknown potential). No JAS questionnaire was 

completed for a non-agricultural tract and these tracts were assigned a non-farm status in JAS. 

Based on the FNRP results, a tract in one of the first five categories may be misclassified for 

farm/non-farm JAS status. The primary purpose of the ALUS is to obtain revised JAS 

indications by adjusting for JAS misclassification.

Table 5: JAS tract category

Category JAS Type of Agricultural Tract Farm Status

1 Agricultural tract-estimated Farm

2 Agricultural tract-estimated Non-Farm

3 Non-agricultural tract-with potential Non-Farm

4 Non-agricultural tract-without potential Non-Farm

5 Non-agricultural tract-with unknown potential Non-Farm

6 Agricultural tract-completed interview Farm

7 Agricultural tract-completed interview Non-Farm
13

3.1 JAS Estimation

First, consider the annual JAS indication of the number of farms, and then the indication 

incorporating the information obtained during the ALUS (second-phase sample) will be 

developed. Under stratified simple random sampling, the JAS estimator is

where

is the index of stratum, is the number of land-use strata;

is the index of substratum, is the number of substrata in stratum ;

is the index of segment, is the number of segments in substratum within stratum ;

is the expansion factor or the inverse of the probability of the selection for each 

segment in substratum in land-use stratum ;

is the index of tract;

 is the number of farm tracts in the segment;

is the tract-to-farm ratio, which is .

For the JAS estimator, only agricultural tracts in categories 1 (JAS farm) and 6 (JAS estimated 

farm) are included in the summation. Tracts within the same segment have the same expansion 

factor because the JAS design is based on segments. The unbiasedness of this estimator can be 

seen by first rewriting it as

.

Here, N is the total number of tracts in the population;

is the tract-to-farm ratio of the tract (ti = 0 for non-farm tracts); is the expansion factor or 

the inverse of the probability of selection of the tract, i.e.,

Under the assumption that JAS provides accurate information for all tracts, is unbiased 

because which is the total number of farms (the quantity 

being estimated). 
14

The variance is 

where 

However, the JAS indication is biased because some tracts from categories 2 to 5 are 

misclassified as non-farms.

3.2 JAS-ALUS Estimation without Non-response

Before presenting the estimation methodology that allows non-response in ALUS, we discuss the 

estimation methodology without non-response. The combined JAS-ALUS represents a twophase design using stratified simple random subsampling at each phase. First, assume all ALUS 

tracts have an associated completed interview; that is, non-response is not present in ALUS. 

Then, the JAS-ALUS two-phase sampling design-based estimator for the number of farms is

where 

is the number of ALUS segments in substratum j within stratum ; is the expansion factor 

or the inverse of the probability of selection in the second phase for each segment in substratum 

, land-use stratum ; is the number of farm tracts in the given ALUS selected segment. 

Here, the first term has the same form as the JAS estimator (1), but the tracts in are from 

JAS category 6 only. Unlike T1, does not include JAS category 1 tracts because these tracts 

are part of the second-phase ALUS population. The tracts in the second term are from JAS 

categories 1 to 5. Category 7 tracts are excluded from the estimator because they have clear nonfarm status from the JAS (i.e., not eligible for ALUS). All tracts within the same ALUS segment 

have the same expansion factor because the segment is the primary sampling unit for ALUS. 

The two expansion factors, and are known from the JAS-ALUS two phase design. 

Again, to verify the unbiasedness of this two-phase estimator, the estimator may be rewritten 

in the following form 
15

where 

is the vector of random variables , i.e., the JAS selection 

information;

and is the expansion factor or the inverse of the probability of selection of the tract 

in ALUS, i.e.,

Under the assumption that JAS provides accurate information for the tracts from categories 6 and 

7 and ALUS provides accurate information for the tracts from categories 1 to 5, is an unbiased 

estimator because

Assuming that tracts from different segments are independent, the variance is

An unbiased estimator of the variance of T2 is

The first term is

where Notice that the values of these 

quantities, and are different from (2) although the same notation is used.
16

The second term is 

This can be calculated from the first term of (4.4) in Sarndal and Swensson (1987). See 

Appendix D for details.

The third term is 

where is the total number of qualified ALUS segments within stratum , substratum j (ALUS 

population), is the ALUS sample variance within stratum , substratum . Formula (5) can be 

calculated from the second term of (4.4) in Sarndal and Swensson (1987). See Appendix D for 

details.

3.3 JAS-ALUS Estimation with Non-Response

Although no non-response was assumed above, ALUS would have non-response. We have 

studied three-phase sampling design estimation and put the general result in Appendix E. The 

results derived in Appendix E are now applied to the JAS-ALUS two phase estimator with nonresponse adjustment within the second phase. Under the assumption of the probability of 

response being well modeled by the Bernoulli distribution within each stratum, the two-phase 

estimator of the number of farms with non-response adjustment, is

Here is the expansion factor or the inverse of the response probability of tract in 

segment substratum land-use stratum is unknown from the two-phase design 

but it can be estimated by modeling. As in the tracts in the first term are from JAS category 

6 only. The tracts in the second term are from JAS categories 1 to 5. Another way to write this 

estimator is
17

Here 

is the vector of random variables , i.e., the ALUS selection 

information; is the response expansion factor or the inverse of the probability of response, i.e.,

is an unbiased estimator because 

Assuming that tracts from different segments are independent, the variance is

Again, the first term is given by Let be the probability of response of the tracts in 

segment and be the probability that two tracts have response in segments . We 

assume that all tracts in a same segment have same response mechanism. As shown in Appendix 

F, one design unbiased estimator of the second term in (6) is 
18

3.4 Estimation of Any Farm-Level Items

Although the methodology presented here has focused on the estimation of farm numbers, it can 

also be used to estimate farm-level items of interest (i.e., commodity total or presence/absence of 

agricultural items). To estimate farm-level items, two additional variables, and , must 

be defined. Their definitions depend on whether the farm-level item being estimated is the 

proportion of farms with a specific agricultural commodity or a total for an agricultural 

commodity.

Case 1: Proportion of Farms

First suppose that the proportion of farms with a specific agricultural commodity is to be 

estimated. Define if that agricultural commodity is present on the farm associated 

with the tract in segment within substratum of stratum; otherwise, . 

Similarly, 

Case 2: Commodity Total

Now suppose that the item to be estimated is an inventory. Define as the farm-level 

quantity of the agricultural commodity from the tract in segment within substratum of 

stratum. Similarly, is the farm-level quantity of that item from the tract.

Now, for the farm-level items, all estimators previously considered are revised by using 

and instead of the respective tract-to-farm ratios and considered earlier. 

For example, the farm numbers estimator for the JAS-ALUS, assuming full response (see 

(3)), becomes 
19

And the sum of the tract-to-farm ratios of tracts in segment k in substratum j of stratum i as 

in becomes 

If a quantity does not involve the tract-to-farm ratio, such as or the definition remains 

unchanged.

3.5 Alternative Method

An alternative, less costly approach to adjusting JAS for misclassification error by matching JAS 

records to the annual sampling list frame has also been proposed. See Lopiano et al. (2011) and 

Abreu et al. (2011) for details of this methodology. However, as compared to this proposed 

approach, the results of ALUS would likely provide lower CVs for indications of the number of 

farms, would provide annual monitoring of classification error that may inform the data 

collection process, and would provide a means for producing improved indications for other 

variables. However, the cost of conducting ALUS is non-trivial. Researchers intend to pursue 

further comparisons of these two approaches.

4. DISCUSSION & CONCLUSIONS

We have discussed the sampling design of the proposed ALUS. ALUS can be treated as a second 

phase to the JAS. All segments that have estimated or non-agricultural JAS tracts in the JAS 

sample are ALUS-eligible. The ALUS sample allocation of segments to each state-stratum 

combination considers two factors: the proportion of the ALUS population in the stratum and the 

proportion of the FNRP adjustment from non-agricultural tracts in the stratum. The ALUS 

sample design has been demonstrated using a sample allocation.

If there is full response in ALUS, an unbiased estimator of the number of farms based on the 

two-phase survey design (JAS-ALUS) has been developed. An unbiased estimator for the 

variance has also been developed. In reality, ALUS will have non-response. Thus, a general 

three-phase design-based unbiased estimator with an unbiased estimator of its variance has been 

developed. Then we applied this method to the JAS-ALUS, with the third phase addressing nonresponse in ALUS.

5. RECOMMENDATIONS

1) A follow-on survey to the JAS, called ALUS, should be conducted so that the farm number 

indications (as well as any included commodities) from the JAS can be adjusted for 

misclassification.

2) The ALUS questionnaire should be a modified version of the JAS-FNRP questionnaire. 

While essentially a shortened version of the JAS questionnaire, the FNRP questionnaire was also

designed to target misclassification of farms and capture data on the type of farms that were 

believed to have been misclassified via multiple modes of data collection (face-to-face, phone, or 

mail). Additional JAS variables added to this questionnaire would allow the ability to adjust 

these variables as well. 
20

3). The methodology for a two-phase study with nonresponse considered to be a third phase, as 

developed in this report, should be used for the analysis of the JAS-ALUS.

Note: The recommendations above were discussed at the January 2011 Senior Executive Team

meeting where the decision was made to not conduct the ALUS in 2011 due to budget 

constraints.

6. REFERENCE

Abreu, Denise A., Andrea C. Lamas, Hailin Sang, Kenneth K. Lopiano, Pam Arroway, Linda J. 

Young (2011). On the Feasibility of Using NASS’s Sampling List Frame to Evaluate 

Misclassification Errors of the June Area Survey. Research and Development Division. RDD 

Research Report # RDD-11-01. 

Abreu, Denise A., J. S. McCarthy, L. A. Colburn (2010). Impact of the Screening Procedures of 

the June Area Survey on the Number of Farms Estimates. Research and Development Division. 

RDD Research Report # RDD-10-03.

Kott, P. S. (1990). Mathematical Formulae for the 1989 Survey Processing System (SPS) 

Summary. NASS Staff Report, SRB-90-08, National Agricultural Statistics Service, USDA.

Lopiano, Kenneth K., Andrea C. Lamas, Denise A. Abreu, Hailin Sang, Pam Arroway, Young J.

Linda (2011). Proposal for Using the List Frame to Adjust for Misclassification in the June Area 

Survey. Research and Development Division. RDD Research Report # RDD-11-YY. 

Sarndal, C-E. and B. Swensson (1987). A General View of Estimation for Two Phases of 

Selection with Applications to Two-Phase Sampling and Nonresponse. International Statistical 

Review. v 55(3):279-294
APPENDIX A -- FNRP Questionnaire

21
APPENDIX A -- FNRP Questionnaire

22
APPENDIX A -- FNRP Questionnaire

23

 
APPENDIX A -- FNRP Questionnaire

24
APPENDIX B

25

Example allocations by state and stratum using 2010 JAS data

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

AL 145 132 120 106 89 76 63 29 15

13 24 20 17 14 12 10 9 4 2

20 61 52 43 37 31 27 22 10 5

40 60 60 60 55 46 39 32 15 8

AZ 48 46 45 42 35 28 22 12 9

13 10 8 7 6 5 4 4 2 1

14 4 4 4 4 3 2 1 1 1

18 4 4 4 4 3 2 2 1 1

28 4 4 4 4 3 2 2 1 1

38 4 4 4 4 3 2 1 1 1

42 11 11 11 10 9 8 6 3 2

43 11 11 11 10 9 8 6 3 2

AR 162 150 140 124 105 89 73 34 18

11 52 44 37 32 27 23 19 9 5

21 26 22 19 16 14 12 10 5 3

42 84 84 84 76 64 54 44 20 10

CA 65 59 53 49 41 35 27 14 9

11 4 4 4 4 3 2 1 1 1

21 40 34 28 25 21 18 14 7 4

41 19 19 19 18 15 13 10 5 3

45 2 2 2 2 2 2 2 1 1

CO 117 101 86 76 64 52 43 22 14

13 4 4 4 4 3 2 2 1 1

15 28 24 20 17 15 12 10 5 3

20 4 4 4 4 3 2 2 1 1

25 47 39 33 28 24 20 17 8 4
APPENDIX B

26

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

28 17 15 12 11 9 8 6 3 2

34 7 6 5 4 4 3 3 2 1

35 6 5 4 4 3 3 2 1 1

38 4 4 4 4 3 2 1 1 1

49 0 0 0 0 0 0 0 0 0

CT 1 1 1 1 1 1 1 1 1

40 1 1 1 1 1 1 1 1 1

DE 85 72 60 51 43 37 31 14 7

13 24 20 17 14 12 10 9 4 2

20 61 52 43 37 31 27 22 10 5

FL 66 62 59 54 45 37 30 15 9

13 24 20 17 14 12 10 9 4 2

21 4 4 4 4 3 2 2 1 1

31 4 4 4 4 3 2 1 1 1

40 15 15 15 14 12 10 8 4 2

42 19 19 19 18 15 13 10 5 3

GA 169 156 144 127 107 91 75 34 17

13 24 20 17 14 12 10 9 4 2

20 61 52 43 37 31 27 22 10 5

40 84 84 84 76 64 54 44 20 10

ID 44 38 33 29 24 22 18 9 5

10 10 9 7 6 5 5 4 2 1

15 18 15 13 11 9 8 7 3 2

22 11 9 8 7 6 5 4 2 1

43 5 5 5 5 4 4 3 2 1

IL 157 135 116 102 86 72 58 28 15

11 80 67 56 48 41 35 28 13 7

12 23 19 16 14 12 10 8 4 2
APPENDIX B

27

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

20 35 30 25 22 18 15 13 6 3

31 4 4 4 4 3 2 1 1 1

40 15 15 15 14 12 10 8 4 2

IN 111 97 83 75 63 53 42 21 12

11 40 34 28 25 21 18 14 7 4

12 13 11 9 8 7 6 5 2 1

20 39 33 27 24 20 17 14 7 4

31 4 4 4 4 3 2 1 1 1

40 15 15 15 14 12 10 8 4 2

IA 64 54 45 39 33 28 23 11 6

13 24 20 17 14 12 10 9 4 2

20 40 34 28 25 21 18 14 7 4

KS 157 135 114 99 85 71 59 28 15

11 58 49 41 35 30 25 21 10 5

12 28 24 20 17 15 12 10 5 3

20 59 50 41 36 30 26 21 10 5

40 12 12 12 11 10 8 7 3 2

KY 107 95 85 76 63 53 43 21 11

13 24 20 17 14 12 10 9 4 2

20 49 41 34 30 25 21 17 8 4

31 4 4 4 4 3 2 1 1 1

40 30 30 30 28 23 20 16 8 4

LA 74 67 61 54 45 39 32 16 8

13 24 20 17 14 12 10 9 4 2

20 20 17 14 12 10 9 7 4 2

40 30 30 30 28 23 20 16 8 4

ME 40 40 40 37 31 26 21 10 5

40 40 40 40 37 31 26 21 10 5
APPENDIX B

28

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

MD 36 33 29 27 22 19 16 8 5

13 5 5 4 4 3 3 2 1 1

20 21 18 15 13 11 9 8 4 2

40 10 10 10 10 8 7 6 3 2

MA 28 28 28 26 23 19 15 7 4

31 3 3 3 3 3 2 1 1 1

40 25 25 25 23 20 17 14 6 3

MI 80 70 62 56 47 39 32 16 9

11 20 17 14 13 11 9 7 4 2

20 40 33 28 24 20 17 14 7 4

40 20 20 20 19 16 13 11 5 3

MN 163 144 127 112 94 79 65 30 15

11 63 53 44 38 32 27 22 10 5

12 25 21 18 15 13 11 9 4 2

20 35 30 25 22 18 15 13 6 3

40 40 40 40 37 31 26 21 10 5

MS 172 157 144 128 109 91 76 35 19

11 27 23 19 16 14 12 10 5 3

12 10 8 7 6 5 4 4 2 1

20 56 47 39 34 29 24 20 9 5

40 79 79 79 72 61 51 42 19 10

MO 175 154 135 119 100 84 69 32 16

11 63 53 44 38 32 27 22 10 5

12 23 19 16 14 12 10 8 4 2

20 49 42 35 30 25 21 18 8 4

40 40 40 40 37 31 26 21 10 5

MT 100 89 78 70 59 51 41 21 11

13 33 28 23 20 17 14 12 6 3
APPENDIX B

29

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

20 40 34 28 25 21 18 14 7 4

43 5 5 5 5 4 4 3 2 1

44 22 22 22 20 17 15 12 6 3

NE 174 152 131 116 98 83 68 32 17

11 82 69 58 50 42 36 29 14 7

12 24 21 17 15 13 11 9 4 2

20 39 33 27 24 20 17 14 7 4

40 29 29 29 27 23 19 16 7 4

NV 8 7 6 6 5 5 3 2 2

28 2 2 2 2 2 2 1 1 1

38 6 5 4 4 3 3 2 1 1

NH 12 12 12 12 10 8 7 4 3

14 4 4 4 4 3 2 2 1 1

31 2 2 2 2 2 2 1 1 1

40 6 6 6 6 5 4 4 2 1

NJ 36 32 28 25 21 16 12 7 5

13 4 4 4 4 3 2 1 1 1

20 28 24 20 17 15 12 10 5 3

31 4 4 4 4 3 2 1 1 1

NM 31 30 29 28 22 19 15 10 7

13 8 7 6 5 4 4 3 2 1

18 4 4 4 4 3 2 2 1 1

19 1 1 1 1 1 1 1 1 1

28 4 4 4 4 3 2 2 1 1

38 4 4 4 4 3 2 1 1 1

40 5 5 5 5 4 4 3 2 1

48 5 5 5 5 4 4 3 2 1

NY 44 37 31 27 23 19 16 8 4
APPENDIX B

30

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

13 9 7 6 5 5 4 3 2 1

20 35 30 25 22 18 15 13 6 3

NC 232 208 188 166 139 117 96 45 23

13 10 8 7 6 5 4 4 2 1

20 138 116 97 83 70 59 49 22 11

31 4 4 4 4 3 2 1 1 1

40 80 80 80 73 61 52 42 20 10

ND 162 138 113 99 83 71 57 27 14

11 74 63 52 45 38 32 26 12 6

12 29 25 20 18 15 13 10 5 3

20 59 50 41 36 30 26 21 10 5

OH 116 101 89 79 68 57 46 23 13

11 35 29 24 21 18 15 12 6 3

12 12 10 8 7 6 5 4 2 1

20 30 25 21 18 15 13 11 5 3

31 9 7 6 5 5 4 3 2 1

32 2 2 2 2 2 2 1 1 1

40 28 28 28 26 22 18 15 7 4

OK 131 115 101 88 75 63 53 25 13

11 40 34 28 24 20 17 14 7 4

12 13 11 9 8 7 6 5 2 1

20 47 39 33 28 24 20 17 8 4

40 31 31 31 28 24 20 17 8 4

OR 70 63 57 51 43 37 30 14 8

13 11 9 8 7 6 5 4 2 1

20 32 27 22 19 16 14 11 5 3

43 27 27 27 25 21 18 15 7 4

PA 120 108 97 87 73 61 49 24 13
APPENDIX B

31

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

13 7 6 5 4 4 3 3 2 1

20 69 58 48 42 35 30 24 11 6

31 4 4 4 4 3 2 1 1 1

40 40 40 40 37 31 26 21 10 5

RI 12 12 12 12 10 8 6 4 3

14 4 4 4 4 3 2 1 1 1

31 2 2 2 2 2 2 1 1 1

40 6 6 6 6 5 4 4 2 1

SC 104 95 87 79 66 56 45 21 11

13 6 5 4 4 3 3 2 1 1

20 49 41 34 30 25 21 17 8 4

40 49 49 49 45 38 32 26 12 6

SD 135 119 105 92 79 67 55 25 13

11 52 44 37 32 27 23 19 9 5

12 13 11 9 8 7 6 5 2 1

20 37 31 26 22 19 16 13 6 3

40 33 33 33 30 26 22 18 8 4

TN 211 190 172 153 129 109 89 42 22

13 30 25 21 18 16 13 11 5 3

20 95 80 67 58 49 41 34 16 8

31 6 5 4 4 3 3 2 1 1

40 80 80 80 73 61 52 42 20 10

TX 363 324 291 259 219 185 149 74 44

10 27 22 19 16 14 12 10 5 3

13 13 11 9 8 7 6 5 2 1

14 4 4 4 4 3 2 1 1 1

15 28 24 20 17 15 12 10 5 3

16 6 5 4 4 3 3 2 1 1
APPENDIX B

32

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

18 10 8 7 6 5 4 4 2 1

20 40 33 28 24 20 17 14 7 4

21 2 2 2 2 2 2 1 1 1

24 41 35 29 25 21 18 15 7 4

25 47 39 33 28 24 20 17 8 4

26 4 4 4 4 3 2 1 1 1

27 7 6 5 5 4 3 3 2 1

28 17 15 12 11 9 8 6 3 2

31 6 5 4 4 3 3 2 1 1

32 4 4 4 4 3 2 1 1 1

40 52 52 52 47 40 34 28 13 7

41 44 44 44 40 34 29 23 11 6

42 11 11 11 10 9 8 6 3 2

UT 29 26 24 22 19 17 13 8 6

13 8 7 6 5 4 4 3 2 1

20 11 9 8 7 6 5 4 2 1

31 4 4 4 4 3 2 1 1 1

32 2 2 2 2 2 2 1 1 1

46 2 2 2 2 2 2 2 1 1

48 2 2 2 2 2 2 2 1 1

VT 20 20 20 19 16 13 11 5 3

40 20 20 20 19 16 13 11 5 3

VA 131 119 107 97 81 69 56 26 14

13 5 5 4 4 3 3 2 1 1

20 70 59 49 43 36 30 25 12 6

31 6 5 4 4 3 3 2 1 1

40 50 50 50 46 39 33 27 12 6

WA 137 126 116 103 88 74 61 30 17
APPENDIX B

33

State 

Stratum

Approximate Total Sample Size (segments)

5000 4500 4000 3500 3000 2500 2000 1000 500

10 32 27 23 20 17 14 12 6 3

11 15 13 11 9 8 7 6 3 2

20 28 24 20 17 15 12 10 5 3

41 19 19 19 18 15 13 10 5 3

43 43 43 43 39 33 28 23 11 6

WV 71 70 68 63 52 44 35 18 11

13 4 4 4 4 3 2 2 1 1

20 10 9 7 6 5 5 4 2 1

31 4 4 4 4 3 2 1 1 1

32 2 2 2 2 2 2 1 1 1

40 51 51 51 47 39 33 27 13 7

WI 115 102 90 79 67 56 45 22 12

11 23 19 16 14 12 10 8 4 2

12 8 7 6 5 4 4 3 2 1

20 56 48 40 34 29 24 20 9 5

31 4 4 4 4 3 2 1 1 1

40 24 24 24 22 19 16 13 6 3

WY 103 103 103 95 80 67 53 27 17

11 4 4 4 4 3 2 1 1 1

12 4 4 4 4 3 2 1 1 1

20 4 4 4 4 3 2 2 1 1

31 2 2 2 2 2 2 1 1 1

32 2 2 2 2 2 2 1 1 1

42 84 84 84 76 64 54 44 20 10

44 1 1 1 1 1 1 1 1 1

45 2 2 2 2 2 2 2 1 1

Total 4933 4424 3965 3536 2980 2513 2045 991 550

Actual Total Sample Size (segments)
APPENDIX C

34

Anticipated cost by state for each example allocation

Approximate Total Sample Size (segments)

State 5000 4500 4000 3500 3000 2500 2000 1000 500

AL $23,629 $21,511 $19,555 $17,274 $14,503 $12,385 $10,266 $4,726 $2,444

AZ $3,548 $3,400 $3,326 $3,105 $2,587 $2,070 $1,626 $887 $665

AR $32,931 $30,492 $28,459 $25,207 $21,344 $18,092 $14,839 $6,912 $3,659

CA $14,524 $13,183 $11,842 $10,949 $9,161 $7,820 $6,033 $3,128 $2,011

CO $13,694 $11,821 $10,065 $8,895 $7,491 $6,086 $5,033 $2,575 $1,639

CT $244 $244 $244 $244 $244 $244 $244 $244 $244

DE $20,706 $17,539 $14,616 $12,424 $10,475 $9,013 $7,552 $3,410 $1,705

FL $18,887 $17,742 $16,883 $15,453 $12,877 $10,588 $8,585 $4,292 $2,575

GA $31,137 $28,741 $26,531 $23,398 $19,714 $16,766 $13,818 $6,264 $3,132

ID $3,080 $2,660 $2,310 $2,030 $1,680 $1,540 $1,260 $630 $350

IL $37,014 $31,828 $27,348 $24,048 $20,275 $16,975 $13,674 $6,601 $3,536

IN $18,897 $16,513 $14,130 $12,768 $10,725 $9,023 $7,150 $3,575 $2,043

IA $12,580 $10,614 $8,845 $7,666 $6,486 $5,504 $4,521 $2,162 $1,179

KS $20,925 $17,993 $15,194 $13,195 $11,329 $9,463 $7,864 $3,732 $1,999

KY $19,474 $17,290 $15,470 $13,832 $11,466 $9,646 $7,826 $3,822 $2,002

LA $9,034 $8,179 $7,447 $6,592 $5,494 $4,761 $3,907 $1,953 $977

ME $9,744 $9,744 $9,744 $9,013 $7,552 $6,334 $5,116 $2,436 $1,218

MD $8,770 $8,039 $7,064 $6,577 $5,359 $4,628 $3,898 $1,949 $1,218

MA $6,821 $6,821 $6,821 $6,334 $5,603 $4,628 $3,654 $1,705 $974

MI $16,397 $14,347 $12,708 $11,478 $9,633 $7,993 $6,559 $3,279 $1,845

MN $31,218 $27,579 $24,323 $21,450 $18,003 $15,130 $12,449 $5,746 $2,873

MS $33,423 $30,508 $27,982 $24,873 $21,181 $17,683 $14,768 $6,801 $3,692

MO $34,888 $30,701 $26,914 $23,724 $19,936 $16,746 $13,756 $6,380 $3,190

MT $11,760 $10,466 $9,173 $8,232 $6,938 $5,998 $4,822 $2,470 $1,294

NE $27,088 $23,663 $20,394 $18,059 $15,257 $12,921 $10,586 $4,982 $2,647

NV $502 $439 $376 $376 $314 $314 $188 $125 $125

NH $2,675 $2,675 $2,675 $2,675 $2,229 $1,783 $1,560 $892 $669
APPENDIX C

35

Approximate Total Sample Size (segments)

State 5000 4500 4000 3500 3000 2500 2000 1000 500

NJ $7,741 $6,881 $6,021 $5,376 $4,516 $3,441 $2,580 $1,505 $1,075

NM $4,184 $4,049 $3,914 $3,779 $2,969 $2,564 $2,024 $1,350 $945

NY $7,589 $6,382 $5,347 $4,657 $3,967 $3,277 $2,760 $1,380 $690

NC $45,992 $41,234 $37,269 $32,908 $27,555 $23,194 $19,031 $8,921 $4,560

ND $22,861 $19,475 $15,947 $13,971 $11,713 $10,020 $8,044 $3,810 $1,976

OH $16,045 $13,970 $12,310 $10,927 $9,406 $7,884 $6,363 $3,181 $1,798

OK $29,197 $25,631 $22,511 $19,613 $16,716 $14,041 $11,813 $5,572 $2,897

OR $10,466 $9,420 $8,523 $7,626 $6,429 $5,532 $4,486 $2,093 $1,196

PA $26,208 $23,587 $21,185 $19,001 $15,943 $13,322 $10,702 $5,242 $2,839

RI $2,923 $2,923 $2,923 $2,923 $2,436 $1,949 $1,462 $974 $731

SC $13,395 $12,236 $11,206 $10,175 $8,501 $7,213 $5,796 $2,705 $1,417

SD $15,498 $13,661 $12,054 $10,562 $9,069 $7,692 $6,314 $2,870 $1,492

TN $33,557 $30,218 $27,355 $24,333 $20,516 $17,335 $14,155 $6,680 $3,499

TX $85,174 $76,023 $68,280 $60,772 $51,386 $43,408 $34,961 $17,363 $10,324

UT $3,995 $3,582 $3,306 $3,031 $2,617 $2,342 $1,791 $1,102 $827

VT $4,872 $4,872 $4,872 $4,628 $3,898 $3,167 $2,680 $1,218 $731

VA $19,294 $17,526 $15,759 $14,286 $11,930 $10,162 $8,248 $3,829 $2,062

WA $19,180 $17,640 $16,240 $14,420 $12,320 $10,360 $8,540 $4,200 $2,380

WV $15,984 $15,758 $15,308 $14,183 $11,706 $9,905 $7,879 $4,052 $2,476

WI $26,404 $23,419 $20,664 $18,138 $15,383 $12,858 $10,332 $5,051 $2,755

WY $23,187 $23,187 $23,187 $21,386 $18,010 $15,083 $11,931 $6,078 $3,827

Total $897,335 $806,408 $724,620 $646,563 $544,842 $458,884 $373,442 $180,855 $100,402
APPENDIX D Variance Estimation of JAS-ALUS without Non-response

1

The first term in Sarndal and Swensson (1987) is For JAS-ALUS, 

 If but they are in a same substratum, 

, 

. If are from different substratum, If 

The second term in Sarndal and Swensson (1987) is For JAS-ALUS, 

 Therefore, 
APPENDIX E Variance of a General Three-Phase Sampling Design Estimation

2

Sarndal and Swensson (1987) studied the estimation from a two-phase sampling design. Their 

method can be applied to two-phase sampling design estimation without considering 

response/non-response in the second phase. It can also be used to adjust for non-response in a 

one-phase survey. Here we extend their work to a general three-phase survey design. 

To be consistent and complete, we state the notations in the first two phases as in Sarndal and 

Swensson (1987). Then we continue to the third-phase.

Let be the cardinality of the finite population and be the value of study. is the value of 

for the unit. The population total is . We allow a general sampling design in 

each phase.

(a) The first-phase sample of size not necessarily fixed, is drawn according to a 

sampling design such that is the probability of choosing . The inclusion 

probabilities are defined by 

with Set We assume that for all 

for all in variance estimation. is the probability of selection of the unit 

in the first phase sampling. is the probability of selection both the unit and the unit 

in the first phase sampling.

(b) Given the second-phase sample of size , not necessarily fixed, is drawn 

according to a sampling design such that is the conditional probability of 

choosing . The inclusion probabilities given are defined by 

Set We assume that for any for all 

for all in variance estimation. is the probability of selection of the 

unit in the second phase sampling given the result of the first phase sampling. is the 

probability of selection both the unit and the unit in the second phase sampling given the 

result of the first phase sampling.

(c) Given the third-phase sample of size , not necessarily fixed, is drawn 

according to a sampling design such that is the conditional probability of 

choosing . is the set of response for the second phase in a two-phase sampling design. The 

inclusion probabilities given are defined by 
APPENDIX E Variance of a General Three-Phase Sampling Design Estimation

3

Set is the probability when the unit has 

response for the second phase in a two-phase sampling design. is the probability when 

both the unit and the unit have response for the second phase in a two-phase sampling 

design.

Now for all and any define is the 

probability of selection of the unit in a two-phase sampling design. is the probability of 

selection both the unit and the unit in a two-phase sampling design. Set 

Next, define

.

For all and any R, is the probability such that the unit is chosen in a 

two-phase sampling design and has response. is probability such that both the unit and 

the unit are chosen in a two-phase sampling design and have response.

Set . Then the first-phase expanded -value is The second-phase 

expanded -value is The third-phase expanded -value is

The expanded values are 

Now we define the basic estimator in three-phase sampling, estimator.

Recall is the population total. To provide the variance formula for this estimator, we 

first decompose as

Now let refer to conditional expectation or variance in 

phase two, given the outcome of phase one. refer to 
APPENDIX E Variance of a General Three-Phase Sampling Design Estimation

4

conditional expectation or variance in phase three, given the outcome of phase two. Then the 

variance

Now we calculate the first term of (7). Given the first phase sample, is constant, the second 

and third phase estimators are unbiased. Therefore 

For the second term of (7), we have

Given the second phase sample, and are constants; the third phase estimator is unbiased. 

Therefore in the first term of (9), 

Hence,

On the other hand,

From (9), (11) and (12), 

From (7), (8) and (13), we have 

Here,
APPENDIX E Variance of a General Three-Phase Sampling Design Estimation

5

But this variance formula (14) cannot be applied directly. Therefore we need provide a design 

unbiased estimator of the variance. For arbitrary constant ,

Let in the above argument (18), a design unbiased estimator of the first term of 

(14) is 

Let by using the first two equations of (18), a design unbiased estimator of 

the second term of (14) is 

Let by using the first equation of (18), a design unbiased estimator of the 

first term of the third term of (14) is 

Put (19), (20) and (21) together, we have a design unbiased estimator of (14),
APPENDIX F Variance Estimation of JAS-ALUS with Non-response

1

In this Appendix we apply (22) to obtain a design unbiased estimator of the second term

In the JAS-ALUS sampling design, the unit is segment. One unit in Appendix E is one segment 

in substratum within stratum It includes all tracts in that segment. Recall that all segments 

within a same substratum have the same expansion factor. The first phase expansion factor is 

and the second phase expansion factor is for all segments in substratum within stratum 

Therefore, and 

If but they are in a same substratum, , 

. If are from different substratum , 

If 

Therefore, 

if 

if are from different substratum. 

if 
APPENDIX F Variance Estimation of JAS-ALUS with Non-response

2

In the second phase, ALUS, recall that , if the two 

segments are in a same substratum. Otherwise, Therefore, 

 and if the two segments are in a same 

substratum. if the two segments are in different substratum. 

 and if If we have and 

then we can have and the third-phase expanded -value 

by (24). 

Together with all the analysis in this Appendix, assuming that all tracts in a same segment have 

same response mechanism, we can obtain the design unbiased estimator (22) of the second term 

in (6). In our case, is the probability of response of the tracts in segment . is the 

probability that two tracts have response in segments .
APPENDIX F Variance Estimation of JAS-ALUS with Non-response

3

It can be simplified to