An Empirical Investigation Into The Effects Of Replicate 

Reweighting On Variance Estimates For The Annual Capital 

Expenditures Survey 

Katherine J. Thompson1

U.S. Census Bureau 

Room 3108-4,Washington, D.C. 20233, U.S.A/ katherine.j.thompson@census.gov

Introduction

Many of the business surveys conducted by the U.S. Census Bureau use a one-stage stratified simple random sample 

without replacement (SRS-WOR) design. When the sampling unit is an establishment or even an EIN (tax-reporting 

entity), the sampling rates are generally negligible (e.g., less than 20 percent in all strata). However, if the ultimate

sampling unit is the company, the size of the universe is much smaller, the number of companies in a strata decrease, 

and often sampling fractions can no longer be ignored in computation of variance estimates. The Annual Capital 

Expenditures Survey (ACES) conducted by the U.S. Census Bureau is one such survey: sampling rates in 

approximately one-fifth of all non-certainty strata are greater than 0.20.

The ACES estimation procedure is fairly straightforward. Sample weights are adjusted for unit non-response, and 

these non-response adjusted weights are used for subsequent estimation of (annual) expansion estimates and year-toyear change estimates. The ACES variance estimation procedure is equally straightforward: ACES uses a “simple” 

delete-a-group jackknife variance estimator (Kott 2001) to produce variance estimates of totals and uses Taylor 

linearization methods to produce variances of change estimates. The “simple” refers to the replicate weights, which 

are constructed from the full-sample non-response adjusted weights; currently ACES does not replicate their nonresponse adjustment procedure. This variance estimator is computationally quick and easy and does not require 

excessive overhead in terms of computer storage, storing only 16 replicate weights per sample unit. 

 

The decision to use simple delete-a-group jackknife variance estimation for ACES was based on research presented 

in Thompson, Sigman, and Goodwin (2003). Somewhat surprisingly, that study found few statistical benefits in 

replicating the non-response adjustment procedures with those data. I hypothesized that our unexpected results 

could be caused by a variety of factors, such as high unit response in large company strata, non-negligible sampling 

fractions in large company strata, an inappropriate choice of variance estimator (with a highly stratified survey, the 

stratified jackknife might be more appropriate), the method of non-response weight adjustment, or characteristics 

specific to capital expenditures data. 

This paper considers all of the factors above while examining the effects of replicating non-response weight 

adjustment procedures (the fully replicated procedure) on variance estimates in comparison with corresponding 

“simple” replicated variance estimates (“shortcut” procedure variances). I first provide background on the ACES 

design and estimation methodology and briefly introduce a model-assisted interpretation of unit non-response in 

ACES. The section that follows describes the capital expenditures data characteristics. After discussing the survey 

design and data characteristics, I provide definitions of the delete-a-group jackknife and the stratified jackknife 

variance estimates, then present and discuss empirical variance estimates of capital expenditures statistics from three 

years’ of ACES data: the first two data sets (from survey years 2002 and 2003) are the full collection of final 

tabulated ACES data and the third data set (survey year 2003) contains a mid-survey collection of data. Following 

the empirical comparisons, I provide some simulation study results that pursue some of the possible issues 

uncovered in the empirical comparisons, considering only the delete-a-group jackknife variance estimator. The 

paper concludes with a few specific comments about ACES applications and some general observations. 

 

1

This paper reports the results of research and analysis undertaken by the U.S. Census Bureau staff. It has 

undergone a Census Bureau review more limited in scope than that given to official Census Bureau publications. 

This report is released to inform interested parties of ongoing research and to encourage discussion of work in 

progress. 
 

Annual Capital Expenditures Survey (Design And Estimation)

The ACES survey collects data about the nature and level of capital expenditures in non-farm businesses operating 

within the United States. Respondents report capital expenditures for the calendar year in all subsidiaries and 

divisions for all operations within the United States. ACES respondents report total capital expenditures, broken 

down by type (expenditures on Structures and expenditures on Equipment). In all subsequent sections and tables, I 

refer to these characteristics as Total, Structures, and Equipment. 

The ACES universe contains two sub-populations: employer companies and non-employer companies2

. Different 

forms are mailed to sample units depending on whether they are employer (ACE-1) companies or non-employer 

(ACE-2) companies. New ACE-1 and ACE-2 samples are selected each year, both with stratified SRS-WOR 

designs. The ACE-1 sample comprises approximately seventy-five percent of the ACES sample (roughly 45,000 

companies selected per year for ACE-1, and 15,000 selected per year for ACE-2). 

The ACE-1 frame is stratified first by primary industry activity found in the Census Bureau’s Business Register. 

Five separate strata are formed within industry: one certainty stratum consisting of companies with 500 or more 

employees, and four non-certainty strata determined using a modified Lavallee-Hidiriglou method with payroll as a 

measure of size (Slanta and Krenzke, 1996). In the ACE-1 nomenclature, the stratum label indicates the relative size 

of the companies, with stratum 2A containing the largest non-certainty companies in the industry, decreasing to 

stratum 2D, which contains the smallest non-certainty companies in the industry. Stratum 10 units are certainty 

units and are not included in variance estimation (see Section 3). Sampling fractions in the large size non-certainty 

ACE-1 (2A) strata can be quite high, as shown in Table 1. In each survey year, at least sixty percent of the sizeclass 2A strata have non-negligible sampling fractions. Consequently, the sample units from the large-unit noncertainty strata contribute relatively little to the variance estimates (See Section 3). 

Table 1: Distribution of Sampling Fractions (fh = nh/Nh) in ACE-1 Non-Certainty Strata 

 Number of Industry-Level Strata 

Survey Year Size Strata fh < .20 (.20 <= fh < .5) (.50 <= fh < 1) fh ≡ 1 Total Strata 

2A 39 32 28 32 131 

2B 103 18 7 0 128 

2C 123 3 0 0 126 

2001 

2D 121 0 0 0 121 

2A 50 28 23 29 130 

2B 101 18 10 0 129 

2C 121 5 0 0 126 

2002 

2D 123 0 0 0 123 

2A 44 34 25 27 130 

2B 103 15 11 0 129 

2C 124 2 0 0 126 

2003 

2D 122 0 0 0 122 

In contrast to the ACE-1 design, sampling fractions in all ACE-2 strata are quite low (all less than 0.01). 

Table 2 presents the size strata percentage contribution to ACE-1 capital expenditures estimates over all sampled 

industries, both overall and as a median percentage within industry size-stratum. Regardless of survey year, the 

certainty cases – which are excluded from the variance estimation – contribute the most to each estimate; the 

contributions from the non-certainty size strata are relatively equal within industry for both total capital expenditures 

and equipment. 

 

2

 A nonemployer business is one that has no paid employees, has annual business receipts of $1,000 or more ($1 or 

more in the construction industries), and is subject to federal income taxes. Most nonemployers are self-employed 

individuals operating very small unincorporated businesses, which may or may not be the owner’s principal source 

of income. 
Table 2: Percentage Contribution to ACE-1 Survey Estimates by Size Stratum 

Over All Industries Median Percentage within Industry SizeStratum 

Survey Year Size Strata Total Structures Equipment Total Structures Equipment 

10 0.74 0.69 0.77 0.66 0.72 0.63 

2A 0.07 0.09 0.06 0.08 0.08 0.09 

2B 0.07 0.08 0.07 0.10 0.07 0.10 

2C 0.07 0.08 0.06 0.09 0.04 0.07 

2001 

2D 0.05 0.05 0.05 0.03 0.01 0.04 

10 0.73 0.68 0.76 0.65 0.72 0.64 

2A 0.08 0.11 0.06 0.09 0.08 0.08 

2B 0.07 0.09 0.07 0.08 0.06 0.09 

2C 0.07 0.06 0.07 0.08 0.04 0.08 

2002 

2D 0.06 0.06 0.05 0.04 0.01 0.04 

10 0.67 0.38 0.68 0.60 0.48 0.57 

2A 0.09 0.20 0.09 0.10 0.03 0.09 

2B 0.07 0.09 0.07 0.09 0.00 0.09 

2C 0.09 0.22 0.08 0.08 0.00 0.09 

2003 

2D 0.08 0.11 0.08 0.06 0.00 0.07 

ACES uses adjustment-to-sample models described in Kalton and Flores-Cervantes (2002) to reduce bias due to unit 

non-response, i.e., all sampling weights in a weighting class l are multiplied by a factor derived from data 

corresponding to sample units. The ACE-1 non-response adjustment procedure controls sampling weights to 

independently obtained estimates of payroll; that is, the non-response weighting adjustment factor for a weighting 

cell l is the sum of the sample-weighted administrative payroll data for units in the weighting cell divided by the 

sum of the sample-weighted administrative payroll data for all responding units in the weighting cell. Under 

complete non-response in an industry’s certainty stratum or in the large company (2A) stratum, the two strata are 

combined into one weighting cell (within the sample industry). Presently, there is no collapsing procedure in place 

for complete non-response in the three remaining non-certainty strata. In general, stratum collapsing is a very rare 

occurrence and is hereafter ignored in this paper. I refer to the ACE-1 weight adjustment procedure as the ratioadjustment procedure. 

Ignoring collapsing, the non-response adjusted ACE-1 estimator is a separate ratio estimator. Let yhj be the value of 

the survey-data item for sampled unit j in strata h, xhj be the corresponding auxiliary-variable data (known for all 

sample units). With unit non-response, the sampled units subdivide into two disjoint sets: respondents (complete 

data observed) and non-respondents (only auxiliary data observed). In a sample of n units, there are r respondents 

and nr = (n-r) non-respondents. The ACE-1 ratio estimator assumes the following model 

, ,0(~ ) hj hj h hj hj hy = x β + ε ε σ (2.1) 

where xhj is the value of payroll for sampled unit j. The estimator of characteristic is given by 1

ˆYACE−

h

hr

hr

h h

h

hr

hr

hj

hj

h

h

h h r

h hr

hr

h hr

hj

hjh

hj

h h j

hjh

hj

hjh hj

hj

hjh

x

x

y

N

x

y

x

n

N

W y

W x

W x

W Iy

W Ix

W x

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

=

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎝

⎛

∑ ∑ ∑ ∑ ∑ ∑

∑

∑ ∑ ∑

∑

∈ ∈

∈

∈

∈

∈

∈

(2.2) 

where Ihj is a unit-response indicator variable for unit j, Wh = Nh/nh, and an “hr” index indicates the summation for 

all r respondents in a given strata. 

The ACE-1 estimator is equivalent to the simple regression imputation estimator: 

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎥ +

⎦

⎤

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

− = ∑ ∑ ∑

hr

hr

h

nrhh

hr

hr

h

h h hr

hr

hr

h

ACE h

x

y

W x

x

y

x W x

x

y

Y 1 N ,

ˆ

 (2.3) 

 
where hr y and hr x are the sample means of the respondent and auxiliary data from the respondent units, xhr is the 

total auxiliary data for respondent units, and x,h,nr is the total auxiliary data for non-respondent units. Thus, the 

variance of the ACE-1 capital expenditures estimates has two components, one for the respondent data and one for 

the imputed portion. 

The ACE-2 estimates are controlled to sample counts within strata; that is, the non-response weighting adjustment 

factor for a weighting cell is calculated as the number of sampled units in the weighting cell divided by the number 

of responding units in the weighting cell, i.e., the unweighted cell inverse response rate, as recommended by 

Vartivarian and Little (2002). I refer to the ACE-2 weight adjustment procedure as the count adjustment. The 

estimator of characteristic is given by 2

ˆYACE−

hr

h h

h

h

hr

h

h

h

h h r

h hr

hr

hr

hj

hj

N y

r

y

n

n

N

W y

I

I

∑ ∑ ∑ ∑ ∑

∑

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

⎟

⎟

⎟

⎟

⎠

⎞

⎜

⎜

⎜

⎜

⎝

⎛

∈

∈

∈

 (2.4)

where Ihj is a sample-inclusion indicator variable and Ihr is the unit-response indicator variable defined above. 

Again, this estimator has a response data component and an imputed data component:

⎥

⎦

⎤

⎢

⎣

⎡

+ − ⎥

⎦

⎤

⎢

⎣

⎡

− = ∑ = ∑ ∑

h

h h h hr

h

hr h hr

h

ACE h Y N y W y W (n r ) y

ˆ

2

 (2.5)

See Caldwell (1999b) for more details on the ACE-1 and ACE-2 non-response weighting adjustment procedures. 

A model-assisted perspective on non-response considers the “selection” of non-responding units as another phase of 

sample selection from g disjoint response homogeneity groups (RHG)3

, where the true response distribution exists 

but is unknown (S@rndal, Swenssen, Wretmen, 1992, Ch.15 and Kott 1994). The literature refers to this model as 

the quasi-randomization model since the second phase of sample selection depends on an unknown response 

distribution. Under quasi-randomization models, the ACE-1 and ACE-2 stratified SRS-WOR samples are the first 

stage of sample selection, the ACE-1 and ACE-2 strata are the response homogeneity groups (RHG), and the 

respondents are Bernouilli samples within RHG. In this framework, the final-weighted ACE-1 estimates are twophase sample regression (ratio) estimators and the final-weighted ACE-2 estimates are two-phase sample estimates, 

with sampling weight equal to Nh/ rh (=Nh/nh × nh/rh), i.e., are Horvitz-Thompson estimates based on sample 

respondents. Kott and Stukel (1997) provide evidence that the stratified jackknife variance estimator yields 

conditionally unbiased variance estimates (with respect to the assumed model) from a two-phase sample regression 

estimator (c.f. the ACE-1 ratio adjustment estimator) if the sampling is performed with replacement at the first stage, 

but not for the two-phase sample estimator (c.f. the ACE-2 count adjustment estimator). A design-based 

perspective for these same estimators requires that the variance estimate include components for the variance from 

responding units, variance due to imputation, and covariance of the two terms. 

Wolter (1985, Ch.2, pp. 31-32) provides guidelines for dividing the parent sample of either a two-stage or two-phase 

sampling into random groups for variance estimation, advocating allocation of entire first stage units (not allocation 

of ultimate sampling units or respondents) to random groups. The applications described follow these guidelines. 

Characteristics Of Capital Expenditures Data 

Capital expenditures data are fairly atypical business data. First, they often are characterized by low year-to-year 

correlation for the same reporting unit: for example, a company that spends a large amount of capital expenditures 

on structural (building) improvements one year is unlikely to invest much in structural improvements in the 

following year. Second, they are often poorly correlated with auxiliary data such as payroll or receipts, especially 

for small companies. Finally, company purchasing patterns are more correlated with the size of the company than 

the sample industry. For example, in some industries, capital expenditures on structures and equipments are 

negatively correlated for small companies and positively correlated for large companies. For these reasons, the 

ACES survey does not perform imputation for item non-response. 

 

 

3

 Units in RHG groups have the same response probability, with data missing at random within RHG group. 
 

In addition, while the ACE-1 and ACE-2 data are characterized by fairly high unit respondent rates, a reported zero 

value for most data items is generally legitimate, and the item reported-zero rate can be quite high. Table 3 presents 

median unit response rates within industry size-class strata (ACE-1) or strata (ACE-2), along with median 

unweighted item reported-zero rates for the three variables of interest. Note that the item reported-zero rate for total 

capital expenditures is often much smaller than the corresponding rate for the other two items: a company may not 

have the expenditures data available in the requested breakdown and may report only a total value. 

Table 3: Median Unit Response Rate and Item Reported Zero Rates 

Item Reported-Zero Rates 

Survey Year Population Size Strata Unit Response Rate

Total Structures Equipment 

10 0.88 4.41 20.42 5.08 

2A 0.83 4.84 25.11 6.13 

2B 0.84 5.19 90.89 5.27 

2C 0.80 9.94 43.7 10.61 

ACE-1 

2D 0.71 10.81 51.72 12.15 

2001 

ACE-2 All Strata 0.80 10.84 97.69 12.13 

10 0.87 16.61 63.26 18.28 

2A 0.84 18.08 72.39 18.92 

2B 0.83 17.68 100.00 20.36 

2C 0.78 31.64 84.72 32.29 

ACE-1 

2D 0.71 31.48 85.86 31.71 

2002 

ACE-2 All Strata 0.72 31.06 100.00 33.84 

10 0.92 57.85 96.48 59.02 

2A 0.74 59.00 95.62 60.05 

2B 0.75 55.96 100.00 58.99 

2C 0.71 59.45 91.81 61.44 

ACE-1 

2D 0.66 71.01 97.19 72.94 

2003 

ACE-2 All Strata 0.58 66.55 91.15 69.16 

For ACE-1, the item reported-zero rate tends to increase as the size of the company decreases, particularly for 

structures. ACE-2 companies rarely report capital expenditures on structures. Notice that all of the item reportedzero rates increase each year, and that the item reported-zero rates are extremely high for all items and strata in the 

2003 survey year. Recall that the 2003 ACES data are an incomplete collection (the 2001 and 2002 data sets 

comprise the complete survey collection), and consequently, the 2003 rates reported in Table 3 are very different 

from those in the final data sets. 

Variance Estimation Methodology

I consider two variance estimators: the delete-a-group jackknife variance estimator and the stratified jackknife 

variance estimator. To perform delete-a-group jackknife variance estimation, the non-certainty portion of the survey 

sample is divided into K random groups. The delete-a-group jackknife replicate estimate is then computed for each 

replicate k by removing the k

th random group from the full sample and multiplying each replicate estimate by K/(K1). Note that these replicate factors differ from those presented in Kott (2001), which recommends developing 

stratum-specific replicate factors from sampled units in stratum h assigned to delete-a-group jackknife replicate k. 

The delete-a-group jackknife variance estimator used in this paper is equivalent to the random group variance 

estimator for Horvitz-Thompson estimates (total estimates) and is an unconditionally unbiased estimate of the true 

variance, given a systematic assignment of sample units to random group (see Appendix). Certainty units are 

included in each delete-a-group jackknife replicate estimate. Thus, for delete-a-group jackknife replication, K

replicate weights are assigned to each sample unit j. If unit j is in a non-certainty stratum, the k

th replicate weight is 

zero when unit j is in random group k. In a certainty stratum, all K replicate weights are equal to the sampling or 
final weight (which may not be equal to 1 with a non-response adjustment). The full sample estimation procedure is 

then applied to each of the replicate weights (e.g., non-response adjustments, post-stratification) or to the replicate 

estimates themselves. The delete-a-group variance for any estimate is θ i

ˆ

 ∑( − )

−

=

k

DAG i ki DAG i

K

K

v

2

( ) 0

ˆ ˆ

1

)

ˆ

ˆ (θ θ θ (4.1) 

where is the random group k replicate estimate and is the full-sample estimate. ACES uses K = 15 

random groups. To account for the non-negligible sampling fractions, I multiply all replicate weights by

( )

ˆθ ki DAG 0i

ˆθ

h 1 − f as 

proposed by Wolter (1985, Ch. 2, pp. 36). Thus in my applications, the and (the tabulated full-sample 

estimate) are not equivalent. 

0i

ˆθ θ iˆ

Kott (2001) states that appropriate use of the delete-a-group jackknife requires that the number of sample units per 

first-phase stratum be large in all strata. This requirement is always met in the ACE-2 sample. However, it is 

sometimes violated in the ACE-1 strata; for example, in the 2001 survey year design, 11 of the 506 designated noncertainty ACE-1 strata contain less than 15 sampled units. ACES does not employ the extended delete-a-group 

jackknife procedure proposed for these small strata described in Kott (2001). 

The stratified jackknife variance estimator uses considerably more replicates than the delete-a-group jackknife 

variance estimator. Replicate estimates for the stratified jackknife are constructed by dropping one observation at a 

time from the strata and multiplying the remaining observations by nh/(nh –1). Thus, the stratified jackknife 

procedure constructs one replicate estimate per sample unit. The stratified jackknife variance estimator is given by 

∑ ∑ ( )

= =

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ −

= −

H

h

n

r

rhi hi

h

h

STRJ i h

h

n

n

v f

1 1

2

)

ˆ ˆ

(

1

) 1

ˆ

ˆ (θ θ θ (4.2) 

The purpose of this study was to investigate the effect of replicating the non-response adjustment procedure on 

ACE-1 and ACE-2 variance estimates. I expected that the “shortcut” approach used by ACES should 

underestimate the true variance. There are, however, some arguments against this. The shortcut variance estimator 

is not the naive variance estimator, which treats imputed values as though they were reported values and which has 

been shown repeatedly to underestimate the variance. The naive variance estimator replaces the missing item 

responses with imputed values, yielding a dataset with no (visible) item non-response, and the replicate values do 

not contain any missing values. My replicate assignment procedures assign sample units to replicates. 

Consequently, replicate estimates contain both responding and non-responding units, thus incorporating secondphase effects (model-assisted interpretation) or non-sampling error effects (design-based interpretation). The 

difference between the shortcut and fully replicated procedures are that the shortcut method estimates conditional

variance (conditioned on the sample respondents), whereas the fully replicated procedure estimates an unconditional 

variance for the model-assisted interpretation of two-phase sampling. The fully replicated weight adjustment 

procedure has varying strata weight adjustment factors by replicate and is conditionally model unbiased (Kott and 

Stukel, 1997). A derivation of the conditional expectation of the shortcut procedure variance is beyond the scope of 

this paper. 

Intuition notwithstanding, the shortcut procedure could overestimate the variability induced by the ACE-1 ratio 

estimation procedure, since it treats the ACE-1 estimate as an expansion estimator and does not include covariance 

between numerator and denominator in calculations (the fully replicated procedure accounts for this). Note that 

Yung and Rao (1996) found that using a shortcut procedure with a post-stratified estimator from a multi-stage 

stratified design with the stratified jackknife severely overestimated the variance. 

There is limited support in the literature for this shortcut approach. Wolter (1985, pp. 83-84) cites results from two 

studies that showed the slight improvements in random group variance estimates using full replication versus the 

shortcut approach did not offset the additional computing costs. In a similar vein, Schindler (2002) found trivial 

differences between the variance computed with a fully reweighted stratified jackknife procedure versus those 

obtained with a simple jackknife that used final weights in all estimates (shortcut procedure) for selected dual 

system estimates from the Census 2000 Accuracy and Coverage Enumeration Survey. Finally, Canty and Davison 

 
 

(1999) found better statistical properties with a partially replicated procedure on a post-stratified estimator (one 

iteration of a multi-stage ratio raking procedure) using the stratified jackknife than with the fully replicated 

procedure. 

 

In the following section, I use “DAG” to indicate delete-a-group jackknife and “STRJ” to indicate stratified 

jackknife, combined with “S” (simple/shortcut) and “R” (reweighted). Certainty cases are excluded from all of the 

discussed replicate variance estimates via the fpc-adjustment (all certainty cases have replicate weights of zero) as 

are the non-certainty cases in size-strata 2A (within industry) that have sampling rates equal to 1. 

Empirical Data Results

This section presents empirical comparisons of standard error estimates using the delete-a-group jackknife and the 

stratified jackknife with and without replicating the non-response adjustments. Within variance estimation method, 

Table 4 compares the fully replicated standard errors to shortcut standard errors. Although the ACE-1 component 

requires a variance estimation adjustment due to non-negligible sampling fractions in many strata, I present results 

both with and without the fpc-adjustment to determine whether the fpc-adjustments have impact on the effects of 

replicating the weight adjustment procedure. 

Table 4: Ratios of Reweighted/Simple Standard Errors Within Variance Estimation Method 

ACE-1 ACE-2 

Survey Year Data Item 

DAGR/DAGS STRJR/STRJS DAGR/DAGS STRJR/STRJS 

FPC No FPC FPC No FPC FPC No FPC FPC No FPC 

2001 Total 0.97 1.00 0.98 0.98 1.04 1.04 0.99 0.99 

Structures 1.00 1.00 0.99 0.99 1.00 1.00 1.00 1.00 

Equipment 0.92 0.92 0.98 0.98 1.04 1.04 0.99 0.99 

2002 Total 0.99 1.00 0.97 0.97 1.03 1.03 0.99 0.99 

Structures 1.00 1.01 0.98 0.98 0.99 0.99 1.00 1.00 

Equipment 0.90 0.91 0.96 0.96 1.08 1.08 0.99 0.99 

2003 Total 1.02 1.02 0.99 0.98 0.97 0.97 1.00 1.00 

Structures 1.04 1.04 1.00 1.00 1.02 1.02 1.00 1.00 

Equipment 0.99 1.00 0.99 0.98 0.98 0.98 1.00 1.00 

Within replication method (delete-a-group jackknife or stratified jackknife), the full replication effects are generally 

the same regardless of fpc correction. This result is expected: most of the ACE-1 unit non-response occurs in the 

smaller company size strata, which have negligible sampling fractions; and all of the ACE-2 sampling fractions are 

negligible. In fact, the fpc corrections to the stratified jackknife standard error estimates are generally ignorable. 

Notice that the full replicate standard error estimates are very close to the shortcut standard error estimates for 

capital expenditures on structures, the characteristic with the highest proportion of reported zero values in all years. 

Also, the delete-a-group and stratified jackknife ACE-1 standard error estimates of total capital expenditures and 

equipment decrease when the full replication procedure is used for the 2001 and 2002 survey years (the two high 

unit-response years). This pattern continues for the stratified jackknife standard error estimates with the 2003 data, 

but changes for the delete-a-group jackknife standard error estimates, perhaps further illustrating the effect of high 

item non-response. 

In contrast to the ACE-1 estimates, the effects of full replication on ACE-2 standard error estimates are quite 

different for the two variance estimators: the stratified jackknife standard errors are virtually unchanged, whereas 

almost all of the 2001 and 2002 delete-a-group jackknife standard errors increase slightly with full replication. 

Given the relatively low unit response rates coupled with the high reported-zero rate for the ACE-2 samples, it 

seems unlikely that the full replication would have little effect on the standard error estimates, c.f., the stratified 

jackknife estimates. 

Table 5 assesses the effect of the fpc-correction on standard error estimates by presenting standard error ratios 

within variance estimation method. [Note: ACE-2 sampling fractions are all less than 0.01, so all standard error 

ratios are 1, as expected]. 
 

 

Table 5: Standard Error Ratios (Without FPC-Correction/With FPC Correction) 

ACE-1 ACE-2 

Survey Year Data Item Delete-a-Group Jackknife Stratified Jackknife Delete-a-Group Jackknife Stratified Jackknife 

Simple Reweighted Simple Reweighted Simple Reweighted Simple Reweighted 

2001 Total 1.05 1.07 1.14 1.14 1.00 1.00 1.00 1.00 

Structures 1.03 1.03 1.19 1.18 1.00 1.00 1.00 1.00 

Equipment 1.09 1.08 1.10 1.10 1.00 1.00 1.00 1.00 

2002 Total 1.02 1.03 1.03 1.03 1.00 1.00 1.00 1.00 

Structures 1.03 1.03 1.03 1.03 1.00 1.00 1.00 1.00 

Equipment 1.06 1.07 1.04 1.04 1.00 1.00 1.00 1.00 

2003 Total 1.27 1.26 1.23 1.23 1.00 1.00 1.00 1.00 

Structures 1.11 1.11 1.10 1.10 1.00 1.00 1.00 1.00 

Equipment 1.33 1.33 1.30 1.30 1.00 1.00 1.00 1.00 

Regardless of variance estimation method, the fpc-correction reduces the ACE-1 standard error estimates. This 

effect is most pronounced with total capital expenditures and equipment. This is in line with the data characteristics 

and the fpcs, i.e., large fpcs in strata with positive (non-zero) capital expenditures estimates and small fpcs in strata

with high item non-response rates (small or approximately zero expenditures estimates). Table 6 compares the size 

of corresponding stratified jackknife and delete-a-group jackknife standard error estimates. 

Table 6: Ratios of Stratified Jackknife Standard Error estimates/Delete-a-Group Jackknife Standard Error 

Estimates

ACE-1 ACE-2 

Survey Year Data Item 

STRJS/DAGS STRJR/DAGR STRJS/DAGS STRJR/DAGR 

FPC No FPC FPC No FPC FPC No FPC FPC No FPC 

2001 Total 1.07 1.17 1.08 1.15 1.00 1.00 0.96 0.96 

Structures 1.18 1.36 1.17 1.34 0.98 0.98 0.98 0.98 

Equipment 0.82 0.83 0.87 0.89 0.83 0.83 0.79 0.79 

2002 Total 1.11 1.12 1.09 1.09 1.00 1.00 0.97 0.97 

Structures 1.28 1.28 1.25 1.24 1.01 1.01 1.01 1.01 

Equipment 1.04 1.02 1.11 1.07 0.99 0.99 0.90 0.90 

2003 Total 0.97 0.94 0.94 0.92 1.09 1.09 1.11 1.11 

Structures 0.87 0.87 0.84 0.84 1.33 1.33 1.30 1.30 

Equipment 1.01 0.99 1.00 0.98 0.73 0.73 0.74 0.74 

Again, the 2001/2002 survey year results are quite different from the 2003 survey year results for ACE-1, with the 

stratified jackknife standard error estimates generally larger in the first two years (complete data collections), and 

smaller or the same in 2003 (incomplete data collection). The ACE-2 ratios are quite variable, again perhaps 

showing the effect of the high item reported-zero rates on the different variance estimators. 

To gauge the effect of the non-response weight adjustment procedure on the ACE-1 standard error estimates, I 

computed ACE-1 variances using the count adjustment procedure (the ACE-2 non-response adjustment procedure). 

Table 7 presents these results. Again, there is very little difference between shortcut and full replication estimates 

within the same variance estimation method. When the count adjustment procedure is used instead of the ratio 

adjustment procedure, the stratified jackknife standard error estimates are much smaller than the corresponding 

delete-a-group jackknife standard error estimates. 

So far, the empirical results show the following: 

• The effects of full replication are not affected by the fpc-adjustment, for either ACE-1 or ACE-2; 
 

• Regardless of variance estimation method, shortcut procedure standard errors are as likely to be smaller

than corresponding fully replicated standard errors as larger (neither procedure yields consistently larger or 

smaller standard error estimates); 

• Corresponding shortcut and full replication standard error estimates are very close in value; 

• For characteristics with high item response rates, the ACE-1 ratio adjustment procedure (for unit nonresponse) generally yields larger stratified jackknife standard error estimates than the corresponding deletea-group jackknife standard error estimates. The stratified jackknife standard error estimates may more 

correctly reflect the variability due to the ratio imputation. However, the stratified jackknife standard error 

estimates for ACE-1 are much smaller than the corresponding delete-a-group jackknife variance estimation 

when the count adjustment procedure (mean imputation) is used; 

• Finally, there appears to be an interaction between the effects of replicating the non-response weight 

adjustment procedure and the unit response rates. Because ACES has high unit response rates in all strata, 

it is impossible to assess the effect of unit response rate on variance estimates from these data sets. 

 

Table 7: ACE-1 Standard Error Ratios Using Ratio-to-Sample Count Non-Response Adjustment 

Reweighted/Simple (Same Variance Estimator) Stratified Jackknife/Delete-a-Group Jackknife 

Survey Year Data Item 

DAGR/DAGS STRJR/STRJS STRJS/DAGS STRJR/DAGR 

FPC No FPC FPC No FPC FPC No FPC FPC No FPC 

2001 Total 0.98 1.02 1.00 1.00 0.59 0.65 0.59 0.64 

Structures 1.00 1.00 1.00 1.00 0.77 0.90 0.77 0.90 

Equipment 0.96 0.98 0.99 0.99 0.47 0.47 0.48 0.48 

2002 Total 1.00 1.00 1.00 1.00 0.65 0.67 0.65 0.67 

Structures 1.00 1.01 1.00 1.00 0.70 0.72 0.70 0.72 

Equipment 0.92 0.93 0.98 1.02 0.74 0.73 0.79 0.77 

2003 Total 1.04 1.03 1.06 1.06 0.63 0.63 0.64 0.65 

Structures 1.02 1.02 1.03 1.03 0.62 0.62 0.63 0.63 

Equipment 1.02 1.02 1.06 1.07 0.64 0.65 0.66 0.68 

ACES uses the delete-a-group jackknife variance estimator, as does a variety of other business surveys administered 

by the Census Bureau. The empirical results shown above provide some evidence that the stratified jackknife 

variance estimator might be more appropriate for the ACE-1 component of the survey, if the current non-response 

weight adjustment procedure is retained. On the other hand, there is very little to recommend the stratified jackknife

variance estimator for ACE-2, or for ACE-1 if the non-response weight adjustment procedure is changed to an 

inverse response-rate procedure. Furthermore, the confusing results vis-à-vis the fully replicated versus shortcut 

standard errors are not ameliorated by the choice of variance estimator. 

The Census Bureau’s Standardized Economic Processing System (StEPS) software offers delete-a-group jackknife 

variance estimation, but does not include the stratified jackknife. Additional testing of computing resources would 

be necessary before adding this variance estimator to StEPS. In the meantime, there are several studies that report 

excellent results with the delete-a-group jackknife on a variety of sample designs (including stratified SRS-WOR) 

for expansion, ratio, and restricted regression estimators (e.g., Kott (1998), Kott (2001) and Smith (2001)). 

The empirical results point to factors that might account for the differences in the shortcut and fully replicated 

standard errors, namely percentage unit non-response, item reported-zero rates, and choice of non-response weight 

adjustment procedure. The next section explores some of these factors via simulation studies, using only the deletea-group jackknife variance estimator. 

Simulation Study Results 

The simulation studies described below assume a two-phase design, where the first phase is a SRS-WOR of units 

from a frame, and the second phase uses a Bernouilli sampling mechanism to assign response status. 
The empirical results indicated a possible interaction effect between unit and item response rates and full replication 

on standard error estimates. To assess this, I simulated twelve different populations: six normally distributed 

populations (similar to income measurements) and six gamma distributed populations (similar to payroll or 

employment data). Within distribution, three of the six populations had “small” population c.v.’s (less than 0.50) 

and three had “large” population c.v.’s (greater than 1.4), and each of the three populations within c.v. category had 

different levels of item reported-zero rate (90%, 70%, and 50%). Then, I selected three sets of 5,000 SRS-WOR 

samples from each population using a 1-in-20 sampling rate, and in each set, I randomly assigned unit non-response 

within sample using a missing-at-random model with the following response proportions – 90%, 70%, and 50%. In 

1,000 of the 5,000 samples, I assigned sample units to 15 random groups and computed fully replicated and shortcut 

procedure variance estimates using the count adjustment (inverse response rate) procedure for unit non-response. 

To summarize, I ran 18 simulations per population, systematically varying patterns of unit response and item 

reported-zero rates. 

To examine the statistical properties of the shortcut and full replication procedures, I used my 5,000 random samples 

within population i and response proportion j to construct empirical variances as 

,5 000

)

ˆ

(

)

ˆ

(

2

000,5

1

∑=

−

=

r

rij ij

ij

X X

V X (6.1) 

where is the estimate from sample r, and Xrij

ˆ Xij is the mean of the . Xrij

ˆ

Then, I calculated two variance estimates (vmeth) per sample r from 1,000 of the 5,000 samples. I compared these 

variance estimates in terms of 

Relative Bias 1

)

ˆ

(

)

ˆ

(

,1 000

1

000,1

1 −

∑=

ij

rij

r

meth

V X

v X

 (6.2)

MSE 2

1000

1

2 2

) )

ˆ

)) ] ( (

ˆ

) (

ˆ

[ ( (

1000

1

∑=

− + −

r

meth Xrij V Xij V Xij ij v σ (6.3) 

where is the population variance. Table 8 presents the results from this first analysis. For consistency with the 

earlier sections, fully replicated delete-a-group jackknife results are labeled “DAGR” and shortcut procedure deletea-group jackknife results are labeled “DAGS.” 

2 σ ij

Both sets of biases are generally quite comparable, as are the MSEs (in fact, the two sets of MSEs are generally 

indistinguishable). Within the gamma-distribution generated populations, the fully replicated variances are usually 

slightly less biased than the shortcut procedure variances (in 15 out of 18 simulations). Within the normaldistribution generated populations, in two-thirds of the simulations, this pattern is repeated. Here, the shortcut 

standard errors appear to be less biased when the unit response rates are low; population c.v.’s and item response 

rates do not seem to affect the outcome. Capital expenditures data resemble the simulated data in the gammadistributed (plus item response) populations, so these results provide some supporting evidence for the reduction in 

variance levels with full replication. 

Table 8: Combined Effects of Unit Non-response, Item Reported-zero Rate, and Population Distribution 

Population Relative Bias 

Distribution 

CV Category Unit Response 

Rate 

Item ReportedZero Rate DAGS DAGR 

Ratio 

DAGS /DAGR MSE

90 0.0215 0.012 1.00 

90 70 0.0383 0.0276 1.00 

50 -0.1302 -0.145 1.00 

90 0.1551 0.1399 0.99 

Normal Large 

70 

70 0.0797 0.07 0.99 

 
 

50 -0.1379 -0.1404 1.00 

90 0.0467 0.0244 1.00 

50 70 0.0469 0.0314 1.00 

50 -0.0872 -0.0905 1.00 

90 0.0817 0.0795 1.00 

90 70 0.0544 0.0523 1.00 

50 -0.0327 -0.0348 1.00 

90 0.058 0.0534 1.00 

70 70 -0.0082 -0.0117 1.00 

50 0.0825 0.0803 1.00 

90 0.0298 0.0237 1.00 

70 -0.0568 -0.0595 1.00 

Small 

50 

50 0.0102 0.0054 1.00 

90 -0.0619 -0.0775 1.00 

90 70 0.0893 0.0788 1.00 

50 0.0716 0.0647 0.99 

90 -0.0273 -0.0481 1.00 

70 70 0.1338 0.1171 0.99 

50 0.0382 0.0311 1.00 

90 0.0505 0.0175 0.99 

70 0.0557 0.0363 0.99 

Large 

50 

50 0.0138 0.0033 0.99 

90 -0.0635 -0.0672 1.00 

90 70 0.0821 0.0796 1.00 

50 0.0319 0.0306 1.00 

90 0.0573 0.053 1.00 

70 70 0.0565 0.0522 1.00 

50 0.0938 0.0914 1.00 

90 0.0715 0.0662 1.00 

70 0.0759 0.0719 1.00 

Gamma 

Small 

50 

50 0.0389 0.0367 1.00 

With the empirical comparisons, the effects of full replication for the ACE-1 variance estimates differed depending 

on the choice of non-response weight adjustment procedure. Unable to draw a direct link between unit nonresponse, item reported-zero rate, and the statistical properties of the shortcut and full replicated procedures, I 

hypothesized that the differences could be a combined effect of the weight adjustment procedure and the unit 

response rate. In addition, the empirical results obtained using a ratio non-response adjustment procedure could 

depend on the correlation between payroll and the capital expenditures characteristic. My second simulation study 

examines the effect of correlation between characteristic and auxiliary variable in conjunction with unit response 

rates. These simulations use three bivariate lognormal populations with differing levels of correlation between the 

two variables (ρ =90%, ρ = 70%, and ρ = 60%). As in the other simulation study, I selected three sets of SRS-WOR 

samples from the three populations and assigned MAR unit response in each sample, then computed shortcut and 

fully replicated variance estimates using both non-response weight adjustment procedures. Table 9 presents these 

results. 
 

Table 9: Comparison of Variance Properties Considering Correlation Between Item and Auxiliary Data and 

Unit Response Rate 

Population Unit 

Response 

Rate 

Weight Adjustment Bias of Shortcut SEs Bias of Fully 

Replicated SEs 

Ratio of Shortcut 

MSE to Fully 

Replicated MSE 

Ratio 0.013 -0.021 1.01 

90 

Count 0.0122 -0.0211 1.01 

Ratio 0.101 0.0556 1.02 

70 

Count 0.0986 0.0556 1.02 

Ratio 0.1574 0.078 1.05 

ρ = 90% 

50 

Count 0.1532 0.078 1.04 

Ratio 0.0181 -0.0197 1.01 

90 

Count 0.0174 -0.0197 1.01 

Ratio 0.0905 0.04 1.02 

70 

Count 0.0883 0.0401 1.02 

Ratio 0.1377 0.0709 1.03 

ρ = 70% 

50 

Count 0.1337 0.0707 1.03 

Ratio 0.1129 0.0706 1.02 

90 

Count 0.1123 0.0706 1.02 

Ratio 0.073 0.0276 1.03 

70 

Count 0.0714 0.0276 1.02 

Ratio 0.1159 0.0462 1.03 

ρ = 60% 

50 

Count 0.1131 0.0461 1.03 

In all three populations, the biases of the ratio-adjusted and count-adjusted variance estimates are very close within 

replication procedure (full versus shortcut) and item response rate category. In all three, the bias of the shortcut 

procedure variance estimates is generally at least twice as large as that of the fully replicated variance estimates 

when the response rate is less than 90%. In the high response rate scenarios (response rates = 90%) in high and 

medium correlation populations, the fully replicated procedure variance estimates are negatively biased and the 

shortcut procedure variance estimates are positively biased, and the magnitude of the two sets of biases is about the 

same. In the low correlation population, both procedures overestimate the variance, but the shortcut biases are 

usually twice as large as the fully replicated procedure (consistent with the results presented in Table 8). Here again, 

the shortcut MSEs are always larger than the corresponding fully replicated MSEs. 

Both simulation studies presented show one consistent relationship between the bias of the shortcut procedure 

variance estimates and the unit response rate, namely that the shortcut variance estimates are overestimates when 

the unit response rate is low (50%), but are often comparable to the fully replicated procedure variance estimates 

when the unit response rates are high (70% or greater). The two sets of one-strata sample simulation study results 

show that the fully replicated procedure variances are often (slightly) less biased than corresponding shortcut 

procedure variance estimates, but that this frequency is highly dependent on the underlying population distribution. 

The effect of underlying population distribution is further illustrated by the second set of simulation results (with

stratified sampling), where the different strata population distributions clearly have an aggregated effect on the level 

of the biases for the shortcut and fully replicated procedures. 

Conclusion 

This study examines in detail differences between variance estimates that fully replicate a non-response weight 

adjustment procedure and those that use a shortcut procedure. The empirical comparison considers the delete-agroup jackknife variance estimator and the stratified jackknife. With both variance estimators, I found very few 

differences between the fully replicated and shortcut procedure variance estimates, regardless of weight adjustment 

method. Somewhat surprisingly, I also found that the fully replicated standard errors were often smaller than the 

corresponding shortcut procedure standard errors. This finding was reinforced by my simulation study results, 
 

where the fully replicated variance estimates were often less biased than those from the corresponding shortcut 

procedure. Moreover, the MSE of the fully replicated variance estimates was almost always less than that of the 

shortcut procedure variance estimates, providing further evidence of the “closer to unbiasedness” properties of the 

fully replicated variance estimates. 

The empirical results indicated a relationship between the combined percentage of unit and item reported-zero rates 

and the size of the full replicated and shortcut procedure variance estimates. The simulation study results confirmed 

this, demonstrating underestimation with the shortcut procedure under very low unit response rates and 

overestimation with the shortcut procedure otherwise. It is worth noting that when response rates are high, the 

biases of the shortcut and fully replicated standard errors were comparable. 

In interpreting the empirical results, recall that capital expenditures data are somewhat atypical from other business 

data in their proportion of legitimate zero responses and that consequently, small differences in variance estimate 

level may appear as “large” ratio differences (and may yield equivalent coefficients of variation with all standard 

error estimates). Moreover, the ACES unit response rates are quite high in all strata, which may explain the 

generally small differences between corresponding shortcut and fully replicated standard errors. 

The simulation study results attempt to quantify the empirical results using a model-assisted interpretation of unit 

non-response. To guard against this model assumption, I use a variety of response propensities. However, the view 

of unit non-response as a second stage of sampling is not necessarily realistic for a voluntary economic survey: it is 

equally likely that several non-respondents are fixed in the population, so that the unit non-response is an 

estimation-bias problem, not a variance estimation problem. 

If there is a moral to this research, it is that our intuition can often lead us astray. Without evaluation, I assumed that 

the shortcut procedure variance estimates must always be smaller than their fully replicated counterparts. Upon 

reflection, there is no reason to assume that this must be true, especially when unit response rates are high. In this 

case, while there are clearly theoretical advantages to fully replicating the weight adjustment procedure, there may 

be little or no practical advantage. If time is of the essence or computing resources are scarce, the shortcut 

procedure variances – which are very close to the fully replicated variances for the ACES data – are a quite viable 

alternative. 

Finally, my original suspicion that the effects of full replication were hidden by the fpc-adjustment turned out to be

unfounded since the strata used for variance estimation generally have negligible fpc’s. This brings up an 

unaddressed issue. If we truly believe that the unit non-response is another stage of sampling, then we should 

consider including a “certainty” component (for the certainty strata with unit non-response) in our variance 

estimates. Figuring out how to do this correctly is, however, the topic for future research. 

Acknowledgements 

The author thanks Amy Newman-Smith, Rita Petroni, and Richard Sigman for their contributions to this research 

and for their useful comments on an earlier version of this manuscript, and Phillip Kott for his excellent discussion 

of this research. 
 

References 

Caldwell, C. (1999a). Response Characteristics for the 1997 ACES. Unpublished internal 

memorandum. Washington, DC: U.S. Bureau of the Census. 

Caldwell, C. (1999b). Non-response Weighting for the 1999 Annual Capital Expenditures 

Survey (ACES). Unpublished internal memorandum. Washington, DC: U.S. Bureau of the Census. 

Canty, A.J. and Davison, A.C. (1999). Resampling-based Variance Estimation for Labour Force Surveys. The 

Statistician, 48, pp. 379-391. 

Kalton, G. and Flores-Cervantes, I. (2003). Weighting Methods. JOS, 19, pp. 81-97. 

Kott, P. (1994). A Note on Handling Nonresponse in Sample Surveys. Journal of the American Statistical 

Association, 89, pp. 693 –696. 

Kott, P. (1998). Using the Delete-A-Group Jackknife Variance Estimator in NASS Surveys. RD 

Research Report Number RD-98-01. Washington, DC: National Agricultural Statistics Service. 

Kott, P. (2001). The Delete-A-Group Jackknife. Journal of Official Statistics, 17, pp. 521-526. 

Kott, P. and Stukel, D. (1997). Can the Jackknife Be Used With a Two-Phase Sample? Survey Methodology, 23, 

pp. 81-89. 

S@rndal, C., Swensson, B., and Wretman, J. (1992). Model Assisted Survey Sampling. New York: Springer 

Verlag. 

Schindler, E. (2002). Simplified Variance Estimation for Complex Surveys. Proceedings of the Section on Survey 

Research Methods, American Statistical Association, pp. 3095-3100. 

Slanta, J.G. and Krenzke, T.R. (1996). Applying the Lavallee and Hidiroglou Method to Obtain 

Stratification Boundaries for the Census Bureau’s Annual Capital Expenditures Survey. Survey 

Methodology, 22, pp.65-76. 

Smith, H. (2001). Investigation of the Delete-a-Group Jackknife Variance Estimator for the 

HLFS. Research report, Statistics New Zealand Paper 13, Te Tari Tatau. Wellington, New Zealand. 

Thompson, K., Sigman, R. and Goodwin, R. (2003). Investigation of Alternative Replicate Variance Estimators for 

the Annual Capital Expenditures Survey. Proceedings of the Section on Survey Research Methods, 

American Statistical Association, pp. 4226-4233. 

Wolter, Kirk M. (1985). Introduction to Variance Estimation. New York: Springer-Verlag. 

Vartivarian, S. and Little, R.J. (2002). On the Formation of Weighting Adjustment Cells for Unit Non-response. 

Proceedings of the Section of Survey Research Methods, American Statistical Association, pp. 3553-3558. 

Yung, W. and Rao, J.N.K. (1996). Jackknife Linearization Variance Estimators Under Stratified Multi-Stage 

Sampling. Survey Methodology, 22, pp. 23-31. 
Appendix 

Postulate 1: Given an unbiased systematic assignment of sampled units to K random groups within all strata and 

delete-a-group jackknife replicate factors of (K/K-1), the delete-a-group jackknife variance estimator is an 

unbiased estimator of the variance of an estimate of a total. 

Let = ∑∑ = a full sample estimate, 

h i

hi hi θ w y

ˆ ∑∑∉

⎟

⎠

⎞

⎜

⎝

⎛

−

=

h k i

k hi hi w y

K

K

1

ˆθ )(

= the delete-a-group jackknife 

replicate k estimate, ∑∑∈

= =

h k i

k hi hi θ w y

(

the random-group k estimate, and θ k = Kθ k =

(

ˆ

the randomgroup replicate k estimate. 

)

ˆ

(

ˆ

)

ˆ ˆ

(

( )1

1

)

ˆ ˆ

(

1

1 1

ˆ

)

ˆ

(

1

1

)

ˆ ˆ

(

1

)

ˆ

(

ˆ

1

2

1

2

2

1

2

1

2

)(

θ

θ θ

θ θ

θ θ θ

θ θ θ

RG

K

k

k

K

k

k

K

k

k

K

k

DAG k

V

K K

K K

K

K

K

K

K

K

K

V

=

⎟

⎠

⎞

⎜

⎝

⎛

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

=

⎟ −

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛ −

=

⎥

⎦

⎤

⎢

⎣

⎡

⎟ − −

⎠

⎞

⎜

⎝

⎛

−

⎟

⎠

⎞

⎜

⎝

⎛ −

=

⎟ −

⎠

⎞

⎜

⎝

⎛ −

=

∑

∑

∑

∑

=

=

=

=

(

Wolter (1985), pp. 21, proves that ) is an unbiased estimator of ˆ

( Vˆ

RG θ ). ˆ V (θ

Postulate 2: There are two valid approaches to replicating the non-response adjustment procedure for the count 

estimator. 

 

Consider the full-sample estimate in stratum h for a stratified SRS-WOR design with the count adjustment applied 

for unit non-response: ,

ˆ

h rh

hi

hi

h

h

hi

hi

h

h

h

h

h

y N y

r

N

y

r

n

n

N

Y = ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

= ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

= ∑ ∑ ∈ ∈

where rh are the respondent units in 

stratum h.

Approach 1 (used in this paper): 

Use replicate factors of (K/K-1) to produce replicate estimates, and replicate non-response adjustment as .

)(

)(

hg

hg

r

n

Advantage: 

• Unconditionally unbiased variance estimates, even if weighting cells are collapsed. 

Disadvantage: 

• Form of replicate estimates is different from full-sample estimator (replicate estimates are not HorvitzThompson estimates based on sample respondents). 

Approach 2 (advocated in Kott 2001) 

Use strata-specific replicate factors of nh/( nh- nhg). In the case of no collapsing of weighting cells, the replicate 

estimate k is given by 

 
.

ˆ

)(

 and 

)( gr

hi gi

hi h

h hg

h hg

h hg

h

h

h

gh

y N y

r r

n n

n n

n

n

N

Y ∑∈ ∉

=

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

Advantage: 

• Replicate estimates are Horvitz-Thompson estimates based on sample respondents (in the event of no 

weighting cell collapse), leading to a “pure” replication of the non-response adjustment procedure. 

Disadvantage: 

• Variance estimates are not unbiased (follows from postulate 1), although they are conditionally unbiased 

given the realized sample; 

• If weighting cells are collapsed, the form of the replicate estimate is no longer the same as the full-sample 

estimate.