Part A, Attachment B - Source and Accuracy Statement

Source of the Data and Accuracy of the Estimates for the
Household Pulse Survey – Phase 3.1

Interagency Federal Statistical Rapid Response Survey to Measure Household
Experiences during the Coronavirus (COVID-19) Pandemic

SOURCE OF THE DATA

The Household Pulse Survey (HPS), an experimental data product, is an Interagency
Federal Statistical Rapid Response Survey to Measure Household Experiences during the
Coronavirus (COVID-19) Pandemic, conducted by the United States Census Bureau in
partnership with 12 other agencies from the Federal Statistical System:
• Bureau of Labor Statistics (BLS)
• National Center for Health Statistics (NCHS)
• Department of Agriculture Economic Research Service (ERS)
• National Center for Education Statistics (NCES)
• Department of Housing and Urban Development (HUD)
• Social Security Administration (SSA)
• Bureau of Transportation Statistics (BTS)
• Centers for Disease Control (CDC)
• Maternal and Child Health Bureau (MCHB)
• Department of Defense (DOD)
• National Institute for Occupational Safety and Health (NIOSH)
• Office of Management and Budget (OMB)

These agencies collaborated on the design and provided content for the HPS, which was
also reviewed and approved by the Office of Management and Budget (OMB). (OMB #
0607-1013; expires October 31, 2023.)

The Household Pulse Survey (HPS) ended Phase 3 on March 31, 2021. We entered Phase
3.1 to continue collecting information on how the coronavirus pandemic is impacting
households across the country with modifications to the questions. Working with the Office
of Management and Budget (OMB), the HPS has been approved to continue collecting the
HPS with an expiration date of October 31, 2023. In Phase 3.1, we continue to collect and
disseminate data on a bi-weekly basis.
Phase 3.1 has introduced new content to the questionnaire. We added questions related to
telehealth, essential worker occupations, armed forces status, childcare, child preventive
health, and disability and dropped the questions in several sections that were deemed no
longer necessary. We continue to collect and disseminate data on employment status, food
security, housing, physical and mental health, access to health care, educational disruption,
application and receipt of benefits, spending patterns, availability of financial resources,
capacity to telework, and travel practices. The distinction between the phases is the change
in questionnaire content.
The HPS asks individuals about their experiences regarding employment status, spending
patterns, food security, housing, physical and mental health, access to health care, program

2
receipt, and educational disruption. The ability to understand how individuals are
experiencing this period is critical to governmental and non-governmental response in
light of business curtailment and closures, stay-at-home and safer-at-home orders, school
closures, changes in consumer patterns and the availability of consumer goods, and other
abrupt and significant changes to American life.

The HPS is designed to produce estimates at three different geographical levels. The first
level, the lowest geographical area, is for the 15 largest Metropolitan Statistical Areas
(MSAs). The second level of geography is state-level estimates for each of the 50 states plus
the District of Columbia, and the final level of geography is national-level estimates.
The U.S. Census Bureau conducted Phase 1 of the HPS every week starting April 23, 2020.
For details of Phase 1 see the Source and Accuracy Statements at:
https://www.census.gov/programs-surveys/household-pulse-survey/technicaldocumentation.html. Phase 1 of the Household Pulse Survey was collected and
disseminated on a weekly basis. Phase 1 collection ended July 21, 2020.

The first data collection of Phase 2 of the HPS was conducted over 13 days starting August
19, 2020 and ending August 31, 2020. Phase 2 data collection ended with Week 17 when
Census received full approval from OPM. Phase 3 data collection began with reference
Week 18 and ended March 31, 2023 (Week 27). Phase 3.1, with the new questionnaire
content began on April 14, 2021 with reference Week 28. Despite going to a two-week
collection period (in Phases 2 – 3.1), the Household Pulse Survey continues to call these
collection periods "weeks" for continuity with Phase 1. The table below provides the data
collection periods. This document will be updated after each data collection period until the
end of the survey.
Table 1. Data Collection Periods for Phase 3.1 of the
Household Pulse Survey
Data Collection Period
Phase 3.1 - Week* 28
Phase 3.1 - Week* 29
Phase 3.1 - Week* 30

Start Date
April 14, 2021
April 28, 2021
May 12, 2021

Finish Date
April 26, 2021
May 10, 2021
May 24, 2021

* For Phase 3.1 the Household Pulse Survey continues to call these
collection periods "weeks" for continuity with Phase 1.

Sample Design
The HPS utilizes the Census Bureau’s Master Address File (MAF) as the source of sampled
housing units (HUs). Phases 1-3 utilized the January 2020 MAF. Phase 3.1 takes advantage
of the latest MAF updates as of January 2021.

The sample design was a systematic sample of all eligible HUs, with adjustments applied to
the sampling intervals to select a large enough sample to create state level estimates 1 and
estimates for the top 15 MSAs. Sixty-six independent sample areas were defined. For each
1

Including the District of Columbia as a state.

3
data collection period, independent samples were selected and each sampled HU was
interviewed once, unlike the Phase 1 of the HPS.

Sample sizes were determined such that a three percentage coefficient of variation (CV) for
an estimate of 40 percent of the population would be achieved for all sample areas with the
exception of the 11 smallest states. In these smaller states, the sample size was reduced to
produce a 3.5 percent CV. The overall sample sizes within the sampling areas were
adjusted for an anticipated response rate of nine percent. For those counties in one of the
top MSAs, the sampling interval was adjusted to select the higher of the sampling rate for
either the state or MSA.
To enable the use of a rapid deployment internet response system, we added email and
mobile telephone numbers from the Census Bureau Contact Frame to the MAF. Since 2013,
the Census Bureau has maintained contact frames to allow appended contact information
onto sample units within household sample frames to aid in contacting respondents at
those households. The primary motivation for creating this contact frame was to support
research on potential contact strategies for the 2020 Census.

The Contact Frame information is maintained in two separate files – one containing phone
numbers (both landline and cell phones) and the other containing email addresses.
Information is obtained primarily from commercial sources, with additions from
respondents to the American Community Survey (ACS) and Census tests, as well as
participants in food and other assistance programs from a few states, as well as from the
Alaska Permanent Fund Division. Commercial sources were evaluated against respondent
reported phone numbers to determine which sources would be acquired, after determining
which vendors provided the best value for the government.

Commercial, survey, and administrative record data providers link phone numbers and
email addresses to physical addresses before providing them for the Contact Frame.
Addresses were matched to the MAF. For addresses matched with confidence, the contact
information was added to the frame along with the unique identifier from the MAF. In
Phase 3.1 we are using an updated Contact Frame. Approximately 145,000,000 housing
units are represented in the MAF and were considered valid for sampling. The phone
frame contains over a billion phone/address pairs, with over three quarters of those phone
numbers being acquired in the past two years. The email frame contains over 686 million
well-formed email/address pairs, and two-thirds of those emails were acquired in past two
years. Of the 145 million addresses in the MAF, 75 percent of valid addresses are associated
with at least one email, and 62 percent of valid addresses with at least one cell-phone
number, and a combined coverage of 81 percent with at least an email or cell-phone and
represent the frame for the Household Pulse Survey. Unique phone numbers and email
addresses were identified and assigned to only one HU. The HUs on MAF were then limited
to these addresses on the Contact Frame as the final eligible HUs for the HPS. Table 2
shows the number of addresses with updated contact information.

4
Table 2. Number of Addresses on the Master Address File
with Updated Contact Information
Total Addresses
Addresses with any contact information:
Addresses with cell phone
Addresses with email

144,755,000
117,691,000
89,119,000
108,064,000

Source: U.S. Census Bureau Master Address File Extracts and Contact Frame

Sampled households were contacted by both email if an email was available, and by text if a
phone number was available. Contact email addresses and phone numbers were only sent
on weekdays and reminders were sent to nonrespondents.
The Census Bureau conducted the HPS online using Qualtrics as the data collection
platform. Qualtrics is currently used at the Census Bureau for research and development
surveys and provides the necessary agility to deploy the HPS quickly and securely. It
operates in the Gov Cloud, is FedRAMP authorized at the moderate level, and has an
Authority to Operate from the Census Bureau to collect personally-identifiable and Titleprotected data.

Approximately 1,035,000 housing units were selected from the sampling frame for the first
collection period of Phase 3.1. Approximately 59,000 respondents answered the online
questionnaire. Table 3 shows the sample sizes and the number of responses by collection
period for Phase 3.1 of the HPS.
Table 3. Sample Size and Number of Respondents at the
National Level
Data Collection Period
Phase 3.1 - Week* 28
Phase 3.1 - Week* 29
Phase 3.1 - Week* 30

Sample Size
1,040,387
1,040,864
1,041,261

Number of Respondents
68,913
78,467
72,897

Source: U.S. Census Bureau, Household Pulse Survey
* For Phase 3.1 the Household Pulse Survey continues to call these
collection periods "weeks" for continuity with Phase 1.

State-level sample sizes and number of responses can be found in Table A1 on the
Appendix A1 tab in the State-level Quality Measures spreadsheet at
https://www.census.gov/programs-surveys/household-pulse-survey/technicaldocumentation.html under the Source and Accuracy Statements section.

Estimation Procedure
The final HPS weights are designed to produce biweekly estimates for the total persons age
18 and older living within HUs. These weights were created by adjusting the householdlevel sampling base weights by various factors to account for nonresponse, adults per
household, and coverage.
The sampling base weights for each incoming sample in each of the 66 sample areas are
calculated as the total eligible HUs in the sampling frame divided by the number of eligible

5
HUs selected for interviews each week. Therefore, the base weights for all sampled HUs
sum to the total number of HUs for which contact information is known.

The final HPS person weights are created by applying the following adjustments to the
sampling base weights:
1. Nonresponse adjustment – the weight of all sample units that did not respond to the
HPS are evenly allocated to the units that did respond within the same sample
collection period, sample area (MSA or balance of state) and state. After this step, the
weights of all respondents sum to the total HUs with contact information in the
sampling frame.
2. Occupied HU ratio adjustment – this adjustment corrects for undercoverage in the
sampling frame by inflating the HU weights after the nonresponse adjustment to
match independent controls for the number of occupied HUs within each state. Each
sampled respondent was assigned to the state where they reported their current
address, which may be different from the selected state. For this adjustment, the
independent controls are the 2019 American Community Survey (ACS) one-year,
state-level estimates available at www.census.gov 2. These controls were updated for
Phase 3.1.
3. Person adjustment – this adjustment converts the HU weights into person weights by
multiplying them by the number of persons age 18 and older that were reported to
live within the household. The number of adults is based on the subtracting the
number of children under 18 in the household from the number of total persons in
the household. This number was capped at 10 adults. If the number of total persons
and number of children was not reported, then it is imputed.
4. Iterative Raking Ratio to Population Estimates – this procedure controls the person
weights to independent population controls by various demographics within each
state. The ratio adjustment is done through an iterative raking procedure to
simultaneously control the sample estimates to two sets of population controls (both
updated for Phase 3.1) -- Educational attainment estimates from the 2019 1-year ACS
estimates (Table B15001) 3 by age and sex, and the July 1, 2021 Hispanic origin/race
by age and sex estimates from the Census Bureau’s Population Estimates Program
(PEP). PEP provided July 1, 2021 household population estimates by single year of
age (0-84, 85+), sex, race (31 groups), and Hispanic origin for states from the Vintage
2020 estimates series 4. The ACS 2019 estimates were adjusted to match the 2021
pop controls within states by sex, and the five age categories in the ACS educational
attainment estimates. Tables 4 and 5 show the demographic groups formed.
The one-year estimates are at this URL: https://www2.census.gov/programs-surveys/acs/summary_file/2019/data/
The1-year state-level detailed table B15001 is located at this URL:
https://data.census.gov/cedsci/table?q=B15001&g=0100000US.04000.001&tid=ACSDT1Y2019.B15001&hidePreview=t
rue
4 The Vintage 2019 estimates methodology statement is available at this URL: https://www2.census.gov/programssurveys/popest/technical-documentation/methodology/2010-2019/natstcopr-methv2.pdf The Modified Race Summary
File methodology statement is available at this URL: https://www2.census.gov/programs-surveys/popest/technicaldocumentation/methodology/modified-race-summary-file-method/mrsf2010.pdf
2
3

6
Before the raking procedure was applied, cells containing too few responses were
collapsed to ensure all cells met the minimum response count requirement. The cells
after collapsing remained the same throughout the raking. These collapsed cells
were used in the calculation of replicate weights.

Table 4: Educational Attainment Population Adjustment Cells within State

Age

No HS
diploma
Male

No HS
diploma
Female

HS
diploma
Male

18-24
25-34
35-44
45-64
65+

HS
diploma
Female

Some
college or
Associate’s
degree
Male

Some
college or
Associate’s
degree
Female

Bachelor’s
degree or
higher
Male

Bachelor’s
degree or
higher
Female

NonHispanic
Other
Races Male

NonHispanic
Other
Races
Female

Table 5: Race/Ethnicity Population Adjustment Cells within State
Age
18-24
25-29
30-34
35-39
40-44
45-49
50-54
55-64
65+

Hispanic
Any Race
Male

Hispanic
Any Race
Female

NonHispanic
WhiteAlone
Male

NonHispanic
WhiteAlone
Female

NonHispanic
BlackAlone
Male

NonHispanic
BlackAlone
Female

Starting in week 13, the microdata file also contains a household weight for creating
estimates of household-level characteristics. The final HPS household weights are created
by applying the following adjustments to the final HPS person weights:
1. Housing Unit adjustment – this adjustment converts the person level weight back
into a HU weight by dividing the person level weight by the number of persons age
18 and older that were reported to live within the household. The number of adults
is the same value used to create the person adjustment.
2. Occupied HU ratio adjustment – this adjustment ensures that the final HPS
household weights will sum to the 2019 American Community Survey (ACS) oneyear, state-level estimates available at www.census.gov2. This ratio adjustment is
the same adjustment applied to the person weights, but is needed again because
state totals may have changed as a result of the iterative raking adjustment in the
final step of the person weight creation.

7
The detailed tables released for this experimental Household Pulse Survey show frequency
counts rather than percentages. Showing the frequency counts allows data users to see the
count of cases for each topic and variable that are in each response category and in the ‘Did
Not Report’ category. This ‘Did Not Report’ category is not a commonly used data category
in U.S. Census Bureau tables. Most survey programs review these missing data and
statistically assign them to one of the other response categories based on numerous
characteristics.
In these tables, the Census Bureau recommends choosing the numerators and
denominators for percentages carefully, so that missing data are deliberately included or
excluded in these counts. In the absence of external information, the percentage based on
only the responding cases will most closely match a percentage that would result from
statistical imputation. Including the missing data in the denominator for percentages will
lower the percentages that are calculated.

Microdata will be available by FTP in the future. Users may develop statistical imputations
for the missing data but should ensure that they continue to be deliberate and transparent
with their handling of these data.
ACCURACY OF THE ESTIMATES

A sample survey estimate has two types of error: sampling and nonsampling. The accuracy
of an estimate depends on both types of error. The nature of the sampling error is known
given the survey design; the full extent of the nonsampling error is unknown.

Sampling Error
Since the HPS estimates come from a sample, they may differ from figures from an
enumeration of the entire population using the same questionnaires, instructions, and
enumeration methods. For a given estimator, the difference between an estimate based on
a sample and the estimate that would result if the sample were to include the entire
population is known as sampling error. Standard errors, as calculated by methods
described below in “Standard Errors and Their Use,” are primarily measures of the
magnitude of sampling error. However, the estimation of standard errors may include
some nonsampling error.
Nonsampling Error
For a given estimator, the difference between the estimate that would result if the sample
were to include the entire population and the true population value being estimated is
known as nonsampling error. There are several sources of nonsampling error that may
occur during the development or execution of the survey. It can occur because of
circumstances created by the respondent, the survey instrument, or the way the data are
collected and processed. Some nonsampling errors, and examples of each, include:
•

Measurement error: The respondent provides incorrect information, the
respondent estimates the requested information, or an unclear survey question
is misunderstood by the respondent.

8
•
•
•

Coverage error: Some individuals who should have been included in the survey
frame were missed.
Nonresponse error: Responses are not collected from all those in the sample or
the respondent is unwilling to provide information.
Imputation error: Values are estimated imprecisely for missing data.

To minimize these errors, the Census Bureau applies quality control procedures during all
stages of the production process including the design of the survey, the wording of
questions, and the statistical review of reports.

Two types of nonsampling error that can be examined to a limited extent are nonresponse
and undercoverage.
Nonresponse
The effect of nonresponse cannot be measured directly, but one indication of its potential
effect is the nonresponse rate. Table 6 shows the unit response rates by collection period.
Table 6. National Level Weighted Response
Rates by Collection Period for the Household
Pulse Survey
Data Collection Period
Phase 3.1 - Week* 28
Phase 3.1 - Week* 29
Phase 3.1 - Week* 30

Response Rate (Percent)
6.6
7.4
6.8

Source: U.S. Census Bureau, Household Pulse Survey
* For Phase 3.1 the Household Pulse Survey continues to call
these collection periods "weeks" for continuity with Phase 1.

State-level response rates can be found in Table A1 on the Appendix A1 tab in the Statelevel Quality Measures spreadsheet at https://www.census.gov/programssurveys/household-pulse-survey/technical-documentation.html under the Source and
Accuracy Statements section.
In accordance with Census Bureau and Office of Management and Budget Quality
Standards, the Census Bureau will conduct a nonresponse bias analysis to assess
nonresponse bias in the HPS.

Responses are made up of complete interviews and sufficient partial interviews. A
sufficient partial interview is an incomplete interview in which the household or person
answered enough of the questionnaire to be considered a complete interview. Some
remaining questions may have been edited or imputed to fill in missing values. Insufficient
partial interviews are considered to be nonrespondents.
Undercoverage
The concept of coverage with a survey sampling process is defined as the extent to which
the total population that could be selected for sample “covers” the survey’s target
population. Missed housing units and missed people within sample households create

9
undercoverage in the HPS. A common measure of survey coverage is the coverage ratio,
calculated as the estimated population before poststratification divided by the independent
population control. The national household-level coverage ratio is 0.96. State householdlevel coverage ratios can be found in Table A1 on the Appendix A1 tab in the State-level
Quality Measures spreadsheet at https://www.census.gov/programs-surveys/householdpulse-survey/technical-documentation.html under the Source and Accuracy Statements
section.

HPS person coverage varies with age, sex, Hispanic origin/race, and educational
attainment. Generally, coverage is higher for females than for males and higher for nonBlacks than for Blacks. This differential coverage is a general problem for most householdbased surveys. The HPS weighting procedure tries to mitigate the bias from undercoverage
within the raking procedure. However, due to small sample sizes, some demographic cells
needed collapsing to increase sample counts within the raking cells. In this case
convergence to both sets of the population controls was not attained. Therefore, the final
coverage ratios are not perfect for some demographic groups. Table 7 shows the coverage
ratios for the person demographics of age, sex, Hispanic origin/race, and educational
attainment before and after the raking procedure is run.
Table 7. Person-Level Coverage Ratios at the National Level
for Household Pulse Survey Before and After Raking for
Collection Week* 30: May 12 – May 24, 2021
Demographic Characteristic
Total Population
Male
Female
Age 18-24
Age 25-29
Age 30-34
Age 35-39
Age 40-44
Age 45-49
Age 50-54
Age 55-64
Age 65+
Hispanic
Non-Hispanic white-only
Non-Hispanic black-only
Non-Hispanic other races
No high-school diploma
High-school diploma
Some college or associate’s degree
Bachelor’s degree or higher

Before Raking
1.05
0.89
1.20
0.35
0.49
0.74
1.08
1.24
1.22
1.36
1.35
1.24
0.75
1.18
0.72
1.08
0.23
0.47
1.11
1.81

After Raking
1.00
1.00
1.00
0.80
0.91
1.05
1.08
1.14
1.00
1.05
1.02
1.02
1.03
1.00
0.95
1.05
0.71
1.12
0.99
1.02

Source: U.S. Census Bureau, Household Pulse Survey
* For Phase 3.1 the Household Pulse Survey continues to call these
collection periods "weeks" for continuity with Phase 1.

The previous data collection’s national person-level coverage ratios and state person-level
coverage ratios can be found in Table A2 on the Appendix A2 tab in the State-level Quality

10
Measures spreadsheet at https://www.census.gov/programs-surveys/household-pulsesurvey/technical-documentation.html under the Source and Accuracy Statements section.

Biases may also be present when people who are missed by the survey differ from those
interviewed in ways other than age, sex, Hispanic origin/race, educational attainment, and
state of residence. How this weighting procedure affects other variables in the survey is
not precisely known. All of these considerations affect comparisons across different
surveys or data sources.
Comparability of Data
Data obtained from the HPS and other sources are not entirely comparable. This is due to
differences in data collection processes, as well as different editing procedures of the data,
within this survey and others. These differences are examples of nonsampling variability
not reflected in the standard errors. Therefore, caution should be used when comparing
results from different sources.
A Nonsampling Error Warning
Since the full extent of the nonsampling error is unknown, one should be particularly
careful when interpreting results based on small differences between estimates. The
Census Bureau recommends that data users incorporate information about nonsampling
errors into their analyses, as nonsampling error could impact the conclusions drawn from
the results. Caution should also be used when interpreting results based on a relatively
small number of cases.

Standard Errors and Their Use
A sample estimate and its standard error enable one to construct a confidence interval. A
confidence interval is a range about a given estimate that has a specified probability of
containing the average result of all possible samples. For example, if all possible samples
were surveyed under essentially the same general conditions and using the same sample
design, and if an estimate and its standard error were calculated from each sample, then
approximately 90 percent of the intervals from 1.645 standard errors below the estimate
to 1.645 standard errors above the estimate would include the average result of all possible
samples.
A particular confidence interval may or may not contain the average estimate derived from
all possible samples, but one can say with the specified confidence that the interval
includes the average estimate calculated from all possible samples.

The context and meaning of the estimate must be kept in mind when creating the
confidence intervals. Users should be aware of any “natural” limits on the bounds of the
confidence interval for a characteristic of the population when the estimate is near zero –
the calculated value of the lower bound of the confidence interval may be negative. For
some estimates, a negative lower bound for the confidence interval does not make sense,
for example, an estimate of the number of people with a certain characteristic. In this case,
the lower confidence bound should be reported as zero. For other estimates such as

11
income, negative confidence bounds can make sense; in these cases, the lower confidence
interval should not be adjusted. Another example of a natural limit is 100 percent as the
upper bound of a percent estimate.

Standard errors may also be used to perform hypothesis testing, a procedure for
distinguishing between population parameters using sample estimates. The most common
type of hypothesis is that the population parameters are different. An example of this
would be comparing the percentage of adults in households where someone had a loss in
employment income since March 13, 2020 from week 1 to week 2.

Tests may be performed at various levels of significance. A significance level is the
probability of concluding that the characteristics are different when, in fact, they are the
same. For example, to conclude that two characteristics are different at the 0.10 level of
significance, the absolute value of the estimated difference between characteristics must be
greater than or equal to 1.645 times the standard error of the difference.
The Census Bureau uses 90-percent confidence intervals and 0.10 levels of significance to
determine statistical validity. Consult standard statistical textbooks for alternative criteria.
Estimating Standard Errors
The Census Bureau uses successive difference replication to estimate the standard errors
of HPS estimates. These methods primarily measure the magnitude of sampling error.
However, they do measure some effects of nonsampling error as well. They do not
measure systematic biases in the data associated with nonsampling error. Bias is the
average over all possible samples of the differences between the sample estimates and the
true value.
Eighty replicate weights were created for the HPS. Using these replicate weights, the
variance of an estimate (the standard error is the square root of the variance) can be
calculated as follows:
80

4
2
𝑉𝑉𝑉𝑉𝑉𝑉�𝜃𝜃�� =
��𝜃𝜃𝑖𝑖 − 𝜃𝜃��
80
𝑖𝑖=1

(1)

where 𝜃𝜃� is the estimate of the statistic of interest, such as a point estimate, ratio of domain
means, regression coefficient, or log-odds ratio, using the weight for the full sample and 𝜃𝜃𝑖𝑖
are the replicate estimates of the same statistic using the replicate weights. See reference
Judkins (1990).

Creating Replicate Estimates
Replicate estimates are created using each of the 80 weights independently to create 80
replicate estimates. For point estimates, multiply the replicate weights by the item of
interest to create the 80 replicate estimates. You will use these replicate estimates in the
formula (1) to calculate the total variance for the item of interest. For example, say that the
item you are interested in is the difference in the number of people with a loss in

12
employment income in week 1 compared to the number of people with a loss in
employment income in week 2. You would create the difference of the two estimates using
the sample weight, 𝑥𝑥�0 , and the 80 replicate differences, 𝑥𝑥𝑖𝑖 , using the 80 replicate weights.
You would then use these estimates in the formula to calculate the total variance for the
difference in the number of people with a loss in employment income from week 1 to week
2.
80

4
𝑉𝑉𝑉𝑉𝑉𝑉(𝑥𝑥�0 ) =
�(𝑥𝑥𝑖𝑖 − 𝑥𝑥�0 )2
80
𝑖𝑖=1

Where 𝑥𝑥𝑖𝑖 is the ith replicate estimate of the difference and 𝑥𝑥�0 is the full estimate of the
difference using the sample weight.

Users may want to pool estimates over multiple weeks by creating averages for estimates
with small sample sizes. For pooled estimates, where two or more weeks of data are
combined to make one estimate for a longer time period, one would divide the unit-level
weights that formed 𝑥𝑥�0 and 𝑥𝑥𝑖𝑖 (for each of the 80 replicate weights) for each week by the
number of weeks that are combined. Then, form 80 replicate pooled estimates, 𝑥𝑥�𝑖𝑖,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
and the estimate, 𝑥𝑥�0,𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 . Then use the pooled estimates in formula (1) to calculate the
pooled variance for the item of interest.

Example for Variance of Regression Coefficients
Variances for regression coefficients 𝛽𝛽0 can be calculated using formula (1) as well. By
calculating the 80 replicate regression coefficients 𝛽𝛽𝑖𝑖 ′𝑠𝑠 for each replicate and plugging in
the replicate 𝛽𝛽𝑖𝑖 estimates and the 𝛽𝛽0 estimate into the above formula,
80

4
2
𝑉𝑉𝑉𝑉𝑉𝑉�𝛽𝛽̂0 � =
��𝛽𝛽𝑖𝑖 − 𝛽𝛽̂0 �
80
𝑖𝑖=1

gives the variance estimate for the regression coefficient 𝛽𝛽0 .
TECHNICAL ASSISTANCE

If you require assistance or additional information, please contact the Demographic
Statistical Methods Division via e-mail at [email protected].
REFERENCES

Judkins, D. (1990) “Fay’s Method for Variance Estimation,” Journal of Official
Statistics, Vol. 6, No. 3, 1990, pp.223-239.