UNITED NATIONS SECRETARIAT ESA/STAT/AC.93/5 

Statistics Division 03 November 2003 

English only

Expert Group Meeting to 

Review the Draft Handbook on

Designing of Household Sample Surveys

3-5 December 2003

D R A F T

Construction and use of sample weights*

by

Ibrahim S. Yansaneh**

 

*

This document is being issued without formal editing.

** The views expressed in this paper are those of the author and do not imply the expression of any opinion on the 

part of the United Nations Secretariat.
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Table of contents

Chapter Five: Construction and use of sample weights.................................................................3

5.1. The need for sampling weights........................................................................................3

5.2. The development of sampling weights ............................................................................3

5.2.1. Adjustments of sample weights for unknown eligibility .........................................4

5.2.2. Adjustments of sample weights for duplicates ........................................................4

5.3. Weighting for unequal probabilities of selection.............................................................4

5.3.1. A case study in construction of weights: the Viet Nam National Health Survey...6

5.3.2. Self weighting samples ............................................................................................6

5.4. The adjustment of sample weights for non-response.......................................................6

5.4.1. Reducing non-response bias in household surveys..................................................7

5.4.2. Compensating for non-response bias.......................................................................7

5.4.3. Non-response adjustment of sample weights...........................................................8

5.5. The adjustment of sample weights for non-coverage ....................................................10

5.5.1. Sources of non-coverage in household surveys .....................................................10

5.5.2. Compensating for non-coverage in household surveys .........................................11

5.6. Increase in variance due to weighting............................................................................12

5.7. Concluding remarks .......................................................................................................12

References and further reading ......................................................................................................14

Abstract

This chapter provides a brief overview of the various stages in the construction and adjustment 

of sample weights to be used in the analysis of survey data. In particular, the adjustment of 

sample weights to compensate for non-coverage and non-response is described. In addition, the 

chapter discusses how sample weights are used in the development of estimates of characteristics 

of interest. The important ideas presented are illustrated using real examples of current surveys 

conducted in developing countries, or ones that mimic real survey situations. 

Key Words. Base weight; non-response adjustment; post-stratification; domain estimation
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Chapter Five: Construction and use of sample weights

5.1. The need for sampling weights

1. Sampling weights are needed to correct for imperfections in the sample that might lead to 

bias and other departures between the sample and the reference population. Such impe rfections 

include the selection of units with unequal probabilities, non-coverage of the population, and 

non-response. In other words, the purposes of weighting are:

a. To compensate for unequal probabilities of selection.

b. To compensate for (unit) non-response.

c. To adjust the weighted sample distribution for key variables of interest (for example, 

age, race, and sex) to make it conform to a known population distribution. 

2. We shall discuss in detail the procedures underlying each of these scenarios in the 

sections that follow. Once the imperfections in the sample are compensated for, weights can 

then be used in the estimation of population characteristics of interest and also in the estimation 

of the sampling errors of the survey estimates generated.

(Give an example to illustrate what happens when weights are not used)

5.2. The development of sampling weights

3. The development of sampling weights usually starts with the construction of the base 

weight for each sampled unit, to correct for their unequal probabilities of selection. In general, 

the base weight of a sampled unit is the reciprocal of its probability of selection into the sample. 

In mathematical notation, if a unit is included in the sample with probability Pi, then its base 

weight, denoted by wi, is given by

wi = 1/ pi.

4. For example, a sampled unit selected with probability 1/50 represents 50 units in the 

population from which the sample was drawn. Thus sample weights act as inflation factors to 

represent the number of units in the survey population that are accounted for by the sample unit 

to which the weight is assigned. The sum of the sample weights provides an unbiased estimate 

of the total number of individuals in the target population.

5. For multi-stage designs, the base weights must reflect the probabilities of selection at 

each stage. For instance, in the case of a two-stage design in which the i-th PSU is selected with 

probability pi at the first stage, and the j-th household is selected within a selected PSU with 

probability pj(i) at the second stage, then the overall probability of selection of the every

household in the sample is given by

pij = pi* pj(i)
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and the overall base weight the household is obtained as before, by taking the reciprocal of its 

overall probability of selection. Correspondingly, if the base weight for the j-th household is 

wij,b, and the weight attributable to compensation for non-response is wij,nr, and the weight 

attributable to the compensation for non-coverage is wij,nc, then the overall weight of the

household is given by:

wij = wij,b* wij,nr * wij,nc

5.2.1. Adjustments of sample weights for unknown eligibility

(Discuss weighting for unknown eligibility)

5.2.2. Adjustments of sample weights for duplicates

6. If it is known a priori that some units have duplicates on the frame, then increased 

probability of selection of such units can be compensated for by assigning to them weighting 

factors that are reciprocals of the number of duplicate listings on the frame if such units end up in 

the sample. Often however, duplicates are discovered only after the sample is selected, and the 

probabilities of selection of such sampled units need to be adjusted to account for the

duplication. This adjustment is implemented as follows: 

Suppose the i-th sampled unit has a probability of selection, denoted by pi1 and suppose there 

are k-1 additional records on the sampling frame that are identified by this sampled unit as 

duplicates, each with selection probabilities given by pi2,. . ., pik. Then, the adjusted probability 

of selection of the sampled unit in question is given by 

pi= 1 - (1 - pi 1)(1 - pi2) ... (1 - pik)

The sampled unit is then weighted accordingly, that is, by 1/pi.

7. We now illustrate the procedures for constructing sample weights under scenarios

outlined above, with specific examples.

5.3. Weighting for unequal probabilities of selection

8. An epsem sample of 5 households is selected from 250. One adult is selected at random 

in each sampled household. The monthly income (yij) and the level of education (zij= 1, if 

secondary or higher; 0 otherwise) of the j-th sampled adult in the i-th household are recorded. 

Let Mi denote the number of adults in household i. Then, the overall probability of selection of a 

sampled adult is given by:

i i

ij i j i M M

p p p

1

50

1 1

250

5

= × ( ) = × = ×

Therefore, the weight of a sampled adult is given by:
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i

ij

i M

p

w = = 50×

1

ß Example

9. To illustrate the estimation procedure, let us assume a first-stage sample of 5 households 

with data obtained from the single sampled adult for each household as given in the table below:

Sampled Household Mi wi yij zij wiyij wizij wizijyij

1 3 150 70 1 10,500 150 10,500

2 1 50 30 0 1,500 0 0

3 3 150 90 1 13,500 150 13,500

4 5 250 50 1 12,500 250 12,500

5 4 200 60 0 12,000 0 0

TOTAL 16 800 300 3 50,000 550 36,500

10. Estimates of various characteristics can then be obtained from the above table as follows:

a. The estimate of monthly income is

62.5

800

50,000

= = =

∑

∑

i

i ij

w w

w y

y

If weights were not used, this estimate would be 60 (=300/5) 

b. The estimate of the proportion of people with secondary or higher education is

0.6875

800

550

= = =

∑

∑

i

i ij

w w

w z

y or 68.75%

If weights are not used, this estimate would be 3/5 or 0.60 or 60%.

c. The estimate of the total number of people with secondary or higher education is

t

ˆ =∑wi

zij = 550

d. The estimate of the mean monthly income of adults with secondary or higher

education is

66.36

550

36,500

= = =

∑

∑

i ij

i ij ij

w w z

wz y

y

11. Note that for estimating totals, sampled elements need to be weighted by the reciprocal of 

their selection probabilities. For estimating means and proportions, the weights need only be 
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proportional to the reciprocals of the selection probabilities. Thus, in the preceding example, 

the weights wi’s are proportional to Mi (wi=50* Mi). If we use Mi as the weights, then the 

estimate of the proportion with secondary or higher education is 

0.6875

16

11

3 1 3 5 4

3 1 1 0 3 1 5 1 4 0

ˆ = =

+ + + +

× + × + × + × + ×

=

∑

∑

i

i ij

M

M z

p or 68.75%, 

as before. However, the estimate of the total number of adults with secondary or higher 

education is 

pˆ

s = 50∑ Mi

zij = 50×11= 550

5.3.1. A case study in construction of weights: the Viet Nam National Health Survey

12. We now proceed to illustrate the construction of the sampling weights for an actual 

survey, the National Health Survey conducted in Viet Nam in 2001.

(Insert a case study of weight construction for the VNHS)

5.3.2. Self weighting samples

13. When the weights of all sampled units are the same, the sample is referred to as selfweighting. Samples are rarely self-weighting at the national level for several reasons. First, 

sampling units are selected with unequal probabilities of selection. Indeed, even though the 

PSUs are often selected with probability proportional to size, and households selected at an 

appropriate rate within PSUs to yield a self-weighting design, this may be nullified by the 

selection of one person for interview in each sampled household. Second, the selected sample 

often has deficiencies including non-response and under-coverage (see sections 5.4 and 5.5). 

Third, the need for precise estimates for domains and special subpopulations often requires oversampling these domains (see section 5.5).

5.4. The adjustment of sample weights for non-response

14. It is rarely the case that all desired information is obtained from all sampled units in 

surveys. For instance, some households may provide no data at all while other households may 

provide only partial data, that is, data on some but not all questions in the survey. The former 

type of non-response is called unit or total non-response, while the latter is called item nonresponse. If there are any systematic differences between the respondents and non-respondents, 

then naïve estimates based solely on the respondents will be biased. It is important to keep 

survey non-response as low as possible, in order to reduce the possibility that the survey

estimates could be biased in some way by failing to include (or including a disproportionately 

small percentage of) a particular portion of the population. For example, persons who live in 
5-7

urban areas and have relatively high incomes might be less likely to participate in a multipurpose survey that includes income modules. Failure to include a large segment of this portion 

of the population could affect national estimates of average household income, educational

attainment, literacy, etc.

5.4.1. Reducing non-response bias in household surveys

15. The size of the non-response bias for a sample mean, for instance, is a function of two 

factors:

− The proportion of the population that does not respond.

− The size of the difference in population means between respondent and nonrespondent groups.

16. Reducing the bias due to non-response therefore requires that either the non-response rate 

be small, or that there are small differences between responding and non-responding households 

and persons. With proper record keeping of every sampled unit that is selected for the survey, it 

is possible to estimate directly from the survey data, the non-response rate for the entire sample 

and for sub-domains of interest. Furthermore, special carefully designed studies can be carried 

out to evaluate the differences between respondents and non-respondents (Groves and Couper, 

1998).

17. For panel surveys (in which data are collected from the same panel of sampled units 

repeatedly over time) the survey designer has access to more data for studying and adjusting for 

the effects of potential non-response bias than in one time or cross-sectional surveys. Here, nonresponse may arise from units being lost over the course of the survey, or refusing to participate 

in the survey after a while due to respondent fatigue or other reasons, and so on. Data collected 

on previous panel waves can then be used to learn more about differences between respondents 

and non-respondents, and to serve as the basis for the kind of adjustments described below. 

More details on various techniques used for compensating for non-response in survey research 

are provided in Brick and Kalton (1996), Lepkowski (1988), and references cited therein.

5.4.2. Compensating for non-response bias 

18. A number of techniques can be employed to reduce the potential for non-response bias in 

household surveys. The standard method of compensating for partial or item non-response is 

imputation, which is not discussed in this volume. A good introduction to imputation methods 

for large complex datasets is provided by Yansaneh et. al. (1998). For unit or total non-response, 

there are three basic procedures for compensation:

a. Non-response adjustment of the weights.

b. Drawing a larger sample than needed and creating a reserve sample from which 

replacements are selected in case of non-response.

c. Substitution, the process of replacing a non-responding household with another

household that was not sampled which is in close proximity to the non-responding 

household with respect to the characteristic of interest.
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19. It is advisable that unit non-response in household surveys be always handled by

adjusting the sample weights to account for non-responding households. In many surveys in 

developing countries, substitution is frequently used. However, this procedure increases the 

probabilities of selection for the potential substitutes, because non-sampled households close to 

non-responding sampled households have a higher probability of selection than those close to 

responding sampled households. Furthermore, attempts to substitute for non-responding 

households are time-consuming, prone to errors and bias, and very difficult to check or monitor. 

For example, a substitution may be made using a convenient household rather than the household 

specifically designated to serve as the substitute or replacement for a non-responding household, 

thereby introducing bias. 

5.4.3. Non-response adjustment of sample weights

20. The procedure of adjusting sample weights for non-response is the preferred practice in 

major household surveys throughout the world. Essentially, the adjustment transfers the base 

weights of all eligible non-responding sampled units to the responding units, and is implemented 

in the following steps:

Step 1: Apply the initial weights (for unequal selection probabilities and other adjustments 

discussed in Section 5.2, if applicable);

Step 2: Partition the sample into subgroups and compute weighted response rates for each 

subgroup;

Step 3: Use the reciprocal of the subgroup response rates for non-response adjustments; and

Step 4: Calculate the non-response adjusted weight for the i-th unit as:

wi = w1i * w2i, 

where w1i is the initial weight and w2i is the non-response adjustment weight. Note that the 

weighted non-response rate can be defined as the ration of the weighted number of interviews 

completed with eligible sampled cases to the weighted number of eligible sampled cases.

21. We now illustrate the ideas presented in this section with an example.

ß Example

22. A stratified multi-stage sample of 1000 households is selected from two regions (North 

and South) of a country. Households in the North are sampled at a rate of 1/100 and those in the 

south at a rate of 1/200. Response rates in urban areas are lower that those in rural areas. Let nh

denote the number of households sampled in stratum h, let rh denote the number of eligible 

households that responded to the survey, and let th denote the number of responding households 
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with access to primary health care. Then, the non-response adjusted weight for the households in 

stratum h is given by:

wh = w1h*w2h,

where w2h= nh/ rh. Assume that the stratum-level data are as given in the following table:

Stratum nh rh th w1h w2h wh wh rh wh th

North-Urban 100 80 70 100 1.25 125 10,000 8,750

North-Rural 300 120 100 100 2.50 250 30,000 25,000

South-Urban 200 170 150 200 1.18 236 40,120 35,400

South-Rural 400 360 180 200 1.11 222 79,920 39,960

Total 1,000 730 500 160,040 109,110

23. Therefore, the estimated proportion of households with access to primary health care is:

0.682

160,040

109,110 ˆ = = =

∑

∑

h h

h h

w r

w t

p or 68.2%

The estimated number of households with access to primary health care is

t

ˆ =∑wh

t

h =109,110 = 68.2% of 160,040

Note that the unweighted estimated proportion of households with access to primary health care, 

using only the respondent data is 

0.685

730

500 ˆ = = =

∑

∑

h

h

uw

r

t

p or 68.5%,

and the estimated proportion using the initial weights without non-response adjustment is

0.659

126,000

83,000 ˆ

1

1

1 = = =

∑

∑

h h

h h

w r

w t

p or 65.9%.

24. Note also this example is provided for the purpose of illustrating how initial weights are 

adjusted to compensate for non-response. The results show considerable disparity between the 

estimated proportion using only the initial weights compared to that using non-response adjusted 

weights, but the difference between the unweighted proportion and the non-response-adjusted 

proportion appears to be negligible.
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25. After non-response adjustments of the weights, further adjustments can be made to the 

weights as appropriate. In the next section, we consider adjustment of the weights to account for 

non-coverage.

5.5. The adjustment of sample weights for non-coverage

26. Non-coverage refers to the failure of the sampling frame to cover all of the target 

population and thus some sampling units have no probability of selection into the sample 

selected for the household survey. This is just one of many possible deficiencies of sampling 

frames used to select samples for surveys in developing countries. See Yansaneh (2003), and 

references cited therein, for a detailed discussion of sampling frame problems and some possible 

solutions. 

5.5.1. Sources of non-coverage in household surveys

27. Most household surveys in developing countries are based on stratified multi-stage area 

probability designs. The first-stage units, or primary sampling units, are usually geographic area 

units. At the second stage, a list of households or dwelling units is created, from which the 

sample of households is selected. At the last stage, a list of house members or residents is 

created, from which the sample of persons is selected. Thus non-coverage may occur at three 

levels: the PSU level, the household level, and the person level. 

28. Since PSUs are generally based on enumeration areas identified and used in a preceding 

population and housing census, they are expected to cover the entire geographic extent of the 

target population. Thus, the size of PSU non-coverage is generally small. For household

surveys in developing countries, PSU non-coverage is not as serious as non-coverage at 

subsequent stages of the design. However, non-coverage of PSUs does occur in most surveys. 

For instance, a survey may be designed to provide estimates for the entire population in a 

country, or a region of a country, but some PSUs may be excluded on purpose at the design 

stage, because some regions of a country are inaccessible due to civil war or unrest, a natural 

disaster, or other reasons. Also, remote areas with very few households or persons are

sometimes removed from the sampling frames for household surveys because they are too costly 

to cover, and they represent a small proportion of the population and so have very little effect on 

the population figures. In reporting results for such a survey, the exclusion of these areas must 

be explicitly stated. The impression should not be created that survey results apply to the entire 

country or region, when in fact a portion of the population is not covered. The non-coverage 

properties of the survey must be fully reported in the survey report.

29. Non-coverage becomes a more serious problem at the household level. Most surveys 

consider households to be the collection of persons who are usually related in some way, and 

who usually reside in a dwelling or housing unit. There are important definitional issues to 

resolve, such as who is a usual resident; and what is a dwelling unit? How are multi-unit 

structures (such as apartment buildings) and dwelling units with multiple households handled? It 

may be easy to identify the dwelling unit, but complex social structures may make it difficult to 

identify the households within the dwelling unit. There is thus a lot of potential for

misinterpretation or inconsistent interpretation of these concepts by different interviewers, or in 
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different countries or cultures. In any event, strict operational instructions are needed to guide 

interviewers on whom to consider a household member or what to consider a dwelling unit.

30. Other factors that contribute to non-coverage include the inadvertent omission of

dwelling units from listings prepared during field operations, or sub-populations of interest (for 

example, young children or the elderly), and omissions due to errors in measurement, noninclusion of absent household members, and omissions due to misunderstanding of survey

concepts. There is also a temporal dimension to the problem, that is, dwelling units may be 

unoccupied or under construction at the time of listing, but become occupied at the time of data 

collection. For household surveys in developing countries, the non-coverage problem is

exacerbated by the fact that most censuses in developing countries, the unique basis for

constructing sampling frames, do not provide detailed addresses of sampling units at the

household and person levels. Frequently, out of date or inaccurate administrative household 

listings are used, and individuals within a household are deliberately or accidentally omitted 

from a household listing of residents. More details on sources of non-coverage are provided in 

Lepkowski (2003) and references cited therein.

5.5.2. Compensating for non-coverage in household surveys

31. Non-coverage is a major concern for household surveys conducted in developing

countries. Evidence of the impact of non-coverage can be seen from the fact that sample 

estimates of population counts based on most developing-country surveys fall well short of 

population estimates from other sources. There are several procedures for handling the problem 

of non-coverage in household surveys (Lepkowski, 2003). These include:

a. Improved field procedures such as the use of multiple frames and improved listing 

procedures.

b. Compensating for the non-coverage through a statistical adjustment of the weights.

32. In this chapter, we shall concentrate on the second procedure. If reliable control totals 

are available for the entire population and for specified subgroups of the population, one could 

attempt to adjust the weights of the sample units in such a way as to make the sum of weights 

match the control totals within the specified subgroups. The subgroups are called post-strata, 

and the statistical adjustment procedure is called post-stratification. This procedure

simultaneously compensates for non-response and non-coverage. It adjusts the weighted

sampling distribution for certain variables so as to conform to a known population distribution. 

See Lehtonen and Pahkinen (1995) for some practical examples of how to analyze survey data 

with poststratification. A simple example is provided below, just to aid understanding of the 

procedure.

ß Example

33. In the preceding example, suppose that the number of households is known to be 45,025 

in the North and 115,800 in the South. Suppose further that the weighted sample totals are 

respectively 40,000 and 120,040.
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Step 1: Compute the post-stratification factors. 

For the North region, we have: 1.126

40,000

45,025

w3h = = ; and

For the South region, we have: 0.965

120,040

115,800

w3h = = .

Step 2: Compute final, adjusted weight: wf = wh ×w3h

34. The numerical results are summarized in the following table:

Stratum rh th wh wf wf*rh wf*th

North-Urban 80 70 125 140.75 11,256 9,849

North-Rural 120 100 250 281.40 33,768 28,140

South-Urban 170 150 236 227.77 38,709 34,155

South-Rural 360 180 222 214.20 77,112 38,556

Total 730 500 160,845 110,700

Therefore, the estimated proportion of households with access to primary health care is:

0.688

160,845

110,700 ˆ = = =

∑

∑

f h

f h

f w r

w t

p or 68.8%

35. Note that with the weights adjusted by post-stratification, the weighted sample counts for 

the North and South regions are respectively 45,024 (11,256+33,768) and 115,821

(38,709+77,112), which closely match the control totals given above. 

5.6. Increase in variance due to weighting

(...)

5.7. Concluding remarks

36. Sample weights have now come to be regarded as an integral part of the analysis of 

household survey data in developing countries, as in the rest of the world. Most survey

programmes now advocate the use of weights even in the rare situations involving self-weighting 

samples (in which case the weights would be 1). In the past, tremendous efforts were expended 

by survey designers for the virtually unattainable goal of achieving self-weighting samples and 

making weights unnecessary in the analysis of survey data. The conventional wisdom was that 

the use of weights made the analyses too complicated, and that there was very little, if any, 

computing infrastructure for weighted analysis of survey data. However, advances in computer 

technology in past decade have invalidated this argument. Computer hardware and software are 
5-13

now affordable and available in many developing countries. In addition, many specialized

computer software packages are now available specifically for the analysis of survey data. These 

are reviewed and compared in chapter 6. More details can be obtained from the references cited 

in chapter 6. 

37. As discussed, the use of weights reduces biases due to imperfections in the sample related 

to non-coverage and non-response. Non-response and non-coverage are different types of error 

due to the failure of a designed survey to obtain information from some units in the target 

population. For household surveys in developing countries, non-coverage is a more serious 

problem than non-response. The chapter provides examples of procedures for developing and 

statistically adjusting the basic weights to compensate for these unavoidable problems of

household surveys, and for using the adjusted weights in the estimation of parameters of interest. 

The advent of fast-speed computers and affordable of free statistical software should make the 

use of weights a routine aspect of the analysis of household survey data even in developing 

countries.
5-14

References and further reading

Brick, J.M. and Kalton, G. (1996). “Handling missing data in survey research,” Statistical 

Methods in Medical Research, Vol. 5, 215-238.

Cochran, W.G. 1977. Sampling Techniques, 3rd edition. New York: John Wiley & Sons.

Groves, R.M., Dillman, D.A., Eltinge, J.L., and Little, R.J.A. 2002. Survey Non-response. New 

York: John Wiley & Sons

Groves, R.M., and Couper, M.P. 1998. Non-response in Household Interview Surveys. New 

York: John Wiley & Sons.

Kalton, G, and Kasprzyk, D. 1986. “The treatment of missing survey data,” Survey 

Methodology, Vol. 12, pp. 1-16

Kalton, G. (1983). Introduction to Survey Sampling. Sage Publications Series, No. 35. 

Kish, Leslie. 1965. Survey Sampling. New York: Wiley.

Kish, L., and Hess, I. 1950. “On noncoverage of sample dwellings,” Journal of the American 

Statistical Association, Vol. 53, pp. 509-524.

Lehtonen, R., and E. J. Pahkinen (1995). Practical Methods for Design and Analysis of Complex 

Surveys. New York: Wiley.

Lepkowski, James M. Non-observation Error in Household Surveys in Developing Countries, 

Technical Report on Surveys in Developing and Transition Countries, United Nations, 

2003.

Lessler, J., and Kalsbeek, W. 1992. Nonsampling Error in Surveys. New York: John Wiley & 

Sons.

Levy, Paul S. and Stanley Lemeshow (1999). Sampling of Populations: Methods and

Applications. Third edition. John Wiley & Sons, New York.

Lohr, Sharon. 1999. Sampling: Design and Analysis. Pacific Grove: Duxbury Press.

Yansaneh, I. S. An Overview of Sample Design Issues for Household Surveys in Developing 

Countries, Technical Report on Surveys in Developing and Transition Countries, United 

Nations, 2003.