Weighting and estimation

Results

All (638)

All (638) (0 to 10 of 638 results)

1. Exact and approximation formulas for second-order inclusion probabilities in randomized systematic sampling with unequal probabilities and without replacement
Articles and reports: 12-001-X202600100003
Description: Probability-proportional-to-size sampling is widely used by national statistical offices. Here population units are selected with probabilities proportional to an auxiliary variable. Variance formulas in such designs require both first- and second-order inclusion probabilities. The computation of second-order inclusion probabilities is particularly challenging for large populations, and has been the subject of extensive research. This article presents some new exact and approximation formulas for second-order inclusion probabilities in randomized systematic sampling with unequal probabilities and without replacement.
Release date: 2026-06-29
2. Improving the coverage of confidence intervals with respect to degrees of freedom: Application to the Canadian Census
Articles and reports: 12-001-X202600100005
Description: Confidence intervals are very often constructed based on a probability distribution that uses a certain number of degrees of freedom as a parameter. This is the case with the Student and the modified Wilson confidence intervals, discussed in this article, which use quantiles from the Student distribution where the number of degrees of freedom is generally unknown. For the length of a confidence interval to be representative of the reliability of an estimate, the actual coverage rate must match the nominal rate. To that end, the number of degrees of freedom in the probability distribution used in practice to calculate the confidence interval must be estimated as precisely as possible. An approximate rule is often used, although it tends to overestimate the actual number of degrees of freedom. In this article, a more precise version of degrees of freedom, derived from the Satterthwaite approximation, is obtained in the context of the Canadian Census of Population. The sampling design is equivalent to a simple random design without replacement, cluster-stratified, and the variance estimation method is an adaptation of the balanced repeated replication method. An explicit expression of the degrees of freedom is obtained under these conditions, enabling the factors influencing them to be identified. For comparison, the degree of freedom formula is also established for the conventional variance estimator. A simulation study shows that using this version of degrees of freedom corrects the undercoverage problem observed with the approximate rule, showing the importance of accurately assessing this number.
Release date: 2026-06-29
3. The inverse-variance trap: A simple ratio fix for combining skewed data
Articles and reports: 12-001-X202600100009
Description: Combining estimates from independent surveys via inverse-variance weights can lead to negative bias when unknown variances are estimated and the target variable is non-negative and positively skewed. In such cases, strong positive correlations typically arise between the estimators and their corresponding variance estimators, causing standard linear combinations with inverse-variance weights to exhibit negative bias. We introduce a strikingly simple method to reduce bias: replace the standard weight with the ratio of the estimator to the variance estimator. Under a linear model linking the two, we show that the new ratio-weighted estimator is approximately unbiased, whereas the conventional inverse-variance combination exhibits downward bias. Through simulations, we demonstrate that the new method brings both the bias and the mean squared error closer to the optimum for a wide range of different target variables. As our method uses only standardly reported summary statistics, it can be immediately adopted to reduce this widespread bias and improve the reliability of scientific findings in various fields.
Release date: 2026-06-29
4. Variance of the generalized regression estimator under measurement error
Articles and reports: 12-001-X202600100010
Description: With the exception of two-phase sampling, the standard variance approximation of the generalized regression (GREG) estimator assumes that the population totals in the weighting scheme are observed without error. If the weighting model of the GREG estimator contains population totals that are observed with measurement error sources other than the sampling error of first-phase estimates, then this uncertainty will be ignored by the variance approximation of the GREG estimator. This paper proposes a variance approximation for the GREG estimator that accounts for additional uncertainty arising from measurement error in one or more of the population totals used in the weighting scheme. This approach has been developed for, and is being applied to, the Dutch Labour Force Survey (DLFS). The monthly publications of the DLFS are obtained with a time series model, which corrects for rotation group bias and discontinuities caused by major redesigns and the loss of face-to-face interviews during COVID-19. The GREG estimates for the quarterly figures are benchmarked to the average of the monthly publications to enforce numerical consistency between monthly and quarterly publication tables. The standard variance approximation of the GREG estimator assumes that these population totals are observed without error. This results in an underestimation of the variance of the GREG estimator. The variance approximation proposed in this paper results in more realistic standard errors for the quarterly GREG estimates.
Release date: 2026-06-29
5. Bayesian differential privacy for small counties and individual commodities
Articles and reports: 12-001-X202600100011
Description: We construct a hybrid Bayesian method, which includes a differentially private mechanism, to mask Census county totals for a U.S. state on acreage of a commodity. We use surrogates for data collected at the farm level from a past U.S. Census of Agriculture to illustrate our procedure. We use two Bayesian small area models (parametric and mixture) to accommodate the smaller counties with fewer farms and some counties with large acres. In these models, the Laplace distribution provides a differentially private mechanism. In pre-processing, we also incorporate the Census weights to form the observed total acreage, a scaling factor to the Laplace mechanism for each county, a square-root transformation of the observed total acreage to avoid negative masked estimates especially for small counties, and the p-percent rule and the 3+ rule to partition the counties into suppressed counties, non-sensitive counties and sensitive counties. Because of difficulties in specifying and tuning the privacy budget (an unknown parameter), to balance security and utility, we specify a prior for the privacy budget, where the values are not specified, and the Gibbs sampler is used to fit the hierarchical Bayesian models. In post-processing, we use Bayesian predictive inference to obtain masked county acreages, and this includes a benchmarking so that the masked state total matches the observed state total. As a measure of reliability of the Bayesian procedure, we use the posterior coefficients of variation for the masked posterior means of the counties. As a measure of utility, we use the absolute relative errors for the individual counties, together with other global measures. For the sensitive counties, there are some differences between the two small area models but both are much better than an individual area model; the mixture model being the best compromise for security and utility.
Release date: 2026-06-29
6. Small area estimation of general indicators in off-census years
Articles and reports: 12-001-X202600100012
Description: We propose small area estimators of general indicators in off-census years, which avoid the use of deprecated census microdata, but are nearly optimal in census years. The procedure is based on replacing the obsolete census file with a larger unit-level survey that adequately covers the areas of interest and contains the values of useful auxiliary variables. However, the minimal data requirement of the proposed method is a single survey with microdata on the target variable and suitable auxiliary variables for the period of interest. We also develop an estimator of the mean squared error (MSE) that accounts for the uncertainty introduced by the large survey used to replace the census of auxiliary information. Our empirical results indicate that the proposed predictors perform clearly better than the alternative predictors when census data are outdated, and are very close to optimal ones when census data are correct. They also illustrate that the proposed total MSE estimator corrects for the bias of purely model-based MSE estimators that do not account for the large survey uncertainty.
Release date: 2026-06-29
7. A Framework for Projecting Housing Use Associated with Immigrants and Non-Permanent Residents
Surveys and statistical programs – Documentation: 11-633-X2026002
Description: Recent changes in Canada’s immigration levels have heightened interest in understanding how immigration affects housing demand. This article develops a methodological framework for projecting housing use associated with permanent residents (PRs) and non-permanent residents (NPRs) under alternative immigration scenarios. The framework applies observed per capita housing use rates from the Census of Population to estimate incremental housing use by tenure over time.
Release date: 2026-04-24
8. Effects of model misspecification on small area estimators
Articles and reports: 12-001-X202500200001
Description: Nested error regression models are commonly used to incorporate unit specific auxiliary variables to improve small area estimates. When the mean structure of the model is misspecified, the design-based mean squared prediction error (MSPE) of Empirical Best Linear Unbiased Predictors (EBLUP) generally increases. The Observed Best Prediction (OBP) method has been proposed with the intent to improve on the design-based MSPE over EBLUP. In this paper, we conduct a Monte Carlo simulation experiments to understand the effect of misspsecification of mean structures on different small area estimators. Our findings suggest that the OBP using unit-level auxiliary variables does not outperform the EBLUP in terms of design-based MSPE, unless the number of small areas m is extremely large. Conversely, the performance of OBP significantly improves when area-level auxiliary variables are employed. This paper includes both analytical and numerical evidence to demonstrate these observations, providing practical insights for addressing model misspecification in small area estimation (SAE).
Release date: 2025-12-23
9. Estimating monthly indicators for Consumer Confidence using Structural Time Series Models
Articles and reports: 12-001-X202500200003
Description: In this paper a model-based inference procedure based on a multivariate structural time series model is developed for the production of monthly figures about consumer confidence. The input for the model are five series of direct estimates for the indices that measure consumer confidence, which are derived from the Dutch Consumer Survey. The model improves the accuracy of the direct estimates, since it provides a better separation of measurement errors and sampling errors from estimated target parameters. The standard errors for the month-to-month changes are clearly smaller under the time series model. A second problem addressed in this paper is related to the transition to a new survey process in 2017. Structural time series models in combination with a parallel run are applied to estimate discontinuities induced by the redesign. An algorithm designed for the consumer confidence variables is developed to construct uninterrupted input series for the aforementioned structural time series model. This inference method facilitated a smooth transition to a new survey design and resulted in uninterrupted series about consumer confidence that date back to 1986. The method is implemented for the production of official monthly figures on consumer confidence in the Netherlands.
Release date: 2025-12-23
10. Quantile balancing inverse probability weighting for non-probability samples
Articles and reports: 12-001-X202500200005
Description: The use of non-probability data sources for statistical purposes and for official statistics has become increasingly popular in recent years. However, statistical inference based on non-probability samples is made more difficult by nature of their biasedness and lack of representativity. In this paper we propose quantile balancing inverse probability weighting estimator (QBIPW) for non-probability samples. We apply the idea of Harms and Duchesne (2006) allowing the use of quantile information in the estimation process to reproduce known totals and the distribution of auxiliary variables. We discuss the estimation of the QBIPW probabilities and its variance. Our simulation study has demonstrated that the proposed estimators are robust against model mis-specification and, as a result, help to reduce bias and mean squared error. Finally, we applied the proposed methods to estimate the share of job vacancies aimed at Ukrainian workers in Poland using an integrated set of administrative and survey data about job vacancies.
Release date: 2025-12-23

Data (0)

Data (0) (0 results)

No content available at this time.

Analysis (610)

Analysis (610) (40 to 50 of 610 results)

41. Comments by Takumi Saegusa on “Exchangeability assumption in propensity-score based adjustment methods for population mean estimation using non-probability samples”: Causal inference, non-probability sample, and finite population
Articles and reports: 12-001-X202400100006
Description: In some of non-probability sample literature, the conditional exchangeability assumption is considered to be necessary for valid statistical inference. This assumption is rooted in causal inference though its potential outcome framework differs greatly from that of non-probability samples. We describe similarities and differences of two frameworks and discuss issues to consider when adopting the conditional exchangeability assumption in non-probability sample setups. We also discuss the role of finite population inference in different approaches of propensity scores and outcome regression modeling to non-probability samples.
Release date: 2024-06-25
42. Comments by Jae Kwang Kim and Yonghyun Kwon on “Exchangeability assumption in propensity-score based adjustment methods for population mean estimation using non-probability samples”
Articles and reports: 12-001-X202400100007
Description: Pseudo weight construction for data integration can be understood in the two-phase sampling framework. Using the two-phase sampling framework, we discuss two approaches to the estimation of propensity scores and develop a new way to construct the propensity score function for data integration using the conditional maximum likelihood method. Results from a limited simulation study are also presented.
Release date: 2024-06-25
43. Exchangeability assumption in propensity-score based adjustment methods for population mean estimation using non-probability samples
Articles and reports: 12-001-X202400100008
Description: Nonprobability samples emerge rapidly to address time-sensitive priority topics in different areas. These data are timely but subject to selection bias. To reduce selection bias, there has been wide literature in survey research investigating the use of propensity-score (PS) adjustment methods to improve the population representativeness of nonprobability samples, using probability-based survey samples as external references. Conditional exchangeability (CE) assumption is one of the key assumptions required by PS-based adjustment methods. In this paper, I first explore the validity of the CE assumption conditional on various balancing score estimates that are used in existing PS-based adjustment methods. An adaptive balancing score is proposed for unbiased estimation of population means. The population mean estimators under the three CE assumptions are evaluated via Monte Carlo simulation studies and illustrated using the NIH SARS-CoV-2 seroprevalence study to estimate the proportion of U.S. adults with COVID-19 antibodies from April 01-August 04, 2020.
Release date: 2024-06-25
44. Enhancing data for rural Canada: Small area estimation of remote work opportunities
Articles and reports: 18-001-X2024001
Description: This study applies small area estimation (SAE) and a new geographic concept called Self-contained Labor Area (SLA) to the Canadian Survey on Business Conditions (CSBC) with a focus on remote work opportunities in rural labor markets. Through SAE modelling, we estimate the proportions of businesses, classified by general industrial sector (service providers and goods producers), that would primarily offer remote work opportunities to their workforce.
Release date: 2024-04-22
45. Study: Enhancing data for rural Canada: Small area estimation of remote work opportunities Archived
Stats in brief: 11-001-X202411338008
Description: Release published in The Daily – Statistics Canada’s official release bulletin
Release date: 2024-04-22
46. A proposal for the problem of matching probabilities estimation in record linkage Archived
Articles and reports: 11-522-X202200100001
Description: Record linkage aims at identifying record pairs related to the same unit and observed in two different data sets, say A and B. Fellegi and Sunter (1969) suggest each record pair is tested whether generated from the set of matched or unmatched pairs. The decision function consists of the ratio between m(y) and u(y),probabilities of observing a comparison y of a set of k>3 key identifying variables in a record pair under the assumptions that the pair is a match or a non-match, respectively. These parameters are usually estimated by means of the EM algorithm using as data the comparisons on all the pairs of the Cartesian product ?=A×B. These observations (on the comparisons and on the pairs status as match or non-match) are assumed as generated independently of other pairs, assumption characterizing most of the literature on record linkage and implemented in software tools (e.g. RELAIS, Cibella et al. 2012). On the contrary, comparisons y and matching status in ? are deterministically dependent. As a result, estimates on m(y) and u(y) based on the EM algorithm are usually bad. This fact jeopardizes the effective application of the Fellegi-Sunter method, as well as automatic computation of quality measures and possibility to apply efficient methods for model estimation on linked data (e.g. regression functions), as in Chambers et al. (2015). We propose to explore ? by a set of samples, each one drawn so to preserve independence of comparisons among the selected record pairs. Simulations are encouraging.
Release date: 2024-03-25
47. A Model-based Disaggregation Method for Estimation of Adult Competency Archived
Articles and reports: 11-522-X202200100003
Description: Estimation at fine levels of aggregation is necessary to better describe society. Small area estimation model-based approaches that combine sparse survey data with rich data from auxiliary sources have been proven useful to improve the reliability of estimates for small domains. Considered here is a scenario where small area model-based estimates, produced at a given aggregation level, needed to be disaggregated to better describe the social structure at finer levels. For this scenario, an allocation method was developed to implement the disaggregation, overcoming challenges associated with data availability and model development at such fine levels. The method is applied to adult literacy and numeracy estimation at the county-by-group-level, using data from the U.S. Program for the International Assessment of Adult Competencies. In this application the groups are defined in terms of age or education, but the method could be applied to estimation of other equity-deserving groups.
Release date: 2024-03-25
48. Labour Force Survey initiatives under Statistics Canada’s Disaggregated Data Action Plan Archived
Articles and reports: 11-522-X202200100004
Description: In accordance with Statistics Canada’s long-term Disaggregated Data Action Plan (DDAP), several initiatives have been implemented into the Labour Force Survey (LFS). One of the more direct initiatives was a targeted increase in the size of the monthly LFS sample. Furthermore, a regular Supplement program was introduced, where an additional series of questions are asked to a subset of LFS respondents and analyzed in a monthly or quarterly production cycle. Finally, the production of modelled estimates based on Small Area Estimation (SAE) methodologies resumed for the LFS and will include a wider scope with more analytical value than what had existed in the past. This paper will give an overview of these three initiatives.
Release date: 2024-03-25
49. Application of sampling variance smoothing methods for small area proportion estimation Archived
Articles and reports: 11-522-X202200100005
Description: Sampling variance smoothing is an important topic in small area estimation. In this paper, we propose sampling variance smoothing methods for small area proportion estimation. In particular, we consider the generalized variance function and design effect methods for sampling variance smoothing. We evaluate and compare the smoothed sampling variances and small area estimates based on the smoothed variance estimates through analysis of survey data from Statistics Canada. The results from real data analysis indicate that the proposed sampling variance smoothing methods work very well for small area estimation.
Release date: 2024-03-25
50. Measuring the number of food aid recipients Archived
Articles and reports: 11-522-X202200100013
Description: Respondents to typical household surveys tend to significantly underreport their potential use of food aid distributed by associations. This underreporting is most likely related to the social stigma felt by people experiencing great financial difficulty. As a result, survey estimates of the number of recipients of that aid are much lower than the direct counts from the associations. Those counts tend to overestimate due to double counting. Through its adapted protocol, the Enquête Aide alimentaire (EAA) collected in late 2021 in France at a sample of sites of food aid distribution associations, controls the biases that affect the other sources and determines to what extent this aid is used.
Release date: 2024-03-25

Reference (28)

Reference (28) (20 to 30 of 28 results)

21. Sampling and Weighting (Reference Products: Technical Reports: 1996 Census of Population) Archived
Surveys and statistical programs – Documentation: 92-371-X
Description:
This report deals with sampling and weighting, a process whereby certain characteristics are collected and processed for a random sample of dwellings and persons identified in the complete census enumeration. Data for the whole population are then obtained by scaling up the results for the sample to the full population level. The use of sampling may lead to substantial reductions in costs and respondent burden, or alternatively, can allow the scope of a census to be broadened at the same cost.
Release date: 1999-12-07
22. Random effects models for longitudinal data from complex samples Archived
Surveys and statistical programs – Documentation: 11-522-X19980015017
Description:
Longitudinal studies with repeated observations on individuals permit better characterizations of change and assessment of possible risk factors, but there has been little experience applying sophisticated models for longitudinal data to the complex survey setting. We present results from a comparison of different variance estimation methods for random effects models of change in cognitive function among older adults. The sample design is a stratified sample of people 65 and older, drawn as part of a community-based study designed to examine risk factors for dementia. The model summarizes the population heterogeneity in overall level and rate of change in cognitive function using random effects for intercept and slope. We discuss an unweighted regression including covariates for the stratification variables, a weighted regression, and bootstrapping; we also did preliminary work into using balanced repeated replication and jackknife repeated replication.
Release date: 1999-10-22
23. Weighting versus modeling in adjusting for non-response in the British Labour Force Survey: An application to gross flows estimation Archived
Surveys and statistical programs – Documentation: 11-522-X19980015019
Description:
The British Labour Force Survey (LFS) is a quarterly household survey with a rotating sample design that can potentially be used to produce longitudinal data, including estimates of labour force gross flows. However, these estimates may be biased due to the effect of non-response. Weighting adjustments are a commonly used method to account for non-response bias. We find that weighting may not fully account for the effect of non-response bias because non-response may depend on the unobserved labour force flows, i.e., the non-response is non-ignorable. To adjust for the effects of non-ignorable non-response, we propose a model for the complex non-response patterns in the LFS which controls for the correlated within-household non-response behaviour found in the survey. The results of modelling suggest that non-response may be non-ignorable in the LFS, causing the weighting estimates to be biased.
Release date: 1999-10-22
24. Calculation of weights for the European panel of households: A comparison of some indicators based on variables selected to correct for homogenous categorical non-response Archived
Surveys and statistical programs – Documentation: 11-522-X19980015020
Description:
At the end of 1993, Eurostat lauched a 'community' panel of households. The first wave, carried out in 1994 in the 12 countries of the European Union, included some 7,300 households in France, and at least 14,000 adults 17 years or over. Each individual was then followed up and interviewed each year, even if they had moved. The individuals leaving the sample present a particular profile. In the first part, we present a sketch of how our sample evolves and an analysis of the main characteristics of the non-respondents. We then propose 2 models to correct for non-response per homogeneous category. We then describe the longitudinal weight distribution obtained from the two models, and the cross-sectional weights using the weight share method. Finally, we compare some indicators calculated using both weighting methods.
Release date: 1999-10-22
25. Mobility matrices and computation of associated precision Archived
Surveys and statistical programs – Documentation: 11-522-X19980015023
Description:
The study of social mobility, between labour market statuses or between income levels, for example, is often based on the analysis of mobility matrices. When comparing these transition matrices, with a view to evaluating behavioural changes, one often forgets that the data derive from a sample survey and are therefore affected by sampling variances. Similarly, it is assumed that the responses collected correspond to the ' true value.'
Release date: 1999-10-22
26. Evaluating nonresponse adjustment in the Current Population Survey (CPS) using longitudinal data Archived
Surveys and statistical programs – Documentation: 11-522-X19980015026
Description:
The purpose of the present study is to utilize panel data from the Current Population Survey (CPS) to examine the effects of unit nonresponse. Because most nonrespondents to the CPS are respondents during at least one month-in-sample, data from other months can be used to compare the characteristics of complete respondents and panel nonrespondents and to evaluate nonresponse adjustment procedures. In the current paper we present analyses utilizing CPS panel data to illustrate the effects of unit nonresponse. After adjusting for nonresponse, additional comparisons are also made to evaluate the effects of nonresponse adjustment. The implications of the findings and suggestions for further research are discussed.
Release date: 1999-10-22
27. Variance estimation in longitudinal studies of income dynamics Archived
Surveys and statistical programs – Documentation: 11-522-X19980015028
Description:
We address the problem of estimation for the income dynamics statistics calculated from complex longitudinal surveys. In addition, we compare two design-based estimators of longitudinal proportions and transition rates in terms of variability under large attrition rates. One estimator is based on the cross-sectional samples for the estimation of the income class boundaries at each time period and on the longitudinal sample for the estimation of the longitudinal counts; the other estimator is entirely based on the longitudinal sample, both for the estimation of the class boundaries and the longitudinal counts. We develop Taylor linearization-type variance estimators for both the longitudinal and the mixed estimator under the assumption of no change in the population, and for the mixed estimator when there is change.
Release date: 1999-10-22
28. Weighting and variance estimation for the exploration of possible time trends in data from the U.S. Third National Health and Nutrition Examination Survey Archived
Surveys and statistical programs – Documentation: 11-522-X19980015031
Description:
The U.S. Third National Health and Nutrition Examination Survey (NHANES III) was carried out from 1988 to 1994. This survey was intended primarily to provide estimates of cross-sectional parameters believed to be approximately constant over the six-year data collection period. However, for some variable (e.g., serum lead, body mass index and smoking behavior), substantive considerations suggest the possible presence of nontrivial changes in level between 1988 and 1994. For these variables, NHANES III is potentially a valuable source of time-change information, compared to other studies involving more restricted populations and samples. Exploration of possible change over time is complicated by two issues. First, there was of practical concern because some variables displayed substantial regional differences in level. This was of practical concern because some variables displayed substantial regional differences in level. Second, nontrivial changes in level over time can lead to nontrivial biases in some customary NHANES III variance estimators. This paper considers these two problems and discusses some related implications for statistical policy.
Release date: 1999-10-22

Date modified:: 2026-09-12