# Weighting and estimation

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- Articles and reports: 12-001-X201800154925Description:
This paper develops statistical inference based on super population model in a finite population setting using ranked set samples (RSS). The samples are constructed without replacement. It is shown that the sample mean of RSS is model unbiased and has smaller mean square prediction error (MSPE) than the MSPE of a simple random sample mean. Using an unbiased estimator of MSPE, the paper also constructs a prediction confidence interval for the population mean. A small scale simulation study shows that estimator is as good as a simple random sample (SRS) estimator for poor ranking information. On the other hand it has higher efficiency than SRS estimator when the quality of ranking information is good, and the cost ratio of obtaining a single unit in RSS and SRS is not very high. Simulation study also indicates that coverage probabilities of prediction intervals are very close to the nominal coverage probabilities. Proposed inferential procedure is applied to a real data set.

Release date: 2018-06-21 - Articles and reports: 12-001-X201800154959Description:
Small area models handling area level data typically assume normality of random effects. This assumption does not always work. The present paper introduces a new small area model with t random effects. Along with this, this paper also considers joint modeling of small area means and variances. The present approach is shown to perform better than other methods.

Release date: 2018-06-21 - Articles and reports: 12-001-X201800154963Description:
The probability-sampling-based framework has dominated survey research because it provides precise mathematical tools to assess sampling variability. However increasing costs and declining response rates are expanding the use of non-probability samples, particularly in general population settings, where samples of individuals pulled from web surveys are becoming increasingly cheap and easy to access. But non-probability samples are at risk for selection bias due to differential access, degrees of interest, and other factors. Calibration to known statistical totals in the population provide a means of potentially diminishing the effect of selection bias in non-probability samples. Here we show that model calibration using adaptive LASSO can yield a consistent estimator of a population total as long as a subset of the true predictors is included in the prediction model, thus allowing large numbers of possible covariates to be included without risk of overfitting. We show that the model calibration using adaptive LASSO provides improved estimation with respect to mean square error relative to standard competitors such as generalized regression (GREG) estimators when a large number of covariates are required to determine the true model, with effectively no loss in efficiency over GREG when smaller models will suffice. We also derive closed form variance estimators of population totals, and compare their behavior with bootstrap estimators. We conclude with a real world example using data from the National Health Interview Survey.

Release date: 2018-06-21 - Articles and reports: 11-626-X2017077Description:
On April 13, 2017, the Government of Canada tabled legislation to legalize the recreational use of cannabis by adults. This will directly impact Canada’s statistical system. The focus of this Economic Insights article is to provide experimental estimates for the volume of cannabis consumption, based on existing information on the prevalence of cannabis use. The article presents experimental estimates of the number of tonnes of cannabis consumed by age group for the period from 1960 to 2015. The experimental estimates rely on survey data from multiple sources, statistical techniques to link the sources over time, and assumptions about consumption behaviour. They are subject to revision as improved or additional data sources become available.

Release date: 2017-12-18 - Articles and reports: 12-001-X201700114819Description:
Structural time series models are a powerful technique for variance reduction in the framework of small area estimation (SAE) based on repeatedly conducted surveys. Statistics Netherlands implemented a structural time series model to produce monthly figures about the labour force with the Dutch Labour Force Survey (DLFS). Such models, however, contain unknown hyperparameters that have to be estimated before the Kalman filter can be launched to estimate state variables of the model. This paper describes a simulation aimed at studying the properties of hyperparameter estimators in the model. Simulating distributions of the hyperparameter estimators under different model specifications complements standard model diagnostics for state space models. Uncertainty around the model hyperparameters is another major issue. To account for hyperparameter uncertainty in the mean squared errors (MSE) estimates of the DLFS, several estimation approaches known in the literature are considered in a simulation. Apart from the MSE bias comparison, this paper also provides insight into the variances and MSEs of the MSE estimators considered.

Release date: 2017-06-22 - Articles and reports: 12-001-X201700114823Description:
The derivation of estimators in a multi-phase calibration process requires a sequential computation of estimators and calibrated weights of previous phases in order to obtain those of later ones. Already after two phases of calibration the estimators and their variances involve calibration factors from both phases and the formulae become cumbersome and uninformative. As a consequence the literature so far deals mainly with two phases while three phases or more are rarely being considered. The analysis in some cases is ad-hoc for a specific design and no comprehensive methodology for constructing calibrated estimators, and more challengingly, estimating their variances in three or more phases was formed. We provide a closed form formula for the variance of multi-phase calibrated estimators that holds for any number of phases. By specifying a new presentation of multi-phase calibrated weights it is possible to construct calibrated estimators that have the form of multi-variate regression estimators which enables a computation of a consistent estimator for their variance. This new variance estimator is not only general for any number of phases but also has some favorable characteristics. A comparison to other estimators in the special case of two-phase calibration and another independent study for three phases are presented.

Release date: 2017-06-22 - Articles and reports: 12-001-X201600214660Description:
In an economic survey of a sample of enterprises, occupations are randomly selected from a list until a number r of occupations in a local unit has been identified. This is an inverse sampling problem for which we are proposing a few solutions. Simple designs with and without replacement are processed using negative binomial distributions and negative hypergeometric distributions. We also propose estimators for when the units are selected with unequal probabilities, with or without replacement.

Release date: 2016-12-20 - Articles and reports: 12-001-X201600214663Description:
We present theoretical evidence that efforts during data collection to balance the survey response with respect to selected auxiliary variables will improve the chances for low nonresponse bias in the estimates that are ultimately produced by calibrated weighting. One of our results shows that the variance of the bias – measured here as the deviation of the calibration estimator from the (unrealized) full-sample unbiased estimator – decreases linearly as a function of the response imbalance that we assume measured and controlled continuously over the data collection period. An attractive prospect is thus a lower risk of bias if one can manage the data collection to get low imbalance. The theoretical results are validated in a simulation study with real data from an Estonian household survey.

Release date: 2016-12-20 - Articles and reports: 12-001-X201600214664Description:
This paper draws statistical inference for finite population mean based on judgment post stratified (JPS) samples. The JPS sample first selects a simple random sample and then stratifies the selected units into H judgment classes based on their relative positions (ranks) in a small set of size H. This leads to a sample with random sample sizes in judgment classes. Ranking process can be performed either using auxiliary variables or visual inspection to identify the ranks of the measured observations. The paper develops unbiased estimator and constructs confidence interval for population mean. Since judgment ranks are random variables, by conditioning on the measured observations we construct Rao-Blackwellized estimators for the population mean. The paper shows that Rao-Blackwellized estimators perform better than usual JPS estimators. The proposed estimators are applied to 2012 United States Department of Agriculture Census Data.

Release date: 2016-12-20 - Articles and reports: 12-001-X201600214677Description:
How do we tell whether weighting adjustments reduce nonresponse bias? If a variable is measured for everyone in the selected sample, then the design weights can be used to calculate an approximately unbiased estimate of the population mean or total for that variable. A second estimate of the population mean or total can be calculated using the survey respondents only, with weights that have been adjusted for nonresponse. If the two estimates disagree, then there is evidence that the weight adjustments may not have removed the nonresponse bias for that variable. In this paper we develop the theoretical properties of linearization and jackknife variance estimators for evaluating the bias of an estimated population mean or total by comparing estimates calculated from overlapping subsets of the same data with different sets of weights, when poststratification or inverse propensity weighting is used for the nonresponse adjustments to the weights. We provide sufficient conditions on the population, sample, and response mechanism for the variance estimators to be consistent, and demonstrate their small-sample properties through a simulation study.

Release date: 2016-12-20

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## Analysis (447)

## Analysis (447) (0 to 10 of 447 results)

- Articles and reports: 12-001-X201800154925Description:
This paper develops statistical inference based on super population model in a finite population setting using ranked set samples (RSS). The samples are constructed without replacement. It is shown that the sample mean of RSS is model unbiased and has smaller mean square prediction error (MSPE) than the MSPE of a simple random sample mean. Using an unbiased estimator of MSPE, the paper also constructs a prediction confidence interval for the population mean. A small scale simulation study shows that estimator is as good as a simple random sample (SRS) estimator for poor ranking information. On the other hand it has higher efficiency than SRS estimator when the quality of ranking information is good, and the cost ratio of obtaining a single unit in RSS and SRS is not very high. Simulation study also indicates that coverage probabilities of prediction intervals are very close to the nominal coverage probabilities. Proposed inferential procedure is applied to a real data set.

Release date: 2018-06-21 - Articles and reports: 12-001-X201800154959Description:
Small area models handling area level data typically assume normality of random effects. This assumption does not always work. The present paper introduces a new small area model with t random effects. Along with this, this paper also considers joint modeling of small area means and variances. The present approach is shown to perform better than other methods.

Release date: 2018-06-21 - Articles and reports: 12-001-X201800154963Description:
The probability-sampling-based framework has dominated survey research because it provides precise mathematical tools to assess sampling variability. However increasing costs and declining response rates are expanding the use of non-probability samples, particularly in general population settings, where samples of individuals pulled from web surveys are becoming increasingly cheap and easy to access. But non-probability samples are at risk for selection bias due to differential access, degrees of interest, and other factors. Calibration to known statistical totals in the population provide a means of potentially diminishing the effect of selection bias in non-probability samples. Here we show that model calibration using adaptive LASSO can yield a consistent estimator of a population total as long as a subset of the true predictors is included in the prediction model, thus allowing large numbers of possible covariates to be included without risk of overfitting. We show that the model calibration using adaptive LASSO provides improved estimation with respect to mean square error relative to standard competitors such as generalized regression (GREG) estimators when a large number of covariates are required to determine the true model, with effectively no loss in efficiency over GREG when smaller models will suffice. We also derive closed form variance estimators of population totals, and compare their behavior with bootstrap estimators. We conclude with a real world example using data from the National Health Interview Survey.

Release date: 2018-06-21 - Articles and reports: 11-626-X2017077Description:
On April 13, 2017, the Government of Canada tabled legislation to legalize the recreational use of cannabis by adults. This will directly impact Canada’s statistical system. The focus of this Economic Insights article is to provide experimental estimates for the volume of cannabis consumption, based on existing information on the prevalence of cannabis use. The article presents experimental estimates of the number of tonnes of cannabis consumed by age group for the period from 1960 to 2015. The experimental estimates rely on survey data from multiple sources, statistical techniques to link the sources over time, and assumptions about consumption behaviour. They are subject to revision as improved or additional data sources become available.

Release date: 2017-12-18 - Articles and reports: 12-001-X201700114819Description:
Structural time series models are a powerful technique for variance reduction in the framework of small area estimation (SAE) based on repeatedly conducted surveys. Statistics Netherlands implemented a structural time series model to produce monthly figures about the labour force with the Dutch Labour Force Survey (DLFS). Such models, however, contain unknown hyperparameters that have to be estimated before the Kalman filter can be launched to estimate state variables of the model. This paper describes a simulation aimed at studying the properties of hyperparameter estimators in the model. Simulating distributions of the hyperparameter estimators under different model specifications complements standard model diagnostics for state space models. Uncertainty around the model hyperparameters is another major issue. To account for hyperparameter uncertainty in the mean squared errors (MSE) estimates of the DLFS, several estimation approaches known in the literature are considered in a simulation. Apart from the MSE bias comparison, this paper also provides insight into the variances and MSEs of the MSE estimators considered.

Release date: 2017-06-22 - Articles and reports: 12-001-X201700114823Description:
The derivation of estimators in a multi-phase calibration process requires a sequential computation of estimators and calibrated weights of previous phases in order to obtain those of later ones. Already after two phases of calibration the estimators and their variances involve calibration factors from both phases and the formulae become cumbersome and uninformative. As a consequence the literature so far deals mainly with two phases while three phases or more are rarely being considered. The analysis in some cases is ad-hoc for a specific design and no comprehensive methodology for constructing calibrated estimators, and more challengingly, estimating their variances in three or more phases was formed. We provide a closed form formula for the variance of multi-phase calibrated estimators that holds for any number of phases. By specifying a new presentation of multi-phase calibrated weights it is possible to construct calibrated estimators that have the form of multi-variate regression estimators which enables a computation of a consistent estimator for their variance. This new variance estimator is not only general for any number of phases but also has some favorable characteristics. A comparison to other estimators in the special case of two-phase calibration and another independent study for three phases are presented.

Release date: 2017-06-22 - Articles and reports: 12-001-X201600214660Description:
In an economic survey of a sample of enterprises, occupations are randomly selected from a list until a number r of occupations in a local unit has been identified. This is an inverse sampling problem for which we are proposing a few solutions. Simple designs with and without replacement are processed using negative binomial distributions and negative hypergeometric distributions. We also propose estimators for when the units are selected with unequal probabilities, with or without replacement.

Release date: 2016-12-20 - Articles and reports: 12-001-X201600214663Description:
We present theoretical evidence that efforts during data collection to balance the survey response with respect to selected auxiliary variables will improve the chances for low nonresponse bias in the estimates that are ultimately produced by calibrated weighting. One of our results shows that the variance of the bias – measured here as the deviation of the calibration estimator from the (unrealized) full-sample unbiased estimator – decreases linearly as a function of the response imbalance that we assume measured and controlled continuously over the data collection period. An attractive prospect is thus a lower risk of bias if one can manage the data collection to get low imbalance. The theoretical results are validated in a simulation study with real data from an Estonian household survey.

Release date: 2016-12-20 - Articles and reports: 12-001-X201600214664Description:
This paper draws statistical inference for finite population mean based on judgment post stratified (JPS) samples. The JPS sample first selects a simple random sample and then stratifies the selected units into H judgment classes based on their relative positions (ranks) in a small set of size H. This leads to a sample with random sample sizes in judgment classes. Ranking process can be performed either using auxiliary variables or visual inspection to identify the ranks of the measured observations. The paper develops unbiased estimator and constructs confidence interval for population mean. Since judgment ranks are random variables, by conditioning on the measured observations we construct Rao-Blackwellized estimators for the population mean. The paper shows that Rao-Blackwellized estimators perform better than usual JPS estimators. The proposed estimators are applied to 2012 United States Department of Agriculture Census Data.

Release date: 2016-12-20 - Articles and reports: 12-001-X201600214677Description:
How do we tell whether weighting adjustments reduce nonresponse bias? If a variable is measured for everyone in the selected sample, then the design weights can be used to calculate an approximately unbiased estimate of the population mean or total for that variable. A second estimate of the population mean or total can be calculated using the survey respondents only, with weights that have been adjusted for nonresponse. If the two estimates disagree, then there is evidence that the weight adjustments may not have removed the nonresponse bias for that variable. In this paper we develop the theoretical properties of linearization and jackknife variance estimators for evaluating the bias of an estimated population mean or total by comparing estimates calculated from overlapping subsets of the same data with different sets of weights, when poststratification or inverse propensity weighting is used for the nonresponse adjustments to the weights. We provide sufficient conditions on the population, sample, and response mechanism for the variance estimators to be consistent, and demonstrate their small-sample properties through a simulation study.

Release date: 2016-12-20

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## Reference (87)

## Reference (87) (0 to 10 of 87 results)

- 1. Revisions to 2006 to 2011 income dataArchivedSurveys and statistical programs – Documentation: 75F0002M2015003Description:
This note discusses revised income estimates from the Survey of Labour and Income Dynamics (SLID). These revisions to the SLID estimates make it possible to compare results from the Canadian Income Survey (CIS) to earlier years. The revisions address the issue of methodology differences between SLID and CIS.

Release date: 2015-12-17 - Surveys and statistical programs – Documentation: 91-528-XDescription:
This manual provides detailed descriptions of the data sources and methods used by Statistics Canada to estimate population. They comprise Postcensal and intercensal population estimates; base population; births and deaths; immigration; emigration; non-permanent residents; interprovincial migration; subprovincial estimates of population; population estimates by age, sex and marital status; and census family estimates. A glossary of principal terms is contained at the end of the manual, followed by the standard notation used.

Until now, literature on the methodological changes for estimates calculations has always been spread throughout various Statistics Canada publications and background papers. This manual provides users of demographic statistics with a comprehensive compilation of the current procedures used by Statistics Canada to prepare population and family estimates.

Release date: 2015-11-17 - Surveys and statistical programs – Documentation: 13-605-X201500414166Description:
Estimates of the underground economy by province and territory for the period 2007 to 2012 are now available for the first time. The objective of this technical note is to explain how the methodology employed to derive upper-bound estimates of the underground economy for the provinces and territories differs from that used to derive national estimates.

Release date: 2015-04-29 - Surveys and statistical programs – Documentation: 99-002-X2011001Description:
This report describes sampling and weighting procedures used in the 2011 National Household Survey. It provides operational and theoretical justifications for them, and presents the results of the evaluation studies of these procedures.

Release date: 2015-01-28 - Surveys and statistical programs – Documentation: 99-002-XDescription: This report describes sampling and weighting procedures used in the 2011 National Household Survey. It provides operational and theoretical justifications for them, and presents the results of the evaluation studies of these procedures.Release date: 2015-01-28
- Surveys and statistical programs – Documentation: 12-001-X201400111886Description:
Bayes linear estimator for finite population is obtained from a two-stage regression model, specified only by the means and variances of some model parameters associated with each stage of the hierarchy. Many common design-based estimators found in the literature can be obtained as particular cases. A new ratio estimator is also proposed for the practical situation in which auxiliary information is available. The same Bayes linear approach is proposed for obtaining estimation of proportions for multiple categorical data associated with finite population units, which is the main contribution of this work. A numerical example is provided to illustrate it.

Release date: 2014-06-27 - Surveys and statistical programs – Documentation: 12-001-X201300211869Description:
The house price index compiled by Statistics Netherlands relies on the Sale Price Appraisal Ratio (SPAR) method. The SPAR method combines selling prices with prior government assessments of properties. This paper outlines an alternative approach where the appraisals serve as auxiliary information in a generalized regression (GREG) framework. An application on Dutch data demonstrates that, although the GREG index is much smoother than the ratio of sample means, it is very similar to the SPAR series. To explain this result we show that the SPAR index is an estimator of our more general GREG index and in practice almost as efficient.

Release date: 2014-01-15 - Surveys and statistical programs – Documentation: 12-001-X201300211888Description:
When the study variables are functional and storage capacities are limited or transmission costs are high, using survey techniques to select a portion of the observations of the population is an interesting alternative to using signal compression techniques. In this context of functional data, our focus in this study is on estimating the mean electricity consumption curve over a one-week period. We compare different estimation strategies that take account of a piece of auxiliary information such as the mean consumption for the previous period. The first strategy consists in using a simple random sampling design without replacement, then incorporating the auxiliary information into the estimator by introducing a functional linear model. The second approach consists in incorporating the auxiliary information into the sampling designs by considering unequal probability designs, such as stratified and pi designs. We then address the issue of constructing confidence bands for these estimators of the mean. When effective estimators of the covariance function are available and the mean estimator satisfies a functional central limit theorem, it is possible to use a fast technique for constructing confidence bands, based on the simulation of Gaussian processes. This approach is compared with bootstrap techniques that have been adapted to take account of the functional nature of the data.

Release date: 2014-01-15 - 9. Combining cohorts in longitudinal surveysArchivedSurveys and statistical programs – Documentation: 12-001-X201300111828Description:
A question that commonly arises in longitudinal surveys is the issue of how to combine differing cohorts of the survey. In this paper we present a novel method for combining different cohorts, and using all available data, in a longitudinal survey to estimate parameters of a semiparametric model, which relates the response variable to a set of covariates. The procedure builds upon the Weighted Generalized Estimation Equation method for handling missing waves in longitudinal studies. Our method is set up under a joint-randomization framework for estimation of model parameters, which takes into account the superpopulation model as well as the survey design randomization. We also propose a design-based, and a joint-randomization, variance estimation method. To illustrate the methodology we apply it to the Survey of Doctorate Recipients, conducted by the U.S. National Science Foundation.

Release date: 2013-06-28 - Surveys and statistical programs – Documentation: 12-001-X201200111683Description:
We consider alternatives to poststratification for doubly classified data in which at least one of the two-way cells is too small to allow the poststratification based upon this double classification. In our study data set, the expected count in the smallest cell is 0.36. One approach is simply to collapse cells. This is likely, however, to destroy the double classification structure. Our alternative approaches allows one to maintain the original double classification of the data. The approaches are based upon the calibration study by Chang and Kott (2008). We choose weight adjustments dependent upon the marginal classifications (but not full cross classification) to minimize an objective function of the differences between the population counts of the two way cells and their sample estimates. In the terminology of Chang and Kott (2008), if the row and column classifications have I and J cells respectively, this results in IJ benchmark variables and I + J - 1 model variables. We study the performance of these estimators by constructing simulation simple random samples from the 2005 Quarterly Census of Employment and Wages which is maintained by the Bureau of Labor Statistics. We use the double classification of state and industry group. In our study, the calibration approaches introduced an asymptotically trivial bias, but reduced the MSE, compared to the unbiased estimator, by as much as 20% for a small sample.

Release date: 2012-06-27

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