Analysis

In 2009, two major surveys in the Governments Division of the U.S. Census Bureau were redesigned to reduce sample size, save resources, and improve the precision of the estimates (Cheng, Corcoran, Barth and Hogue 2009). The new design divides each of the traditional state by government-type strata with sufficiently many units into two sub-strata according to each governmental unit’s total payroll, in order to sample less from the sub-stratum with small size units. The model-assisted approach is adopted in estimating population totals. Regression estimators using auxiliary variables are obtained either within each created sub-stratum or within the original stratum by collapsing two sub-strata. A decision-based method was proposed in Cheng, Slud and Hogue (2010), applying a hypothesis test to decide which regression estimator is used within each original stratum. Consistency and asymptotic normality of these model-assisted estimators are established here, under a design-based or model-assisted asymptotic framework. Our asymptotic results also suggest two types of consistent variance estimators, one obtained by substituting unknown quantities in the asymptotic variances and the other by applying the bootstrap. The performance of all the estimators of totals and of their variance estimators are examined in some empirical studies. The U.S. Annual Survey of Public Employment and Payroll (ASPEP) is used to motivate and illustrate our study.

Nonresponse in longitudinal studies often occurs in a nonmonotone pattern. In the Survey of Industrial Research and Development (SIRD), it is reasonable to assume that the nonresponse mechanism is past-value-dependent in the sense that the response propensity of a study variable at time point t depends on response status and observed or missing values of the same variable at time points prior to t. Since this nonresponse is nonignorable, the parametric likelihood approach is sensitive to the specification of parametric models on both the joint distribution of variables at different time points and the nonresponse mechanism. The nonmonotone nonresponse also limits the application of inverse propensity weighting methods. By discarding all observed data from a subject after its first missing value, one can create a dataset with a monotone ignorable nonresponse and then apply established methods for ignorable nonresponse. However, discarding observed data is not desirable and it may result in inefficient estimators when many observed data are discarded. We propose to impute nonrespondents through regression under imputation models carefully created under the past-value-dependent nonresponse mechanism. This method does not require any parametric model on the joint distribution of the variables across time points or the nonresponse mechanism. Performance of the estimated means based on the proposed imputation method is investigated through some simulation studies and empirical analysis of the SIRD data.

Business surveys often use a one-stage stratified simple random sampling without replacement design with some certainty strata. Although weight adjustment is typically applied for unit nonresponse, the variability due to nonresponse may be omitted in practice when estimating variances. This is problematic especially when there are certainty strata. We derive some variance estimators that are consistent when the number of sampled units in each weighting cell is large, using the jackknife, linearization, and modified jackknife methods. The derived variance estimators are first applied to empirical data from the Annual Capital Expenditures Survey conducted by the U.S. Census Bureau and are then examined in a simulation study.

In longitudinal surveys nonresponse often occurs in a pattern that is not monotone. We consider estimation of time-dependent means under the assumption that the nonresponse mechanism is last-value-dependent. Since the last value itself may be missing when nonresponse is nonmonotone, the nonresponse mechanism under consideration is nonignorable. We propose an imputation method by first deriving some regression imputation models according to the nonresponse mechanism and then applying nonparametric regression imputation. We assume that the longitudinal data follow a Markov chain with finite second-order moments. No other assumption is imposed on the joint distribution of longitudinal data and their nonresponse indicators. A bootstrap method is applied for variance estimation. Some simulation results and an example concerning the Current Employment Survey are presented.

In surveys under cluster sampling, nonresponse on a variable is often dependent on a cluster level random effect and, hence, is nonignorable. Estimators of the population mean obtained by mean imputation or reweighting under the ignorable nonresponse assumption are then biased. We propose an unbiased estimator of the population mean by imputing or reweighting within each sampled cluster or a group of sampled clusters sharing some common feature. Some simulation results are presented to study the performance of the proposed estimator.

Like most other surveys, non-response often occurs in the Current Employment Survey conducted monthly by the U.S. Bureau of Labor Statistics (BLS). In a given month, imputation using reported data from previous months generally provides more efficient survey estimators than ignoring non-respondents and adjusting survey weights. However, imputation also has an effect on variance estimation: treating imputed values as reported data and applying a standard variance estimation method lead to negatively biased variance estimators. In this article, we propose some variance estimators using the Grouped Balanced Half Sample method and re-imputation to take imputation into account. Some simulation results for the finite sample performance of the imputed survey estimators and their variance estimators are presented.

In this paper, we discuss the application of the bootstrap with a re-imputation step to capture the imputation variance (Shao and Sitter 1996) in stratified multistage sampling. We propose a modified bootstrap that does not require rescaling so that Shao and Sitter's procedure can be applied to the case where random imputation is applied and the first-stage stratum sample sizes are very small. This provides a unified method that works irrespective of the imputation method (random or nonrandom), the stratum size (small or large), the type of estimator (smooth or nonsmooth), or the type of problem (variance estimation or sampling distribution estimation). In addition, we discuss the proper Monte Carlo approximation to the bootstrap variance, when using re-imputation together with resampling methods. In this setting, more care is needed than is typical. Similar results are obtained for the method of balanced repeated replications, which is often used in surveys and can be viewed as an analytic approximation to the bootstrap. Finally, some simulation results are presented to study finite sample properties and various variance estimators for imputed data.

Imputation is a common procedure to compensate for nonresponse in survey problems. Using auxiliary data, imputation may produce estimators that are more efficient than the one constructed by ignoring nonrespondents and re-weighting. We study and compare the mean squared errors of survey estimators based on data imputed using three difference imputation techniques: the commonly used ratio imputation method and two cold deck imputation methods that are frequently adopted in economic area surveys conducted by the U.S. Census Bureau and the U.S. Bureau of Labor Statistics.

In 2009, two major surveys in the Governments Division of the U.S. Census Bureau were redesigned to reduce sample size, save resources, and improve the precision of the estimates (Cheng, Corcoran, Barth and Hogue 2009). The new design divides each of the traditional state by government-type strata with sufficiently many units into two sub-strata according to each governmental unit’s total payroll, in order to sample less from the sub-stratum with small size units. The model-assisted approach is adopted in estimating population totals. Regression estimators using auxiliary variables are obtained either within each created sub-stratum or within the original stratum by collapsing two sub-strata. A decision-based method was proposed in Cheng, Slud and Hogue (2010), applying a hypothesis test to decide which regression estimator is used within each original stratum. Consistency and asymptotic normality of these model-assisted estimators are established here, under a design-based or model-assisted asymptotic framework. Our asymptotic results also suggest two types of consistent variance estimators, one obtained by substituting unknown quantities in the asymptotic variances and the other by applying the bootstrap. The performance of all the estimators of totals and of their variance estimators are examined in some empirical studies. The U.S. Annual Survey of Public Employment and Payroll (ASPEP) is used to motivate and illustrate our study.

Nonresponse in longitudinal studies often occurs in a nonmonotone pattern. In the Survey of Industrial Research and Development (SIRD), it is reasonable to assume that the nonresponse mechanism is past-value-dependent in the sense that the response propensity of a study variable at time point t depends on response status and observed or missing values of the same variable at time points prior to t. Since this nonresponse is nonignorable, the parametric likelihood approach is sensitive to the specification of parametric models on both the joint distribution of variables at different time points and the nonresponse mechanism. The nonmonotone nonresponse also limits the application of inverse propensity weighting methods. By discarding all observed data from a subject after its first missing value, one can create a dataset with a monotone ignorable nonresponse and then apply established methods for ignorable nonresponse. However, discarding observed data is not desirable and it may result in inefficient estimators when many observed data are discarded. We propose to impute nonrespondents through regression under imputation models carefully created under the past-value-dependent nonresponse mechanism. This method does not require any parametric model on the joint distribution of the variables across time points or the nonresponse mechanism. Performance of the estimated means based on the proposed imputation method is investigated through some simulation studies and empirical analysis of the SIRD data.

Business surveys often use a one-stage stratified simple random sampling without replacement design with some certainty strata. Although weight adjustment is typically applied for unit nonresponse, the variability due to nonresponse may be omitted in practice when estimating variances. This is problematic especially when there are certainty strata. We derive some variance estimators that are consistent when the number of sampled units in each weighting cell is large, using the jackknife, linearization, and modified jackknife methods. The derived variance estimators are first applied to empirical data from the Annual Capital Expenditures Survey conducted by the U.S. Census Bureau and are then examined in a simulation study.

In longitudinal surveys nonresponse often occurs in a pattern that is not monotone. We consider estimation of time-dependent means under the assumption that the nonresponse mechanism is last-value-dependent. Since the last value itself may be missing when nonresponse is nonmonotone, the nonresponse mechanism under consideration is nonignorable. We propose an imputation method by first deriving some regression imputation models according to the nonresponse mechanism and then applying nonparametric regression imputation. We assume that the longitudinal data follow a Markov chain with finite second-order moments. No other assumption is imposed on the joint distribution of longitudinal data and their nonresponse indicators. A bootstrap method is applied for variance estimation. Some simulation results and an example concerning the Current Employment Survey are presented.

In surveys under cluster sampling, nonresponse on a variable is often dependent on a cluster level random effect and, hence, is nonignorable. Estimators of the population mean obtained by mean imputation or reweighting under the ignorable nonresponse assumption are then biased. We propose an unbiased estimator of the population mean by imputing or reweighting within each sampled cluster or a group of sampled clusters sharing some common feature. Some simulation results are presented to study the performance of the proposed estimator.

Like most other surveys, non-response often occurs in the Current Employment Survey conducted monthly by the U.S. Bureau of Labor Statistics (BLS). In a given month, imputation using reported data from previous months generally provides more efficient survey estimators than ignoring non-respondents and adjusting survey weights. However, imputation also has an effect on variance estimation: treating imputed values as reported data and applying a standard variance estimation method lead to negatively biased variance estimators. In this article, we propose some variance estimators using the Grouped Balanced Half Sample method and re-imputation to take imputation into account. Some simulation results for the finite sample performance of the imputed survey estimators and their variance estimators are presented.

In this paper, we discuss the application of the bootstrap with a re-imputation step to capture the imputation variance (Shao and Sitter 1996) in stratified multistage sampling. We propose a modified bootstrap that does not require rescaling so that Shao and Sitter's procedure can be applied to the case where random imputation is applied and the first-stage stratum sample sizes are very small. This provides a unified method that works irrespective of the imputation method (random or nonrandom), the stratum size (small or large), the type of estimator (smooth or nonsmooth), or the type of problem (variance estimation or sampling distribution estimation). In addition, we discuss the proper Monte Carlo approximation to the bootstrap variance, when using re-imputation together with resampling methods. In this setting, more care is needed than is typical. Similar results are obtained for the method of balanced repeated replications, which is often used in surveys and can be viewed as an analytic approximation to the bootstrap. Finally, some simulation results are presented to study finite sample properties and various variance estimators for imputed data.

Imputation is a common procedure to compensate for nonresponse in survey problems. Using auxiliary data, imputation may produce estimators that are more efficient than the one constructed by ignoring nonrespondents and re-weighting. We study and compare the mean squared errors of survey estimators based on data imputed using three difference imputation techniques: the commonly used ratio imputation method and two cold deck imputation methods that are frequently adopted in economic area surveys conducted by the U.S. Census Bureau and the U.S. Bureau of Labor Statistics.