Inference and foundations

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  • Articles and reports: 12-001-X201800154928
    Description:

    A two-phase process was used by the Substance Abuse and Mental Health Services Administration to estimate the proportion of US adults with serious mental illness (SMI). The first phase was the annual National Survey on Drug Use and Health (NSDUH), while the second phase was a random subsample of adult respondents to the NSDUH. Respondents to the second phase of sampling were clinically evaluated for serious mental illness. A logistic prediction model was fit to this subsample with the SMI status (yes or no) determined by the second-phase instrument treated as the dependent variable and related variables collected on the NSDUH from all adults as the model’s explanatory variables. Estimates were then computed for SMI prevalence among all adults and within adult subpopulations by assigning an SMI status to each NSDUH respondent based on comparing his (her) estimated probability of having SMI to a chosen cut point on the distribution of the predicted probabilities. We investigate alternatives to this standard cut point estimator such as the probability estimator. The latter assigns an estimated probability of having SMI to each NSDUH respondent. The estimated prevalence of SMI is the weighted mean of those estimated probabilities. Using data from NSDUH and its subsample, we show that, although the probability estimator has a smaller mean squared error when estimating SMI prevalence among all adults, it has a greater tendency to be biased at the subpopulation level than the standard cut point estimator.

    Release date: 2018-06-21

  • Articles and reports: 12-001-X201700254872
    Description:

    This note discusses the theoretical foundations for the extension of the Wilson two-sided coverage interval to an estimated proportion computed from complex survey data. The interval is shown to be asymptotically equivalent to an interval derived from a logistic transformation. A mildly better version is discussed, but users may prefer constructing a one-sided interval already in the literature.

    Release date: 2017-12-21

  • Articles and reports: 12-001-X201700114822
    Description:

    We use a Bayesian method to infer about a finite population proportion when binary data are collected using a two-fold sample design from small areas. The two-fold sample design has a two-stage cluster sample design within each area. A former hierarchical Bayesian model assumes that for each area the first stage binary responses are independent Bernoulli distributions, and the probabilities have beta distributions which are parameterized by a mean and a correlation coefficient. The means vary with areas but the correlation is the same over areas. However, to gain some flexibility we have now extended this model to accommodate different correlations. The means and the correlations have independent beta distributions. We call the former model a homogeneous model and the new model a heterogeneous model. All hyperparameters have proper noninformative priors. An additional complexity is that some of the parameters are weakly identified making it difficult to use a standard Gibbs sampler for computation. So we have used unimodal constraints for the beta prior distributions and a blocked Gibbs sampler to perform the computation. We have compared the heterogeneous and homogeneous models using an illustrative example and simulation study. As expected, the two-fold model with heterogeneous correlations is preferred.

    Release date: 2017-06-22

  • Articles and reports: 12-001-X201600214662
    Description:

    Two-phase sampling designs are often used in surveys when the sampling frame contains little or no auxiliary information. In this note, we shed some light on the concept of invariance, which is often mentioned in the context of two-phase sampling designs. We define two types of invariant two-phase designs: strongly invariant and weakly invariant two-phase designs. Some examples are given. Finally, we describe the implications of strong and weak invariance from an inference point of view.

    Release date: 2016-12-20

  • Articles and reports: 12-001-X201600114545
    Description:

    The estimation of quantiles is an important topic not only in the regression framework, but also in sampling theory. A natural alternative or addition to quantiles are expectiles. Expectiles as a generalization of the mean have become popular during the last years as they not only give a more detailed picture of the data than the ordinary mean, but also can serve as a basis to calculate quantiles by using their close relationship. We show, how to estimate expectiles under sampling with unequal probabilities and how expectiles can be used to estimate the distribution function. The resulting fitted distribution function estimator can be inverted leading to quantile estimates. We run a simulation study to investigate and compare the efficiency of the expectile based estimator.

    Release date: 2016-06-22

  • Articles and reports: 12-001-X201400114004
    Description:

    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.

    Release date: 2014-06-27
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  • Articles and reports: 12-001-X201800154928
    Description:

    A two-phase process was used by the Substance Abuse and Mental Health Services Administration to estimate the proportion of US adults with serious mental illness (SMI). The first phase was the annual National Survey on Drug Use and Health (NSDUH), while the second phase was a random subsample of adult respondents to the NSDUH. Respondents to the second phase of sampling were clinically evaluated for serious mental illness. A logistic prediction model was fit to this subsample with the SMI status (yes or no) determined by the second-phase instrument treated as the dependent variable and related variables collected on the NSDUH from all adults as the model’s explanatory variables. Estimates were then computed for SMI prevalence among all adults and within adult subpopulations by assigning an SMI status to each NSDUH respondent based on comparing his (her) estimated probability of having SMI to a chosen cut point on the distribution of the predicted probabilities. We investigate alternatives to this standard cut point estimator such as the probability estimator. The latter assigns an estimated probability of having SMI to each NSDUH respondent. The estimated prevalence of SMI is the weighted mean of those estimated probabilities. Using data from NSDUH and its subsample, we show that, although the probability estimator has a smaller mean squared error when estimating SMI prevalence among all adults, it has a greater tendency to be biased at the subpopulation level than the standard cut point estimator.

    Release date: 2018-06-21

  • Articles and reports: 12-001-X201700254872
    Description:

    This note discusses the theoretical foundations for the extension of the Wilson two-sided coverage interval to an estimated proportion computed from complex survey data. The interval is shown to be asymptotically equivalent to an interval derived from a logistic transformation. A mildly better version is discussed, but users may prefer constructing a one-sided interval already in the literature.

    Release date: 2017-12-21

  • Articles and reports: 12-001-X201700114822
    Description:

    We use a Bayesian method to infer about a finite population proportion when binary data are collected using a two-fold sample design from small areas. The two-fold sample design has a two-stage cluster sample design within each area. A former hierarchical Bayesian model assumes that for each area the first stage binary responses are independent Bernoulli distributions, and the probabilities have beta distributions which are parameterized by a mean and a correlation coefficient. The means vary with areas but the correlation is the same over areas. However, to gain some flexibility we have now extended this model to accommodate different correlations. The means and the correlations have independent beta distributions. We call the former model a homogeneous model and the new model a heterogeneous model. All hyperparameters have proper noninformative priors. An additional complexity is that some of the parameters are weakly identified making it difficult to use a standard Gibbs sampler for computation. So we have used unimodal constraints for the beta prior distributions and a blocked Gibbs sampler to perform the computation. We have compared the heterogeneous and homogeneous models using an illustrative example and simulation study. As expected, the two-fold model with heterogeneous correlations is preferred.

    Release date: 2017-06-22

  • Articles and reports: 12-001-X201600214662
    Description:

    Two-phase sampling designs are often used in surveys when the sampling frame contains little or no auxiliary information. In this note, we shed some light on the concept of invariance, which is often mentioned in the context of two-phase sampling designs. We define two types of invariant two-phase designs: strongly invariant and weakly invariant two-phase designs. Some examples are given. Finally, we describe the implications of strong and weak invariance from an inference point of view.

    Release date: 2016-12-20

  • Articles and reports: 12-001-X201600114545
    Description:

    The estimation of quantiles is an important topic not only in the regression framework, but also in sampling theory. A natural alternative or addition to quantiles are expectiles. Expectiles as a generalization of the mean have become popular during the last years as they not only give a more detailed picture of the data than the ordinary mean, but also can serve as a basis to calculate quantiles by using their close relationship. We show, how to estimate expectiles under sampling with unequal probabilities and how expectiles can be used to estimate the distribution function. The resulting fitted distribution function estimator can be inverted leading to quantile estimates. We run a simulation study to investigate and compare the efficiency of the expectile based estimator.

    Release date: 2016-06-22

  • Articles and reports: 12-001-X201400114004
    Description:

    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.

    Release date: 2014-06-27
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