<abstract xmlns="http://www.w3.org/1999/xhtml"><p>When analyzing data sampled with unequal inclusion probabilities, correlations between the probability of selection and the sampled data can induce bias if the inclusion probabilities are ignored in the analysis. Weights equal to the inverse of the probability of inclusion are commonly used to correct possible bias. When weights are uncorrelated with the descriptive or model estimators of interest, highly disproportional sample designs resulting in large weights can introduce unnecessary variability, leading to an overall larger mean square error compared to unweighted methods.</p><p>We describe an approach we term ‘weight smoothing’ that models the interactions between the weights and the estimators as random effects, reducing the root mean square error (RMSE) by shrinking interactions toward zero when such shrinkage is allowed by the data. This article adapts a flexible Laplace prior distribution for the hierarchical Bayesian model to gain a more robust bias-variance tradeoff than previous approaches using normal priors. Simulation and application suggest that under a linear model setting, weight-smoothing models with Laplace priors yield robust results when weighting is necessary, and provide considerable reduction in RMSE otherwise. In logistic regression models, estimates using weight-smoothing models with Laplace priors are robust, but with less gain in efficiency than in linear regression settings.</p></abstract>

When analyzing data sampled with unequal inclusion probabilities, correlations between the probability of selection and the sampled data can induce bias if the inclusion probabilities are ignored in the analysis. Weights equal to the inverse of the probability of inclusion are commonly used to correct possible bias. When weights are uncorrelated with the descriptive or model estimators of interest, highly disproportional sample designs resulting in large weights can introduce unnecessary variability, leading to an overall larger mean square error compared to unweighted methods.We describe an approach we term ‘weight smoothing’ that models the interactions between the weights and the estimators as random effects, reducing the root mean square error (RMSE) by shrinking interactions toward zero when such shrinkage is allowed by the data. This article adapts a flexible Laplace prior distribution for the hierarchical Bayesian model to gain a more robust bias-variance tradeoff than previous approaches using normal priors. Simulation and application suggest that under a linear model setting, weight-smoothing models with Laplace priors yield robust results when weighting is necessary, and provide considerable reduction in RMSE otherwise. In logistic regression models, estimates using weight-smoothing models with Laplace priors are robust, but with less gain in efficiency than in linear regression settings.

Weight Smoothing for Generalized Linear Models Using a Laplace Prior

Dept. of Biostatistics, University of Michigan School of Public Health, 1415 Washington Heights, Ann Arbor, MI 48109, USA.

Survey Methodology Program, Institute for Social Research, University of Michigan, 426 Thompson St., Ann Arbor, MI 48109, USA.

Journal of Official Statistics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Weight Smoothing for Generalized Linear Models Using a Laplace Prior"}

When analyzing data sampled with unequal inclusion probabilities, correlations between the probability of selection and the sampled data can induce bias if the...

Weight Smoothing for Generalized Linear Models Using a Laplace Prior

Published Online: May 28, 2016

Page range: 507 - 539

Received: Jun 01, 2014

Accepted: Nov 01, 2015

DOI: https://doi.org/10.1515/jos-2016-0026

Keywords
Weight trimming, winsorization, Bayesian finite population inference, Hierarchical models

© 2016 Xi Xia et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Weight Smoothing for Generalized Linear Models Using a Laplace Prior

Published Online: May 28, 2016

Page range: 507 - 539

Received: Jun 01, 2014

Accepted: Nov 01, 2015

DOI: https://doi.org/10.1515/jos-2016-0026

KeywordsWeight trimming, winsorization, Bayesian finite population inference, Hierarchical models

© 2016 Xi Xia et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Weight trimming, winsorization, Bayesian finite population inference, Hierarchical models