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Abstract Many large‐scale surveys collect both discrete and continuous variables. Small‐area estimates may be desired for means of continuous variables, proportions in each level of a categorical variable, or for domain means defined as the mean of the continuous variable for each level of the categorical variable. In this paper, we introduce a conditionally specified bivariate mixed‐effects model for small‐area estimation, and provide a necessary and sufficient condition under which the conditional distributions render a valid joint distribution. The conditional specification allows better model interpretation. We use the valid joint distribution to calculate empirical Bayes predictors and use the parametric bootstrap to estimate the mean squared error. Simulation studies demonstrate the superior performance of the bivariate mixed‐effects model relative to univariate model estimators. We apply the bivariate mixed‐effects model to construct estimates for small watersheds using data from the Conservation Effects Assessment Project, a survey developed to quantify the environmental impacts of conservation efforts. We construct predictors of mean sediment loss, the proportion of land where the soil loss tolerance is exceeded, and the average sediment loss on land where the soil loss tolerance is exceeded. In the data analysis, the bivariate mixed‐effects model leads to more scientifically interpretable estimates of domain means than those based on two independent univariate models.
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Empirical Bayes small area prediction under a zero‐inflated lognormal model with correlated random area effects
Abstract Many variables of interest in agricultural or economical surveys have skewed distributions and can equal zero. Our data are measures of sheet and rill erosion called Revised Universal Soil Loss Equation‐2 (RUSLE2). Small area estimates of mean RUSLE2 erosion are of interest. We use a zero‐inflated lognormal mixed effects model for small area estimation. The model combines a unit‐level lognormal model for the positive RUSLE2 responses with a unit‐level logistic mixed effects model for the binary indicator that the response is nonzero. In the Conservation Effects Assessment Project (CEAP) data, counties with a higher probability of nonzero responses also tend to have a higher mean among the positive RUSLE2 values. We capture this property of the data through an assumption that the pair of random effects for a county are correlated. We develop empirical Bayes (EB) small area predictors and a bootstrap estimator of the mean squared error (MSE). In simulations, the proposed predictor is superior to simpler alternatives. We then apply the method to construct EB predictors of mean RUSLE2 erosion for South Dakota counties. To obtain auxiliary variables for the population of cropland in South Dakota, we integrate a satellite‐derived land cover map with a geographic database of soil properties. We provide an R Shiny application calledviscover(available athttps://lyux.shinyapps.io/viscover/) to visualize the overlay operations required to construct the covariates. On the basis of bootstrap estimates of the mean square error, we conclude that the EB predictors of mean RUSLE2 erosion are superior to direct estimators.
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- Award ID(s):
- 1733572
- PAR ID:
- 10455695
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Biometrical Journal
- Volume:
- 62
- Issue:
- 8
- ISSN:
- 0323-3847
- Page Range / eLocation ID:
- p. 1859-1878
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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