A small area procedure for a two-way table of proportions is developed, where the estimated proportions are from a complex survey. Estimation is difficult because the observed proportions do not have multinomial distributions, the observed proportions are correlated with estimated variances, benchmarking is required, and mean models are nonlinear. A predictor based on a nonlinear mixed model is specified for the proportions. No transformation of the observations is involved, and the estimation procedure gives predictions that are in the parameter space. A bootstrap estimator of the mean squared error of a benchmarked predictor is suggested and performed well in simulations. The procedure is applied to the proportions in the two-way table defined by occupations crossed with Canadian provinces. The direct estimators are from the Canadian Labour Force Survey (LFS), and the corresponding two-way table from the previous Canadian Census of Population provides auxiliary information. The application of the prediction procedure to the LFS data leads to gains in estimated mean squared errors relative to the direct estimators between approximately 30 percent and 80 percent. A comparison of the predictors to the Census 2006 proportions further supports the suggested procedures.

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 (CEAP), 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.

We examine the incorporation of analyst input into the constrained estimation process. In the calibration literature, there are numerous examples of estimators with “optimal” properties. We show that many of these can be derived from first principles. Furthermore, we provide mechanisms for injecting user input to create user-constrained optimal estimates. We include derivations for common deviance measures with linear and nonlinear constraints and we demonstrate these methods on a contingency table and a simulated survey data set. R code and examples are available at https://github.com/mwilli/Constrained-estimation.git.