Estimation and inference of domain means subject to qualitative constraints
Section 1. Introduction

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For many large-scale surveys, a goal is to produce estimates for a large number of domains, many of which might have small sample size. These domains are typically created by cross-classifying categorical variables such as demographic, geographic or other similar characteristics of interest. For instance, the U.S. Current Population Survey releases estimates for domains defined by sex, age, race and/or educational attainment. Similarly, the U.S. American Community Survey produces detailed estimates by sex, age, race/ethnicity for different levels of geography (depending on the release). In another example we will discuss further below, the U.S. National Survey of College Graduates is interested in estimates defined by crossing level and field of degree, occupation and gender. Depending on the survey program, such “granular” estimates are often as important as the higher-level or population estimates.

However, although the overall sample size of such surveys might be very large, samples sizes for numerous domains are often too small for reliable estimates. One possible approach to avoid this problem could be to aggregate small domains into bigger scales so that more reliable direct estimators can be produced for those scales, leading to the generation of more aggregated information than the actual desired scale. An alternative to producing small domain estimates could be changing from a design-based to a model-based estimation methodology such as small area models. While that is certainly a statistically valid approach for creating precise estimates at small scales, it is labor-intensive and sensitive to potential model misspecification. It also replaces the sampling error by model error, so that the mode of inference changes. For those reasons, statistical agencies prefer to stay within the design-based approach, which offers robustness and also allows to stay with the standard mode of inference for surveys.

In this paper, we present an estimation approach that is applicable when “natural” or qualitative relationships are expected to hold among the domain means at the population level. These relationships can be used to stabilize the sample domain estimates, while staying within the design-based mode of estimation and inference. The type of relationships we are considering here lead to inequalities among population domain means. For instance, certain job types might be expected to receive better salaries than others, or individuals with graduate degrees in a given discipline are expected to have higher salaries than those without graduate degrees in that discipline. However, given that small domains tend to produce estimates with high variability, such expected population-level relationships are often violated at the sample level. While such violations should be expected by data users due to statistical variability, they might lead them to question the overall reliability of the survey, by producing “absurd” estimates.

There is a large literature in survey statistics related to calibrating survey estimates, see e.g. Särndal, Swensson and Wretman (1992) for an overview. While these estimators also rely on constraints, there are important differences, including the fact that the constraints are equality constraints and that they are applied to the survey weights, not the estimates themselves. While we do not explore this here, it would be possible to combine calibration and constrained estimation, since the latter could use calibrated domain estimates as the starting point for constructing constrained domain estimates. In the model-based setting, Rueda and Lombardía (2012) adapted methods in small area estimation for the case of monotonically ordered domain means.

Recently, Wu, Meyer and Opsomer (2016) proposed a domain mean estimation methodology that relies on the assumption of monotone population domain means along a single domain-defining categorical variable (e.g., age classes). By combining the monotonicity information of domain means and design-based estimators in the estimation stage, they proposed a constrained estimator that respects the monotone assumption. Such an estimator was shown to improve precision and variability of domain mean estimates in comparison with direct estimators, given that the assumption of monotonicity is reasonable.

We generalize this work here by allowing a much larger class of constraints between domain means, applicable to the multi-dimensional setting. Many other types of constraints beyond monotonicity may be expected to hold between population domain means in real surveys, especially in the presence of domains defined by the cross-classifications of many categorical variables. In general, any set of linear inequality constraints can be represented through a constraint matrix, where each row defines a constraint and each column a domain mean. For illustration of a constraint matrix, suppose the variable of interest is the annual average salary of faculty in land-grant universities of a certain size. Further, consider domains generated from the cross-classification of the variables job position $(x_1;$ 1 = Untenured and 2 = Tenured) and three specific departments $(x_2;$ 1 = Anthropology, 2 = English and 3 = Engineering). Under the assumptions that, on average within a discipline, tenured faculty have higher salaries than untenured faculty; and that, within tenured and untenured, Engineering faculty members are expected to have higher salaries than those in either the Anthropology or English departments, then we can express the corresponding restrictions as,

$$A\mu \ge 0\mathrm{,}\text{where}A=\left(\begin{array}{rrrrrr}\hfill -1& \hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 1\\ \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill -1& \hfill 0& \hfill 0& \hfill 0& \hfill 1\\ \hfill 0& \hfill 0& \hfill 0& \hfill -1& \hfill 0& \hfill 1\end{array}\right)\mathrm{,}(1.1)$$

$\mu \mathrm{<mo>=</mo>}{\left({\mu }_{11}\mathrm{,}{\mu }_{21}\mathrm{,}{\mu }_{12}\mathrm{,}{\mu }_{22}\mathrm{,}{\mu }_{13}\mathrm{,}{\mu }_{23}\right)}^T,$ with ${\mu }_{ij}$ representing the mean of the domain that corresponds to $x_1\mathrm{<mo>=</mo>}i$ and $x_2\mathrm{<mo>=</mo>}j;$ $0$ being the zero vector, and the inequality being element-wise. This paper describes a new constrained estimator for population domain means that respect constraints that can be expressed with matrix inequalities of the form given in (1.1). By combining design-based domain mean estimators with these shape constraints, we propose a broadly applicable estimator that improves precision and variability of the most common direct estimators.

The remainder of the paper is organized as follows. In Section 2 we formally introduce the constrained estimator and propose a linearization-based method for variance estimation. This section also contains some scenarios of interest where shape constraints can naturally arise for survey data. Section 3 states the main theoretical properties of the constrained estimator. The necessary assumptions used in these theoretical derivations are also stated in this section. Proofs of main theorems and auxiliary lemmas are provided in the Appendix. Section 4 shows through simulations that the constrained estimator improves domain mean estimation and variability in comparison with the unconstrained estimator, even when the assumed shape holds only approximately at the population level. Section 5 demonstrates the advantages of the proposed methodology on real survey data through an application to the 2015 National Survey of College Graduates. A few concluding remarks are provided in Section 6.

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