1 Introduction

Jun Shao, Eric Slud, Yang Cheng, Sheng Wang, and Carma Hogue

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The U.S. Annual Survey of Public Employment and Payroll (ASPEP) provides current estimates for full- and part-time state and local government employment and payroll classified by government functions (such as: elementary and secondary education, higher education, police protection, fire protection, financial administration, judicial and legal, etc.). This survey covers state and local government units (89,526 according to the 2007 Census of Governments), which include counties, cities, townships, units called "special districts", and school districts. ASPEP is the only source of public employment data by government function and job category, providing data on numbers of full- and part-time employees and payroll, as well as on hours worked by part-time employees. Data collection usually begins in March and continues for about seven months, with the pay period containing March 12 as reference period.

Let $U$ denote the finite population of $N$ units stratified into $H$ strata, $U_1\mathrm{,}\dots \mathrm{,}{U}_H,$ where $U_h$ contains $N_h$ units and $N_1+\cdots +N_H=N.$ The traditional sampling design for the ASPEP is a stratified probability proportional to size (PPS) design, where the strata are constructed using state and the government types, which are county, subcounty (city or town), special district, or school district. The size of each unit is the total payroll, and sampling across strata is independent. In 2009, a modified sampling design was developed, which cuts some strata $U_h$ into two sub-strata, $U_{h1}$ and $U_{h2}$ with $N_{h1}$ and $N_{h2}$ units, respectively, where $U_{h1}$ contains smaller-size units (Cheng et al. 2009). The idea was to save resources and reduce respondent burden by selecting a sample from $U_{h1}$ with smaller sample size under the modified than under the traditional design. Let $S_{hj}$ be a PPS sample of size $n_{hj}$ from $U_{hj},$ $j=1,2,$ $n_{h1}+n_{h2}=n_h.$ Note that $n_{h1}$ may still be larger than $n_{h2}$ because $N_{h1}$ is usually much larger than $N_{h2}.$

For unit $i\in U,$ let $y_i$ be a key survey variable (e.g., the full-time employment, full-time payroll, part-time employment, part-time payroll, part-time hours), $x_i$ be an auxiliary variable, say the same variable as $y_i$ from the most recent census, and let $z_i$ be the covariate used as the size variable in PPS sampling. The covariate values $x_i$ and $z_i$ are observed for all $i\in U,$ whereas $y_i$ is observed only for each sampled unit $i.$

The Horvitz-Thompson estimator of the unknown total $Y={\displaystyle {\sum }_{i\in U}}{y}_i$is

$${\widehat{Y}}_{\text{HT}}={\displaystyle \sum _h}{\displaystyle \sum _j}{\displaystyle \sum _{i\in S_{hj}}}{y}_i/{\pi }_i\mathrm{,} (1.1)$$

where ${\pi }_i$ is the first-order inclusion probability of unit $i$ in $S_{hj},$ a known function of $z_i’\text{s}.$ To utilize the auxiliary variable $x_i$ and increase the accuracy of estimation of $Y,$the model-assisted approach (Särndal, Swensson and Wretman 1992) is adopted. Applying regression within each $S_{hj}$ leads to the regression estimator of $Y$ as

$${\widehat{Y}}_{\text{reg}\mathrm{,2}}={\displaystyle \sum _h}{\displaystyle \sum _j}\left[\frac{N_{hj}{\widehat{Y}}_{hj}}{{\widehat{N}}_{hj}}+{\widehat{\beta }}_{hj}\left(X_{hj}-\frac{N_{hj}{\widehat{X}}_{hj}}{{\widehat{N}}_{hj}}\right)\right]\mathrm{,}(1.2)$$

where $X_{hj}={\displaystyle {\sum }_{i\in U_{hj}}}{x}_i,$ ${\widehat{Y}}_{hj}={\displaystyle {\sum }_{i\in S_{hj}}}{y}_i/{\pi }_i,$ ${\widehat{X}}_{hj}={\displaystyle {\sum }_{i\in S_{hj}}}{x}_i/{\pi }_i,$ ${\widehat{N}}_{hj}={\displaystyle {\sum }_{i\in S_{hj}}}1/{\pi }_i,$ and

$${\widehat{\beta }}_{hj}=\frac{{\displaystyle {\sum }_{i\in S_{hj}}\left(x_i-{\widehat{X}}_{hj}/{\widehat{N}}_{hj}\right)y_i/{\pi }_i}}{{\displaystyle {\sum }_{i\in S_{hj}}{\left(x_i-{\widehat{X}}_{hj}/{\widehat{N}}_{hj}\right)}^2/{\pi }_i}}\mathrm{.}$$

Alternatively, combining the two sub-strata $S_{h1}$ and $S_{h2}$ results in the following regression estimator. (A referee correctly points out that ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ in (1.3) is not the pooled estimator one would use if regression lines in stratum $h$ were combined but the two sub-strata were not; however, it is the natural estimator when not only regression lines but also sub-strata are combined.)

$${\widehat{Y}}_{\text{reg}\mathrm{,1}}={\displaystyle \sum _h}\left[\frac{N_h{\widehat{Y}}_h}{{\widehat{N}}_h}+{\widehat{\beta }}_h\left(X_h-\frac{N_h{\widehat{X}}_h}{{\widehat{N}}_h}\right)\right]\mathrm{,}(1.3)$$

where ${\widehat{Y}}_h={\displaystyle {\sum }_j}{\widehat{Y}}_{hj},$ ${\widehat{X}}_h={\displaystyle {\sum }_j}{\widehat{X}}_{hj},$ ${\widehat{N}}_h={\displaystyle {\sum }_j}{\widehat{N}}_{hj},$ and

$${\widehat{\beta }}_h=\frac{{\displaystyle {\sum }_j{\displaystyle {\sum }_{i\in S_{hj}}\left(x_i-{\widehat{X}}_h/{\widehat{N}}_h\right)y_i/{\pi }_i}}}{{\displaystyle {\sum }_j{\displaystyle {\sum }_{i\in S_{hj}}{\left(x_i-{\widehat{X}}_h/{\widehat{N}}_h\right)}^2/{\pi }_i}}}\mathrm{.}$$

Since both ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ and ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ are model-assisted estimators, they are consistent with respect to repeated sampling, whether or not the regression model holds. If the least-squares regression lines in two sub-strata $U_{hj}’\text{s}$ are the same, ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ may be more efficient than ${\widehat{Y}}_{\text{reg}\mathrm{,2}}.$ On the other hand, if the regression lines are different, ${\widehat{Y}}_{\text{reg}\mathrm{,2}}$ may be more efficient than ${\widehat{Y}}_{\text{reg}\mathrm{,1}}.$

A decision-based method was proposed in Cheng et al. (2010), which applies hypothesis testing to decide whether we combine $S_{h1}$ and $S_{h2}.$ Within stratum $h,$the slopes of the regression lines in $U_{h1}$ and $U_{h2}$are tested for equality. Let

$$\begin{array}{lll}{\widehat{\alpha }}_{hj}\hfill & =\hfill & \frac{{\widehat{Y}}_{hj}-{\widehat{\beta }}_{hj}{\widehat{X}}_{hj}}{{\widehat{N}}_{hj}}\mathrm{,}{\widehat{\sigma }}_{xe\mathrm{,}hj}^2=\frac{n_{hj}}{{\widehat{N}}_{hj}^2}{\displaystyle \sum _{i\in S_{hj}}}{\left(x_i-\frac{{\widehat{X}}_{hj}}{{\widehat{N}}_{hj}}\right)}^2\frac{{\left(y_i-{\widehat{\alpha }}_{hj}-{\widehat{\beta }}_{hj}{x}_i\right)}^2}{{\pi }_i^2}\mathrm{,}\hfill \\ {\widehat{\sigma }}_{xhj}^2\hfill & =\hfill & {\displaystyle \sum _{i\in S_{hj}}}\frac{{\left(x_i-{\widehat{X}}_{hj}/{\widehat{N}}_{hj}\right)}^2}{{\pi }_i{\widehat{N}}_{hj}}\mathrm{,}{t}_h=\sqrt{n_h-4}\left({\widehat{\beta }}_{h1}-{\widehat{\beta }}_{h2}\right)/\sqrt{n_h{\displaystyle \sum _{j=1}^2}\frac{{\widehat{\sigma }}_{xe\mathrm{,}hj}^2}{n_{hj}{\widehat{\sigma }}_{xhj}^4}}\mathrm{.}\hfill \end{array}$$

If $\left|t_h\right|\text{>}{t}_{1-\tau /\mathrm{2,}{n}_h-4}$, where $t_{1-\tau /\mathrm{2,}\nu }$ is the $1-\tau /2$ quantile of the t-distribution with $\nu$ degrees of freedom, then we reject the hypothesis of common slope and use ${\widehat{\beta }}_{hj}$ (and set ${\zeta }_h=1$ ). Here $\tau$ is a nominal significance level set by default to 0.05, although we will consider other choices of $\tau$ in the simulations section. The test-statistic definition involving $n_h-4$ degrees of freedom is a slightly artificial choice designed to make the moderate-sample rejection probabilities closer to nominal, but the large-sample asymptotic distribution theory justifying this test is given in part (c) of Theorem 1. If $\left|t_h\right|\le t_{1-\tau /\mathrm{2,}{n}_h-4},$ then we accept the hypothesis of common slope, combine sub-strata $S_{h1}$ and $S_{h2},$ and use ${\widehat{\beta }}_h$ $\left(\text{setting}{\zeta }_h=0\right).$ Tests are performed independently across strata $h=\mathrm{1,}\dots \mathrm{,}H.$ The decision-based estimator of $Y$ is then

$${\widehat{Y}}_{\text{dec}}={\displaystyle \sum _h}{\displaystyle \sum _j}{\zeta }_h\left[\frac{N_{hj}{\widehat{Y}}_{hj}}{{\widehat{N}}_{hj}}+{\widehat{\beta }}_{hj}\left(X_{hj}-\frac{N_{hj}{\widehat{X}}_{hj}}{{\widehat{N}}_{hj}}\right)\right]+{\displaystyle \sum _h}\left(1-{\zeta }_h\right)\left[\frac{N_h{\widehat{Y}}_h}{{\widehat{N}}_h}+{\widehat{\beta }}_h\left(X_h-\frac{N_h{\widehat{X}}_h}{{\widehat{N}}_h}\right)\right]\mathrm{.}(1.4)$$

Since two regression lines with a common slope can have different intercepts, one might test a further hypothesis regarding intercepts to decide whether to combine the two sub-strata. However, population points $\left(x_i\mathrm{,}{y}_i\right)$ falling on two parallel but not identical substratum regression lines would be discontinuous around the cut-off point between the two sub-strata $U_{h1}$ and $U_{h2},$which seems to occur only rarely in practical situations. In ASPEP, for example, Cheng et al. (2010) investigated the slopes and intercepts for sub-strata in 2002 and 2007 Census data sets, noting that the hypothesis of a common intercept could never be rejected when the hypothesis of a common slope could not be rejected. Thus, the decision-based estimator in (1.4) depends only on hypothesis testing for equality of sub-stratum regression slopes.

The two-stage estimators studied here are particular instances of procedures previously termed estimators following preliminary testing. There is a large literature on such procedures in surveys, including a bibliography by Bancroft and Han (1977), a book by Saleh (2006), and a treatment by Fuller (2009, Section 6.7). An idea from Saleh (2006) is to estimate coefficients by a convex combination of the estimated coefficients from the separate strata with proportions depending on a test statistic. Such smoothed estimators might be more efficient than our decision-based procedures. If the stratum-specific intercepts and slopes were regarded as random, then a model-based empirical-Bayes approach to survey estimation might also be tried.

The decision-based estimators (1.4) are novel because they are model-assisted design-consistent in the survey-sampling context, making explicit use of the known substratum population sizes. In a somewhat similar spirit, Rao and Ramachandran (1974) previously made an exact comparison of the separate and combined ratio estimators under a ratio model similar to the regression model of this paper.

The purpose of this paper is to show some asymptotic and empirical properties of the estimators of $Y$ described above and their variance estimators. Consistency and asymptotic normality of ${\widehat{Y}}_{\text{reg}\mathrm{,1}},$ ${\widehat{Y}}_{\text{reg}\mathrm{,2}},$ and ${\widehat{Y}}_{\text{dec}}$ are established in Section 2, in terms of either design-based or model-assisted asymptotic theory. Although the first-order asymptotics favor ${\widehat{Y}}_{\text{reg}\mathrm{,2}},$ ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ may be better when some substratum sample sizes $n_{h2}$ are moderate, a second-order asymptotic effect. The virtue of the decision-based estimator ${\widehat{Y}}_{\text{dec}}$ is in adapting to be close to the better of ${\widehat{Y}}_{\text{reg}\mathrm{,1}}$ and ${\widehat{Y}}_{\text{reg}\mathrm{,2}}.$ As the discussion in paragraph (III) of Section 4.4 indicates, simulations show that the benefit of this adaptivity is to reduce MSE up to a few percent under reasonable parameter settings, and by larger amounts in stranger settings.

Variance estimation for the decision-based estimator is treated in Section 3. While the asymptotic theory in Section 2 suggests that consistent variance estimators are obtained by substituting for unknown quantities in the asymptotic variance formulas, we also study bootstrap variance estimators suggested in Cheng et al. (2010), which are generally found to have better finite sample performance than the substitution estimators. Empirical results are presented in Section 4, with Section 4.4 providing interpretations and concluding remarks. All technical proofs are given in the Appendix.

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