Semiparametric quantile regression imputation for a complex survey with application to the Conservation Effects Assessment Project
Section 4. Application to Conservation Effects Assessment Project

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The cropland component of the Conservation Effects Assessment Project (CEAP) consists of a series of surveys meant to measure soil and nutrient loss from crop fields. The first cropland assessment was a national survey conducted over the period 2003-2006. Data collection for a second national survey, planned for 2015-2016, was on-going at the time of writing this paper. Each of the time periods 2003-2006 and 2015-2016 is considered one time point for estimation. Data are collected over multiple years (i.e., 2003-2006 or 2015-2016) for operational reasons, and no unit is in the sample for two years in the same time period. Temporal changes of interest are changes between the two time periods, rather than changes between two years in the same time period. The temporal structure leads to unbalanced data because some units respond in both time periods, some units never respond, and some units respond in only one of the two time periods. Providing the data user with a complete, imputed data set with a single set of weights simplifies analyses involving more than one time point.

We investigate the feasibility of imputation for CEAP using a subset of the data collected during 2003-2005. We omit the data collected in 2006 because the sample design changed, and we do not have the information required to compute sampling weights for the 2006 survey. The data from the 2015-2016 survey are not yet collected. This analysis is considered an investigation of the feasibility of using QRI to impute missing data in CEAP in the direction of addressing the broader problem of estimation of change over time.

An understanding of the CEAP sample design requires an understanding of the design of the National Resources Inventory (NRI). The NRI monitors status and trends in land use, land cover, and erosion, with emphasis on characteristics related to natural resources and agriculture. Primary sampling units in the NRI are land areas called segments, which are approximately 160 acres. Each segment contains approximately three secondary sampling units, which are randomly selected locations called points. From 1982-1997, the same sample of approximately 300,000 segments, referred to as the foundation sample, was revisited every five years. The foundation sample is a stratified sample of segments, with a typical sampling rate of approximately 4%. See Nusser and Goebel (1997) for details of the design of the NRI foundation sample. In 2000, the NRI transitioned to annual sample design. Because revisiting every sampled segment in the foundation sample on an annual basis is infeasible, a rotating panel design is used. A subsample of the foundation segments, called the core panel, is revisited annually. The core panel is supplemented with a rotation panel, which changes each year. In essence, the core and rotation panels are stratified samples of the foundation sample. The strata, called sample classes, depend on the characteristics of the NRI segment observed in 1982-1997, such as presence of wetlands, cropland, and forest. See Nusser (2006) and Breidt and Fuller (1999) for further detail on the NRI annual samples.

For the Conservation Effects Assessment Project (CEAP), data collectors visit a subset of the NRI points that are located in sampled crop fields and collect more detailed information on crop choices and conservation practices. The sample for the 2003-2005 CEAP survey essentially consists of segments in the NRI core panel, 2002 rotation panel, and 2003 rotation panel that contain at least one cropland point. For segments containing more than one cropland point, one cropland point was selected randomly. The selection of one point per segment is an effort to improve geographic spread and reduce the number of instances in which a farm operator associated with multiple sampled points is selected into the sample, thereby reducing the respondent burden.

Because the first phase sampling rate for the NRI is small $\left(\approx \mathrm{4\; <mi>\%</mi>}\right),$ we approximate the CEAP sample as a probability proportional to size with replacement sample. The selection probabilities for CEAP largely reflect the sample design for the NRI. Details of construction of first and second order selection probabilities for CEAP are provided in Section C of the online supplement https://github.com/emilyjb/Semiparametric-QRI-Supplement/blob/master/SupplementToQRI.pdf, (Berg and Yu, 2016).

Data collection for crop fields sampled for the CEAP survey consists of multiple components. An important component is a farmer interview survey that collects detailed information on farming managements and conservation practices. Nonresponse can occur in CEAP if a farmer refuses to participate in the interview.

Response variables in CEAP are measurements of different types of soil and nutrient loss, obtained from a physical process model called the Agricultural Policy Environmental Extender (APEX). The APEX model converts data from the farmer surveys as well as information from administrative sources and the NRI to numerical measures of erosion. For this study, we consider a measure of soil loss due to sheet and rill erosion called RUSLE2, discussed further in Section 4.1.

The NRI survey provides a convenient source of auxiliary information for imputing CEAP response variables. Because the NRI survey data are collected through aerial photographs of sampled segments, nonresponse due to refusals does not occur in the NRI. As a consequence, NRI data are available for all sampled points in CEAP. Furthermore, the NRI collects data related to land use, conservation practices and erosion $–$ characteristics that are expected to be correlated with outputs of the APEX model. As an auxiliary variable, we use USLE, a measure of sheet and rill erosion collected in the NRI.

Domains of interest in CEAP are ten “CEAP production regions”. We focus on estimation of mean RUSLE2 for seven states (Iowa, Illinois, Indiana, Michigan, Minnesota, Ohio, and Wisconsin) that comprise the majority of the CEAP production region called the Corn Belt. We use semiparametric quantile regression to impute missing values for RUSLE2 using USLE as an auxiliary variable for each of these seven states in the Corn Belt region.

4.1  Imputation model and procedures

The variable of interest, RUSLE2, is a measure of sheet and rill erosion obtained from the APEX model. Because interest is in mean erosion on a per acre basis, the parameter of interest $\theta ,$ the mean RUSLE2 erosion in the state, is defined as a ratio by,

$$\theta =\frac{E\left[{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^{m_{ek}}{R}_{ik}{D}_k{m}_k^{-1}}\right]}{E\left[m_{ek}{D}_k{m}_k^{-1}\right]}\mathrm{,}(4.1)$$

where $R_{ik}$ is the RUSLE2 erosion for point $i$ in segment $k$ sampled in the period 2003-2005, $D_k$ is the area of segment $k,$ $m_k$ is the total number of points in segment $k,$ and $m_{ek}$ is the number of points in segment $k$ that are eligible for the CEAP survey. As discussed above, the period 2003-2005 is considered one time point, and no point is sampled more than once in this collection of years. Therefore, each sampled unit has one value $R_{ik}$ for this set of years, and $R_{ik}$ does not need a subscript of $t$ for year.

The RUSLE2 erosion is an advancement of a simpler measure of sheet and rill erosion called USLE. The USLE is a product of five numerical indexes associated with slope steepness and length, rainfall, soil erodibility, conservation practices, and crop managements. While RUSLE2 is only observed for respondents to the CEAP survey, USLE is available from the main NRI sample for all points in the CEAP sample. We use the average USLE across years 2003-2005 as the covariate in the imputation model. Specifically, for point $i$ in segment $k,$ we define, $U_{ik}=3^{-1}{\displaystyle {\sum }_{t\mathrm{<mo>=</mo>2003}}^{2005}}{U}_{tik},$ where $U_{tik}$ is the USLE soil loss in the NRI for point $i$ in segment $k$ for year $t.$

Because the RUSLE2 and USLE are highly skewed, the quantile regression model is applied after transforming both $R_{ik}$ and $U_{ik}$ by a power of 0.2. The quantile regression model postulated for the superpopulation can be expressed as, $P\left(y_{ik}\le \tau |x_{ik}\right)\mathrm{<mo>=</mo>}{q}_{\tau }\left(x_{ik}\right),$ where $y_{ik}\mathrm{<mo>=</mo>}{R}_{ik}^{0.2},$ and $x_{ik}\mathrm{<mo>=</mo>}{U}_{ik}^{0.2}.$ The unknown function $q_{\tau }\left(x_{ik}\right)$ is approximated by a linear combination of B-spline basis functions generated from $x_{ik}.$ To define the penalized B-spline, we set $p\mathrm{<mo>=</mo>\; 3},$ $m\mathrm{<mo>=</mo>\; 2},$ $K_n\mathrm{<mo>=</mo>\; 16},$ and $\lambda \mathrm{<mo>=</mo>\; 0.004.}$

Because the quantity of interest is erosion on a per acre basis, the estimator $\widehat{\theta }$ of $\theta$ defined in (4.1) is a ratio of two estimators. That is, $\widehat{\theta }\mathrm{<mo>=</mo>}{\widehat{\theta }}_2^{-1}{\widehat{\theta }}_1,$ where ${\widehat{\theta }}_1$ is an estimator of ${\theta }_1\mathrm{<mo>=</mo>}E\left[D_k{U}_{ik}\right],$ and ${\theta }_2\mathrm{<mo>=</mo>}E\left[D_k\right].$ The estimator of ${\theta }_2$ is the Hájek estimator, ${\widehat{\theta }}_2\mathrm{<mo>=</mo>}\left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_{ik}^{-1}{D}_k\right){\left({\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^n}{\pi }_{ik}^{-1}\right)}^{-1},$ where ${\pi }_{ik}$ is the probability of selecting point $i$ in segment $k$ into the CEAP sample. The estimator ${\widehat{\theta }}_1$ of ${\theta }_1$ is obtained from GMM with $g\left(y\mathrm{,}{\theta }_1\right)\mathrm{<mo>=</mo>}\left(D_k{y}^5-{\theta }_1\right).$

4.2  Estimates and variance estimates

Table 4.1 contains estimates of average RUSLE2 soil loss based on QRI, along with estimated standard errors for seven states in the Corn Belt CEAP region. For comparison, the complete case estimator $\left({\overline{R}}_{cc}\right)$ and corresponding estimated standard error is also provided in Table 4.1. The complete case estimator is the ratio of Hájek estimators constructed using only the units that provide a usable response for RUSLE2.

For each of the seven states, the complete case estimator is larger than the estimator based on the imputed data. The imputation procedure reduces the estimator of $\theta ,$ relative to the complete case estimator, because the weighted mean of $U_{ik}$ among sampled units is smaller than the mean of $U_{ik}$ among respondents, as shown in the last two rows of Table 4.1.

As expected, the estimated standard error for $\widehat{\theta }$ is smaller than the estimated standard error for the complete case estimator. The ratios of the estimated variances for the complete case estimator to the estimated variances of $\widehat{\theta }$ range from 1.103 for MN to 1.252 for IN. This comparison demonstrates the potential for efficiency gain due to the use of imputation. The reduction in estimated standard deviation occurs because the imputation procedure uses $U_{ik}$ for the full sample, while the complete case estimator is based only on $R_{ik}$ for the subset of respondents.

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