Design-based regression estimators for spatial means of soil properties: the use of two-phase sampling when the means of the auxiliary variables are unknown

Abstract

The accuracy of design-based sampling strategies can be increased by using regression models at the estimation stage. The usual regression estimator requires that the means of the auxiliary variables are known. If these means are unknown, then it sometimes pays to estimate these from a large preliminary sample that is subsampled in the second phase. Two types of regression estimator, the simple regression estimator and the ratio estimator are compared with the π* estimator in two case studies. In one study, a simulated, bivariate field (correlation coefficient 0.841) was repeatedly sampled by Simple Random Sampling in both phases. The standard error of the simple regression estimator was 61% to 81% of the standard error of the π* estimator; for the ratio estimator, this percentage varied from 78% to 90%. Given the size of the subsample, the gain in precision depends on the size of the first phase sample and the quality of the model. For a sample size ratio of four, the gain was 70% to 75% of the gain that would be achieved if the mean of the auxiliary variable were known. The gain in precision achieved by the ratio estimator, assuming incorrectly that the intercept is 0, was 50% to 60% of the gain achieved by the regression estimator. Despite the incorrect model assumptions, the ratio estimator was approximately design-unbiased, and the confidence intervals were valid. For small sample sizes in the second phase, the ratio estimator performs even slightly better than the regression estimator with respect to validity. This stresses the importance of using simple models. In a second study, the estimated spatial means of the Mean Highest Water table and Mean Lowest Water table of six map units were estimated by Stratified Simple Random Sampling in both phases in combination with the regression estimator that uses the starting depth of hydromorphic properties in the soil profile as an auxiliary variable. The gain in precision did not outweigh the extra costs of collecting the auxiliary data.