A method to combine non-probability sample data with probability sample data in estimating spatial means of environmental variables

In estimating spatial means of environmental variables of a region from datacollected by convenience or purposive sampling, validity of the results canbe ensured by collecting additional data through probability sampling. Theprecision of the estimator that uses the probability sample can beincreased by interpolating the values at the nonprobability sample points tothe probability sample points, and using these interpolated values as anauxiliary variable in the difference or regression estimator. Theseestimators are (approximately) unbiased, even when the nonprobability sampleis severely biased such as in preferential samples. The gain in precisioncompared to the estimator in combination with Simple Random Samplingis controlled by the correlation between the target variable andinterpolated variable. This correlation is determined by the size (density)and spatial coverage of the nonprobability sample, and the spatialcontinuity of the target variable. In a case study the average ratio of thevariances of the simple regression estimator and estimator was 0.68for preferential samples of size 150 with moderate spatial clustering, and0.80 for preferential samples of similar size with strong spatialclustering. In the latter case the simple regression estimator wassubstantially more precise than the simple difference estimator.