Geostatistical Estimation Variance for the Spatial Mean in Two-Dimensional Systematic Sampling

Abstract

Many ecologists use two-dimensional systematic sampling to estimate mean density of individuals over the domain sampled. They usually calculate the variance of the mean as if the sample were a simple random sample, using the unbiased estimator under this sampling design, that is, σ2/n. This practice leads to a selection bias, i.e., incorrect inclusion probabilities of population units in the sample are used in the estimator of variance of the mean. The magnitude of the bias varies with the underlying spatial autocorrelation structure. Design-based inference and model-based inference are two conceptual frameworks for tackling estimation of variance of the mean in a systematic sample. This paper reports use of the geostatistical estimation variance σ2E with the model-based approach. This variance of the overall spatial mean depends on the spatial autocorrelation structure of the data. We illustrate the method by considering the density of acorns fallen under a sessile oak during one season. Acorns were numbered in square quadrats of 0.25 m2; the data set was exhaustive. We drew nine one-start systematic samples from the whole population of quadrats. We computed semivariograms for the whole population and each sample and fitted them to exponential models without nugget. Using geostatistical theoretical results, we calculated a variance of the mean density of acorns by Monte Carlo integration. We show that our variance estimate depends on (1) the origin of the systematic sample, the central sample being the most accurate, (2) quadrat size, with a decrease in the variance when quadrat size increases, (3) the semivariogram model, (4) discretization of the domain used in Monte Carlo integration, and (5) the random number generator. Considering all sources of variation, the variance estimate we calculated ranged from ∼2 to 36 in our example, for an overall mean equal to 95 acorns per quadrat. Geostatistical variance mainly reflects the locations and size of the sampled quadrats, and the spatial autocorrelation function. In our example, using the variance estimator of a simple random sample leads to a high selection bias. The bias can be neglected only in the absence of significant spatial autocorrelation in the sample. Otherwise, we advocate a geostatistical model-based approach that accounts for spatial autocorrelation.