Estimation of the Mean From a Two-Dimensional Sample: The Geostatistical Model-Based Approach

Three common sampling methods for spatial data are simple random sampling (SRS), systematic sampling (Syst), and stratified random sampling (Strat). When values are spatially correlated, SRS usually leads to estimates having higher variances than with the other two methods. However, the SRS-based variance can be “design-based,” using hypothetical repetitions of the sampling process: it needs no assumptions about the unobserved values. Syst and Strat methods do need “model-based” assumptions. As a result, variances of estimates based on Syst or Strat are often estimated by the SRS formula, e.g., s2/n for the variance of a mean. In a previous paper, we showed that these estimates are misleading, and we described another way to obtain improved estimates. It assumes that the pattern being sampled arose from a random process and considers the expected squared error, E([estimate − parameter]2), that would arise if this process were run repeatedly and sampled at the same places each time. The unconditional variance uses all such hypothetical runs of the process. This paper extends the previous one to a conditional variance; it uses only those hypothetical runs whose values at the sampled points are the same as those in the actual sample. We show that the conditional variance is more reliable and that the other two overestimate uncertainty, often greatly. We calculate confidence and prediction intervals of the mean from the unconditional and the conditional variances. Intervals based on the conditional variance are usually shorter, and they better describe the true uncertainty. We illustrate the methods by case studies on three populations: one using data on the spatial population of acorns under an isolated oak, and the other two simulated. We emphasize that, although our approach applies to any two-dimensional sample, model-based inference requires that the sample be sufficiently representative of the population and that the probabilistic spatial model be specified with caution.