Spatial correlations at different spatial scales are themselves highly correlated in isolation by distance processes.

Although many properties of spatial autocorrelation statistics are well characterized, virtually nothing is known about possible correlations among values at different spatial scales, which ultimately would influence how inferences about spatial genetics are made at multiple spatial scales. This article reports the results of stochastic space-time simulations of isolation by distance processes, having a very wide range of amounts of dispersal for plants or animals, and analyses of the correlations among Moran's I-statistics for different mutually exclusive distance classes. In general, the stochastic correlations are extremely large (>0.90); however, the correlations bear a complex relationship with level of dispersal, spatial scale and spatial lag between distance classes. The correlations are so large that any existing or conceived statistical method that employs more than one distance class (or spatial scale) should not ignore them. This result also suggests that gains in statistical power via increasing sample size are limited, and that increasing numbers of assayed loci generally should be preferred. To the extent that sampling error for real data sets can be treated as white noise, it should be possible to account for stochastic correlations in formulating more precise statistical methods. Further, while the current results are for isolation by distance processes, they provide some guidance for some more complex stochastic space-time processes of landscape genetics. Moreover, the results hold for several popular measures other than Moran's I. In addition, in the results, the signal to noise ratios strongly decreased with distance, which also has several implications for optimal statistical methods using correlations at multiple spatial scales.