Many studies correlate geographic variation of biotic variables (e.g., species ranges, species richness, etc.) with variation in environmental variables (climate, topography, history). Often, the resulting correlations are interpreted as evidence of causal links. However, both the dependent and independent variables in these analyses are strongly spatially structured. Several studies have suggested that spatially structured variables may be significantly correlated with one another despite the absence of a causal link between them. In this study we ask: if two variables are spatially structured, but causally unrelated, how strong is the expected correlation between them? As a specific example, we consider the correlations between broad-scale variation in gamma species richness and climatic variables. Are these correlations likely to be statistical artefacts? To answer these questions, we randomly generated pseudo-climatic variables that have the same range and spatial autocorrelation as temperature and precipitation in the Americas. We related mammal and bird species richness both to the real and the pseudo-climatic variables. We also observed the correlations among pseudo-climate simulations. Correlations among randomly generated, spatially unstructured, variables are very small. In contrast, the median correlations between spatially structured variables are near r²=0.1 – 0.3, and the 95% confidence limits extend to r²=0.6 – 0.7. Viewing this as a null expectation, given spatially structured variables, it is worth nothing that published richness–climate correlations are typically marginally stronger than these values. However, many other published richness–environment correlations would fail this test. Tests of the “predictive ability” of a correlation cannot reliably distinguish correlations due to spatial structure from causal relationships. Our results suggest a three-part update of Tobler’s “First Law of Geography”: #1) Everything in geography that is spatially structured will be collinear. #2) Near things are more related than distant things. #3) The more strongly spatially structured two variables are, the stronger the collinearity between them will be.