Estimating extinction from species--area relationships: why the numbers do not add up

Researchers commonly use species-area relationships (SAR) to estimate extinction rates caused by habitat loss by reversing the SAR, extrapolating backward from area to calculate expected species loss. We have previously shown that the backward SAR method considerably overestimates extinction rates due to a previously unrecognized sampling artifact. Jacob Bock Axelsen, Uri Roll, Lewi Stone, and Andrew Solow recently argued that the backward SAR method is correct and the method does not overestimate extinction rates. In this paper, we further elaborate and clarify our previous results. We show that the backward SAR method gives the correct extinction rate only under a strict complementary-area sampling design, which is not used in practice because it requires knowing which species are endemic to the area of destroyed habitat, or the number of species in the complementary area. Because of this problem, researchers substitute a power-law model for the SAR in the backward SAR equation. However, this substitution violates the backward SAR method's requirement for complementary sampling. With this model substitution, the backward SAR equation is no longer correct, except in the special case of randomly distributed species. For the complementary sampling or random distribution of species, the first individual of a species to be encountered and the last individual to be encountered to lose the species are exchangeable (or the same individual). But this is not the case for other sampling designs or if species are not randomly distributed and explains why the backward SAR method fails to correctly estimate extinction rates. Our proofs and results are general and explain the widely recognized overestimation of extinction by the backward SAR method. We suggest future directions for developing general theory for estimating species extinction from species-area relationships. Until then, however, the backward SAR method should not be used to estimate species extinction in practice.