The gamma can only predict the variance of richness and diversity under phylogenetic coarse-graining when covariance is included.
a,b) In contrast with the mean, the variance of richness and diversity estimates predicted by the gamma (Eq. 15) assuming independence among species fails to capture empirical estimates from the human gut. Predictions are only comparable when empirical estimates of covariance are included in the predictions of the gamma distribution. c,d) This lack of predictive success is constant across environments, e,f) though the addition of covariance consistently improves our analytic predictions.