Gut microbiota ecology: Biodiversity estimated from hybrid neutral-niche model increases with health status and aging

the number of accepted parameters was highly variable from sample to sample. For some sam- ples the number of parameters accepted in the first step was very small or null. For other sam- ples, however, thousands of parameters were identified as acceptable. For this reason, we chose to limit the number of accepted parameters obtained from each sample to 5. This allowed to avoid high unbalances in the number of parameters derived from each sample, as well as to improve the computational efficency of the posterior fitting. Finally, the posterior distributions obtained in the first step were used as data set-specific prior distributions and the ABC algo- rithm was run again to obtain the final model estimates (second step). In both fitting steps, for 107 times we randomly sampled a set of parameters from their prior distributions and we sim- ulated a number of data equal to the number of OTUs according to the selected model (1NB, 2NB or 3NB). Then, we evaluated the accordance between the simulated and the empirical RSA comparing the two Preston’s plots. This choice was made to reduce the numerical issues related to the sparsity of the data in the heavy tailed distributions [30]. Specifically, each set of parameters was accepted if both the conditions described in the following were satisfied. First, we set the maximum acceptable absolute difference between data and simulation counts in each bin to 30% of the data counts. Secondly, we constructed a variant of the chi-squared test that was appropriate for the comparison of the two observed samples (real and simulated

data). Our null hypothesis, here, is that the difference between the observed values for each bin of the Preston’s plot is distributed as a Skellam’s distribution (the difference between two sta- tistically independent random variables, each Poisson-distributed with respective expected val- ues μ1 and μ2). Given that this distribution converges to the Normal distribution, we can use a variant of the chi-squared test obtained by summing the standardized Skellam’s distributions of each bin of the Preston’s plot: w2 ¼ PN i¼1 ðmi1 mi2Þ2 ðmi1 þ mi2Þ ð12Þ where i indicates the i-th bin of the Preston’s plot. The number of degrees of freedom was set equal to the number of non-zero bins and the set of simulated parameters was accepted when the probability that the simulated and true bins of the Preston’s plot come from the same distribution was higher than the probability that they did not, i.e. when the chi-squared cumulative distribution at χ2 was lower than 0.5. This crite- ria was designed according the samples acceptance criteria in ABC. Finally, for each sample, the posterior distributions of the parameters were computed considering all accepted set of parameters. Model selection was achieved comparing the posterior probability of each model, estimated computing the ratio between the number of accepted parameters sets over the total number of simulated parameters (107), obtaining a median number of accepted parameters of ~2000 sam- ples (acceptance rate of ~2�10−4, relative error in the rate estimation of 2%). Specifically, for

each pair of models we computed the logarithm of the ratio of the two posterior probabilities of each sample. This method computes an approximation of the Logarithmic Bayes Factor, defined as the log ratio of the probabilities that each model is the actual true model, without incorporating prior beliefs about the plausibility of each mode [47]. This method is akin to a Bayesian equivalent of the likelihood ratio tests such as BIC (Bayes- ian Information Criterion) and AIC (Akaike Information Criterion) [48], but including an implicit penalization for the number of parameters that each model possesses and the shape of the prior for each parameter: wider priors (encoding for less certain parameters) cause a greater penalization than narrower ones (encoding for more information available about them). PLOS ONE Gut microbiota biodiversity and healthy aging PLOS ONE | https://doi.org/10.1371/journal.pone.0237207 October 30, 2020 16 / 23