Estimates, by maximum likelihood, the beta of Aldous' beta-splitting model
from the topology and the c of the lineage-dependent pure-birth process
from the branching times, which places an empirical tree on the same scale
as the trees of sim_tree_shape(). Optionally, a parametric bootstrap with
the package's own generators gives percentile intervals.
Arguments
- tree
A binary, rooted, ultrametric
phyloobject with at least four tips.- boot
Number of parametric bootstrap replicates (0 for none).
- level
Coverage of the bootstrap percentile intervals.
- beta_range, c_range
Search intervals for
beta(lower bound above -2) andc.- tol
Tolerance of the ultrametricity check, as in
ape::is.ultrametric()(scaled range of root-to-tip distances). Trees with rounded branch lengths may need a larger value, or can be made ultrametric first (for instance withphytools::force.ultrametric()).
Value
An object of class shape_fit: a list with n, beta, c, r
(speciation rate per lineage at one lineage, in the time units of the
tree), loglik_beta (log-likelihood of beta up to an additive constant,
so not comparable with other software), loglik_c (profile log-likelihood
of c, equal to the log-likelihood of laser's DDX model at its maximum),
beta_at_bound and c_at_bound (estimate at a bound of the search
interval), beta_range, c_range, shape (output of tree_shape()) and,
with boot > 0, boot (data frame of bootstrap estimates) and ci
(matrix of percentile intervals).
Details
beta. Each internal node with a clade of \(m\) tips split into
\(i\) and \(m - i\) tips contributes the probability of that split under
the beta-splitting model (see rtopology_beta()), and the log-likelihood is
the sum over nodes (Aldous 1996; this is the estimator applied by Blum &
Francois 2006 and implemented in maxlik.betasplit() of the archived package
apTreeshape). It is maximized over beta_range. In the package's validation
runs (data-raw/validate_estimators.R in the source repository), the
estimate was nearly unbiased for beta of -1 or less, but had a positive
bias and a large variance for balanced trees with few tips (beta of 0 or
more with 50 to 100 tips). An estimate at a bound of beta_range only says
that the topology is extreme for the model.
c. With \(k\) lineages, each lineage splits at rate \(r k^c\), which is the density-dependent exponential (DDX) model of the R package laser (Rabosky 2006), whose per-lineage rate is \(r k^{-x}\), so that \(c = -x\). As in laser, the splits at 2 to \(n - 1\) lineages are events and the last interval, from the most recent split to the present, is not; \(r\) is profiled out (its maximum likelihood value is \((n - 2) / \sum_k k^{1 + c} g_k\), where \(g_k\) is the time with \(k\) lineages). The estimate describes the shape of the lineage-through-time curve, not its cause: branching times alone do not separate diversity-dependent from time-dependent diversification (Pannetier et al. 2021).
Bootstrap. With boot > 0, topologies are drawn with
rtopology_beta() at the estimated beta and intervals with
intervals_lineage_rate() at the estimated c, and both parameters are
re-estimated, following the recommendation of a parametric bootstrap for
diversification models by Etienne et al. (2016). The spread of the bootstrap
estimates around the point estimate includes the small-sample bias of the
estimators, and, for c, a small additional bias because
intervals_lineage_rate() draws the last interval from the same law,
whereas the likelihood treats it as censored. Percentile intervals need a
few hundred replicates.
References
Aldous, D. (1996). Probability distributions on cladograms. In D. Aldous & R. Pemantle (Eds.), Random discrete structures (pp. 1–18). Springer. doi:10.1007/978-1-4612-0719-1_1
Blum, M. G. B., & Francois, O. (2006). Which random processes describe the tree of life? A large-scale study of phylogenetic tree imbalance. Systematic Biology, 55, 685–691. doi:10.1080/10635150600889625
Etienne, R. S., Pigot, A. L., & Phillimore, A. B. (2016). How reliably can we infer diversity-dependent diversification from phylogenies? Methods in Ecology and Evolution, 7, 1092–1099. doi:10.1111/2041-210X.12565
Pannetier, T., Martinez, C., Bunnefeld, L., & Etienne, R. S. (2021). Branching patterns in phylogenies cannot distinguish diversity-dependent diversification from time-dependent diversification. Evolution, 75, 25–38. doi:10.1111/evo.14124
Rabosky, D. L. (2006). LASER: a maximum likelihood toolkit for detecting temporal shifts in diversification rates from molecular phylogenies. Evolutionary Bioinformatics, 2. doi:10.1177/117693430600200024
Examples
set.seed(1)
tr <- sim_tree_shape(200, beta = -1, c = -0.8)
fit_shape(tr)
#> Shape estimates for a tree with 200 tips
#> beta c
#> -0.864 -0.895
#> gamma = -7.59 | normalized Colless = 3.03
fit <- fit_shape(tr, boot = 50)
fit$ci
#> 2.5% 97.5%
#> beta -1.102879 -0.5561590
#> c -1.046498 -0.7468201