probAnc: Probability of being a sampled ancestor of another sampled taxon

Description

This function uses models from Foote (1996) to calculate the probability of sampling a descendant of a morphotaxon in the fossil record, given the sampling probability and estimates of origination and extinction rates.

Usage

probAnc(
  p,
  q,
  R,
  mode = "budding",
  analysis = "directDesc",
  Mmax = 85,
  nrep = 10000
)

Arguments

p: Instantaneous rate of speciation (lambda). If the underlying model assumed is anagenetic (e.g. taxonomic change within a single lineage, 'phyletic evolution') with no branching of lineages, then p will be used as the rate of anagenetic differentiation.
q: Instantaneous rate of extinction (mu)
R: Per-interval probability of sampling a taxon at least once.
mode: Mode of morphotaxon differentiation, based on definitions in Foote, 1996. Can be pure cladogenetic budding ("budding"), pure cladogenetic bifurcating ("bifurcating") or pure anagenetic within-lineage change ("anagenesis"; i.e. Foote's 'phyletic change'). Default mode is "budding".
analysis: The type of analysis to be performed, either the probability of sampling direct descendants ("directDesc") or of sampling indirect descendants ("indirectDesc").
Mmax: The maximum number of direct descendants (M) to sum over in the function, which is ideally meant to be a sum from zero to infinity, like nrep. Unfortunately, (2*M) is used in a factorial, which means we are limited to a relatively small upper bound on M.
nrep: Number of repetitions to run in functions which are meant to sum over infinity. Default is arbitrarily high.

Details

These values are always calculated assuming infinite time for the potential ancestor to produce daughter taxa (assuming it lives that long) and under homogenous birth, death and sampling rates/probabilities, which is a situation that may be overly ideal relative to many real fossil records.

These probabilities can be calculated for either direct descendants, i.e. the probability of sampling any morphotaxa that arise immediately from the particular morphotaxon that could be an ancestor, or indirect descendants, i.e. the probability for any morphotaxon that has the morphotaxon of question as an ancestor, no matter how distant. See the argument analysis for details. Mode of differentiation can also be varied for three different models, see the argument mode.

The probability of sampling a taxon's ancestor is calculated while accounting for the probability that extinction might occur before any descendants are produced. Thus, if p = q, the probability of a taxon going extinct before it produces any descendants will be 0.5, which means that even when sampling is perfect (R = 1, meaning completeness of 100 can be no higher than 0.5. See Foote (1996) for a graphic depiction of this non-intuitive ceiling. For reasons (probably?) having to do with finite approximations of infinite summations, values close to perfect sampling may have values slightly higher than this ceiling, which is also apparent visually in the figures in Foote (1996). Thus, values higher than 0.5 when p = q should be discounted, and in general when sampling rate is high, results should be treated cautiously as overestimates.

References

Foote, M. 1996 On the Probability of Ancestors in the Fossil Record. Paleobiology 22(2):141--151.

Examples

Run this code

# examples, run at very low nrep for sake of speed (examples need to be fast)

# default options 
    # probability of sampling a direct descendant
probAnc(p = 0.1, q = 0.1, R = 0.5, 
        mode = "budding", 
        analysis = "directDesc",
        nrep = 100)

# other modes
probAnc(p = 0.1, q = 0.1, R = 0.5, 
        mode = "bifurcating", 
        analysis = "directDesc",
        nrep = 100)
probAnc(p = 0.1, q = 0.1, R = 0.5, 
        mode = "anagenesis", 
        analysis = "directDesc",
        nrep = 100)

# probability of having sampled indirect descendants of a taxon

# first, the default
probAnc(p = 0.1, q = 0.1, R = 0.5, 
        mode = "budding", 
        analysis = "indirectDesc",
        nrep = 100)
        	
probAnc(p = 0.1, q = 0.1, R = 0.5, 
        mode = "bifurcating", 
        analysis = "indirectDesc", 
        nrep = 100)
        
probAnc(p = 0.1, q = 0.1, R = 0.5, 
        mode = "anagenesis", 
        analysis = "indirectDesc",
        nrep = 100)