Calculates Values for Foote's Inverse Survivorship Analyses
Description
This function calculates the intermediary values needed for fitting Foote's inverse survivorship analyses, as listed in the table of equations in Foote (2003), with the analyses themselves described further in Foote (2001) and Foote (2005).
Usage
1 2  footeValues(p, q, r, PA_n = 0, PB_1 = 0, p_cont = TRUE, q_cont = TRUE,
Nb = 1)

Arguments
p 
Instantaneous origination/branching rate of taxa. Under a continuous model, assumed to be per interval, or equal to the product of interval lengths and the rates per lineage time units for each interval. Under a pulsed mode (p_cont=FALSE), p is a perinterval 'rate' which can exceed 1 (because diversity can more than double; Foote, 2003a). Given as a vector with length equal to the number of intervals, so a different value may be given for each separate interval. Must be the same length as q and r. 
q 
Instantaneous extinction rate of taxa. Under a continuous model, assumed to be per interval, or equal to the product of interval lengths and the rates per lineage time units for each interval. Under a pulsed mode (q_cont=FALSE), q is a perinterval 'rate' but which cannot be observed to exceed 1 (because you can't have more taxa go extinct than exist). Given as a vector with length equal to the number of intervals, so a different value may be given for each separate interval. Must be the same length as p and r. 
r 
Instantaneous sampling rate of taxa, assumed to be per interval, or equal to the product of interval lengths and the rates per lineage time units for each interval. Given as a vector with length equal to the number of intervals, so a different value may be given for each separate interval. Must be the same length as p and q. 
PA_n 
The probability of sampling a taxon after the last interval included in a survivorship study. Usually zero for extinct groups, although more logically has the value of 1 when there are still extant taxa (i.e., if the last interval is the Holocene and the group is still alive, the probability of sampling them later is probably 1...). Should be a value of 0 to 1. 
PB_1 
The probability of sampling a taxon before the first interval included in a survivorship study. Should be a value of 0 to 1. 
p_cont 
If TRUE (the default), then origination is assumed to be a continuous time process with an instantaneous rate. If FALSE, the origination is treated as a pulsed discretetime process with a probability. 
q_cont 
If TRUE (the default), then extinction is assumed to be a continuous time process with an instantaneous rate. If FALSE, the extinction is treated as a pulsed discretetime process with a probability. 
Nb 
The number of taxa that enter an interval (b is for 'bottom'). This is an arbitrary constant used to scale other values in these calculations and can be safely set to 1. 
Details
Although most calculations in this function agree with the errata for Foote's 2003 table (see references), there were some additional corrections for Prob(DFL) made as part of a personal communication in 2013 between the package author and Michael Foote.
Value
Returns a matrix with number of rows equal to the number of intervals (i.e. the length of p, q and r) and named columns representing the different values calculated by the function: "Nb", "Nbt", "NbL", "NFt", "NFL", "PD_bt", "PD_bL", "PD_Ft", "PD_FL", "PA", "PB", "Xbt", "XbL", "XFt" and "XFL".
Author(s)
David W. Bapst, with advice from Michael Foote.
References
Foote, M. 2001. Inferring temporal patterns of preservation, origination, and extinction from taxonomic survivorship analysis. Paleobiology 27(4):602630.
Foote, M. 2003a. Origination and Extinction through the Phanerozoic: A New Approach. The Journal of Geology 111(2):125148.
Foote, M. 2003b. Erratum: Origination and Extinction through the Phanerozoic: a New Approach. The Journal of Geology 111(6):752753.
Foote, M. 2005. Pulsed origination and extinction in the marine realm. Paleobiology 31(1):620.
Examples
1 2 3 4 5 6 7 8 9 10 11 12  #very simple example with three intervals, same value for all parameters
#example rates (for the most part)
rate<rep(0.1,3)
#all continuous
footeValues(rate,rate,rate)
#origination pulsed
footeValues(rate,rate,rate,p_cont=FALSE)
#extinction pulsed
footeValues(rate,rate,rate,q_cont=FALSE)
#all pulsed
footeValues(rate,rate,rate,p_cont=FALSE,q_cont=FALSE)
