calc_obs_like: Calculate likelihood of count data given true presence/absence and parameters

Description

This function calculates P(data|presence,parameters), i.e. the probability of some detection and taphonomic control counts, given the true geographic range/state, and parameters such as dp, a detection probability (and, optionally, a false detection probability, fdp).

Usage

calc_obs_like(truly_present = TRUE, obs_target_species, obs_all_species, mean_frequency = 0.1, dp = 1, fdp = 0)

Arguments

truly_present

Is the OTU of interest known/conditionally assumed to be truly present (TRUE) or truly absent (FALSE)?

obs_target_species

A count of detections of your OTU of interest, e.g. as produced from a cell of the matrix output from read_detections.

obs_all_species

A count of detections of your taphonomic controls, e.g. as produced from a cell of the output from read_controls.

mean_frequency

This is the proportion of samples from the taphonomic control group that will truly be from this OTU, GIVEN that the OTU is present. This could be estimated, but a decent first guess is (total # samples of OTU of interest / total # of samples in the taphonomic control group where the OTU is known to be present). All that is really needed is some reasonable value, such that more sampling without detection lowers the likelihood of the data on the hypothesis of true presence, and vice versa. This value can only be 1 when the number of detections = the number of taphonomic control detections, for every OTU and area. This is the implicit assumption in e.g. standard historical biogeography analyses in LAGRANGE or BioGeoBEARS.

The detection probability. This is the per-sample probability that you will correctly detect the OTU in question, when you are looking at it. Default is 1, which is the implicit assumption in standard analyses.

fdp

The false detection probability. This is probability of falsely concluding a detection of the OTU of interest occurred, when in fact the specimen was of something else. The default is 0, which assumes zero error rate, i.e. the assumption being made in all historical biogeography analyses that do not take into account detection probability. These options are being included for completeness, but it may not be wise to try to infer mean_frequency, dp and fdp all at once due to identifiability issues (and estimation of fdp may take a very large amount of data). However, fixing some of these parameters to reasonable values can allow the user to effectively include beliefs about the uncertainty of the input data into the analysis, if desired.

Value

lnlike_allobs_given_absence The natural log-likelihood of the data, given the model & assumption of true presence or absence.

Details

The idea of taphonomic controls dates back at least to work of Bottjer & Jablonski (1988). The basic idea is that if you have taxa of roughly similar detectability, then detections of other taxa give some idea of overall detection effort. Obviously this is a very simple model that can be criticized in any number of ways (different alpha diversity in each region, different detectability of individual taxa, etc.), but it is a useful starting point as there has been no implementation of any detection model in historical/phylogenetic biogeography to date.

One could imagine (a) every OTU and area has a different count of detections and taphonomic control detections, or (b) the taphonomic control detections are specified by area, and shared across all OTUs. Situation (b) is likely more common, but this function assumes (a) as this is the more thorough case. Behavior (b) could be reproduced by summing each column, and/or copying this sum to all cells for a particular area.

References

http://phylo.wikidot.com/matzke-2013-international-biogeography-society-poster

Matzke_2012_IBS

Bottjer_Jablonski_1988

Examples

Run this code

# Example: 10 observations of the species mean dramatically higher likelihood of the
# data on the hypothesis that it is truly present.

# With zero error rate
obs_target_species = 10
obs_all_species = 100
mean_frequency=0.1
dp=1
fdp=0
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence
# Note that the probability of getting detections, under the hypothesis of
# true absence, is -Inf


# With a small error rate, there is some small but positive probability of
# falsely getting 10 detections
obs_target_species = 10
obs_all_species = 100
mean_frequency=0.1
dp=0.99
fdp=0.001
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence
# i.e. the prob. of the data is 1 under the hypothesis of presence, and 0
# under the hypothesis of absence (ln(prob) = 0 & -Inf, respectively)


# Note that with very high error rates, your conclusion could reverse
obs_target_species = 10
obs_all_species = 100
mean_frequency=0.1
dp=0.5
fdp=0.3
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence


# Example #2 -- what if you have ZERO detections, but lots of detections
# of your taphonomic control?
obs_target_species = 0
obs_all_species = 1
mean_frequency=0.1
dp=1
fdp=0
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence

# With a slight error rate
obs_target_species = 0
obs_all_species = 1
mean_frequency=0.1
dp=0.99
fdp=0.001
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence


obs_target_species = 0
obs_all_species = 2
mean_frequency=0.1
dp=1
fdp=0
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence

# With a slight error rate
obs_target_species = 0
obs_all_species = 2
mean_frequency=0.1
dp=0.99
fdp=0.001
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence





# Example #3 -- what if you have ZERO detections, but only a few
# detections of your taphonomic control?
obs_target_species = 0
obs_all_species = 100
mean_frequency=0.1
dp=1
fdp=0
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence

# With a slight error rate
obs_target_species = 0
obs_all_species = 100
mean_frequency=0.1
dp=0.99
fdp=0.001
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence



# Special cases -- e.g., no data
# Prob(data)=1, ln(prob)=0
obs_target_species = 0
obs_all_species = 0
mean_frequency=0.1
dp=0.99
fdp=0.001
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence

obs_target_species = 0
obs_all_species = 0
mean_frequency=0.1
dp=1
fdp=0
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence


# What if, for some reason, you put in identical detections and taphonomic control
# counts? (e.g., you load in a standard tipranges file)
obs_target_species = 1
obs_all_species = 1
mean_frequency=1
dp=1
fdp=0
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence

# What if, for some reason, you put in identical detections and taphonomic control
# counts? (e.g., you load in a standard tipranges file)
obs_target_species = 1
obs_all_species = 1
mean_frequency=1
dp=0.99
fdp=0.001
LnL_under_presence = calc_obs_like(truly_present=TRUE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_absence = calc_obs_like(truly_present=FALSE, obs_target_species,
obs_all_species, mean_frequency, dp, fdp)
LnL_under_presence
LnL_under_absence

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