Transcript Questions from reading material

Questions from reading material
•
What are Houle et al’s 10 “commandments”
and why are they important?
•
How would you help McKinney (1997)
improve on Figure 2? (and hence the
inferences he draws from them?
•
Assignment*Which commandments did
Sepkoski (1984) break, do you think his
inferences hold (if so, to what extent)?
10 commandments
1. Keep theoretical
context in mind
2. Honour your family of
hypotheses
3. Make meaningful
definitions
4. Know what the
numbers mean
5. Remember where the
numbers come from
6. Respect scale type
7. Know the limits of
your model
8. Never substitute a test
for an estimate
9. Clothe estimates in the
modest raiment of
uncertainty
10. Never separate a
number form its unit
McKinney 1997 "Extinction vulnerability and selectivity:
Combining ecological and paleontological views."
Annual Review of Ecology and Systematics 28: 495-516.
Statistical Paleobiology
Remote lecture 9 Sep 2013 Oslo Helsinki
Extinction: When did a taxon
become extinct?
Outcrop
Outcrop
Horizons
• What do we know?
• How confident are we?
• What are our assumptions?
Assessing the Causes of Late Pleistocene Extinctions on the Continents
Barnosky et al. Science 2004
Bradshaw, C. J. A., et al. (2012). "Robust estimates of extinction time
in the geological record." Quaternary Science Reviews 33: 14-19.
Marshall, C. R. (1995). "Distinguishing between Sudden and Gradual Extinctions
in the Fossil Record - Predicting the Position of the Cretaceous-Tertiary Iridium
Anomaly Using the Ammonite Fossil Record on Seymour Island, Antarctica."
Geology 23(8): 731-734.
Why is it important to ‘know’ when a taxon
became extinct?
• In its own right
• Helps to understand WHY it became extinct (or
migrated in case of local extinction)
• Changes in extinction rates (multiple taxa)
• Understanding drivers of extinction
• Temporal correlation of strata
Why is estimating confidence intervals
important?
•
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Assess hypotheses of extinction
Baseline for incompleteness of fossil record
Predict future fossil finds
Predict which fossils species might be extant
Assess phylogenetic hypotheses (and taxonomic
assignments)
Set Refresher
Grey box: the events in which taxon A was alive and
then dead and buried
If A is an event, then
𝑃 𝐴 =1−𝑃 𝐴
A
Not A
Red circle: the events in which taxon A was sampled
Additive Law of Probability
𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵 − 𝑃 𝐴∩𝐵
If A and B are mutually exclusive, then
𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵
Additive Law of Probability
𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵 − 𝑃 𝐴∩𝐵
If A and B are mutually exclusive, then
𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵
Multiplicative Law of Probability
𝑃 𝐴∩𝐵 =𝑃 𝐴 𝑃 𝐵 𝐴 =𝑃 𝐵 𝑃 𝐴 𝐵
If A and B are independent, then
𝑃 𝐴∩𝐵 =𝑃 𝐴 𝑃 𝐵
Class exercise:
There are two events A and B. A is the event where a
fossil deer species is present and B is the event where
its fossil predator is present. The union of A and B is 0.4
while P(A) and P(B) are 0.2 and 0.3 respectively. Find the
following and say what they mean.
i) 𝑷 𝑨 ∩ 𝑩
ii) 𝑷 𝑨 ∪ 𝑩
iii) 𝑷 𝑨 ∩ 𝑩
iv) 𝑷 𝑨|𝑩
Solution
0.6
0.1
0.1 0.2
i) 𝑃 𝐴 ∩ 𝐵 =0.1 (both deer and predator present)
ii) 𝑃 𝐴 ∪ 𝐵 = 0.9 (when one or both of them are absent)
iii) 𝑃 𝐴 ∩ 𝐵 = 0.6 (when either or both is present)
iv) 𝑃 𝐴|𝐵 = 2/3
(the chances of a deer being present given that the predator is present)
Basic Probability Review
Interactive Multimedia Course developed by Rice University, U of Houston
Clear Lake and Tufs University
• http://onlinestatbook.com/
Probability lecture
• http://www.youtube.com/watch?v=F5TDpbPS
y1w
If A is an event, then
𝑃 𝐴 =1−𝑃 𝐴
Marshall 1990 (actually “anglicizing
Sadler and Strauss older papers):
Assuming random preservation/sampling
Stratigraphic range is AWALYS shorter than TRUE duration (barring reworking)
1
α = 1 − 𝐶1
−(
)
n_ 1
𝐶1 = 1 − 1 + α
𝐶2 = 1 − 2 1 + α
− (n_1)
−1
− (n_1)
− 1 + 2α
− (n_1)
Marshall 1990 Assuming random
preservation/sampling
Class exercise
• N=10
• C=0.95
• α=?
• Oldest sample =60 Ma
• Youngest sample
=50Ma
• What is the range of
true extinction?
1
α = 1 − 𝐶1
−(
)
_
n1
−1
• What is the range of
true origination?
• What is the range of
both (simultaneously?)
• R
Marshall 1990 Assuming random
preservation/sampling
Testing assumptions
• Is fossilization random? (is sampling
stochastically constant?)
• Are fossilization events independent?
(multiple records taken as one)
• *Continuous sampling
• R
Marshall 1990 Assuming random
preservation/sampling
Marshall, C. R. (1995). "Distinguishing between Sudden and Gradual Extinctions
in the Fossil Record - Predicting the Position of the Cretaceous-Tertiary Iridium
Anomaly Using the Ammonite Fossil Record on Seymour Island, Antarctica."
Geology 23(8): 731-734.
Stratigraphic range is AWALYS shorter than TRUE duration (barring reworking)
• estimates of θ1 and θ2 and are y and z respectively
but these are biased because
P(Y< θ1 )=1
P(Z> θ2 )=1
(if time in MYA runs from bottom to top )
• Doing nothing is a decision,
doing something is better than doing
nothing in this case
Strauss, D. and P. M. Sadler (1989). "Classical
Confidence-Intervals and Bayesian Probability
Estimates for Ends of Local Taxon Ranges."
Mathematical Geology 21(4): 411-421.
Marshall 1994 Paleoiology
gap size
3.0
1
1
1
2.5
6
5
4
Median = 4.5
24
2
8
1.5
0.5
1.0
3
Frequency
2.0
7
6
5
4
0.0
3
0
2
5
10
15
20
25
30
gap size
1
12
1
Non-random preservation/sampling
Marshall 1994 Paleoiology
Assumes gap duration distribution free
• Any gap has a 50% chance of being larger than
the median
• The chance for all gaps to be larger than the
median of the underlying distribution 0
0.56=0.0156.
• That also means that the probability that the
median gap lies within the range of those
sampled is 1-0.0312 = 0.9688
• Catch: CI’s have own uncertainties
Marshall 1994 Paleoiology
Confidence levels
For N = 6 and for the statement,
that a gap has a 50 % chance of
being greater or smaller than the
median, we have a 0.95 probability
that the next gap is as small as the
first smallest gap and or as large as
the 6th largest gap.
Non-random preservation/sampling
Cheetham, A. H. (1986). "Tempo of Evolution in a Neogene Bryozoan:
Rates of Morphologic Change Within and Across Species Boundaries."
Paleobiology 12(2): 190-202.
Marshall 1994 Paleoiology
Solow, A. R. (2003). "Estimation of stratigraphic
ranges when fossil finds are not randomly
distributed." Paleobiology 29(2): 181-185.
(Based on Robson and Whitlock 1964)
R exercise
Reasons for non randomness
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Sequence stratigraphic architectures
Variation in paleo-environment
Variation in quality of outcrop
Taphonomic regimes
Collecting practices
Ocean circulation
Biotic interactions
(many more reasons for global non-randomness)
Marshall, C. R. (1997). "Confidence intervals on stratigraphic ranges with
nonrandom distributions of fossil horizons." Paleobiology 23(2): 165-173.
Summary of single taxon extinction time
estimation covered
• Assume uniform random sampling (Strauss and
Sadler 1986, Marshall 1990)
• Distribution free gaps (Marshall 1994)
• Non-random distribution of fossil finds (Solow 2003)
• When the fossil recovery potential is known
(Marshall 1997)
• If a paper doesn’t talk about assumptions, think
about the implicit ones
• violating assumptions vs not measuring uncertainty
at all
References
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READING: Marshall 2010 in Quantitative Paleobiology short course
Strauss, D. and P. M. Sadler (1989). "Classical Confidence-Intervals and Bayesian Probability
Estimates for Ends of Local Taxon Ranges." Mathematical Geology 21(4): 411-421.
Marshall, C. R. (1990). "Confidence-intervals on stratigraphic ranges." Paleobiology 16(1): 110.
Marshall, C. R. (1994). "Confidence-intervals on stratigraphic ranges - partial relaxation of the
assumption of randomly distributed fossil horizons." Paleobiology 20(4): 459-469.
Marshall, C. R. (1997). "Confidence intervals on stratigraphic ranges with nonrandom
distributions of fossil horizons." Paleobiology 23(2): 165-173.
Weiss, R. E. and C. R. Marshall (1999). "The uncertainty in the true end point of a fossil's
stratigraphic range when stratigraphic sections are sampled discretely." Mathematical
Geology 31(4): 435-453.
Solow, A. R. (2003). "Estimation of stratigraphic ranges when fossil finds are not randomly
distributed." Paleobiology 29(2): 181-185.
Bradshaw, C. J. A., et al. (2012). "Robust estimates of extinction time in the geological
record." Quaternary Science Reviews 33: 14-19.
Assignment
• Download sampled occurrence data for a taxon of your interest
from the PBDB (can be species or genus or family) (at least 7 data
points)
• Write a short description of the taxon
• Using the data you downloaded, write an R script (annotated) to
organize the data and to estimate the range end points using the
methods presented in Marshall 1990 and Solow 2003.
• Write a summary of your observations
• What assumptions must you make and are these assumptions likely
to have been violated?
• What are the consequences of the violations? Should you use the
method given that assumptions have been violated or would you
rather just report raw or mean values?
http://pbdb.org/
http://www.r-project.org/
Optional Assignments
• Marshall 1990 is based on continuous fossilization. Simulate
both a continuous fossilization process and a discrete
fossilization process and explore how much of an issue it is to
violate the assumption that fossilization is continuous, in R.
• Solow 2003 seems like a dream, so simple and elegant.
Simulate a few probable fossilization processes and apply
Solow 2003 to them to check out how reliable the approach
is, in R.
Summary
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*Question to be answer (Houle and Sepkoski)
Sets
Introduction to R
Single taxon extinction
*Paleobiology Database
– As an example of data units
• *Simulation exercise