Email answers to me
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Assignment*Which commandments did
Sepkoski (1984) break, do you think his
inferences hold (if so, to what extent)?
R assignment(s)
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Keep working on it
Statistical Paleobiology
Remote lecture 9 Sep 2013 Oslo Helsinki
Extinction: When did a taxon
become extinct?
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
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
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)
U = point estimate for
extinction time
L = point estimate for
“speciation” or
migration
X = vector for data of
times of occurrence
where X1 is the oldest
and Xn is the youngest.
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
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 within a genus or genera within a
family) (at least 7 temporal 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?
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.