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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)

• Keep working on it

Extinction: When did a taxon become extinct?

Statistical PaleobiologyRemote lecture 9 Sep 2013 Oslo Helsinki

Marshall 1990 (actually “anglicizing Sadler and Strauss older papers): Assuming random preservation/sampling

If A is an event, then

α=(1−𝐶1 )−(1n1

)−1

𝐶 1=1− (1+α )−(n1)

Marshall 1990 Assuming random preservation/sampling

Stratigraphic range is AWALYS shorter than TRUE duration (barring reworking)

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 timeL = point estimate for “speciation” or migrationX = vector for data of times of occurrence where X1 is the oldest and Xn is the youngest.

Non-random preservation/sampling

7654

3

2

1

654

3

2

1

111

24

8

12

gap size

gap size

Freq

uenc

y0 5 10 15 20 25 30

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Marshall 1994 Paleoiology

Median = 4.5

• 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

Assumes gap duration distribution free

Marshall 1994 Paleoiology

Non-random preservation/sampling

Confidence levelsFor 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.

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

• 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• 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): 1-

10.• 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.