1. Appalachian State University
Rarefaction
Rarefaction: how many is enough?
Steve Hageman
Appalachian State University

2. The Question: Given an assemblage of fossils,
Rarefaction: The Exercise
The Question:
Given an assemblage of fossils,
how many specimens do you need
to collect and identify in order to
adequately represent the diversity
present in the whole?

3. Rationale: • Too few specimens = inadequate
Rarefaction: The Exercise
Rationale:
• Too few specimens = inadequate
representation of diversity.
• Too many specimens = wasted
time and resources.

4. Expected Pattern Number of Species Number of Specimens “Rare Species”
Rarefaction: The Exercise
Expected Pattern
“Rare Species”
Too many specimens =
wasted time and resources.
Number of Species
Too few specimens =
inadequate representation.
Number of Specimens

6. Rarefaction: how many is enough?
Assemblage A

7. 1. Pour out Assemblage-A onto sorting grid (approximately evenly)
Rarefaction: The Exercise
1. Pour out Assemblage-A onto sorting grid (approximately evenly)

8. 2. Start in grid “1” lower left and select the first five specimens.
Rarefaction: The Exercise
2. Start in grid “1” lower left
and select the first five
specimens.
3. Sort the specimens into
discrete species using
your own criteria.

9. 4. Plot the number of species for first five specimens.
Rarefaction: The Exercise
4. Plot the number of species for first five specimens.

10. Rarefaction: The Exercise
5. Repeat steps 2-4 working, your way five specimens at a time through the first 30 specimens (#specimens, #species).
(5, 4)
(10, 5)
(15, 5)
(20, 5)
(25, 5)
(30, 6)

11. 4. Plot the number of species for first 30 specimens.
Rarefaction: The Exercise
4. Plot the number of species for first 30 specimens.

12. 6 maybe 4. Plot the number of species for first 30 specimens.
Rarefaction: The Exercise
4. Plot the number of species for first 30 specimens.
Questions:
For the first 30 specimens,
how many species have you found?
Based on your rarefaction curve,
do you expect to keep finding
new species as you add data? 6
maybe

13. Rarefaction: The Exercise
5. Repeat steps 2-4 working, five specimens at a time through another 30 specimens and plot results.

14. 6 unlikely 4. Plot the number of species for first 30 specimens.
Rarefaction: The Exercise
4. Plot the number of species for first 30 specimens.
Questions:
For the first 60 specimens,
how many species have you found?
Based on your rarefaction curve,
do you expect to keep finding
new species as you add data? 6
unlikely

15. Rarefaction: The Exercise
For any given assemblage, write a description of how you would use a rarefaction cure to determine whether you had observed all (or the majority) of the diversity present.
Go ahead an finish sampling and plotting the data (5 at time) just to be sure you did not miss any species (total of 78 individuals)

16. Rarefaction: The Exercise
7. Plot for all 78 specimens.

17. Rarefaction: is it ever enough?
Assemblage B

18. 1. Pour out Assemblage-B onto sorting grid (approximately evenly)
Rarefaction: The Exercise
1. Pour out Assemblage-B onto sorting grid (approximately evenly)

19. 2. Start in grid “1” lower left and select the first ten specimens.
Rarefaction: The Exercise
2. Start in grid “1” lower left
and select the first ten
specimens.
3. Sort the specimens into
discrete species using
your own criteria.
(#specimens, #species)
(10, 8)

20. 4. Plot the number of species for first ten specimens.
Rarefaction: The Exercise
4. Plot the number of species for first ten specimens. B

21. Rarefaction: The Exercise
5. Repeat steps 2-4 working, your way (10 specimens at a time) through the first 60 specimens.
(20, 16)
(30, 20)
(10, 8)
(60, 31)
(40, 26)
(50, 28)

22. 31 yes 5. First 60 specimens, ten at time. Questions:
Rarefaction: The Exercise
5. First 60 specimens, ten at time. B
Questions:
For the first 60 specimens,
how many species have you found?
Based on your rarefaction curve,
do you expect to keep finding
new species as you add data? 31
yes

23. Rarefaction: The Exercise
6. Continue working, ten specimens at a time until you have sampled 120 specimens.

24. 39 yes 5. First 120 specimens, ten at time. Questions:
Rarefaction: The Exercise
5. First 120 specimens, ten at time. B
Questions:
For the first 120 specimens,
how many species have you found?
Based on your rarefaction curve,
do you expect to keep finding
new species as you add data? 39
yes

25. Rarefaction: The Exercise
6. Continue working, ten specimens at a time until you have sampled a total of 240 specimens.
“Rare Species”
Too many specimens =
wasted time and resources.
Number of Species
Too few specimens =
inadequate representation.
Number of Specimens
As you work, consider when you might pass into the “Rare Species” and “Too Many Specimens” regions of your curve.

26. 130 no! First 240 specimens, ten at time. Majority of common species
Rarefaction: The Exercise
First 240 specimens, ten at time. B
Majority of common species
identified observed by sample#? ___
Oversampling achieved?
130
no!

27. Rarefaction: The Exercise
Continue working, ten specimens at a time until you have sampled all 310 specimens.

28. 47 250 All 310 specimens, ten at time. There are 47 nominative species
Rarefaction: The Exercise
All 310 specimens, ten at time. B
There are 47 nominative species
in this sample. How many species
did you recognize? __ ____
At which sample# did observed
match actual diversity? 47
250

29. All 310 specimens, ten at time.
Rarefaction: The Exercise
All 310 specimens, ten at time. B
If you needed to know every species
present, how many more specimens from
this assemblage should you sample
before you were confident that you had
observed all species?
Can’t predict, depends entirely on the nature of each sample
(# of rare species).
Rarefaction = empirical enterprise

30. 35 species All 310 specimens, ten at time.
Rarefaction: The Exercise
All 310 specimens, ten at time. B
Approximately, how many species would
you have observed in Assemblage B
if you hand only sampled
78 individuals
(like Assemblage A).
35 species

31. 33 species n = 78 All 310 specimens, ten at time.
Rarefaction: The Exercise
All 310 specimens, ten at time. B
But Wait, what if you sampled
again. The curve may be slightly
different.
n = 78
33 species

32. All 310 specimens, ten at time.
Rarefaction: The Exercise
All 310 specimens, ten at time. B
What if, you constructed the curve
100 times? You could calculate an
average for each sample number
and confidence limits.

33. Optional Exercise – Return to Assemblage A
Rarefaction: The Exercise
Optional Exercise – Return to Assemblage A
trial #species
1 4
2 2
3 4
4 4
5 2
6 3
7 2
8 3
9 2
10 2
11 1
12 3
13 2
14 2
15 3
16 3
17 2
18 2
19 3
ave species
cl 95% ± 0.38
Sample 4 at a time
– without replacement
(chosen randomly)
Record # of species each time.
Calculate average and 95% confidence limit of mean for:
n = 4 specimens per sample
N = 78 total
S = 6 species

34. Rarify Assemblage B to compare with A
Rarefaction: The Exercise
Rarify Assemblage B to compare with A B
If you know:
N = total number specimens (310)
n = subsample size (78)
Ni = # of specimens of ith species

35. Rarify Assemblage B to compare with A
Rarefaction: The Exercise
Rarify Assemblage B to compare with A B
Slightly tedious, but good to know
that it can be done when you need
to compare assemblages .

36. uu®®mmqqzv¥¬∆ ®®®®®®mmmmmmµ vs. Part II Comparing Diversity:

37. uu®®mmqqzv¥¬∆ ®®®®®mmmmmmµµ Two concepts: A) # of “kinds” B) evenness
Comparing Diversity
Two concepts: A) # of “kinds”
B) evenness
uu®®mmqqzv¥¬∆
9-kinds,
1 to 2 each
Share 2 kinds
®®®®®mmmmmmµµ
3-kinds,
2 to 6 each

38. total # of taxa in smaller sample (W)
Comparing Diversity
Simpson Coefficient = C / W x 100
emphasizes similarity
(more sensitive to changes)
# of taxa in common (C) x 100
–––––––––––––––––––
total # of taxa in smaller sample (W)
uu®®mmqqzv¥¬∆
vs.
®®®®®mmmmmmµµ
C = 2
W = 13
(2 x 100)/13 = 15.4

39. (# exclusive to A + # exclusive to B - # in common)
Comparing Diversity
Jaccard Coefficient = C / (A + B – C)
emphasizes differences
# of taxa in common (C)
–––––––––––––––––––
(# exclusive to A + # exclusive to B - # in common)
uu®®mmqqzv¥¬∆
C = 2
A = 7
B = 1
vs.
®®®®®mmmmmmµµ
2 / (7 + 1 – 2) = 0.33

40. A: uu®®mmqqzv¥¬∆ B: ®®®®®mmmmmmµµ C: ®®®mmmm¬¬µ D: u®mmqzzzvv
Comparing Diversity
Diversity Coefficients only have meaning
in the context of multiple comparisons
A: uu®®mmqqzv¥¬∆
B: ®®®®®mmmmmmµµ
C: ®®®mmmm¬¬µ
D: u®mmqzzzvv
Calculate Jaccard Coefficients for each pair-wise
comparison (A-B, A-C, A-D, B-C, B-D, C-D)

41. The larger the value, the more similar
Comparing Diversity
The larger the value, the more similar
the assemblages. Rank comparisons:
Assemblage Pair Jaccard Coefficient
Most Similar ______ _____
______ _____
Least Similar ______ _____

42. For Completeness, Calculate the Simpson Coefficient
Rarefaction
For Completeness, Calculate the Simpson Coefficient
and Jaccard Coefficient for comparing your two gastropod
assemblages.
Assemblage A Assemblage B
Number of individuals W = N =
Number of species SA = SB =  
Number of shared
species C =
Number of individuals
unique to each A = B =
Simpson = ______ Jaccard ______ 

43. Rarefaction: Instructor Set Up

44. Rarefaction: Instructor Set Up • Two Assemblages
1 – large # (~300), high diversity,
with many rare specimens.
2 – subset of common species
from first assemblage,
moderate number (50-100),
few species (6-10)

45. Rarefaction: Instructor Set Up • Ideas for Assemblages
1 – “shell collection”
2 – abundant “supplemental”
specimens
pro’s: instructor control, ease of identification,
use otherwise useless collections
con: not a natural assemblage

46. Rarefaction: Instructor Set Up • Natural Assemblages
1 – microfossil suites
2 – sediment
3 – slabs
pro: real world example
con: difficult to gauge student error vs.
natural variation.

48. Rarefaction

49. Rarefaction

50. Rarefaction

52. Rarefaction: Instructor Set Up • Sorting Grid (6x5 cells on Tabloid)
• Graph Paper (generic or: 300 x 50
and 100 x 10)
• Brief Introduction to Rarefaction
and Diversity
• Worksheet