This document summarizes a seminar on detectability in ecological systems. It discusses using genetic data from mosquito blood meals to estimate the population size of American robins, presenting mathematical frameworks including maximum likelihood estimation and Good-Turing estimators. It also briefly outlines using turtle recapture data to estimate their population size. The seminar covers challenges in estimating population sizes when detectability is imperfect and individuals vary in their likelihood of being detected.

1. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Detectability in ecological systems:
two nonstandard examples
Ben Bolker, McMaster University
Departments of Mathematics & Statistics and Biology
Math Bio Research Seminar
3 October 2014
Ben Bolker Math Bio Research Seminar
Detectability

2. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Acknowledgements
Money NSF, NSERC
Computational resources SHARCnet
Data and discussions Aaron Berk, Alan Bolten, Karen Bjorndal,
Leonid Bogachev, Ethan Bolker, Ira Gessel, Marm
Kilpatrick
Ben Bolker Math Bio Research Seminar
Detectability

3. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Outline
1 Introduction
2 Mosquitoes/WNV
3 Turtle surveys
4 Meta- stuff
Ben Bolker Math Bio Research Seminar
Detectability

4. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Outline
1 Introduction
2 Mosquitoes/WNV
3 Turtle surveys
4 Meta- stuff
Ben Bolker Math Bio Research Seminar
Detectability

5. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Detectability in ecological problems
ecological sampling is imperfect;
individuals may vary in detectability
sometimes it matters
sometimes it’s unidentifiable
sampling designs
(e.g. capture-mark-recapture)
statistical methods
(MLE, Bayesian MCMC)
relevance in other fields of math
bio?
Ben Bolker Math Bio Research Seminar
Detectability

6. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Introductory meta- stuff
Working on problems:
the “Pacala method”
http://weedactivist.com/2013/04/26/reinventing-the-wheel/
Ben Bolker Math Bio Research Seminar
Detectability

7. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Outline
1 Introduction
2 Mosquitoes/WNV
3 Turtle surveys
4 Meta- stuff
Ben Bolker Math Bio Research Seminar
Detectability

8. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
The problem
American Robins /
mosquitoes /
West Nile virus
genotyped blood meals
(one per mosquito)
what can we tell
about the robin
population from these
data?
size, heterogeneity?
Turdus migratorius
allaboutbirds.org
Culex spp.
alamel.free.fr
WNV (Wikipedia) Marm Kilpatrick
Ben Bolker Math Bio Research Seminar
Detectability

9. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Mathematical framework
occupancy spectrum:
S = fsig, i = 0; : : : ; imax =
P#of birds sampled P
by i mosquitoes
si = B,
isi = M
V is the (unordered) occupancy:
e.g. for B = 4, M = 5:
V = ff0; 1; 1; 3gg $ S = f1; 2; 0; 1g
s0 = “missing mass”
(how) can we estimate B?
birds mosquitoes
Ben Bolker Math Bio Research Seminar
Detectability

10. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Mathematical framework
occupancy spectrum:
S = fsig, i = 0; : : : ; imax =
P#of birds sampled P
by i mosquitoes
si = B,
isi = M
V is the (unordered) occupancy:
e.g. for B = 4, M = 5:
V = ff0; 1; 1; 3gg $ S = f1; 2; 0; 1g
s0 = “missing mass”
(how) can we estimate B?
birds mosquitoes
Ben Bolker Math Bio Research Seminar
Detectability

11. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Occupancy spectrum
Maxwell-Boltzmann statistics
define the multinomial coefficient
M(S)
(
P
Qsi )!
si !
:
then the likelihood of the occupancy spectrum is
P(SjB;M) =
1
BMM(S)M(V)
zeros are unobserved;
use s0 = B K where K (total birds observed)
P
i0 si
Ben Bolker Math Bio Research Seminar
Detectability

12. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Maximum likelihood estimation
Log-likelihood as a function of B is
L = C M log B + log B! log(B K)!
we know M (# of mosquitoes) and K (# of birds represented)
! K is a sufficient statistic for estimating B
apply standard MLE machinery
Ben Bolker Math Bio Research Seminar
Detectability

13. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Likelihood estimation
20 50 100
25
24
23
22
21
20
19
18
Total number of birds (B)
negative log-likelihood (L) for K = 16, M = 20:
ˆB = 41
CI={21,119}
Ben Bolker Math Bio Research Seminar
Detectability

14. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Reasons to like maximum likelihood estimation
consistent and asymptotically Normal
(= unbiased for large data sets)
asymptotically efficient
(= most statistically powerful unbiased estimator for large data
sets)
. . . a universal “Swiss Army Knife”. When it can do
the job, it’s rarely the best tool for the job but it’s
rarely much worse than the best (at least for large
samples). [Steve Ellner]
Ben Bolker Math Bio Research Seminar
Detectability

15. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Reasons to like maximum likelihood estimation
consistent and asymptotically Normal
(= unbiased for large data sets)
asymptotically efficient
(= most statistically powerful unbiased estimator for large data
sets)
. . . a universal “Swiss Army Knife”. When it can do
the job, it’s rarely the best tool for the job but it’s
rarely much worse than the best (at least for large
samples). [Steve Ellner]
Ben Bolker Math Bio Research Seminar
Detectability

16. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Reasons to like maximum likelihood estimation
consistent and asymptotically Normal
(= unbiased for large data sets)
asymptotically efficient
(= most statistically powerful unbiased estimator for large data
sets)
. . . a universal “Swiss Army Knife”. When it can do
the job, it’s rarely the best tool for the job but it’s
rarely much worse than the best (at least for large
samples). [Steve Ellner]
Ben Bolker Math Bio Research Seminar
Detectability

17. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Simulation results: bias and mean squared error
B: 32 B: 56 B: 100 B: 178 B: 316 B: 562 B: 1000
0.25
0.00
−0.25
−0.50
−0.75
−1.00
0.8
0.6
0.4
0.2
0.0
stat: bias stat: MSE
10 20 10 20 5010 20 50 10200 50 100 50 10020050 100200 501000 200 500
Number of mosquitoes
method
MLE
Strong negative bias for small B/very small M,
slight positive bias 20% for intermediate samples
Ben Bolker Math Bio Research Seminar
Detectability

18. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Good-Turing estimators
alternative approach:
count doublets, W =
P
vi (vi 1): set observed=expected
and solve for ^B:
^B = 1 +
1
2
p
1 + 4M(M 1)=W
Related (loosely) to Good-Turing estimators (Good, 1979)
(estimated frequency distribution of codebook pages)
the Pacala method:
if you’re reinventing important wheels
you’re on the right track!
Ben Bolker Math Bio Research Seminar
Detectability

19. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Good-Turing estimators
alternative approach:
count doublets, W =
P
vi (vi 1): set observed=expected
and solve for ^B:
^B = 1 +
1
2
p
1 + 4M(M 1)=W
Related (loosely) to Good-Turing estimators (Good, 1979)
(estimated frequency distribution of codebook pages)
the Pacala method:
if you’re reinventing important wheels
you’re on the right track!
Ben Bolker Math Bio Research Seminar
Detectability

20. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Good-Turing estimators
alternative approach:
count doublets, W =
P
vi (vi 1): set observed=expected
and solve for ^B:
^B = 1 +
1
2
p
1 + 4M(M 1)=W
Related (loosely) to Good-Turing estimators (Good, 1979)
(estimated frequency distribution of codebook pages)
the Pacala method:
if you’re reinventing important wheels
you’re on the right track!
Ben Bolker Math Bio Research Seminar
Detectability

21. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Estimator comparison
B: 32 B: 56 B: 100 B: 178 B: 316 B: 562 B: 1000
0.25
0.00
−0.25
−0.50
−0.75
−1.00
0.8
0.6
0.4
0.2
0.0
stat: bias stat: MSE
10 20 10 20 5010 20 50 10200 50 100 50 10020050 100200 501000 200 500
Number of mosquitoes
method
MLE
doublets
Doublet method works (much) better:
largely suppresses positive bias
Ben Bolker Math Bio Research Seminar
Detectability

22. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
a bit of data
l
l
l l
l
l l l
l
Baltimore Foggy_Bottom The_Mall
1000
100
10
2008 2010 2004 2005 2006 2008 2011 2004 2005
year
Est. bird population
(N == K)
l
l
FALSE
TRUE
Ben Bolker Math Bio Research Seminar
Detectability

23. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Conclusions open questions
Conclusions
doublet estimator is better
(bias/MSE),
reasonable for M 10 20
estimates effective
population size —
exactly what we want for
vector-borne disease
models!
Open questions
confidence intervals,
K == M estimates for
doublets
estimate coverage?
estimating heterogeneity/
subtler effects of
heterogeneity on disease
dynamics?
combining data from
multiple sites years
Ben Bolker Math Bio Research Seminar
Detectability

24. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Conclusions open questions
Conclusions
doublet estimator is better
(bias/MSE),
reasonable for M 10 20
estimates effective
population size —
exactly what we want for
vector-borne disease
models!
Open questions
confidence intervals,
K == M estimates for
doublets
estimate coverage?
estimating heterogeneity/
subtler effects of
heterogeneity on disease
dynamics?
combining data from
multiple sites years
Ben Bolker Math Bio Research Seminar
Detectability

25. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Outline
1 Introduction
2 Mosquitoes/WNV
3 Turtle surveys
4 Meta- stuff
Ben Bolker Math Bio Research Seminar
Detectability

26. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Green turtles at Tortuguero
green turtles
(Chelonia mydas)
at Tortuguero, Costa Rica
data from
Carr/Bjorndal/Bolten
survey data: 1971–present;
renesting interval data:
1955–present
estimate detection
probability,
recover 1955-1970
population size estimates?
Sea Turtle Conservancy /
http://www.conserveturtles.org
Ben Bolker Math Bio Research Seminar
Detectability

27. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
data
1970s 1980s 1990s
600
400
200
20 40 60 20 40 60 20 40 60
Renesting interval (days)
Counts (square-root scale)
Ben Bolker Math Bio Research Seminar
Detectability

28. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Fit by convolution
true distribution of inter-nesting intervals F(t; )
distribution of turtles observed on their second nesting
attempt is pF,
where p is the detection probability
distribution of nth-nesting-interval times:
n-fold convolution, Fn F F F : : : F
probability of detecting after n intervals is geometric,
p(1 p)n1
overall distribution observed is
F =
X
n
p(1 p)n1Fn()
obst NegBinom(F(t))
Ben Bolker Math Bio Research Seminar
Detectability

29. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Fit by convolution
true distribution of inter-nesting intervals F(t; )
distribution of turtles observed on their second nesting
attempt is pF,
where p is the detection probability
distribution of nth-nesting-interval times:
n-fold convolution, Fn F F F : : : F
probability of detecting after n intervals is geometric,
p(1 p)n1
overall distribution observed is
F =
X
n
p(1 p)n1Fn()
obst NegBinom(F(t))
Ben Bolker Math Bio Research Seminar
Detectability

30. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Fit by convolution
true distribution of inter-nesting intervals F(t; )
distribution of turtles observed on their second nesting
attempt is pF,
where p is the detection probability
distribution of nth-nesting-interval times:
n-fold convolution, Fn F F F : : : F
probability of detecting after n intervals is geometric,
p(1 p)n1
overall distribution observed is
F =
X
n
p(1 p)n1Fn()
obst NegBinom(F(t))
Ben Bolker Math Bio Research Seminar
Detectability

31. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Fit by convolution
true distribution of inter-nesting intervals F(t; )
distribution of turtles observed on their second nesting
attempt is pF,
where p is the detection probability
distribution of nth-nesting-interval times:
n-fold convolution, Fn F F F : : : F
probability of detecting after n intervals is geometric,
p(1 p)n1
overall distribution observed is
F =
X
n
p(1 p)n1Fn()
obst NegBinom(F(t))
Ben Bolker Math Bio Research Seminar
Detectability

32. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Brute force approach
make F a discrete distribution with support from days 7–18
is just 11 parameters describing P
the distribution
(constraints: 0 Fi 1,
Fi = 1)
use distr package in R for numerical convolution calculations
brute-force convolution calculation
(various MCMC/latent-variable strategies also possible,
but probably slower)
Ben Bolker Math Bio Research Seminar
Detectability

33. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Brute force approach
make F a discrete distribution with support from days 7–18
is just 11 parameters describing P
the distribution
(constraints: 0 Fi 1,
Fi = 1)
use distr package in R for numerical convolution calculations
brute-force convolution calculation
(various MCMC/latent-variable strategies also possible,
but probably slower)
Ben Bolker Math Bio Research Seminar
Detectability

34. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Yearly renesting interval estimates
0.3
0.2
0.1
0.0
9 12 15 18
day
proportion
Ben Bolker Math Bio Research Seminar
Detectability

35. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Prediction for 1971
lll
l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
l
l
l
ll
l
l
l
l
lll
l
l
ll
llllllllllllllll
0.3
0.2
0.1
20 40 60
Renesting interval (days)
Proportion
Ben Bolker Math Bio Research Seminar
Detectability

36. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Estimated detection probabilities
1.00
0.75
0.50
0.25
0.00
1960 1970 1980 1990
Year
Est. detection probability (ˆp)
Ben Bolker Math Bio Research Seminar
Detectability

37. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Raw and adjusted counts
l l l
l l
l
l
l
l
l
l
l
l l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l l
l
l
l
l
l
l
l
4000
3000
2000
1000
1960 1970 1980 1990
Year
Total counts
variable
l
l
count
adjcount
Ben Bolker Math Bio Research Seminar
Detectability

38. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
calibration
1.00
probability
0.75
detection 0.50
0.25
year 1960 1970 1980 1990
model
fn_dnbinom
fn_dnbinom1
fn_dpois
method
BFGS
L−BFGS−B
Nelder−Mead
Ben Bolker Math Bio Research Seminar
Detectability

39. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Conclusions open questions
Conclusions
detection probability
60–70%
(highly variable)
seems to recapture
Open questions
check calibration on modern
data
smoother renesting-interval
curve?
Ben Bolker Math Bio Research Seminar
Detectability

40. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Conclusions open questions
Conclusions
detection probability
60–70%
(highly variable)
seems to recapture
Open questions
check calibration on modern
data
smoother renesting-interval
curve?
Ben Bolker Math Bio Research Seminar
Detectability

41. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Outline
1 Introduction
2 Mosquitoes/WNV
3 Turtle surveys
4 Meta- stuff
Ben Bolker Math Bio Research Seminar
Detectability

42. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Cross-citation study
who cares about math biology?
more specifically, what is the information flow
from MB to bio (or math) and vice versa?
extract information from ISI Journal Citation Report
(thanks to Aaron Berk)
find top 100 cited/citing journals for:
(Bull Math Biol, J Theor Biol, Theor Popul Biol, J Math Biol,
Math Biosci, PLoS Comput Biol)
Ben Bolker Math Bio Research Seminar
Detectability

43. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Ordination of (1/(1+avg cites))
LANGMUIR
J_PHYS_CHEM_C
SOFT_MATTER
BBA−BIOMEMBRANES ASTROPHYS_J
PHYS_REV_B
J_CHEM_PHJYCSOMPUT_PHYS
PROTEIN_PEPTIDE_LETT
PROTEINS
PROG_BIOPHYS_MOL_BIO
BIOPHYS_J
MOL_BIOSYST
BIOCHEMISTRY−US
NAT_NEUROSCI
J_PHYSIOL−LONDON
NMDS axis 1 NMDS axis 2
AM_J_PHYSIOL−HEART_C
PHYS_BIOL
J_MOJL__NBEIUORLOPHYSIOL
J_VIROL
SCIENCE
P_NATL_ACAD_SCI_USA
BMC_BIOINFORMATICS
BIOINFORMATICS
PHYS_REV_LETT
BIOSYSTEMS
BMC_GENOMICS BMC_SYST_BIOL
PHILOS_T_R_SOC_B
THEOR_POPUL_BIOL
J_EXP_BIOL
STOCH_PROC_APPL
MATH_MOD_METH_APPL_S
NONLINEAR_DYNAM
MATHMBAITOHS_CBIIOSCI_ENG
AM_NAT
GENETICS
BMC_EVOL_BIOL
ECOL_LETT
TRENDS_ECOL_EVOL
ANIM_BEHAV
APPL_MATH_MODEL
COMPUT_MATH_APPL
APPL_MATH_COMPUT
BEHAV_ECOBL_ESHOACV_IOEBCIOOLL
B_MATH_BIOL
CANCER_RES
CELL
CIRC_RES
COMMUN_NONLINEAR_SCI
CURR_OPIN_STRUC_BIOL
DISCRETE_CONT_DYN−B
ECOLOGY
ECOL_MODEL
EVOLUTION
INT_J_BIOMATH
J_AM_CHEM_SOC
NONLINEAR_ANAL−THEOR
J_APPL_PROBAB
J_BIOL_CHEM
J_BIOL_SYST
J_EVOLUTION_BIOL
J_GEOPHYS_RES
J_IMMUNOL
J_MATH_ANAL_APPL
J_MATH_BIOL
J_NEUROSCI
J_PHYS_CHEM_B
J_THEOR_BIOL
MATH_COMPUT_MODEL
MOL_BIOL_EVOL
MOL_ECOL
NATURE
NAT_GENET
NAT_REV_GENET
NEURAL_COMPUT
NEUROIMAGE
NEURON
NONLINEAR_ANAL−REAL
NUCLEIC_ACIDS_RES
OIKOS
PHYS_REV_A
PHYS_REV_E
PLOS_COMPUT_BIOL
PLOS_GENET
PLOS_ONE
P_ROY_SOC_B−BIOL_SCI
SIAM_J_APPL_MATH
THEOR_ECOL−NETH
cat3
a
a
a
a
a
a
a
biology
chemistry
eco_evo_behav
general
math
mathbio
physics
Ben Bolker Math Bio Research Seminar
Detectability

44. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Biology + math biology only
PROTEIN_PEPTIDE_LETT
MOL_BIOSYST
NAT_NEUROSCI
PROTEINS
PROG_BIOPHYS_MOL_BIO
J_PHYSIOL−LONDON
AM_J_PHYSIOL−HEART_C
BIOPHYS_J
PHYS_BIOL
BIOSYSTEMS
PLOS_COMPUT_BIOL
SCIENCE
P_NATL_ACAD_SCI_USA
PHILOS_T_R_SOC_B
THEOR_POPUL_BIOL
AM_NAT
BBA−BIOMEMBRANES
BIOCHEMISTRY−US
BIOINFORMATICS
BMC_GENOMICS BMC_SYST_BIOL
GENETICS
J_EXP_BIOL
TRENDS_ECOL_EVOL
BEHAV_ECOL_SBOECHIAOVB_IEOCLOL
ECOL_LETT
ANIM_BEHAV
BMC_BIOINFORMATICS
BMC_EVOL_BIOL
B_MATH_BIOL
CANCER_RES
CELL
CIRC_RES
CURR_OPIN_STRUC_BIOL
ECOLOGY
ECOL_MODEL
EVOLUTION
INT_J_BIOMATH
J_BIOL_CHEM
J_BIOL_SYST
J_EVOLUTION_BIOL
J_IMMUNOL
J_MATH_BIOL
J_MOL_BJI_ONLEUROPHYSIOL
J_NEUROSCI
J_THEOR_BIOL
J_VIROL
MATH_BIOMSACTIH_BIOSCI_ENG
MOL_BIOL_EVOL
MOL_ECOL
NATURE
NAT_GENET
NAT_REV_GENET
NEURAL_COMPUT
NEUROIMAGE
NEURON
NUCLEIC_ACIDS_RES
OIKOS
PLOS_GENET
PLOS_ONE
P_ROY_SOC_B−BIOL_SCI
THEOR_ECOL−NETH
NMDS axis 1 NMDS axis 2
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
biochemistry
virology
molbio
cellbio
immunology
neurobio
medicine
physiology
bioinformatics
genetics
evolution
ee
ecology
behavior
biology
general
mathbio
Ben Bolker Math Bio Research Seminar
Detectability

45. Introduction Mosquitoes/WNV Turtle surveys Meta- stuff References
Good, IJ, 1979. Biometrika, 66(2):393–396. ISSN 0006-3444. doi:10.2307/2335677. URL
http://www.jstor.org/stable/2335677.
Platt, JR, 1964. Science, 146:347–353. ISSN 00368075. URL http:
//links.jstor.org/sici?sici=0036-8075%2819641016%293%3A146%3A3642%3C347%3ASI%3E2.0.CO%3B2-K.
Ben Bolker Math Bio Research Seminar
Detectability