Observation Error in CountBased PVAs
Counting few individuals produce
accurate results, but…
N=23
The easy alternative
• Because measured variation in the
population growth rate is a surrogate for
the environmentally driven variation:
• WE JUST ACCEPT THAT OUR
ESTIMATES WILL BE OVERLY
PESIMISTIC
• Because observation error will inflate the
measured variation
Potential sources of observation
error
• Observation error is not strictly
synonymous of sampling variation
(although sampling variation is one potential
source of observation error)
sampling variation
• Considering that growth rate can vary
continuously over a broad range of values,
the population of all possible growth rates
is of infinite size
• However, a limited set of yearly censuses
can yield only a small subsample of
growth rates from this infinite “population”
sampling variation
• Even if we could measure the size of the
ecological population, and therefore its
growth rate, with perfect precision each
year, observing and counting every
individual with complete accuracy, we
would still not expect to get exactly the
same growth rates if we were to census
the population for a different interval of
years
Thus our estimates will differ
• We used the theoretical sampling
distributions of the mean and variance of a
random normal variable to account for this
sampling distribution
However…
• We are essentially never able to observe
and count every individual in an ecological
population.
• Typically we will estimate total population
size at each time by extrapolation from a
sample
N(1)=1
N(2)=1
N(3)=4
3 quadrats
Mean
2*9=18
Median
1*9=9
Obs(mice)=20
N(1)=1
N(2)=2
N(3)=3
3 quadrats
Mean
2*9=18
Median
2*9=18
Obs(mice)=20
N(1)=1
N(2)=4
N(3)=5
3 quadrats
Mean
3.3*9=30
Median
4*9=36
Obs(mice)=20
N=2000
N=2000
N=2000
N=2000
Sampling variation
• the important distinction is that sampling
variation is a consequence of limited
amounts of data (even if those data are
measured accurately), whereas
observation error is any inaccuracy in
estimates of population size
Considerations for reducing
observation error before is initiated
• Explore and test alternative sampling methods
• Carefully describe your sampling protocol
• Establish a stratified randomly selected set of
census plots
• If not possible to count all members of a
population, choose the most easily detected
subset of individuals
• Design data collection and data recording
methods that allow sampling effort to be clearly
understood and quantified
Quantifying observation error while
a census is being conducted
• Try to quantify its magnitude
• Record sampling effort and determine the
effort needed to yield acceptable accurate
data
Hypericum cumulicola
How Does An
Endangered Plant
Become Part of the
Family?
• By Eréndira
Quintana-Morales
• Wednesday, August
4, 2004
• HIGHLANDS TODAY
Hypericum cumulicola:
• Small short-lived perennial herb
• Narrowly endemic and endangered
• Flowers are small and bisexual
• Self-compatible, but requires pollinators to set seed
• High degree
of genetic population
differentiation
(Fst = 0.72, n = 34)
• Low expected
heterozygosity (0.023)
Menges et al. (1999)
Dolan et al. (1999)
Boyle and Menges (2001)
Stages
Growth
se
v
s
m
l
seedlings
vegetative
small reproductive
medium reproductive
large reproductive
Reproductive
effort
Mortality
2
3-6
9-14
>20
Time since fire (years)
Quintana-Ascencio et al. (2003)
Hypericum cumulicola
plants
• Catastrophes & Bonanzas
Fire
Fire killed nearly all plants, but the population
rapidly recovered via recruitment from a persistent
soil seed bank
Hypericum cumulicola
• Correlations
Population growth & fire
Hypericum cumulicola:
Ln (lambda) on time
since fire
Quintana-Ascencio et al. (2003)
Hypericum cumulicola habitat
Sampling
14000
12000
10000
8000
6000
4000
2000
0
0
5
10
15
20
25
30
1000
800
600
400
200
0
0
5
10
15
20
25
20
30
40
50
30
35
15000
10000
5000
0
0
2
x
10
4
10
60
70
1. 8
1. 6
1. 4
1. 2
1
0. 8
0. 6
0. 4
0. 2
0
6
0
x
4
10
20
30
40
50
60
70
80
90
10
5
4
3
2
1
0
0
20
40
60
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120
140
160
180
200
Correcting for observation errors
• This is a complex statistical problem.
• All the methods to correct error have
limitations!!!
Correcting estimates of σ2
when counts represent means of
replicated samples
• to estimate the population size of an
endangered plant, we might count the
number of individual plants encountered
in a set of quadrats or line transects
Typically…
• we will compute the mean of this
replicated counts as a measure of the
average density per sampling unit
• Then use the means multiplied by the area
occupied by the population
• The key point is to recognize that there is
sampling variation associated with each of
those means
σ2
• This sampling variation creates
observation error
• If we compute the variance of the log
population growth rates calculated from
the sample means the resulting value will
represent an amalgamation of the
environmental variance and the sampling
variation
Partial correction of σ2 for sampling
variation
• If we consider a single interval between
censuses t and t+1, our estimate of the log
population growth rate over that interval
will be
• Logλt=log(Nt+1/Nt)
• Because the logλt is a function of the
sample means, we can express the
variances in the logλt as a function of the
sampling variances of the two means.
A useful approximation for the
sampling variance of the logλt is
• Because the Central limit theorem their
variances can be approximated by the
squared standard error of the mean
• Var(logλt)= (1/q)*Σ[(st2/ntNt2)+ [(st+12/nt+1Nt+12)]
• σ2 = σ2 - Var(logλt)