BIOL 4150 Lecture Notes - Lecture 2: Confidence Interval, Transect, Random Search
2 May 2018
School
Department
Course
Professor
Chapter 13
09/12/17
Aerial census techniques
•
Total vs. sample counts
•
Precision, bias, and stratification in sample
counts
•
Population Estimation
Understand interactions between
individuals and with predators (frequency
of encounters)
○
Population viabilities
○
Competition for resources (spatial scales)
○
Need a density estimate?
•
Index -comparing concentrations
(comparing different habitats or areas)
○
Absolute density is harder to determine
○
Index may be simpler and less expensive to
determine
○
Absolute density vs. index of density?
•
Accuracy is not precision
○
Precision -how repeatable is the estimate?
○
Accuracy -how exact is measurement?
○
e.g. for harvesting animals while
maintaining a sustainable population
accuracy is very important
!
*context is important!
○
How precise of an estimate is needed?
•
What options are available?
•
Can become a crucial consideration if there
are limitations
○
How much money/time/effort can I invest?
•
Is this the best investment?
•
Key considerations:
Inaccurate -animals move so they
may be counted twice from separate
individuals
!
Used in Krugar National Park until
recently -too expensive and time
consuming
○
Popular among wildlife managers until
mid-1900s
•
Expensive, and tends to be inaccurate
•
Might be the best method when the entire
population is highly clumped in space (e.g.
hippos)
•
Total Counts:
Count subsets and extrapolate population size to
area in question
•
Accuracy -degree to which the mean of
several sample counts reflects the true
density
○
E.g. cover from trees in moose
sampling causes population to be
underestimates
!
Bias -mean of a set of sample counts is
consistently different from the true density
○
Precision -mean of a set of sample counts
is highly repeatable (low sample error)
○
Key issues:
•
Sample Counts:
Use markers on plane to determine
geographic relationship to predict area of
observation (using time)
○
*see cross section of plane estimations in
slide
○
Need frame of reference
•
48 quadrats sampled with
replacement
!
4 quadrats sampled twice, 1 quadrat
3x
!
523 animals counted
!
This differs from the true total
(1735) due to sample error
□
Estimation: N= (523)*(144) /48 =
1,569
!
Confidence limit = 1741 +/-
300 (1.96*153)
□
1000 repeated estimates gave a mean
of 1741, with a sd=153
!
Method A: Sampling with Replacement
○
1000 repetitions of population
estimate, without counting samples
multiple times (but always sampling
48 quadrats)
!
Confidence limit = 1743 +/-
260 (1.96*131)
□
Estimation: N=1743, sd=131
!
Does not over-represent a
single value (may be extremely
high/low and may not truly
represent actual distribution
patterns/population sizes)
□
Sampling without replacement
always yields a better precision
!
Method B: Sampling without Replacement
○
North/South transects: sd=427
!
East/West transects: sd=69
!
In this case, by having each
transect span the main spatial
gradient
□
Precision is greatly improved when
each sample unit samples as much as
possible of the area
!
Method C: Sampling in Transects
○
E.g. Fake population distribution for barren-
ground caribou (144 quadrats, 1km x 1km)
•
Aerial Census Techniques
How reliable the estimates are to the mean
○
Variability of means of many estimates =
standard error (sd of many estimates*mean)
•
Note: Variability of a single estimate = standard
deviation
Reduced disturbance of animals
○
Tends to more accurately sample the true
variability in abundance across landscape
(sd=48 in a sample of 1000 replicates)
○
Systematic approach makes it easier to do
sampling without replacement
•
Systematic vs. Random Samples
09/14/17
y = 142, 149, 149, and 127
○
Imagine that we sampled without replacement
and drew transects 1,4,5 and 8
•
y -number of animals on a given sample
unit
○
a -the area of a given sample unit
○
A -the total area of the region being
surveyed
○
n -number of units sampled
○
D = (sum of y)/ (a*n) = 11.813
!
D -the estimate of mean density
○
SED= (1/a) * sq.rt( (sum of y^2 -
(sum of y^2/n)) / (n*(n-1))) = 0.432
!
A larger sample size will result
in a smaller standard error
(more precise)
□
A larger area of the given
sample unit will result in a
smaller standard error
□
If 'y' is more variable, the
standard error will be larger
□
Note: the standard error from a
single sample yields an estimate of
the standard deviation in mean
density we might expect if we could
repeat the census many times
!
SED-standard error of estimated mean
density
○
Y = A*D = 1.701*10^3
!
Y -the estimate of the total numbers in the
region of size A
○
SEY= A*SED= 62.233
!
SEY-standard error of the estimate of total
numbers
○
CLD= SED*t = 1.374
!
CLY= A*SED*t = 197.902
!
Confidence limits (for n-1 df and alpha=
0.05, t=3.18)
○
Sample calculation for a single census:
•
See slides!
○
Note: c = concatenation (links variable
with numbers)
○
If you type in y and press enter, 4
values will appear randomly
!
Without the number (4), the full
sample will appear
!
y A VARIABLE<- ASSIGNMENT
VARIABLE/ARGUMENT OPERATOR
sample -A FUNCTION! (rowdata, 4) AN
ARGUMENT
○
Can assign variable a number by…
y3 <-y[3]
!
y[3] = third number stored
○
sumy<-y1+y2+y3+y4
!
Or sumysq<-(y1^2)+(y2^
2)+(y3^2)+(y4^2)
□
sumysq<-(y1*y1)+(y2*y2)+(y3
*y3)+(y4*y4)
!
a<-12
□
n<-4
□
totaln<-12
□
samplearea<-a*n
□
totalarea<-144
□
sumy^1<-sumy^2
□
Assign:
!
D<-sumy/samplearea
!
SED<-(sqrt(1-
(samplearea/totalarea)))*(1/a)*sqrt((s
umysq-(sumy^2)/n)/(n*(n-1)))
!
Calculations:
○
*remove everything: rm(list=ls())
○
Using R Software:
•
SED = (sq.rt(1-(n*a /A))) * [ (1/a)*
sq.rt( (sum of y^2 -(sum of y^
2/n))/(n*(n-1)) ]
○
This is because as the sample size gets
larger, we are converging on an estimate of
true abundance
○
How much of the area was actually
considered
!
As more samples are taken, we
are closer to the actual estimate
□
Corrects estimate of the standard
error
!
Finite population correction factor = sqrt
(1-(n*a/A))
○
**When the sampled area is a meaningful
percentage of the total area, apply the following
correction:
•
Aerial Census Example
In many cases, a species is consistently much
more abundance in some parts of the study
system than in other parts
•
This would typically generate imprecise
population estimates as a result
•
The solution is often stratification into several
regions (called strata) in advance of the census
•
*see slides
○
A = (2000, 7000, 3000)
○
Y = (2000, 35000, 30000)
○
SE = (250, 2400, 1900)
○
Ytotal = sum of Y = 6.7*10^4
○
SEYtotal = sq.rt(sum of (SE^2)) = 3.071*10^
3
○
Example:
•
Hence, we could improve precision by
assigning 3% of the sampling intensity to
the first stratum (because 2000 is roughly
3% of 67,000), then 52% to the second
stratum, and 45% to the third stratum
○
If you have some preliminary information on
relative densities, you can improve precision (but
not necessarily accuracy) by assigning sampling
intensity proportionate to abundance
•
Stratification
Transects preferable to quadrats
•
Sample without replacement
•
Systematic samples rather than pure random
•
Orient each transect to span spatial gradients
(elevation, distance to river, rainfall, vegetation
cover)
•
Stratify samples whenever possible
•
Make observation conditions as consistent as
possible (light, experience, fatigue)
•
Improving Precision
09/19/17
Assume that harvesting is the major source of
mortality for a population and that we know
exactly the number and ages of all harvested
animals
•
N0,t = K0,t + K1,t+1 + K2,t+2 + K3,t+3 +
K4,t+4 + K5,t+5 + K6,t+6 + K7,t+7
!
N7,t = K7,t
!
Population (cohort -group of animals born
around the same time):
○
N0,t = K0,t + (sum of)Ki,t+i /pi
!
N6,t = K6,t + (K7,t+1 /p)
!
N7,t = K7,t
!
We can correct these totals for natural
survival rate (p) in the following manner:
○
E.g. Age Composition of Female White-tailed
Deer in Canonto, Ontario (harvested)
•
Virtual Population Analysis
Assume that a small level of effort (E) samples a
fixed fraction of a population, called the
catchability coefficient (q)
•
q=0.0001
!
N=4
!
E=0..5000
!
H(E) = N*(1 - e-q*E)
!
Ex:
○
Random search theory predicts that the harvest
depends on effort and population size (N)
•
This is because they run out of animals in
the population to harvest
○
Therefore, space is limiting (=
random search theory)
!
There is a slight curve as it reaches the
plateau because the 'empty' space where
individuals have already been harvested (or
gear has been used) cannot be used
○
As hunting effort increases, the number of
harvested individuals will reach a plateau
•
Catch-per-unit-effort
Estimate abundance before and after a removal of
a known number of individuals
•
Initial population estimate: I1 (=301)
○
Subsequent population estimate: I2 (=
76)
○
Number of animals removed: C (=
357)
○
Proportion of removed: p = (I1- I2) / I1
○
Proportion of non-removed: q=1-p
○
Revised estimate of initial numbers: Y1=
(I1*C)/(I1-I2) (=477.587)
○
Estimate of sample variance: Var(Y1) = Y1
^2*(q/p)^2*(1/I1+ 1/I2) (=428.872)
○
Ex:
•
Index-Manipulation-Index
Male proportion prior to manipulation: p1
= 0.47
○
Male proportion after manipulation: p2 =
0.31
○
Male removals: Cx = -137
○
Female removals: Cy = -20
○
Total removals: C = Cx + Cy
○
Population estimate prior to manipulation:
Y1= (Cx -p2*C) / (p2-p1) = 552.063
○
A variation on the index-manipulation-index
method involves sex ratio changes
•
Change-in-ratio Method
Initial number of marked animals: M = 45
○
Marked recaptures: m = 17
○
Re-sample size: n = 178
○
Estimate marked fraction after recapturing a
known number of animals
•
M/Y = m/n
○
Y = (n*M)/m
○
After remixing, you would expect the fraction of
marked animals to be the same as it is in the
whole population
•
Y = (M*(n+1)) / (m+1)
○
SEY= sqrt [ (M^2*(n+1)*(n-m)) / ((m+1)^
2 * (m+2))] = 97.365
○
There is a slight bias in this ratio that is handled
by a modified formula
•
Mark-recapture Methods (Peterson-Lincoln)
*sqrt = a function
(…) = a number
*assumes hunters
are the only source
of mortality
*could be influenced by bias (not exact!)
Estimating Abundance
Tuesday,+ September+ 12,+2017
11:26+AM
Chapter 13
09/12/17
Aerial census techniques
•
Total vs. sample counts
•
Precision, bias, and stratification in sample
counts
•
Population Estimation
Understand interactions between
individuals and with predators (frequency
of encounters)
○
Population viabilities
○
Competition for resources (spatial scales)
○
Need a density estimate?
•
Index -comparing concentrations
(comparing different habitats or areas)
○
Absolute density is harder to determine
○
Index may be simpler and less expensive to
determine
○
Absolute density vs. index of density?
•
Accuracy is not precision
○
Precision -how repeatable is the estimate?
○
Accuracy -how exact is measurement?
○
e.g. for harvesting animals while
maintaining a sustainable population
accuracy is very important
!
*context is important!
○
How precise of an estimate is needed?
•
What options are available?
•
Can become a crucial consideration if there
are limitations
○
How much money/time/effort can I invest?
•
Is this the best investment?
•
Key considerations:
Inaccurate -animals move so they
may be counted twice from separate
individuals
!
Used in Krugar National Park until
recently -too expensive and time
consuming
○
Popular among wildlife managers until
mid-1900s
•
Expensive, and tends to be inaccurate
•
Might be the best method when the entire
population is highly clumped in space (e.g.
hippos)
•
Total Counts:
Count subsets and extrapolate population size to
area in question
•
Accuracy -degree to which the mean of
several sample counts reflects the true
density
○
E.g. cover from trees in moose
sampling causes population to be
underestimates
!
Bias -mean of a set of sample counts is
consistently different from the true density
○
Precision -mean of a set of sample counts
is highly repeatable (low sample error)
○
Key issues:
•
Sample Counts:
Use markers on plane to determine
geographic relationship to predict area of
observation (using time)
○
*see cross section of plane estimations in
slide
○
Need frame of reference
•
48 quadrats sampled with
replacement
!
4 quadrats sampled twice, 1 quadrat
3x
!
523 animals counted
!
This differs from the true total
(1735) due to sample error
□
Estimation: N= (523)*(144) /48 =
1,569
!
Confidence limit = 1741 +/-
300 (1.96*153)
□
1000 repeated estimates gave a mean
of 1741, with a sd=153
!
Method A: Sampling with Replacement
○
1000 repetitions of population
estimate, without counting samples
multiple times (but always sampling
48 quadrats)
!
Confidence limit = 1743 +/-
260 (1.96*131)
□
Estimation: N=1743, sd=131
!
Does not over-represent a
single value (may be extremely
high/low and may not truly
represent actual distribution
patterns/population sizes)
□
Sampling without replacement
always yields a better precision
!
Method B: Sampling without Replacement
○
North/South transects: sd=427
!
East/West transects: sd=69
!
In this case, by having each
transect span the main spatial
gradient
□
Precision is greatly improved when
each sample unit samples as much as
possible of the area
!
Method C: Sampling in Transects
○
E.g. Fake population distribution for barren-
ground caribou (144 quadrats, 1km x 1km)
•
Aerial Census Techniques
How reliable the estimates are to the mean
○
Variability of means of many estimates =
standard error (sd of many estimates*mean)
•
Note: Variability of a single estimate = standard
deviation
Reduced disturbance of animals
○
Tends to more accurately sample the true
variability in abundance across landscape
(sd=48 in a sample of 1000 replicates)
○
Systematic approach makes it easier to do
sampling without replacement
•
Systematic vs. Random Samples
09/14/17
y = 142, 149, 149, and 127
○
Imagine that we sampled without replacement
and drew transects 1,4,5 and 8
•
y -number of animals on a given sample
unit
○
a -the area of a given sample unit
○
A -the total area of the region being
surveyed
○
n -number of units sampled
○
D = (sum of y)/ (a*n) = 11.813
!
D -the estimate of mean density
○
SED= (1/a) * sq.rt( (sum of y^2 -
(sum of y^2/n)) / (n*(n-1))) = 0.432
!
A larger sample size will result
in a smaller standard error
(more precise)
□
A larger area of the given
sample unit will result in a
smaller standard error
□
If 'y' is more variable, the
standard error will be larger
□
Note: the standard error from a
single sample yields an estimate of
the standard deviation in mean
density we might expect if we could
repeat the census many times
!
SED-standard error of estimated mean
density
○
Y = A*D = 1.701*10^3
!
Y -the estimate of the total numbers in the
region of size A
○
SEY= A*SED= 62.233
!
SEY-standard error of the estimate of total
numbers
○
CLD= SED*t = 1.374
!
CLY= A*SED*t = 197.902
!
Confidence limits (for n-1 df and alpha=
0.05, t=3.18)
○
Sample calculation for a single census:
•
See slides!
○
Note: c = concatenation (links variable
with numbers)
○
If you type in y and press enter, 4
values will appear randomly
!
Without the number (4), the full
sample will appear
!
y A VARIABLE<- ASSIGNMENT
VARIABLE/ARGUMENT OPERATOR
sample -A FUNCTION! (rowdata, 4) AN
ARGUMENT
○
Can assign variable a number by…
y3 <-y[3]
!
y[3] = third number stored
○
sumy<-y1+y2+y3+y4
!
Or sumysq<-(y1^2)+(y2^
2)+(y3^2)+(y4^2)
□
sumysq<-(y1*y1)+(y2*y2)+(y3
*y3)+(y4*y4)
!
a<-12
□
n<-4
□
totaln<-12
□
samplearea<-a*n
□
totalarea<-144
□
sumy^1<-sumy^2
□
Assign:
!
D<-sumy/samplearea
!
SED<-(sqrt(1-
(samplearea/totalarea)))*(1/a)*sqrt((s
umysq-(sumy^2)/n)/(n*(n-1)))
!
Calculations:
○
*remove everything: rm(list=ls())
○
Using R Software:
•
SED = (sq.rt(1-(n*a /A))) * [ (1/a)*
sq.rt( (sum of y^2 -(sum of y^
2/n))/(n*(n-1)) ]
○
This is because as the sample size gets
larger, we are converging on an estimate of
true abundance
○
How much of the area was actually
considered
!
As more samples are taken, we
are closer to the actual estimate
□
Corrects estimate of the standard
error
!
Finite population correction factor = sqrt
(1-(n*a/A))
○
**When the sampled area is a meaningful
percentage of the total area, apply the following
correction:
•
Aerial Census Example
In many cases, a species is consistently much
more abundance in some parts of the study
system than in other parts
•
This would typically generate imprecise
population estimates as a result
•
The solution is often stratification into several
regions (called strata) in advance of the census
•
*see slides
○
A = (2000, 7000, 3000)
○
Y = (2000, 35000, 30000)
○
SE = (250, 2400, 1900)
○
Ytotal = sum of Y = 6.7*10^4
○
SEYtotal = sq.rt(sum of (SE^2)) = 3.071*10^
3
○
Example:
•
Hence, we could improve precision by
assigning 3% of the sampling intensity to
the first stratum (because 2000 is roughly
3% of 67,000), then 52% to the second
stratum, and 45% to the third stratum
○
If you have some preliminary information on
relative densities, you can improve precision (but
not necessarily accuracy) by assigning sampling
intensity proportionate to abundance
•
Stratification
Transects preferable to quadrats
•
Sample without replacement
•
Systematic samples rather than pure random
•
Orient each transect to span spatial gradients
(elevation, distance to river, rainfall, vegetation
cover)
•
Stratify samples whenever possible
•
Make observation conditions as consistent as
possible (light, experience, fatigue)
•
Improving Precision
09/19/17
Assume that harvesting is the major source of
mortality for a population and that we know
exactly the number and ages of all harvested
animals
•
N0,t = K0,t + K1,t+1 + K2,t+2 + K3,t+3 +
K4,t+4 + K5,t+5 + K6,t+6 + K7,t+7
!
N7,t = K7,t
!
Population (cohort -group of animals born
around the same time):
○
N0,t = K0,t + (sum of)Ki,t+i /pi
!
N6,t = K6,t + (K7,t+1 /p)
!
N7,t = K7,t
!
We can correct these totals for natural
survival rate (p) in the following manner:
○
E.g. Age Composition of Female White-tailed
Deer in Canonto, Ontario (harvested)
•
Virtual Population Analysis
Assume that a small level of effort (E) samples a
fixed fraction of a population, called the
catchability coefficient (q)
•
q=0.0001
!
N=4
!
E=0..5000
!
H(E) = N*(1 - e-q*E)
!
Ex:
○
Random search theory predicts that the harvest
depends on effort and population size (N)
•
This is because they run out of animals in
the population to harvest
○
Therefore, space is limiting (=
random search theory)
!
There is a slight curve as it reaches the
plateau because the 'empty' space where
individuals have already been harvested (or
gear has been used) cannot be used
○
As hunting effort increases, the number of
harvested individuals will reach a plateau
•
Catch-per-unit-effort
Estimate abundance before and after a removal of
a known number of individuals
•
Initial population estimate: I1 (=301)
○
Subsequent population estimate: I2 (=
76)
○
Number of animals removed: C (=
357)
○
Proportion of removed: p = (I1- I2) / I1
○
Proportion of non-removed: q=1-p
○
Revised estimate of initial numbers: Y1=
(I1*C)/(I1-I2) (=477.587)
○
Estimate of sample variance: Var(Y1) = Y1
^2*(q/p)^2*(1/I1+ 1/I2) (=428.872)
○
Ex:
•
Index-Manipulation-Index
Male proportion prior to manipulation: p1
= 0.47
○
Male proportion after manipulation: p2 =
0.31
○
Male removals: Cx = -137
○
Female removals: Cy = -20
○
Total removals: C = Cx + Cy
○
Population estimate prior to manipulation:
Y1= (Cx -p2*C) / (p2-p1) = 552.063
○
A variation on the index-manipulation-index
method involves sex ratio changes
•
Change-in-ratio Method
Initial number of marked animals: M = 45
○
Marked recaptures: m = 17
○
Re-sample size: n = 178
○
Estimate marked fraction after recapturing a
known number of animals
•
M/Y = m/n
○
Y = (n*M)/m
○
After remixing, you would expect the fraction of
marked animals to be the same as it is in the
whole population
•
Y = (M*(n+1)) / (m+1)
○
SEY= sqrt [ (M^2*(n+1)*(n-m)) / ((m+1)^
2 * (m+2))] = 97.365
○
There is a slight bias in this ratio that is handled
by a modified formula
•
Mark-recapture Methods (Peterson-Lincoln)
*sqrt = a function
(…) = a number
*assumes hunters
are the only source
of mortality
*could be influenced by bias (not exact!)
Estimating Abundance
Tuesday,+ September+ 12,+2017 11:26+AM
Chapter 13
09/12/17
Aerial census techniques
•
Total vs. sample counts
•
Precision, bias, and stratification in sample
counts
•
Population Estimation
Understand interactions between
individuals and with predators (frequency
of encounters)
○
Population viabilities
○
Competition for resources (spatial scales)
○
Need a density estimate?
•
Index -comparing concentrations
(comparing different habitats or areas)
○
Absolute density is harder to determine
○
Index may be simpler and less expensive to
determine
○
Absolute density vs. index of density?
•
Accuracy is not precision
○
Precision -how repeatable is the estimate?
○
Accuracy -how exact is measurement?
○
e.g. for harvesting animals while
maintaining a sustainable population
accuracy is very important
!
*context is important!
○
How precise of an estimate is needed?
•
What options are available?
•
Can become a crucial consideration if there
are limitations
○
How much money/time/effort can I invest?
•
Is this the best investment?
•
Key considerations:
Inaccurate -animals move so they
may be counted twice from separate
individuals
!
Used in Krugar National Park until
recently -too expensive and time
consuming
○
Popular among wildlife managers until
mid-1900s
•
Expensive, and tends to be inaccurate
•
Might be the best method when the entire
population is highly clumped in space (e.g.
hippos)
•
Total Counts:
Count subsets and extrapolate population size to
area in question
•
Accuracy -degree to which the mean of
several sample counts reflects the true
density
○
E.g. cover from trees in moose
sampling causes population to be
underestimates
!
Bias -mean of a set of sample counts is
consistently different from the true density
○
Precision -mean of a set of sample counts
is highly repeatable (low sample error)
○
Key issues:
•
Sample Counts:
Use markers on plane to determine
geographic relationship to predict area of
observation (using time)
○
*see cross section of plane estimations in
slide
○
Need frame of reference
•
48 quadrats sampled with
replacement
!
4 quadrats sampled twice, 1 quadrat
3x
!
523 animals counted
!
This differs from the true total
(1735) due to sample error
□
Estimation: N= (523)*(144) /48 =
1,569
!
Confidence limit = 1741 +/-
300 (1.96*153)
□
1000 repeated estimates gave a mean
of 1741, with a sd=153
!
Method A: Sampling with Replacement
○
1000 repetitions of population
estimate, without counting samples
multiple times (but always sampling
48 quadrats)
!
Confidence limit = 1743 +/-
260 (1.96*131)
□
Estimation: N=1743, sd=131
!
Does not over-represent a
single value (may be extremely
high/low and may not truly
represent actual distribution
patterns/population sizes)
□
Sampling without replacement
always yields a better precision
!
Method B: Sampling without Replacement
○
North/South transects: sd=427
!
East/West transects: sd=69
!
In this case, by having each
transect span the main spatial
gradient
□
Precision is greatly improved when
each sample unit samples as much as
possible of the area
!
Method C: Sampling in Transects
○
E.g. Fake population distribution for barren-
ground caribou (144 quadrats, 1km x 1km)
•
Aerial Census Techniques
How reliable the estimates are to the mean
○
Variability of means of many estimates =
standard error (sd of many estimates*mean)
•
Note: Variability of a single estimate = standard
deviation
Reduced disturbance of animals
○
Tends to more accurately sample the true
variability in abundance across landscape
(sd=48 in a sample of 1000 replicates)
○
Systematic approach makes it easier to do
sampling without replacement
•
Systematic vs. Random Samples
09/14/17
y = 142, 149, 149, and 127
○
Imagine that we sampled without replacement
and drew transects 1,4,5 and 8
•
y -number of animals on a given sample
unit
○
a -the area of a given sample unit
○
A -the total area of the region being
surveyed
○
n -number of units sampled
○
D = (sum of y)/ (a*n) = 11.813
!
D -the estimate of mean density
○
SED= (1/a) * sq.rt( (sum of y^2 -
(sum of y^2/n)) / (n*(n-1))) = 0.432
!
A larger sample size will result
in a smaller standard error
(more precise)
□
A larger area of the given
sample unit will result in a
smaller standard error
□
If 'y' is more variable, the
standard error will be larger
□
Note: the standard error from a
single sample yields an estimate of
the standard deviation in mean
density we might expect if we could
repeat the census many times
!
SED-standard error of estimated mean
density
○
Y = A*D = 1.701*10^3
!
Y -the estimate of the total numbers in the
region of size A
○
SEY= A*SED= 62.233
!
SEY-standard error of the estimate of total
numbers
○
CLD= SED*t = 1.374
!
CLY= A*SED*t = 197.902
!
Confidence limits (for n-1 df and alpha=
0.05, t=3.18)
○
Sample calculation for a single census:
•
See slides!
○
Note: c = concatenation (links variable
with numbers)
○
If you type in y and press enter, 4
values will appear randomly
!
Without the number (4), the full
sample will appear
!
y A VARIABLE<- ASSIGNMENT
VARIABLE/ARGUMENT OPERATOR
sample -A FUNCTION! (rowdata, 4) AN
ARGUMENT
○
Can assign variable a number by…
y3 <-y[3]
!
y[3] = third number stored
○
sumy<-y1+y2+y3+y4
!
Or sumysq<-(y1^2)+(y2^
2)+(y3^2)+(y4^2)
□
sumysq<-(y1*y1)+(y2*y2)+(y3
*y3)+(y4*y4)
!
a<-12
□
n<-4
□
totaln<-12
□
samplearea<-a*n
□
totalarea<-144
□
sumy^1<-sumy^2
□
Assign:
!
D<-sumy/samplearea
!
SED<-(sqrt(1-
(samplearea/totalarea)))*(1/a)*sqrt((s
umysq-(sumy^2)/n)/(n*(n-1)))
!
Calculations:
○
*remove everything: rm(list=ls())
○
Using R Software:
•
SED = (sq.rt(1-(n*a /A))) * [ (1/a)*
sq.rt( (sum of y^2 -(sum of y^
2/n))/(n*(n-1)) ]
○
This is because as the sample size gets
larger, we are converging on an estimate of
true abundance
○
How much of the area was actually
considered
!
As more samples are taken, we
are closer to the actual estimate
□
Corrects estimate of the standard
error
!
Finite population correction factor = sqrt
(1-(n*a/A))
○
**When the sampled area is a meaningful
percentage of the total area, apply the following
correction:
•
Aerial Census Example
In many cases, a species is consistently much
more abundance in some parts of the study
system than in other parts
•
This would typically generate imprecise
population estimates as a result
•
The solution is often stratification into several
regions (called strata) in advance of the census
•
*see slides
○
A = (2000, 7000, 3000)
○
Y = (2000, 35000, 30000)
○
SE = (250, 2400, 1900)
○
Ytotal = sum of Y = 6.7*10^4
○
SEYtotal = sq.rt(sum of (SE^2)) = 3.071*10^
3
○
Example:
•
Hence, we could improve precision by
assigning 3% of the sampling intensity to
the first stratum (because 2000 is roughly
3% of 67,000), then 52% to the second
stratum, and 45% to the third stratum
○
If you have some preliminary information on
relative densities, you can improve precision (but
not necessarily accuracy) by assigning sampling
intensity proportionate to abundance
•
Stratification
Transects preferable to quadrats
•
Sample without replacement
•
Systematic samples rather than pure random
•
Orient each transect to span spatial gradients
(elevation, distance to river, rainfall, vegetation
cover)
•
Stratify samples whenever possible
•
Make observation conditions as consistent as
possible (light, experience, fatigue)
•
Improving Precision
09/19/17
Assume that harvesting is the major source of
mortality for a population and that we know
exactly the number and ages of all harvested
animals
•
N0,t = K0,t + K1,t+1 + K2,t+2 + K3,t+3 +
K4,t+4 + K5,t+5 + K6,t+6 + K7,t+7
!
N7,t = K7,t
!
Population (cohort -group of animals born
around the same time):
○
N0,t = K0,t + (sum of)Ki,t+i /pi
!
N6,t = K6,t + (K7,t+1 /p)
!
N7,t = K7,t
!
We can correct these totals for natural
survival rate (p) in the following manner:
○
E.g. Age Composition of Female White-tailed
Deer in Canonto, Ontario (harvested)
•
Virtual Population Analysis
Assume that a small level of effort (E) samples a
fixed fraction of a population, called the
catchability coefficient (q)
•
q=0.0001
!
N=4
!
E=0..5000
!
H(E) = N*(1 - e-q*E)
!
Ex:
○
Random search theory predicts that the harvest
depends on effort and population size (N)
•
This is because they run out of animals in
the population to harvest
○
Therefore, space is limiting (=
random search theory)
!
There is a slight curve as it reaches the
plateau because the 'empty' space where
individuals have already been harvested (or
gear has been used) cannot be used
○
As hunting effort increases, the number of
harvested individuals will reach a plateau
•
Catch-per-unit-effort
Estimate abundance before and after a removal of
a known number of individuals
•
Initial population estimate: I1 (=301)
○
Subsequent population estimate: I2 (=
76)
○
Number of animals removed: C (=
357)
○
Proportion of removed: p = (I1- I2) / I1
○
Proportion of non-removed: q=1-p
○
Revised estimate of initial numbers: Y1=
(I1*C)/(I1-I2) (=477.587)
○
Estimate of sample variance: Var(Y1) = Y1
^2*(q/p)^2*(1/I1+ 1/I2) (=428.872)
○
Ex:
•
Index-Manipulation-Index
Male proportion prior to manipulation: p1
= 0.47
○
Male proportion after manipulation: p2 =
0.31
○
Male removals: Cx = -137
○
Female removals: Cy = -20
○
Total removals: C = Cx + Cy
○
Population estimate prior to manipulation:
Y1= (Cx -p2*C) / (p2-p1) = 552.063
○
A variation on the index-manipulation-index
method involves sex ratio changes
•
Change-in-ratio Method
Initial number of marked animals: M = 45
○
Marked recaptures: m = 17
○
Re-sample size: n = 178
○
Estimate marked fraction after recapturing a
known number of animals
•
M/Y = m/n
○
Y = (n*M)/m
○
After remixing, you would expect the fraction of
marked animals to be the same as it is in the
whole population
•
Y = (M*(n+1)) / (m+1)
○
SEY= sqrt [ (M^2*(n+1)*(n-m)) / ((m+1)^
2 * (m+2))] = 97.365
○
There is a slight bias in this ratio that is handled
by a modified formula
•
Mark-recapture Methods (Peterson-Lincoln)
*sqrt = a function
(…) = a number
*assumes hunters
are the only source
of mortality
*could be influenced by bias (not exact!)
Estimating Abundance
Tuesday,+ September+ 12,+2017 11:26+AM
Document Summary
Understand interactions between individuals and with predators (frequency of encounters) Index - comparing concentrations (comparing different habitats or areas) Index may be simpler and less expensive to determine. *context is important! e. g. for harvesting animals while maintaining a sustainable population accuracy is very important. Can become a crucial consideration if there are limitations. Used in krugar national park until recently - too expensive and time consuming. Inaccurate - animals move so they may be counted twice from separate individuals. Might be the best method when the entire population is highly clumped in space (e. g. hippos) Count subsets and extrapolate population size to area in question. Accuracy - degree to which the mean of several sample counts reflects the true density. Bias - mean of a set of sample counts is consistently different from the true density. E. g. cover from trees in moose sampling causes population to be underestimates.
