Wlf 315 Wildlife Ecology I Lab
Fall 2008
Population Estimation Concepts
Objectives:
1.
2.
3.
4.
Introduce techniques for analyzing mark-recapture/re-sight data.
Gain an understanding of the assumptions that underlie mark-recapture/re-sight models,
and an understanding of the consequences of violating those assumptions.
Estimate size of our target populations of starlings and fox squirrels.
Analyze data on body mass and measurements to address questions about sex & age
differences.
A brief background on mark-recapture models:
Mark-recapture studies are commonly used by wildlife biologists to estimate population size and to
estimate survival probabilities for individuals. Models used to analyze mark-recapture data are categorized
into those dealing with “open” populations and those dealing with “closed” populations. The difference is
that closed population models assume that there is no change in the population size during the period of
sampling (i.e., no births, deaths, immigration, or emigration). Actually, the sampling is conducted over a
short enough period of time that changes in population size from any of these factors are considered to be
negligible. Models that deal with open populations do not make the assumption that such events do not
occur. Rather, open population models attempt to quantify those factors and to incorporate the
information into the population estimate.
The Lincoln-Peterson model is the most basic model for estimating population size from markrecapture data for a closed population. The number of animals that are captured, marked, and released is
represented by n1. On a second capture event, the total number of animals captured is given by n2, and the
number of previously-marked animals is represented by m2. The basic idea is that the proportion of
marked animals in the second capture should be equal to the proportion of marked animals in the
population. The following equations represent this model (N = the total population size).
m2/n2 = n1/N
Rearranging gives:
N = n1n2/m2
(where N is really “N-hat”, the estimated
population size versus the true size)
Three major assumptions of Lincoln-Peterson model are:
1) that the population is closed;
2) that all animals have the same likelihood of being captured and recaptured; and
3) that all marks are seen and that marks are not lost (e.g., no birds lose leg-bands).
Think about how violating each of these assumptions might affect an estimate of the population size
using this model?
Chapman (1951) developed a modification of the Lincoln-Peterson model that produces estimates
with less bias (more accurate estimates of the true population size). The following is that modification:
Nc =
(n1 + 1)(n2+ 1)
(m2+1)
-1
(where Nc is really “Nc-hat”, the estimated
population size versus the true size)
The variance of the Chapman-modified estimate of population size is:
var(Nc) = (n1 + 1) (n2+ 1) (n1 - m2) (n2 - m2)
(m2 + 1)2 (m2+ 2)
The variance can be used to approximate a 95% confidence interval for the population estimate using the
following equation:
Nc + 1.965 var (Nc)
Many modifications have been made to the basic Lincoln-Peterson model to deal with some of the
assumptions, and to include data from multiple recaptures (often described as k-samples, where k is a
number of recapture events that is greater than one). In the starling exercise, we captured, banded, and
released birds during the first trapping session. In subsequent sessions, we captured birds, noted
recaptures, and then marked any unmarked birds also captured. To make use of the multiple capture and
marking events, we need a more complex model that accounts for multiple sampling periods.
Multiple capture events require more field work, but the population estimates are usually more accurate
and precise than estimates based on only 2 capture events. An extension of the Lincoln-Peterson model for
analyzing data from a closed population that was captured and marked across multiple events is the
Schnabel method (Schnabel 1938). The data are in the following form:
Capture Date
Date
1 Jan 2008
5 Jan 2008
10 Jan 2008
15 Jan 2008
# Captured
Ct
10
27
17
15
# Recaptured
Rt
0
0
3
1
Free Marks
Ft
0
10
37
52
Ct*Ft2
0
2,700
23,273
40,560
Rt*Ft
0
0
111
52
Where: Ct = number captured at each event
Rt = number recaptured at each event
Ft = number of marked individuals in the population available (or free) to be captured (this equals
the sum of all animals marked to date)
The estimate of N (meaning, N-hat):
 Ct*Ft2
 Rt*Ft
What is your estimate for the data above?
All of the assumptions that apply to the Lincoln-Peterson estimate apply to the Schnabel method. In reality,
animals could die during the study, but you would need to adjust the number of “Free Marked” animals to
account for the mortality. Estimates of confidence intervals around the Schnabel population estimate are
not simple to calculate by hand, however, they can be estimated (but we will not do it in this exercise).
References:
Chapman, D.G. 1951. Some properties of the hypergeometric distribution with applications to zoological
censuses. University of California Publications in Statistics. 1:131-160.
Schnabel, Z.E. 1938. The estimation of the total fish population of a lake. American mathematical
monthly. 45:348-352.
Here’s a recent paper that compared the Lincoln-Peterson, Schnabel and several other techniques (just FYI):
Morley, R.C. and R.J van Aarde. 2007. Estimating abundance for a savanna elephant population using
mark-resight methods: a case study for the Tembe Elephant Park, South Africa. Journal of
Zoology. 271: 418-427.
Lab Exercise: Analysis of Fox Squirrel and Starling Data
1.
2.
3.
4.
Exercise Expectations:
Participate in discussion about the techniques, model assumptions, and results.
Perform the calculations during lab to estimate population numbers and address assigned questions.
Complete the calculations, and answer questions below.
Turn in assignment the week after Thanksgiving break (Dec. 1 & 2).
Methods:
Fox Squirrel Data: (closed population with a single recapture event)
Use the Lincoln-Peterson estimate to estimate the population – three different ways that make different
assumptions about the squirrels that were resighted, but whether or not they were marked was unknown:
1. First, ignore the unknown animals (so, n2 = marked + unmarked only). Calculate the population
estimate using the Chapman modification of the Lincoln-Peterson model for the 2002-2008 data. Also,
calculate the variance for the Chapman estimate and use that variance to determine 95% confidence
interval for the population estimate.
a) Do the population estimates differ significantly among years (i.e., do the confidence intervals
overlap or not) from 2002 - 2008?
b) Is there a trend in the population (increasing or decreasing over the past several years?
2. Evaluate how the unknown animals might influence your estimate. Calculate the population estimate,
the variance and 95% confidence intervals using the Chapman modification in two more ways for the 2008
data only:
a) considering the animals in the “unknown” category as "marked"
b) considering the “unknowns” as "unmarked"
Starling Data: (closed population with multiple capture events)
Use the Schnabel method estimate to do the following calculations:
3. Use the data from your starling banding exercises to estimate the population size for 2008, and for 2
other years (you will be assigned which years in class). Only birds banded in the year of recapture will be
counted as "marked" (meaning that captures from previous years' banding efforts are really interesting, but
will not be used in our analyses).
Assignment: Write-up:
Provide the calculations and answer the questions outlined in each sections above (# 1-3). Please label
your calculations and questions clearly and sequentially, and show all intermediate steps.
In addition, provide brief answers to the following questions after you have completed the calculations.
These should be typed, numbered, and answered in complete sentences:
4. How accurate are your population estimates for the fox squirrels (how small or large are the confidence
intervals around the mean)?
5. How do the population estimates for 2008 differ for fox squirrels when you make different assumptions
about the unknown animals? Are the estimates significantly different (meaning do the confidence intervals
overlap or not)?
6. Did you meet the assumptions of the Lincoln-Peterson model in the starling banding exercise? If not,
how might we have violated the assumptions?
7. Look at the recapture data for starlings from this year. Is there any evidence of violation of the
assumption of equal catchability? If so, what evidence?
8. How might violating the assumptions affect the population estimates for starlings?
9. Calculate the mean (average) size and weight measurements for each sex and age class (F adult, F
subadult, M adult, M subadult). Calculate the 95% confidence intervals for each mean.
10. What size and weight measurements differ significantly among the sex and age classes. Briefly discuss
your data – do they differ by sex, age, or both? Do all measurements show the same trends? How does the
size of the confidence intervals differ, or are they fairly similar across sex and age categories?
Formula for a 95% CI for a mean:
Mean + t 0.05,n * SE
notes: 1) SE = var/n
2) If sample size is larger (>450) then t = 1.965, otherwise look up
the t value in a table of t values.