Colorado Cooperative Fish and Wildlife Research Unit, Colorado State University, Fort Collins, Colorado 80523
Patuxent Wildlife Research Center, USGS Laurel, Maryland 20811
A COMPUTER PROGRAM FOR SAMPLE SIZE COMPUTATIONS FOR BANDING STUDIES
Kenneth R. Wilson
Colorado Cooperative Fish and Wildlife Research Unit,
Colorado State University, Fort Collins, Colorado 80523
James D. Nichols and James E. Hines
Patuxent Wildlife Research Center, U.S. Fish and Wildlife Service,
Laurel, Maryland 20811
Abstract
Sample sizes necessary for estimating survival rates of banded birds are derived based on
specified levels of precision. The equations are derived for studies involving adults only, or
adults and young, and the banding study can be new or ongoing. Precision is based on the
desired coefficient of variation (CV) of the survival estimates. The CV for annual survival
estimates, the CV for mean annual survival estimates, and the length of the study must be
specified to compute samples sizes. A computer program, BAND2, has been written to compute
the necessary sample sizes given the desired precision level and length of the study. A
description of the input and output for this program is described.
Wilson, et al. A Computer Program...
Introduction And Rationale
Data from migratory bird banding programs can be used to draw inferences about migration
pathways and geographic distribution patterns. This data can also aid in estimating survival rate
and, in conjunction with data from other sources, harvest rate, recruitment rate and population
size. These banding programs for migratory birds (and marking programs for many other
vertebrates) often entail considerable effort and expense, and it is thus very important to plan
these programs carefully. The investigator must first define study objectives explicitly. Then, the
study design must ensure that objectives have a high probability of achievement. Banding the
correct number of birds is an important component of these studies, and here, we present
methods for calculating sample sizes for banding studies designed to estimate survival rates. The
band recovery models in Brownie et al. (1978, 1985; also see Seber 1970, 1982; Robson and
Youngs 1971; Brownie and Robson 1976) have been used extensively during the last decade to
estimate survival rates of hunted migratory bird populations, and, to a lesser extent, exploited
populations of various fish and mammals. In the migratory bird studies, birds generally are
banded at the same time (season, generally before the hunting season) and location(s) for each of
several consecutive years. Each band is inscribed with a unique number permitting identification
of each banded individual. Banded birds are shot or found dead during the hunting season, and
band numbers of some of these birds are reported to the bander or banding agency (e.g., the U.S.
Fish and Wildlife Service Bird Banding Laboratory). The data resulting from banding operations
of this type consist of the number of birds banded in each year of the study, and from each
banded sample, the number of band recoveries occurring in each subsequent hunting season
included in the study.
In the general estimation approach employed by Brownie et al. (1985), these banding and recovery data are modeled as functions of two kinds of parameters: survival rates and band recovery rates. Survival rate is defined as the probability that a banded bird alive at the time of banding in year i is still alive at the time of banding in year i+1. Recovery rate is defined as the probability that a banded bird alive at the time of banding in year i is recovered (shot or found dead and the band reported to the Bird Banding Laboratory) during the hunting season of year i. The methods of Brownie et al. (1985) permit estimation of survival and recovery rates, and their associated sampling variances and covariances, under a variety of useful and realistic models. These models vary in their assumptions regarding the time and age specificity of the survival and recovery rates.
There are numerous important considerations when designing a banding study to estimate survival rate using band recovery models (Brownie et al. 1985:183-193). Some considerations involve trying to insure that the banded sample is representative of the population about which inferences are to be drawn. Other considerations involve attempts to insure that model assumptions are met or, at least, that failures of the model assumptions are minimized (Brownie et al. 1985:6-7). These considerations are extremely important to the success of any banding study but will not be addressed here.
In this paper, we restrict our attention to issues involving sample size; specifically, how many birds to band each year and how many years to band. To answer these questions the investigator must have some objective in mind. A statistic reflecting precision of the survival rate estimate seems a reasonable choice for such an objective. Here, we follow the approach of Brownie et al. (1985) and define the study objective by specifying the desired coefficient of variation (CV) of the survival estimate, CV(S ~ )=(VAR(S ~ )) 1/2 /S ~ , where S ~ is the bias-adjusted survival rate estimate and VAR denotes sampling variance. The investigator may prefer to specify study objectives in terms of a desired VAR(S ~ ) or approximate 95% confidence interval [S ~ (1.00 1.96CV(S ~ )], and these are easily translated into a desired CV(S ~ ). Guidelines and suggestions for choosing the CV(S ~ ) have been outlined in Brownie et al. (1985:186-187). The investigator must also specify the parameter of interest. For example, a study lasting k years will yield k-1 annual survival estimates, S ~ i , and an arithmetic mean estimate of survival computed over the k-1 years. In some cases the investigator may be interested in annual estimates for one or more specified years of the study (see Youngs and Robson 1975). More frequently, the investigator will be interested in the average survival rate for the entire study (Brownie et al.1985). The choice of the parameter of interest is very important to sample size computations, as annual estimates require larger banded samples than mean annual estimates for a specific coefficient of variation.
We consider banded sample sizes both for new studies and for future years of studies currently
underway. Methods to compute required banded sample sizes for new studies are shown in
Brownie et al. (1985:186-193) and Youngs and Robson (1975), and an unpublished computer
program to carry out the necessary computations has been written (Nichols and Hines, 1979
unpub. memo.). We receive many questions about sample sizes for new studies, and this program
has proven very useful. However, we also receive many questions from investigators who are
currently engaged in a banding study, but who either did not consider study objectives initially or
have chosen to change study objectives. Even in cases where required sample sizes are
determined initially, adaptive adjustment of future sample sizes is possible, and often desirable.
The ease with which birds can be banded often varies from year to year, and realized banded
sample sizes can vary substantially from target values. In addition, it may be found in the course
of a banding study that realized survival and recovery rates may differ substantially from that
anticipated, and upon which initial sample size computations were based. Thus, there are many
circumstances that might cause an investigator to ask how many birds to band in future years of a
study currently underway. Some banding and recovery data are already available for studies
underway, and sample size computations can be conditioned on these accumulated data.
In all of our sample size computations, we assume equal banded sample sizes for each year of
banding in new studies and for each future year of banding in ongoing studies. This approach is
similar to that taken by Brownie et al. (1985) and Youngs and Robson (1975). We also assume
the situation where the number of years of banding, k, equals the number of years of recovery, l.
Again, this approach is similar to that taken by others and seems reasonable, because survival
rates are only estimable through year k-1 and recovery rates are only estimable through year k.
Beyond these years, survival and recovery rates are only estimable as products such as S k f k+1,
and these products are usually not of biological interest (Brownie et al. 1985:15).
To compute a banded sample size needed to yield a specified CV(S ~ ), it is necessary to specify a
priori the true survival and band recovery rates that will operate during the study. Obviously,
these true rates are not known. For new studies the investigator must supply these values, and
they may be based on previous data or simply represent a best guess. For ongoing studies, we
permit the investigator either to use average survival and recovery rate estimates as computed
from the previous years of the study or to supply his best guesses. Our computational methods
permit the investigator to supply different survival and recovery rates for each year of the
projected study. In most cases there will be no basis for anticipating different year-specific
survival and recovery rates, and it will be more reasonable to supply anticipated average rates and
assume that they apply to all future years of the study. However, if a change in recovery or
survival rates is anticipated in future years (e.g. because of a future change in hunting regulations)
then it may be reasonable to use different year-specific survival and recovery rates in sample size
computations.
The next several sections of this report, NOTATION, SAMPLE SIZE COMPUTATIONS FOR
ADULT BIRDS, and SAMPLE SIZE COMPUTATIONS FOR YOUNG BIRDS provide
derivations of the equations used to compute sample size needs. Readers not interested in these
details should turn directly to the final section, COMPUTER PROGRAM FOR SAMPLE SIZE
COMPUTATIONS.
Notation
The notation defined below is very similar to that of Robson and Youngs (1971), Youngs and
Robson (1975), and Brownie et al. (1985).
S i = annual survival rate for adult birds in year i
S i = annual survival rate for young birds in year i
S ~ i = bias adjusted survival estimate of adults for year i
S ~ i = bias adjusted survival estimate of young for year i
f i = recovery rate for adult birds in year i
N i = number of adult birds banded in year i
N = recommended number of adult birds to band annually in future years
M i = number of young birds banded in year i
M = recommended number of young birds to band annually in future years
k = number of years of banding
m = number of years of previous banding (and hence number of years of recoveries from previous banding years)
i = f S f i+1 ... S S i+1 ... S k-1 f
i = N 1 S 1 ... S i-1 N 2 S 2 ... S i-1 ... N
i = S 1 ... S i-1 S 2 ... S i-1 ... 1
d i = N 1 S 1 ... S i-1 N 2 S 2 ... S i-1 ... N m S m ... S i-1
h i = S m+1 ... S i-1 + ... +1
VAR( . ) = the variance of the parameter ( . )
COV . , . = the covariance of the two parameters . , .
CV( . ) = coefficient of variation of the parameter ( . )
Note that we suspect that most investigators will use program BROWNIE and ESTIMATE to
initially analyze their data, thus the bias adjusted survival estimates (e.g. S ~ ) are used for our
computations (see Brownie et al. 1985:8,16).
Sample Size Computations For Adult Birds
VAR(S ~ S ~ 1 N 1- 1 N i+1 1- i+1 i+1 f ( f ) , for i=1,...,k-1 . [1 ]
The covariances of the adult survival estimates can be written as:
COVS ~ ,S ~ 0 , j>i+1 S ~ S ~ j+1 N j+1 1- j+1 j+1 , j=i+1, i=1,...,k-2 . [ ]
For a new study the investigator must specify the number of years of banding (k) the anticipated survival (S i ) and recovery (f i ) rates, and the desired CV for the parameter of interest. Given this information, we write the CV of interest in terms of k, S i , f i , and annual banded sample size (N), and then solve the resulting expression for N.
Annual Survival Estimates: Estimates of specific annual survival rates for a particular level of
precision can be obtained from the equation for the CV(S ~ ):
CV(S ~ ) VAR(S ~ ) S _ . [ ]
Substituting for the right-hand-side of equation 3 results in
CV(S ~ ) 1 N 1- 1 N i+1 1- i+1 i+1 f ( f )
, [ ]
but for a new study we assume all N i =N and so equation 4 becomes
CV(S ~ ) 1 N 1- 1- i+1 i+1 f ( f ) . [ ]
Solving (5) for N results in
N 1- 1- i+1 i+1 f ( f ) [CV(S ~ )] . [ ]
Banding N birds for each of k consecutive years would then result in the desired CV(S ~ )
assuming the guesses or estimates of S i and f i (usually assume S i =S, and f i =f for all i) are
reasonable.
Mean Annual Survival Estimates: Often, the average of the survival estimates over several years
of a study is of interest; this average is referred to as the mean annual survival estimate (S _ ~ ).
Again, the level of precision, CV(S _ ~ ), is specified and the following equation is used to solve
for N:
VAR(S _ ~ ) i=1 k-1 VAR(S ~ ) 2 k-2 i=1 COV(S ~ i ,S ~ i+1
) (k-1) . [ ]
Dividing by S _ results in
[CV(S _ ~ )] i=1 k-1 VAR(S ~ ) 2 k-2 i=1 COV(S ~ i ,S ~ i+1
) S _ (k-1) , [ ]
and, from equations 1,2, and 8, the following equation results
[CV(S _ ~ )] k-1 i=1 S 1 N 1- 1 N i+1 1- i+1 i+1
f ( f )
2 i=1 k-2 S i S i+1 N i+1 1- i+1 i+1 S _ k-1 . [ ]
Again we assume that N i =N, and solving for N, the result is
N= k-1 i=1 S 1- 1- i+1 i+1 f ( f ) 2 i=1 k-2
S S i+1 1- i+1 i+1 [CV(S _ ~ )] S _ k-1 , [ ]
which also depends on the desired precision, the number of years in the study, and the anticipated survival and recovery rates.
Annual Survival Estimates: If we take equation 4 and square both sides the result is
[CV(S ~ )] 1 N 1- 1 N i+1 1- i+1 i+1 f ( f )
. [ ]
Now we solve for N when i=m+1,...,k and N i =N i+1 =N to get
[CV(S ~ )] 1 N 1- 1 N 1- i+1 i+1 f ( f )(d +Nh
i ) , [ ]
where i =N 1 S 1 ... S i-1 + ...+ N i =d i +Nh i , d i =N 1 S 1 ...
S i-1 + ... +N m S m ... S i-1 , and h i =S m+1 ... S i-1 + ... +1.
This results in the following equation which can be solved for N using the quadratic formula:
h i [CV(S ~ i )] N d i [CV(S ~ i )] h i 1- 1- i+1 i+1
f ( f ) Nd i 1- 1- i+1 i+1 0. [ ]
The positive term of the quadratic becomes the correct solution for N, when i=m+1,...,k.
Mean Annual Survival Estimates: If we are interested in the banded sample sizes for mean
survival estimates (S _ ~ ) when previous banding exists for m years, we expand equation 9 to
get
[CVS _ ~ ] (k-1 (S _ ) m-1 i=1 S 1 N 1- 1 N i+1
1- i+1 i+1 f ( f )
S _ 1 N m 1- m m 1 N 1- m+1 m+1 f m m ( m f m ) m 2
i=1 m-1 S S i+1 N i+1 1- i+1 i+1
k-1 i=m+1 S _ 1 N 1- 1 N 1- i+1 i+1 f ( f )
2 k-2 i=m S _ N 1- i+1 i+1 , [ ]
This is equivalent to
[CVS _ ~ ] (k-1 (S _ ) m-1 i=1 S 1 N 1- 1 N i+1
1- i+1 i+1 f ( f )
S _ 1 N m 1- m m f m m ( m f m ) m 2 i=1 m-1 S S i+1
N i+1 1- i+1 i+1
( (S _ ) N 1- m+1 m+1 S _ N k-1 i=m+1 1- 1- i+1
i+1
2S _ ) N k-2 i=m 1- i+1 i+1 S _ k-1 i=m+1 f ( f
) , [ ]
where the last term can be written as (see equation 12)
S _ k-1 i=m+1 f ( f )(d +Nh i ) . [ ]
Equation 15 can be further reduced to get
[CVS _ ~ ] (k-1 (S _ ) m-1 i=1 S 1 N 1- 1 N i+1
1- i+1 i+1 f ( f )
S _ 1 N m 1- m m f m m ( m f m ) m 2 i=1 m-1 S S i+1
N i+1 1- i+1 i+1
(S _ ) N 1- k k S _ k-1 i=m+1 f ( f )(d +Nh i )
. [ ]
Equation 17 does not yield a closed form solution for N and an iterative solution must be
obtained. We used the method of bisection to solve for N (Hamming 1971:36-42).
Sample Size Computations For Young Birds
New Studies
General Approach: The general model H1 (Brownie and Robson 1976, Brownie et al. 1985) is
assumed when computing sample sizes for young birds. Under this model adult and young birds
must be banded each year, and precision of resulting survival estimates is a function of both adult
and young banded sample sizes. As suggested by Brownie et al. (1985:190), we first compute the
number of adults (N) to band under Model 1 to meet the specified precision criterion (see
previous section), and then we use N to compute the number of young (M) to band. The
precision of the adult survival estimate under Model 1 is very similar to that under H1, and this
fact allows the use of the adult sample sizes based on Model 1 survival estimates (Brownie et al.
1985:190). The computed adult banded sample size, N, along with k, S i , S i , and f i are all
used to compute M. Under model H1 (Brownie and Robson 1976, Brownie et al. 1985) the
variance of S ~ i can be written as
VAR(S ~ i ) (S ~ ) M i 1-S ~ i+1 S ~ i+1 (S ~
) N i+1 1- i+1 i+1 [ ]
and the coefficient of variation of S ~ as
CV(S ~ ) VAR(S ~ ) S _ . [ ]
Annual First-Year Survival Estimates: Using the computed value of the corresponding adult
sample size, N of equation 6, substituting equation 18 into 19, and assuming M i =M, we get:
[CV(S ~ )] 1 M 1-S i+1 S i+1 1 N 1- i+1 i+1 ,
[ ]
which leads to the following solution
M 1-S i+1 S i+1 [CV(S ~ )] 1 N 1- i+1 i+1 . [
]
Mean Annual First-Year Survival Estimates: The sampling covariance between nnual first-year
survival estimates is COV(S ~ S ~ )0 , for ij, i=1,...,k-1, so
VAR(S _ ~ ) 1 (k-1) k-1 i=1 VAR(S ~ ) , [ ]
and thus
[CV(S _ ~ )] k-1 i=1 VAR(S ~ i ) (k-1) (S _ ) . [ ]
From equation 18 and 23, the following equation results
[CV(S _ ~ )] (k-1) (S _ ) k-1 i=1 (S ) M 1-S i+1
S i+1 (S ) N 1- i+1 i+1 . [ ]
Equation 24 can be solved for M to get
M k-1 i=1 (S ) 1-S i+1 S i+1 [CV(S _ ~ )] (k-1)
(S _ ) k-1 i=1 (S ) N 1- i+1 i+1 . [ ]
where N is computed from equation 10.
Annual First-Year Survival Estimates: As seen in equation 18, VAR(S ~ i ) depends entirely on
data from bandings in years i. The equation for computing the sample size for annual first-year
survival estimates is thus the same as equation 21 for "New Studies".
Mean Annual First-Year Survival Estimates: Equation 24 can be further partitioned to get
[CV(S _ ~ )] (k-1) (S _ ) m-1 i=1 (S ) M i 1-S
i+1 S i+1 k-1 i=m (S _ ) M 1-S _ i+1 S _ i+1
m-1 i=1 (S ) N i+1 1- i+1 i+1 k-1 i=m (S _ ) N 1-
i+1 i+1 , [ ]
where N is from equation 17. Now, equation 26 can be solved for M and the result is
M k-1 i=m S i 1-S _ i+1 i+1 [CV(S _ ~ )] (k-1) (S _
) m-1 i=1 S M i 1-S i+1 i+1
m-1 i=1 (S ) N i+1 1- i+1 i+1 k-1 i=m (S _ ) N 1-
i+1 i+1 . [ ]
Note that the denominators of equations 21, 25, and 27 can become negative for small N and/or
small k. This means that for certain adult sample sizes, it becomes impossible to band enough
young to achieve the specified level of precision. In these instances, several alternatives exist: 1)
increase the CV for young, 2) increase the number of adults banded (N), or 3) increase the
number of years (k).
Computer Program For Sample Size Computations
To start the program (assumed to be on the C: drive) enter
C:>BAND2 [starts the program].
[Note: to abort the program, type Ctrl-C at any prompt.]
1)Output filename? [name of the output file, cannot be an existing file, LPT1, or CON]
2)Title? [any informative label up to 80 characters]
3)Number of years of proposed banding(4-50): [the number of years in the entire study (the
numbers in parentheses indicate range of acceptable values)]
4)Number of age classes (1 or 2): [1-adults only, 2-young and adults]
5)Enter ADULT annual and mean % coef of var.(CV) (in %): [the desired precision level for
adults, expressed as the percent coefficient of variation of the survival estimates; two rates must
be entered]
6)<IF NUMBER OF AGE CLASSES = 2 > Enter YOUNG annual and mean % coef of var.(CV)
(in %): [the desired precision level for young, expressed as the percent coefficient of variation of
the survival estimates; two rate must be entered]
7)<IF NUMBER OF AGE CLASSES = 2 > Adult population size increment (eg. 1000) ? [used
to increase adult sample size if computations for young are not possible at specified precision (see
pg. 12, the section "SAMPLE SIZE COMPUTATIONS FOR YOUNG BIRDS", subsection
"ONGOING STUDIES", last paragraph for a further explanation)]
8)Do you wish to input previous banding data (y or n)?: [Y-ongoing study with previous
recoveries, N-new study]
BEGIN SECTION FOR <IF PREVIOUS BANDING DATA = N>
9)Are all anticipated surv & recv. rates constant over time (y or n)? [Y-proposed years of banding
have the same survival and recovery rates, N-user will input survival and recovery rates for each
year]
10)<IF RATES CONSTANT = Y> Enter adult survival rate:
11)<IF RATES CONSTANT = Y AND AGE CLASSES = 2> Enter young survival rate:
12)<IF RATES CONSTANT = N> Enter n-1 adult survival rates:
13)<IF RATES CONSTANT = N AND AGE CLASSES = 2> Enter n-1 young survival rates:
14)<IF RATES CONSTANT = Y> Enter adult recovery rate:
15)<IF RATES CONSTANT = N> Enter n adult recovery rates:
END OF SECTION FOR <IF PREVIOUS BANDING DATA = N >
BEGIN SECTION FOR <IF PREVIOUS BANDING DATA = Y >
16)Is a recovery matrix to be read from disk (y or n)? [previous recoveries can be entered in the
form of a recovery matrix and the program will compute survival and recovery estimates from the
data versus entering the survival and recovery estimates directly; the format for the file is
described later]
17)<IF RECOVERY MATRIX = Y> Enter filename for input of recovery matrix: [file with
recovery data from previous banding study]
18)<IF RECOVERY MATRIX = N> Enter # of years of previous banding (m=2-30): [the m
years for which bandings and recoveries already exist from the ongoing study]
19)<IF RECOVERY MATRIX = N> Enter m-1 adult survival rate estimates:
20)<IF RECOVERY MATRIX = N> Enter m adult recovery rate estimates:
21)<IF RECOVERY MATRIX = N> Enter the number of adults banded for each of the m years:
22)<IF RECOVERY MATRIX = N AND AGE CLASSES = 2> Enter m-1 young survival rate
estimates:
23)<IF RECOVERY MATRIX = N AND AGE CLASSES = 2> Enter the number of young
banded for each of the m years:
24)Do you want average anticipated future survival rates estimated from the previous banding
data (y or n): [Y-the program will compute the average of the survival rates from the input data
and use these averages in computations for future years; N-user will input anticipated survival
rates for computations in future years]
25)<IF AVERAGE SURVIVAL ESTIMATED FROM PREVIOUS DATA = N > Enter 1 to
input constant surv. rate for adults & young, enter 2 to input n survival rates (1 or 2): [if the
survival rate for adults/young will be assumed constant for the n years of proposed banding enter
1; entering a 2 will prompt for n survival rates for adults and n survival rates for young]
26)<IF AVERAGE SURVIVAL ESTIMATED FROM PREVIOUS DATA = N AND
QUESTION 24 ABOVE = 1> Enter adult survival rate:
27)<IF AVERAGE SURVIVAL ESTIMATED FROM PREVIOUS DATA = N AND
QUESTION 24 ABOVE = 1 AND AGE CLASSES = 2> Enter young survival rate:
28)<IF AVERAGE SURVIVAL ESTIMATED FROM PREVIOUS DATA = N AND
QUESTION 24 ABOVE = 2> Enter n adult survival rates:
29)<IF AVERAGE SURVIVAL ESTIMATED FROM PREVIOUS DATA = N AND
QUESTION 24 ABOVE = 2 AND AGE CLASSES = 2> Enter n young survival rates:
30)Do you want the average anticipated future adult recovery rate estimated from the previous
banding data (y or n): [Y-the program will compute the average of the recovery rates from the
input data and use the average in computations for future years; N-user will input anticipated
recovery rates for computation in future years]
31)<IF AVERAGE RECOVERY ESTIMATED FROM PREVIOUS DATA = N > Enter 1 to
input a constant recovery rate for adults, enter 2 to input n recovery rates (1 or 2): [if the
recovery rate for adults will be equal for the n years of proposed banding enter 1; entering a 2 will
prompt you for n recovery rates for adults]
32)<IF AVERAGE RECOVERY ESTIMATED FROM PREVIOUS DATA = N AND
QUESTION
30 ABOVE = 1> Enter adult recovery rate: [1 number]
33)<IF AVERAGE RECOVERY ESTIMATED FROM PREVIOUS DATA = N AND
QUESTION 30 ABOVE = 2> Enter n adult recovery rates:
END OF SECTION FOR <IF PREVIOUS BANDING DATA = Y >
Batch Input
As an alternative to interactive input, the user can create an input file with responses in the same
order as in the interactive mode and begin BAND2 in batch mode. The input file can be created
by using an ASCII editor (such as EDLIN, available with MSDOS) or a word processor that can
create an ASCII file. Once the ASCII input file is created the program can be executed by
entering the following statement at the operating system prompt:
BAND2 <INPUT.DAT
The program will then execute without prompting the user. Two examples of inputs for batch
mode are given below:
1) The input file might be called "page58.inp" (the data can be found on page 58 in Brownie et al.
1985).
page58.out
Sample sizes for 5 additional yrs of banding based on pg 58 Brownie et al. 1985
5
2
13,3
13,4.5
1000
y
y
page58.dat
y
y
In addition to the above data the recovery matrix must be located in a file called "page58.dat"; the
format of the recovery matrix is similar to the form described in Brownie et al. 1985:155-157, but
the file must be in free format. The recovery matrix file is described below:
line 1: heading information (80 characters max)
line 2: number of years of banding (k), number of years of recovery, first year of banding study, numeric code (any single digit integer) [number of years of banding and recovery should be equal for program BAND2]
line3: the next lines are for the array of adult recovery numbers; if there are k years of banding then there should be k lines beginning at line 3 [if the recovery matrix is so long that all the numbers will not fit into 80 columns, the lines can be continued past column 80 or on the next line]
next line: the number of adult birds banded in each year
[If young were also banded the following lines would also be included.]
next line: the next lines are for the array of young recovery numbers, if there are k years of banding then there should be k lines beginning at line3
next line: the number of young birds banded in each year
As an example the data from page 58 Brownie et al. 1985, which would be in a data file called
"page58.dat", would look like the following:
Adults and Young banded, San Luis Valley, CO 1963-1971 9,9,1963,1 10,13,6,1,1,3,1,2,0 58,21,16,15,13,6,1,1 54,39,23,18,11,10,6 44,21,22,9,9,3 55,39,23,11,12 66,46,29,18 101,59,30 97,22 21 231,649,885,590,943,1077,1250,938,312 83,35,18,16,6,8,5,3,1 103,21,13,11,8,6,6,0 82,36,26,24,15,18,4 153,39,22,21,16,8 109,38,31,15,1 113,64,29,22 124,45,22 95,25 38 962,702,1132,1201,1199,1155,1131,906,353
If only adults were being analyzed the last 10 lines could be deleted, but this is not necessary
because the program would only read the first 12 lines when computing adults only. The above
data file could have been created with spaces instead of commas. This means that most data files
created using the format of Brownie et al. 1985:155-157 will work with minor adjustments. For
example the last line could also have been written as:
962 702 1132 1201 1199 1155 1131 906 353
Detailed Examples of Program Input and Output
The program will produce the heading information, number of years of proposed banding, desired
annual and mean annual coefficients of variation (CV), and the required number of banded birds.
If it is impossible to band enough birds to achieve the desired mean annual CV, then a message
will be printed along with the minimum possible CV. In addition, if the desired mean annual CV
was not achievable for banded young, then the program will compute possible combinations of
adults and young that would achieve the desired CV. If the number of young computed is greater
than 10,000, the program will also compute additional combinations with larger numbers of
banded adults.
In each of the following examples, the input file is given as well as the computer output. The
program could be run in interactive mode by using each line of the input file below as an answer
to the program questions, or by using the input file in batch mode. Some of the example input
files are included when program BAND2 is distributed. The input portion is in italics.
Example 1: A new five year banding study on adult male mallards is to begin, and a CV of 10 percent for annual survival and a CV of three percent for mean annual survival are desired. From "reliable" sources, the anticipated survival and recovery rates are approximately 0.6 and 0.075, respectively. The input file is as follows:
Tab set at 40
Input File Remarks examp1.out Output data filename Mallard Study,example 1 Title 7 7 years of banding 1 1 age class 10,3 Annual CV=10%, Mean CV=3% n N=no previous banding data y Y=yes, all anticipated surv/recv rates are equal .6 0.6=survival rate .075 0.075=recovery rate
and the output is as follows:
To achieve a mean annual CV of 3%, the investigator should band 1134 adult birds for each of
the seven years. If the investigator wants to achieve an annual CV of 10% in year 4 he should
band 1332 adult birds for each of the seven years. To assure an annual CV of 10% for all years,
the investigator should band the largest suggested number, 2549 .
Example 2: In this example, we hypothesize that a change in hunting regulations will occur after
the first three years of a six year study on Canada geese such that S=.75, S =.65, and f=.06 for
years 1-3, and S=.7, S =.6, and f=.12 for years 4-6. The desired levels of precision for the annual
and mean survival estimates expressed as percent CV are 12 and 4, respectively for both adults
and young. The input would be as follows:
and the output is as follows:
In the printout for example 2, it is impossible to obtain a mean CV of 4% for young by banding
only 396 adults. The lowest possible CV for young is 4.43% with only 396 adults banded. The
last section of the printout shows the results obtained by increasing the number of adults banded
and computing the corresponding number of young needed to achieve the desired mean CV level
for both young and adult. For example, banding 750 adults and 2411 young would result in a CV
of 4% for young and a CV less than or equal to 4% for adults. Other combinations of adults and
young could be examined by running BAND2 again with different adult population size
increments (for example 500 could be used in the input instead of 750, etc.).
Example 3: A five year study on adult and young Canada geese (from page 81-82 of Brownie et
al. 1985) has just been completed, and it is decided to continue the study for another five years.
The survival estimates for the adults were .86, .76, .87, and .74; and for young the survival
estimates were .75, .60, .78, and .80. The adult recovery rate estimates were .072, .070, .065,
.069, and .054. An annual CV of 9% and a mean annual CV of 4% is desired for the next five
years for the adults, while an annual CV of 12% and a mean annual CV of 6% is desired for the
young. The number of adults banded in the first five years was 828, 881, 379, 317, and 358; and
the number of young banded was 662, 596, 573, 676, and 601. We decide to use the mean annual
survival and recovery rate estimates as the anticipated survival and recovery rates for the
remaining years of the study. The input file is as follows:
*************************************************************
Program BAND2 -- Sample Size Computations For Banding Studies
Patuxent Wildl. Res. Center, USFWS, 5/25/1989
*************************************************************
Mallard Study,example 1
7 YEARS BANDING, ADULT - CV(ANNUAL SURVIVAL)=10.0%, CV(MEAN
SURVIVAL)=3.0%
SURV RECV
YEAR RATE RATE % CV REQUIRED BANDS
1 0.6000 0.0750 10.0 1292.
2 0.6000 0.0750 10.0 1190.
3 0.6000 0.0750 10.0 1217.
4 0.6000 0.0750 10.0 1332.
5 0.6000 0.0750 10.0 1621.
6 0.6000 0.0750 10.0 2549.
MEAN 0.6000 0.0750 3.0 1134.
Input File Remarks
examp2.out Output data filename
Adult and Young Canada Goose Study, example 2 Title
6 6 years of banding
2 2 age classes
12,4 ADULT Annual CV=12%, mean CV=4%
12,4 YOUNG Annual CV=12%, mean CV=4% classes
750 Adult pop'n increment=750
n N=no previous banding data
n N=no, anticipated surv/recv rates not
constant
.75 .75 .75 .7 .7 5 anticipated adult survival rates
.65 .65 .65 .6 .6 5 anticipated young survival rates
.06 .06 .06 .12 .12 .12 6 anticipated adult recovery rates
*************************************************************
Program BAND2 -- Sample Size Computations For Banding Studies
Patuxent Wildl. Res. Center, USFWS, 5/24/1989
*************************************************************
Adult and Young Canada Goose Study, example 2
6 YEARS BANDING, ADULT - CV(ANNUAL SURVIVAL)=12.0% CV(MEAN SURVIVAL)= 4.0%
YOUNG - CV(ANNUAL SURVIVAL)= 12.0% CV(MEAN SURVIVAL)= 4.0%
------- ADULT ------ - YOUNG - REQUIRED REQUIRED
YEAR SURV RECV SURV % CV ADLT BNDS YNG BNDS
RATE RATE RATE
1 0.7500 0.0600 0.6500 12.0 502. 597.
2 0.7500 0.0600 0.6500 12.0 455. 619.
3 0.7500 0.0600 0.6500 12.0 431. 615.
4 0.7000 0.1200 0.6000 12.0 547. 987.
5 0.7000 0.1200 0.6000 12.0 947. 1936.
MEAN 0.7300 0.0900 0.6300 4.0 396.
4.0% CV NOT POSSIBLE FOR YOUNG
WITH ONLY 396. ADULTS
SMALLEST POSSIBLE CV(YOUNG) FOR 396. ADULTS= 4.43%
SMALLEST POSSIBLE CV ASSUMES AN INFINITE NUMBER OF YOUNG ARE BANDED
POSSIBLE COMBINATIONS OF ADULTS & YOUNG WITH CV= 4.0% & 4.0%, RESPECTIVELY
ADULTS YOUNG
------ -----
496. 38733.
750. 2411.
1500. 1259.
2250. 1086.
3000. 1016.
Input File Remarks
examp3.out Output data filename
Adult and Young Goose study, example 3 Title
5 5 years of banding
2 2 age classes
9,4 ADULT Annual CV=9%, Mean CV=4%
12,6 YOUNG Annual CV=12%, Mean CV=6%
1000 Adult pop'n increment=1000
y Y=yes, previous banding data exits
n N=No, recovery matrix not read from disk
5 m=5 years of previous banding
.86,.76,.87,.74 4 previous adult survival rates
.072,.070,.065,.069,.054 5 previous adult recovery rates
828,881,379,317,358 Number of adults banded 5 previous years
.75,.60,.78,.80 4 previous young survival rates
662,596,573,676,601 Number of young banded 5 previous years
y Y=yes, mean surv. rate computed from input
y Y=yes, mean recv. rate computed from input
and the output is as follows:
*************************************************************
Program BAND2 -- Sample Size Computations For Banding Studies
Patuxent Wildl. Res. Center, USFWS, 5/24/1989
*************************************************************
Adult and Young Goose study, example 3
5 YEARS BANDING, ADULT - CV(ANNUAL SURVIVAL)=9.0% CV(MEAN SURVIVAL)= 4.0%
YOUNG - CV(ANNUAL SURVIVAL)= 12.0% CV(MEAN SURVIVAL)= 6.0%
--------- ADULT -------- ---- YOUNG ----
YEAR SURV RATE RECV RATE %CV SURV RATE %CV ADLT BNDS YNG BNDS
<< PREVIOUS BANDING DATA >>
1 0.8600 0.0720 **** 0.7500 **** 828. 662.
2 0.7600 0.0700 **** 0.6000 **** 881. 596.
3 0.8700 0.0650 **** 0.7800 **** 379. 573.
4 0.7400 0.0690 **** 0.8000 **** 317. 676.
5 0.8075 0.0540 9.0 0.7325 12.0 358. 601.
<< SAMPLE SIZE RESULTS >>
REQUIRED REQUIRED
ADLT BNDS YNG BNDS
6 0.8075 0.0660 9.0 0.7325 12.0 1028. 568.
7 0.8075 0.0660 9.0 0.7325 12.0 1259. 719.
8 0.8075 0.0660 9.0 0.7325 12.0 1725. 1032.
9 0.8075 0.0660 9.0 0.7325 12.0 3058. 2014.
*** MEAN ADULT BANDED SAMPLE WAS 134. ***
*** BANDING LESS THAN 200 ANIMALS/YEAR IS NOT RECOMMENDED ***
*** SEE PAGE 186, BROWNIE ET AL. (1985) ***
MEAN 0.8075 0.0660 4.0 0.7325 6.0 200. 766.
SMALLEST POSSIBLE CV(YOUNG) FOR 200. ADULTS= 4.95%
SMALLEST POSSIBLE CV ASSUMES AN INFINITE NUMBER OF YOUNG ARE BANDED
POSSIBLE COMBINATIONS OF ADULTS & YOUNG WITH CV=4.0% & 6.0%, RESPECTIVELY
ADULTS YOUNG
------ -----
250. 498.
1000. 242.
2000. 223.
3000. 218.
4000. 215.
In this case the computed number of adult birds to band was 134, but the message indicates that
the minimum the program will allow is 200. Brownie et al. (1985:190) actually recommend a
minimum of 300 birds of each age class. For this example, it is not possible to band enough
young to meet the desired CV(S _ ~ ) of 6% with only 200 adults. The stars (***) under %CV
indicate that the %CV for the previous banding data is not computed. From the printout of the
possible combinations, a suitable combination of adults and young could be obtained. The user
can look at other possible combinations by running the program again with another "adult
population size increment" besides 1000.
Example 4: For the last example, the input file described in the "BATCH INPUT" section is used.
The files are from Brownie et al. 1985:58, and are repeated below.
Input File Remarks
examp4.out Output data filename
5 additional years of banding based on
Brownie et al. 1985:58, example 4 Title (all on one line)
5 5 years of banding
2 2 age classes
13,3 ADULT Annual CV=13%, Mean CV=3%
13,4.5 YOUNG Annual CV=13%, Mean CV=4.5%
1000 Adult pop'n increment=1000
y Y=yes, previous banding data exists
y Y=yes, recovery matrix to be read from disk
examp4.dat Recovery matrix input filename
y Y=yes, mean surv. rate computed from input
y Y=yes, mean recv. rate computed from input
The banding data file "examp4.dat" contains
Note: It may be necessary to enter a return character after the last line above by pressing the
return or enter key.
The results are shown below.
Adults and Young banded, San Luis Valley, CO 1963-1971
9,9,1963,1
10,13,6,1,1,3,1,2,0
58,21,16,15,13,6,1,1
54,39,23,18,11,10,6
44,21,22,9,9,3
55,39,23,11,12
66,46,29,18
101,59,30
97,22
21
231,649,885,590,943,1077,1250,938,312
83,35,18,16,6,8,5,3,1
103,21,13,11,8,6,6,0
82,36,26,24,15,18,4
153,39,22,21,16,8
109,38,31,15,1
113,64,29,22
124,45,22
95,25
38
962,702,1132,1201,1199,1155,1131,906,353
*************************************************************
Program BAND2 -- Sample Size Computations For Banding Studies
Patuxent Wildl. Res. Center, USFWS, 5/24/1989
*************************************************************
5 additional years of banding based on Brownie et al. 1985:58, example 4
5 YEARS OF BANDING, ADULT - CV(ANNUAL SURVIVAL)=13% CV(MEAN SURVIVAL)=3.0%
YOUNG - CV(ANNUAL SURVIVAL)= 13.0% CV(MEAN SURVIVAL)= 4.5%
RECOVERY MATRIX OF
Adults and Young banded, San Luis Valley, CO 1963-1971
--------- ADULT -------- ---- YOUNG ----
YEAR SURV RATE RECV RATE %CV SURV RATE %CV ADLT BNDS YNG BNDS
<< PREVIOUS BANDING DATA >>
1 0.5756 0.0433 **** 0.4709 **** 231. 962.
2 0.6359 0.0856 **** 0.5064 **** 649. 702.
3 0.6665 0.0590 **** 0.5891 **** 885. 1132.
4 0.8051 0.0628 **** 0.5909 **** 590. 1201.
5 0.6496 0.0520 **** 0.4776 **** 943. 1199.
6 0.5525 0.0633 **** 0.6521 **** 1077. 1155.
7 0.5719 0.0789 **** 0.4635 **** 1250. 1131.
8 0.5415 0.0888 **** 0.3926 **** 938. 906.
9 0.6248 0.0673 13.0 0.5179 13.0 312. 353.
<< SAMPLE SIZE RESULTS >>
REQUIRED REQUIRED
ADLT BNDS YNG BNDS
10 0.6248 0.0668 13.0 0.5179 13.0 752. 1253.
11 0.6248 0.0668 13.0 0.5179 13.0 845. 1438.
12 0.6248 0.0668 13.0 0.5179 13.0 1048. 1855.
13 0.6248 0.0668 13.0 0.5179 13.0 1676. 3261.
MEAN 0.6248 0.0668 3.0 0.5179 4.5 209. 2914.
SMALLEST POSSIBLE CV(YOUNG) FOR 209. ADULTS= 3.96%
SMALLEST POSSIBLE CV ASSUMES AN INFINITE NUMBER OF YOUNG ARE BANDED
POSSIBLE COMBINATIONS OF ADULTS & YOUNG WITH CV=3.0% & 4.5%, RESPECTIVELY
ADULTS YOUNG
------ -----
259. 1257.
1000. 455.
2000. 410.
3000. 397.
4000. 390.
In the final example, the investigator would only have to band 209 adults for an additional five
years to achieve a 3% mean annual CV for adults, while 2914 young would need to be banded to
achieve a 4.5% mean annual CV. The smallest possible CV for young would be 3.96%; this is a
theoretical value assuming an infinite number of young are banded. If the interest lies in an annual
CV of 13% for year 12, the investigator would have to band 1048 adults and 1855 young for the
next five years.
If you would like to obtain a copy of the program (including source code and examples) or if you
have questions or problems with the program, write to:
Brownie, C., D. R. Anderson, K. P. Burnham, and D. S. Robson. 1978. Statistical inference from
band recovery data: A handbook. U. S. Fish and Wildl. Serv., Resour. Publ. 130. 212 pp.
Brownie, C., D. R. Anderson, K. P. Burnham, and D. S. Robson. 1985. Statistical inference from
band recovery data: A handbook. 2nd Ed. U. S. Fish and Wildl. Serv., Resour. Publ. 156. 305 pp.
Hamming, R. W. 1971. Introduction to applied numerical analysis. McGraw Hill. New York. 331
pp.
Robson, D. S., and W. D. Youngs. 1971. Statistical analysis of reported tag-recaptures in the
harvest from an exploited population. Biometrics Unit, Cornell Univ., Ithaca, New York.
BU-369-M. 15 pp.
Seber, G. A. F. 1970. Estimating time-specific survival and reporting rates for adult birds from
band returns. Biometrika 57:313-318.
Seber, G. A. F. 1982. Estimation of animal abundance and related parameters. 2nd Ed. MacMillan
Publishing Co., Inc. New York, N. Y. 654 pp.
Youngs, W. D., and D. S. Robson. 1975. Estimating survival rate from tag returns: Model tests
and sample size determination. J. Fish. Res. Board Can. 32:2365-2371. Bibliographic Card
Abstract
Sample sizes necessary for estimating survival rates of banded birds, adults and young, are derived
based on specified levels of precision. The banding study can be new or ongoing. The desired
coefficient of variation (CV) for annual survival estimates, the CV for mean annual survival
estimates, and the length of the study must be specified to compute samples sizes. A computer
program is available for computation of the sample sizes, and a description of the input and output
is described.
Error Condition
An important error condition that will cause program BAND2 to stop is if f > 1-S for a new
study, or if f _ > 1-S _ for a new or an ongoing study. In a new study, this results from choosing
yearly or average survival and recovery values that are illogical, but for an ongoing study this can
also result from poor estimates of the yearly survival and recovery rates, S and f , such that S _
+f _ > 1.0. For an ongoing study, the investigator must manually choose the "best" guess for the
previous yearly survival and recovery rates such that the sum of the average rates, S _ and f _ , is
not greater than 1. In some instances, the data from an ongoing study (and thus, also, the survival
and recovery estimates) may be so "poor" that sample size must be computed as if a new study is
beginning.
Program Details
The program will run on an IBM PC, AT, 80386, or compatible and most runs only take a few
seconds. The program was written in FORTRAN 77, and compiled with the Ryan-McFarland
compiler, RMFORT. Program BAND2 consists of a main routine and 2 subroutines. Currently,
30 years of previous banding can be input, and up to 50 years of proposed banding computed.
This can easily be changed by increasing the dimensions of the program. The program should
compile using most MSDOS FORTRAN compilers, including mainframe compilers, with few
changes, if any.
Jim Hines (jim_hines%40usgs.gov)
BRD-USGS
11510 American Holly Dr. #201
Patuxent Wildlife Research Center
Laurel, MD 20708-4017
or
Ken Wilson
Colorado Cooperative Fish & Wildlife Research Unit
201 Wagar
Colorado State University
Fort Collins, CO 80523
Acknowledgements
We would like to thank John Sauer, Bob Trost, David Anderson, Eric Rexstad, and two
anonymous referees for reviewing the manuscript.
Literature Cited
Brownie, C., and D. S. Robson. 1976. Models allowing for age-dependent survival rates for band
return data. Biometrics 32:305-323.
Key words:
banding, survival rates, sample size, tagging