Project update: 10/15/12
A comparison of the accuracy of analysis methods for mark recapture data
Dan Bachen
Goal: To determine the most effective method of analyzing mark recapture data that increases power
while reduces the necessary number of samples.
Methods:
1.
2.
3.
4.
5.
Set realistic parameters, generate/ assign covariate values to sites
Perform a power analysis on real data using selected methods of analysis
Calculate number of samples needed to resolve a between a -80 to 90% treatment effect
Calculate the bias introduced to estimates of known parameters
Compare results to select best method for use in stepwise analysis
Progress 10/15/12

Updated capture history generator to account for differing probability of capture by
capture history not night
Power Analysis
I performed a power analysis of data I collected in the field to explore the minimum number of
plots needed accurately estimate a realistic treatment effect. The original data covered 24 sites trapped
between mid-May and mid-June 2012. Each was trapped for 5 nights, captured animals were uniquely
marked and released, and subsequent recaptures recorded. On six of the plots there were sparse
captures (=<2 individuals) so these were excluded from analysis.
To analyze the data I first estimated abundance by plot from the capture histories for each plot
using the RCapture package in R (see Figure 1 for results). Some of the sites differed slightly in
vegetation and elevation so I used an ANOVA test to see if a post-hoc stratification design should be
used to more accurately estimate the variance within plots. There was little conclusive evidence that the
plots differed in abundance of deer mice (p-value .25, Figure 2).
Using the estimates of abundance by plot derived from the RCapture analysis I examined the
effect sample size had on the ability to discriminate a gradient of treatment effects. I was specifically
interested in being able to detect a treatment effect of between 80-90% reduction in abundance (as
seen in Ostoja and Schupp 2009) with 20 or 40 total plots, >90% of the time (alpha level .1). The
estimates of abundance were generated through RCapture so the bias introduced by this method is
reflected in these estimates, so I calculated a mean of the abundance and the standard deviation, then
assuming means would be normally distributed used a normal random number generator (rnorm{base}
function in R) to simulate abundances of plots. In a parametric bootstrap framework I simulated
sampling between 2 and 100 sites per treatment and control with treatment effects from between -10%
reduction to -100% reduction. Then I compared the results using a t-test, scoring whether a difference
was detected with a p-value of less than .05 (for results see Figure 3). I found that 40 plots were
adequate to detect a treatment effect of -80%, 99.5% of the time. If 20 plots were used, a treatment
effect would have to exceed -90% to detect it more than 90% of the time. Based on these results the
power to detect a realistic treatment effect is optimized with 40 sites.
Bias
The analysis using RCapture estimated the mean of my sites to be 9.6423 and the SD to be 5.317. These
statistics reflect the actual mean and variance within the population, but through using Rcapture for the
analysis bias is introduced. Fortunately through simulation this bias can be estimate and corrected for.
Using known parameters (fixed probabilities of capture: .2 for first capture and .9 for subsequent
captures, true abundances between 10 and 25) I generated simulated capture histories and analyzed
them with the Rcapture package. In a bootstrap framework I iterated this process 100,000 times and
took the mean and variance of these repetitions. I then calculated the bias of these estimates using the
formula: Bias= Parameter- estimate of parameter. Based on this analysis both the estimate of mean and
variance produced by Rcapture are biased (mean p-value<.0001, SD p-value<.001). Bias also increases
with increasing abundance, where for every additional animal added to the population bias of the mean
increases by .24 and bias of the SD increases by .05 (See Figure 4 for results). Using this information to
correct for bias the true mean of my study plots is 13.96 and the corrected SD is 3.597.
Future progress
The ability to minimize sample size and therefor cost and effort is a real need in ecology, as having
enough resources to gather an adequate sample size is often challenging. Using the process outlined
above I will be able to investigate whether different techniques will give me more power to discriminate
smaller treatment effects for a given sample size or conversely allow me to discriminate a -90%
treatment effect with a smaller sample size. To complete this analysis I will perform power analysis of
my data with 3 additional techniques that estimate abundance from capture histories:
-
Minimum known alive
o The number of unique individuals is used
Lincoln/Peterson estimator in a Bayesian framework
Closed Capture likelihood estimator in a Bayesian framework
I will compare the reduction in sample size needed to resolve the designated treatment effect to select
the most effective method of analysis.
Progress 10/4
Currently I have successfully written code to:






Generate abundances based on defined covariate values
A treatment effect to a subset of these abundances
Generate capture histories based on fixed probabilities
Perform a “black box” analysis to estimate abundance based on capture history using
the Rcapture package with model s(.)p(t) which should correctly account for my
probabilities of capture
Model effect of treatment using covariate values generated for each site and
abundances generated from capture history, generate estimate of treatment effect
Score whether the true value falls within the CI of estimate
What I need to do:


Write code for hierarchical analysis linking abundance model:
Abundance ~ TE+S+C+P+(S*C)+E with likelihood model used in Rcapture analysis
Iterate both analysis types to get an accurate estimate of the proportion of times
correct.
Literature cited
Ostoja, S. M., and E. W. Schupp. 2009. Conversion of sagebrush shrublands to exotic annual grasslands
negatively impacts small mammal communities. Diversity and Distributions 25:863-870.
Appendix
Figure 1. Estimates of abundance by plot estimated with the Rcapture package in R
Plot
LC1
LC2
LC4
LC6
LT1
LT2
LT3
LT4
LT5
LT6
SC1
SC2
SC3
SC4
SC5
SC6
ST1
ST2
ST3
ST4
ST5
ST6
Abundance
9.389161092
3.041862251
12
6.000000002
5.000000003
2.000000002
7.000000003
8.583886749
10.28205028
1
10.4244289
2
6.000000001
1
3
10
12.6243839
25.81725641
13.2219365
5.158047662
6.541381265
13.8778794
Excluded
Y
Y
Y
Y
Figure 2. ANOVA analysis of difference within stratum
Call:
lm(formula = stor.vec.t ~ stratum)
Residuals:
Min 1Q Median 3Q Max
-8.3332 -3.4445 -0.2108 2.7063 16.4840
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.482 1.701 3.809 0.0011 **
stratumU
2.852 2.406 1.185 0.2499
Figure 3. Proportion of reps successfully identifying a treatment effect by sample size
Proportion successful at -20% TE
0.0
0.00
0.05
0.2
0.4
success.sample.size
0.15
0.10
success.sample.size
0.20
0.6
0.25
Proportion successful at -10% TE
20
40
60
80
100
0
40
80
Proportion successful at -30% TE
Proportion successful at -40% TE
1.0
100
0.6
0.2
0.4
success.sample.size
0.8
0.8
0.6
0.4
0.0
0.0
0
20
40
60
80
0
100
20
40
60
80
100
Index
Index
Proportion successful at -60% TE
0.0
0.6
0.4
0.0
0.2
0.2
0.4
0.6
success.sample.size
0.8
0.8
1.0
1.0
Proportion successful at -50% TE
success.sample.size
60
Index
0.2
success.sample.size
20
Index
1.0
0
0
20
40
60
Index
80
100
0
20
40
60
Index
80
100
Proportion successful at -70% TE
0.8
0.6
0.2
0.0
0.0
20
40
60
80
100
0
20
40
60
80
Index
Index
Proportion successful at -90% TE
Proportion successful at -100% TE
100
0.6
0.4
0.0
0.0
0.2
0.2
0.4
0.6
success.sample.size
0.8
0.8
1.0
1.0
0
success.sample.size
0.4
success.sample.size
0.6
0.4
0.2
success.sample.size
0.8
1.0
1.0
Proportion successful at -80% TE
0
20
40
60
80
Index
100
0
20
40
60
80
Index
Summary of percent detecting difference at 10 and 20 plots per treatment or control
TE
-0.6
-0.7
-0.8
-0.9
-1
10
0.63
0.77
0.86
0.92
0.97
20
0.92
0.98
0.995
0.99
0.99
Figure 4. Bias for Rcapture calculations
100
4.0
2.5
3.0
3.5
Bias
4.5
5.0
5.5
Bias by true abundance
5
10
15
Abundance
Output from analysis of bias in abundance with linear model
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.017171 0.035728 -0.481
n
0.638
0.233416 0.001974 118.229 <2e-16 ***
Output from analysis of bias in SD of abundance with linear model
Estimate Std. Error
t value
Pr(>|t|)
(Intercept) 0.827552 0.029430 28.12
1.02e-13 ***
n
9.92e-15 ***
0.054132 0.001626
33.29