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Our example illustrates Lincoln-Petersen method when two different methods are used, one for the capturing and initial marking of animals, the other for detection of marks in the second sample (“recapture”). Rabbits (Sylvilagus nuttali) in central Oregon were captured and marked by dyeing the tails and hind legs with picric acid (batch marking) (Skalski et al. 1983). In the initial sample, 87 animals were captured and dyed with picric acid, then released. In a follow-up sample, 14 rabbits were counted in a drive count, and of these 7 were marked. This yields the following statistics:
n1 = 87 rabbits captured on the first occasion (capture sample),
n2 = 14 rabbits detected on the second occasion (drive count),
m2 = 7 dyed rabbits detected on the second occasion (drive count),
These are used to computed the Lincoln-Petersen, using the bias correction, estimate of
and
for CV(N) = =0.21. This provides an approximate 95% confidence interval on N of 93.8 to 234.2. We can also estimate the capture probabilities by each method, which will be useful in computing sample sizes.
I have written R code that will compute the Lincoln-Petersen estimates, and perform the sample size computations and graphing descibed below.
SAMPLE SIZE ESTIMATION
If we have an estimate (or a guess) for N, we can examine the implications of different amounts of sampling effort-- expresed as combinations of p1 and p2 - or equivalently, numbers of animals marked at each occasion n1 =Np1 and n2 =Np2 . Supposed we design a study to give CV(Ñ) = 0.10. We can use graph produced by the program to get an idea of sampling effort to achieve this result.
In terms of capture probablity
In terms of numbers to capture each occasion
Following the contour line for CV = 0.1 gives us combinations of p1 and p2 that will achieve this result. For instance, suppose that we have pretty much exerted all the effort we can to increase capture rates at the first sample (p1 = 0.5). The figure results show that we need to boost capture rates at the second sample to 0.4 as well, which means in excess of 60 animals at each sample (compared to 14 in our pilot study). More exact results can be obtained by using the sample_cv() function to generate specific results, e.g.,
> sample_cv(164,.3,.3)
[1] 0.1633916
By trial and error we can see that CV=0.1 can be achieved for N =164 with capture probabilities p1=0.5 and p2 = 0.35, for example, or sample sizes of 82 and 57 at the first and second samples, respectively. Other combinations can be explored as well, including some that may be more cost effective. For example, if it is cheaper to check for marks than it is to place marks, it may be better to decrease the first sample in favor of the second. In the present example, effort of p1 = 0.35 and p2 = 0.55 (n1 = 57 and n2 = 82) also provides CV=0.1, and would be favored over the previous design if, as seems likely, re-sampling marks is cheaper than applying them.
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