Estimating the number of fish that return to spawn using capture-recapture methods.

Point Estimate

Recall what is known from this experiment:

Known Statistics	Unknown Parameters
n1 = number of fish initially tagged	p1 = proportion of all fish that are tagged
n2 = number of carcasses examined	p2 = proportion of all carcasses that are searched
m2 = number of tags recovered from carcasses examined	N = total number of fish returning to spawn (the escapement)

We wish to obtain an estimate of N, the total number of spawning fish, which is denoted .

Start by equating the observed ratio in our sample, to an unknown ratio in the population:

 
   m2    number of carcasses with marks   number of tags applied    n1
   -- = ------------------------------- = ----------------------- = --
   n2    number of carcasses examined     number of spawning fish   N

Assumptions

This simple equation depends upon many assumptions. The most important of these are:

• Tagged fish do not lose their tags.
• In each sample, every animal has the same chance of being captured.
• The population is closed, i.e., no other fish from other rivers can enter the system, and none of the fish that pass the tagging site spawn in other rivers.
• The tagged fish mix randomly with the untagged fish between the first and second samples.

If these assumptions hold, the proportion of carcasses in the entire population that are tagged should be equal to the proportion of all fish tagged. Finally, the proportion of carcasses that are tagged in the sample should be roughly equal to the proportion in the population which then gives the left side of the equation.

The point estimate

Rearranging the terms, we find that

                             number of carcasses examined x number of tags applied
   number of spawning fish = -----------------------------------------------------
                                       number of carcasses with marks

or, equivalently,

=	n1n2 ---- m2

To demonstrate that is a sensible estimator of N, we develop the following identities:

n1 = N x p1	Total population x Proportion of fish tagged.
n2 = N x p2	Total population x Proportion of carcasses searched.
m2 ~ N x p1 x p2	Total population x Proportion of fish tagged x Proportion of carcasses searched. Note, that unlike the previous identities, this is only approximate because only a sample of carcasses can be searched

If these identities are substituted into the above equation, we find that

=

n1n2 ---- m2

(Np1)(Np2) = ---------- = N (Np1p2)

Example

=

n1n2 ---- m2

(2,213)(59,665) = --------------- = 261,462 505

Questions

1. Estimate the number of fish that return to spawn if the following statistics were observed:

   1. n₁=3,000; n₂=60,000; m₂=500

   2. n₁=3,000; n₂=30,000; m₂=250

   3. n₁=6,000; n₂=60,000; m₂=1,000

2. Examine the assumptions made for this estimator.

   1. Suppose that a proportion of both tagged and untagged fish "leave" the system. What effect does this have on the point estimate, i.e. express n₂ and m₂ in terms of N, p₁, p₂, and theta (where theta is the proportion that leave the system) and see if the estimator still estimates N.

   2. Suppose that tagging is very stressful, and a certain fraction die before reaching the spawning ground and the carcasses can never be found. What effect does this have on the point estimate, i.e., express n₂ and m₂ in terms of N, p₁, p₂, and the fraction dying of stress and see if the estimator still estimates N.

   3. Suppose that new fish enter the spawning ground without having a chance of being tagged. What effect does this have on the estimator?