Analyzing Band Recovery Data

I. Recovery array

Suppose we had banded adult mallards (Anas platyrhynchos) in August of 1980-1982 and had collected information on band returns from 1980-1984. These data can be organized into a recovery array:

Year Banded (i)	Number Banded (Ni)	Recoveries from Hunters (Rij)
		j = 1980	1981	1982	1983	1984
1980	1603	127	44	37	40	17
1981	1595	.	62	76	44	28
1982	1157	.	.	82	61	24

R_ij = the number of band recoveries in hunting season j from birds originally banded in year i (e.g., R₂₂ = 62 ).

N_i = number of birds banded in year i .

k = number years of banding (e.g., k=3 in this case).

l = number years of recovery (e.g., l = 5 in this case)

i = year banded

j = year of recovery

We usually construct a separate array for different species, sexes, and age classes (e.g., juvenile vs. adult).

II. Expected Recoveries [E(R_ij)]

Expected value of R₁₁ depends on 2 things:

1. N₁ = Total number of birds banded in year 1.

2. F₁ = Band-recovery rate for year i=1

   Fi=	The probability that a banded bird is recovered and reported to the bird banding laboratory in year i , given that it was alive at the beginning of year i.

Expected value for R₁₁ = E(R₁₁) = N₁ F₁

Expected value of R₁₂ depends on 3 things:

• Birds recovered in the second year (j=2) must have survived the first year; therefore, the expected value of R₁₂ is:

E(R₁₂) = (N₁ S₁) F₂ , where S₁ = survival rate for year 1

• Likewise, the expected value of R₁₃ is:

E(R₁₃) = (N₁ S₁ S₂) F₃

If survival and recovery rate were constant (i.e., no changes from year-to-year), then expected recoveries would be:

Year Banded (i)	Number Banded (Ni)	Expected Recoveries by Year [E(Rij)]
		Year j=1	Year 2	Year 3	Year 4
1	N1	N1 F	N1 S F	N1 S S F	N1 S S S F
2	N2	.	N2 F	N2 S F	N2 S S F
3	N3	.	.	N3 F	N3 S F

Here is an example assuming:

F = 0.10 (10% annual recovery rate)

S = 0.50 (50% annual survival rate)

                        ^a R₁₁ = 200 = N₁ F = 2,000 x 0.10

                        ^b R₁₂ = 100 = N₁ S F = 2,000 x 0.50 x 0.10

                  ^c R₁₃ = 50 = N₁ S S F = 2,000 x 0.50 x 0.50 x 0.10

The above model is a bit simplistic to apply to any real situation because survival and recovery rates often vary annually. We can incorporate this annual variability in survival and recovery rates by using a modified model structure:

Parameters of the model are subscripted to indicate the dependence on a calendar year (i.e., the parameters N, S, and F are year-specific).

For purposes of example, let us specify that the underlying population parameters are:

Expected recoveries would be:

                        ^a For example:  R₂₄ = N₂ S₂ S₃ F₄ = 400 × 0.50 × 0.70 × 0.05 = 7 recoveries.

Note:  The expected number of recoveries is quite different under the two models (i.e., this model versus the model with constant survival and recovery rates).

III. Observed versus Expected Recoveries

Expected recoveries depend on assumptions we make about how survival and recovery rates vary over time. Mathematical models have been constructed that represent various combinations of these assumptions.

A. Estimating recovery (F_j) and survival rates (S_j) when these rates vary annually (Model 1):

Banding and recovery data for adult male wood ducks (Aix sponsa) banded preseason in a Midwestern state (Table 2.2 from Brownie et al. 1978):

• R_i = Total recoveries from banding year i .

• C_j = Total recoveries in calendar year j.

• T_j  = Total recoveries in and after year j from all birds banded prior to, and in, year j.

• For model 1, these subtotals are used to estimate F_j and S_j as follows:

T_j = T_j-1 + R_i - C_j-1 , except T₁ = R₁

T₃ = T₂ + R₃ - C₂ = 348 + 167 - 106 = 409

F_j = (R_i / N_i) x (C_j / T_j)

^{^}F₁ = (265 / 1603) x (127 / 265) = 0.0792 or 7.92%

^{^}F₂ = (210 / 1595) x (106 / 348) = 0.0401 or 4.01%

^{^}F₃ = (167 / 1157) x (195 / 409) = 0.0688 or 6.88%

^{^}S_j = [R_i x (T_j - C_j) x (N_i+1 + 1)] / N_i x T_j x (R_i+1 +1)

^{^}S₁ = [265 x (265 - 127) x 1,596] / (1,603 x 265 x 211) = 0.6512 or 65.12%

^{^}S₂ = [210 x (348 - 106) x 1,158] / (1,595 x 348 x 168) = 0.6311 or 63.11%

Note: the subscripts i and j can get confusing, especially when viewing the output from program ESTIMATE. For example subscript j does not appear in the output; however, the survival and recovery estimates provided in the output simply apply to the ith year of the study. Parameter estimates are only given for the ith years of the study (i.e., when banding occurred). However, observed and expected recoveries are provided for all j years in order to perform goodness-of-fit tests, etc.

B. Testing the "fit of the model" to the observed recovery data (example 2.2a from Brownie et al. 1978)

Note: we will use AIC to determine the best-fit model in class and homework exercises.  The following example of a goodness-of-fit test is presented solely for informational purposes.

Model 1 : Analysis under the assumptions of time-specific survival and recovery rates.

Specifically, the model structure is:

Banding and Recovery Input Data : Adult male mallards banded pre-season in the San Luis Valley, Colorado.

Matrix of Expected Values -- Assuming Time-Specific Survival and Recovery Rates:

Matrix of Chi-squared Values -- Assuming Time-Specific Survival and Recovery Rates

Note: frequencies were combined where expected values were small.

A single chi-squared value is computed as: (Observed - Expected)² / Expected

The overall test is made by summing the chi-square values for all cells:

X ² = sum [(O_ij - E_ij)² / E_ij]

Test of the null hypothesis that the recovery data fit Model 1:

Chi-Squared Value (Sample) = 31.59

Theoretical Chi-Square Value at the 5% Level = 37.70

Degrees of Freedom = 25

Probability of a Chi-Square Value Larger than 31.59 = 0.17076620

Let alpha = 0.05

Therefore, we would fail to reject the null hypothesis that the data fit Model 1 (p = 0.171). This does not mean that Model l is the true underlying model for these data. It may very well be the true model, or it may not be the true underlying model and we were unable to reject the hypothesis because of small sample size or other "statistical" problems. We need to look at results of the other goodness-of-fit tests and the results of tests between models before we conclude that Model 1 is the most appropriate model for our data.

The goodness-of-fit tests and the tests between models are more complicated than described here. However, the interpretation and basic idea behind the tests remains the same. For a full description of these tests, see Brownie et al. (1985).

IV. Literature Cited

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 Wildlife Service, Resource Publication 131, Washington, D.C., USA.

Brownie, C., D. R. Anderson, K. P. Burnham, and D. S. Robson. 1985. Statistical inference from band recovery data – a handbook. Second edition. U.S. Fish and Wildlife Service, Resource Publication 156, Washington, D.C., USA.

 

Revised November 28, 2011