Statistical Comparison of Nest Success Rates

Mayfield (1) suggested a method of estimating the success rates of bird nests. The estimator commonly used before that was severely biased in many situations. Mayfield proposed that the number of nests destroyed be divided by the exposure, the number of days a nest was under observation and available to be destroyed. This estimator possesses several desirable properties (1, 2).

Johnson (2) developed a variance estimator for Mayfield's estimated daily mortality rate and indicated how it can be used to compare rates between two groups with a Z test. This note extends the argument to K > 2 groups.

Assume there are K (K ≥ 2) groups of nests and the true daily mortality rate of nests in group i is ρ_i (i = 1, ..., K). Suppose a number of nests in each group are observed and the total exposure for the ith group is e_i, which we assume is fixed in advance. Suppose that d_i of the nests in group i are destroyed, which results in an estimated daily mortality rate of ρ_i = r_i = d_i / e_i. Interest is in testing the hypothesis H_o: ρ₁ = ρ₂ = ... = ρ_K = ρ, say.

Define the sums e_t = Σ e_i and d_t = Σ d_i, along with the pooled estimator of ρ : = r_t = d_t / e_t, where summation is j = 1, ..., K throughout. Consider the test statistic

T = Σ e_j (r_j - r_t)², which we will write in terms of z_j, where

z_j = r_j – ρ.

Because the r_j are independent, so will be the z_j. Then asymptotically each z_j will have a normal distribution with mean zero and variance ρ (1 - ρ) (3).

Since r_j = z_j / + ρ,
we have r_t = Σ e_j r_j / Σ e_j = Σ z_j / e_t + ρ
and T = Σ_j e_j [(z_j / + ρ) - (Σ_i z_i / e_t + ρ)]² = Σ z_j² - (Σ a_j z_j)²,
where a_j = / , j = 1, ..., K.

Hence, writing in vector and matrix notation, z ' = (z₁ z₂ ... z_K) and a ' = (a₁ a₂ ... a_K),

we have T = z ' z - (a ' z)² = z ' (I - aa ')z.

Now

(I - aa ')(I - aa ') = I - aa ' - aa ' + aa 'aa ' = I - aa ',

because a ' a = Σ a_j² = Σ ( / )² = 1. So (I - aa ') is idempotent with rank

rank (I - aa ') = tr (I - aa ') = tr (I) - tr (aa ')

= K - Σ (e_j / e_t) = K - 1.

So from Cochran's theorem (e.g., 3), the quadratic form z ' (I - aa ' )z is distributed as Var(z) × χ² with K-1 degrees of freedom. Also, Var(z) = ρ (1 - ρ). Test statistics such as T result from performing an analysis of variance on daily mortality rates (r_j), using exposure (e_j) as a weight. Instead of using the within-group error as the denominator in an F test, the treatment sum of squares T is divided by r_t(1 - r_t) and referred to a Chi-square distribution.

1. Mayfield, H. (1961) Wilson Bull., 73, 255-261.
   
2. Johnson, D.H. (1979) Auk, 96, 651-661.
   
3. Seber, G.A.F. (1977) Linear Regression Analysis, p. 37. Wiley, New York.