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 ρi (i = 1, ..., K). Suppose a number of nests in each group are observed and the total exposure for the ith group is ei, which we assume is fixed in advance. Suppose that di of the nests in group i are destroyed, which results in an estimated daily mortality rate of ρi = ri = di / ei. Interest is in testing the hypothesis Ho: ρ1 = ρ2 = ... = ρK = ρ, say.
Define the sums et = Σ ei and dt = Σ di, along with the pooled estimator of ρ : = rt = dt / et, where summation is j = 1, ..., K throughout. Consider the test statistic
T = Σ ej (rj - rt)2, which we will write in terms of zj, where
zj = rj ρ.
Because the rj are independent, so will be the zj. Then asymptotically each zj will have a normal distribution with mean zero and variance ρ (1 - ρ) (3).
Since rj = zj /
+ ρ,
we have rt = Σ ej rj
/ Σ ej = Σ
zj
/ et + ρ
and T = Σj ej [(zj
/ + ρ)
- (Σi
zi / et + ρ)]2
= Σ zj2 - (Σ
aj zj)2,
where aj =
/ , j = 1, ...,
K.
Hence, writing in vector and matrix notation, z ' = (z1 z2 ... zK) and a ' = (a1 a2 ... aK),
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 aj2
= Σ (
/ )2
= 1. So (I - aa ') is idempotent with rank
rank (I - aa ') = tr (I - aa ') = tr (I) - tr (aa ')
= K - Σ (ej / et) = 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 (rj), using exposure (ej) 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 rt(1 - rt) and referred to a Chi-square distribution.
- Mayfield, H. (1961) Wilson Bull., 73, 255-261.
- Johnson, D.H. (1979) Auk, 96, 651-661.
- Seber, G.A.F. (1977) Linear Regression Analysis, p. 37. Wiley, New York.
This resource is based on the following source (Northern Prairie Publication 755):
Johnson, Douglas H. 1990. Statistical comparison of nest success rates. North Dakota Academy of Science Proceedings 44:67.This resource should be cited as:
Johnson, Douglas H. 1990. Statistical comparison of nest success rates. North Dakota Academy of Science Proceedings 44:67. Jamestown, ND: Northern Prairie Wildlife Research Center Online. http://www.npwrc.usgs.gov/resource/birds/statcomp/index.htm (Version 31OCT2000).
Downloading Instructions -- Instructions on downloading and extracting files from this site.
statcomp.zip ( 11K ) -- Statistical Comparison of Nest Success RatesInstallation: Extract all files and open index.htm in a web browser.