USGS - science for a changing world

Northern Prairie Wildlife Research Center

  Home About NPWRC Our Science Staff Employment Contacts Common Questions About the Site

Statistical Comparison of Nest Success Rates

Douglas H. Johnson


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 : rho hat = 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 = square root of e sub-j rjsquare root of e sub-j 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 / square root of e sub-j + ρ,
we have rt = Σ ej rj / Σ ej = Σ square root of e sub-j zj / et + ρ
and T = Σj ej [(zj / square root of e sub-j + ρ) - (Σi square root of e sub-i zi / et + ρ)]2 = Σ zj2 - (Σ aj zj)2,
where aj = square root of e sub-j / square root of e sub-t , 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 (square root of e sub-j / square root of e sub-t)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.


  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.

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).


Douglas H. Johnson, U.S. Fish and Wildlife Service, Jamestown, North Dakota 58402


Downloading Instructions -- Instructions on downloading and extracting files from this site.
(Download) statcomp.zip ( 11K ) -- Statistical Comparison of Nest Success Rates
Installation: Extract all files and open index.htm in a web browser.

Accessibility FOIA Privacy Policies and Notices

Take Pride in America logo USA.gov logo U.S. Department of the Interior | U.S. Geological Survey
URL: http://www.npwrc.usgs.gov/resource/birds/statcomp/index.htm
Page Contact Information: Webmaster
Page Last Modified: Friday, 01-Feb-2013 19:57:38 EST
Reston, VA [vaww54]