Northern Prairie Wildlife Research Center

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 __i__th 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}: ρ_{1} = ρ_{2}
= ... = 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_{j}^{2} - (Σ
a_{j} z_{j})^{2},

where a_{j} =
/ , j = 1, ...,
K.

Hence, writing in vector and matrix notation, __z__ ' = (z_{1}
z_{2} ... z_{K}) and __a__ ' = (a_{1} a_{2}
... 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_{j}^{2}
= 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.

- 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):**

**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).

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