Northern Prairie Wildlife Research Center

A purpose of most duck nesting studies is to obtain an estimate of nest success, the probability (P) that a nesting attempt will result in production of one or more ducklings. In the past, nest success was calculated as the percentage of observed nesting attempts that were successful. This estimator (P

where N_{s} and N_{u} are the observed numbers of studies
of successful and unsuccessful nests, respectively. This method is appropriate
provided that successful and unsuccessful nests are discovered with equal probability,
a condition that may be met in studies of artificial nesting structures, small
islands, or other small units of habitat that can be intensively searched for
terminated nests.

Apparent nest success estimates are biased when applied to samples of clutches gathered by flushing hens from their nests. The age of these clutches at discovery can range from newly initiated (one egg) to late incubation. Advanced clutches are more likely to hatch than young clutches because of the shorter interval between discovery and hatching.

To illustrate this concept, consider a population of mallard nests with a
constant daily survival rate (s) of 0.96. The probability of a clutch surviving
1 day is 0.96; 2 days, 0.922 (0.96 X 0.96 or 0.96^2); 3 days, 0.885 (0.96 X
0.96 X 0.96 or 0.96^3), and so on. If the mean laying period plus incubation
period of successful clutches (h) is 35 days, the probability of hatching (P)
for a clutch found on the day it was initiated is **(Equation 2)**:

The probability of hatching for clutches found 5 days after initiation is

because they have already survived 5 days and need to survive only 30 additional days to hatch. With a daily survival rate of 0.96, the probability of hatching for clutches found at various ages is:

Age found Probability of hatchingApparent nest success, applied to a sample of active nests, reflects the true success rate only when all clutches are found on the day they were initiated. Apparent success overestimates true success when clutches are found at a later date.(days) (apparent nest success) Bias0 0.96^35 = 0.24 0.00 5 0 96^30 = 0.29 +0.05 10 0.96^25 = 0.36 +0.12 15 0.96^20 = 0.44 +0.20 20 0.96^15 = 0.54 +0.30 25 0.96^10 = 0.66 +0.42 30 0.96^5 = 0.82 +0.58 34 0.96^1 = 0.96 +0.72

To overcome the bias in apparent success rates, Mayfield (1961, 1975) estimated
daily survival for the interval that nests were exposed to risk while under
observation and used it to estimate nest success from Eq. 2. The exposure interval
(in days) for each nest begins on the day the nest was found and ends on the
day eggs hatched or the clutch was destroyed or abandoned or was no longer under
observation. When the day of destruction or abandonment is unknown, it is assumed
to have occurred midway between the last two visits. For nests observed on 2
or more days but not revisited to determine their fate, the day of the last
visit marks the end of the exposure interval. From a group of nests the daily
mortality rate (m) is estimated by the number of nesting attempts that failed
divided by the number of exposure days. The daily survival rate (s) is the complement
of m, or s = 1-m, and the Mayfield estimator of nest success (P_{2})
from Eq. 2 is **(Equation 3)**:

where E is the total number of exposure days. A basic assumption of the method is that the daily survival rate for the intervals that nests were under observation is the same for all nests and all intervals. This method has gained acceptance by ornithologists. Miller and Johnson (1978) and Johnson (1979) suggested modifications in the calculation of exposure days that improved the method for use in duck nesting studies. Johnson (1979) and Bart and Robson (1982) described a maximum likelihood estimator that is more appropriate when nests are visited periodically and the day nesting attempts fail is unknown. However, computations are difficult and use of a computer is advised. Johnson and Klett (1985) described a method for obtaining quick estimates of duck nest success that is a special application of the maximum likelihood method.

In the following sections we summarize some standard procedures that have been used to estimate success rates for samples of duck nests that were found by flushing hens from their nests. In doing so we repeat this caution by Mayfield (1975): ". . .we should be wary of being lured into a fictitious appearance of precision. Any method we use will give no more than an approximation of the truth, and this method merely helps avoid certain gross errors that are common."

One minor modification in Mayfield's method for calculating exposure days uses the estimated hatch date to determine exposure days for successful nests and help define the interval during which a nest was lost. Another modification involves calculating exposure when the intervals between nest visits are long.

We present a formula for the standard error of the Mayfield estimator with which confidence limits can be constructed for use in estimation and hypothetical testing.

**Essential Data**

A standard form is recommended for recording data in the field (a sample form with instructions appears in Appendix B). The format may vary, but it should be designed for ease in summarizing the data either manually or by computer. Space should be provided to record the species of duck (A.O.U. number, see Appendix Table B-1), a unique nest identification number, and the full clutch size (when known). Essential data recorded each time a nest is inspected are the date, number of eggs and their stage of incubation, and clutch status (still viable; eggs hatched, abandoned, or destroyed; or unknown). Space should also be provided for identifying nests that are not used in calculating success. Examples are clutches that had hatched or were partially or totally destroyed before they were found, and clutches that were destroyed or abandoned because of searching activities on the day they were found. Data from these nests may be useful for other purposes, but are not used to compute nest success.

**Calculations**

All calculations can be made on an inexpensive hand-held calculator with the capability to take the root value of any positive number or raise any positive number to any power.

Mayfield nest success estimates (P_{2}) are calculated from Eq. 3.
Mean ages of clutches at hatching (h) are presented in Appendix Table B-1.

The exposure period is usually the number of days between the date a nest was discovered and the date it was terminated either successfully or unsuccessfully. It is legitimate, however, to use known periods of nest survival observed between nest visits even if the fate of the nest is not determined, or if the nest fails because of investigator disturbance some days after the nest was found.

Exposure for successful clutches is the number of days between discovery and
the estimated hatch date. The hatch date can be estimated from the age of the
clutch when found, determined by candling, and the mean age of clutches at hatching.
The date on which a nesting attempt fails is usually unknown unless the nest
is visited daily. The date on which a clutch is abandoned can sometimes be determined
by noting changes in the number of eggs or stage of incubation since the last
visit. When the exact date on which a clutch is destroyed or abandoned is unknown,
exposure days are derived from two sources: *known exposure* - the number
of days between two visits that the clutch was known to survive, and *probable
exposure* - a percentage of the interval during which the clutch was known
to have been destroyed or abandoned. To estimate probable exposure for short
intervals (< 15 days), we assume that the clutch survived half of the interval
between the last two visits when the last visit (date fate was determined by
the investigator) occurs before the estimated hatch date. When fate was determined
after the estimated hatch date, we assume the clutch survived half of the interval
between the preceding visit and the estimated hatch date.

In actuality the expected date of loss is dependent on the daily survival rate and the length of the interval during which the loss occurred. Johnson (1979) presented a formula for the probable exposure of a destroyed nest. For moderate values of s, the midpoint assumption was reasonable up to about 15 days but overestimated probable exposure when the interval was longer (as is often the practice in waterfowl nesting studies). For intervals longer than 15 days, 40% of the interval was a more reasonable approximation of probable exposure. Because the expected days at risk are difficult to calculate we recommend using 50% of intervals < 15 days and 40% of longer intervals as an estimate of probable exposure.

Detailed instructions for computing nesting success by the modified Mayfield method are presented in Appendix Table B-3.

If desired, the variance (v) and standard error (SE) can be approximated for the estimator (s) of daily survival (Johnson 1979):

The standard error can be used to obtain confidence intervals for s (95% CI = s +/- 2 SE) and to compare daily survival rates of two samples of nests. Confidence limits for s can be used to establish approximate limits for nest success by raising them to a power equal to h (mean age of clutches at hatching, Appendix Table B-1). It should be noted that confidence limits for nest success are asymmetrical because they are derived exponentially.

Because of this exponential relation, minor changes in v generate major changes in the width of the confidence interval for nest success. An increase in the number of exposure days in the sample of nests will decrease the variance and hence, narrow the confidence interval. For studies designed to obtain precise estimates of nest success, (e.g., within homogeneous habitat units), we recommend a minimum of 750 exposure days. On the basis of previous information on the expected range of daily survival rates and exposure days in samples of nests, we believe this goal should be reached with a sample of 50 to 75 nests. Smaller confidence intervals can be obtained if investigators are willing to accept lower confidence levels, such as 90% or even 80%. For those interested in details of sample size requirements, Bart and Robson (1982) present formulas for obtaining specified precision on single estimates and for estimating differences between two populations.

**Advantages and Limitations**

The modification of Mayfield's method presented here is appropriate with samples of nests that were active when found. An important assumption is that the clutch survival rate is constant from day to day. Johnson (1979) reported that the bias introduced by variation in daily survival rates is slight unless the variation is extreme. Klett and Johnson (1982) found that nest success rates were overestimated (by as much as 13%) when daily survival rates are much lower during the early stages of laying than during incubation, and they suggested a method for correcting this bias. Bart and Robson (1982) described a method for checking the assumption of constant survivorship. Large samples of nests, however, are required.

The assumption that all nests are subject to the same mortality is obviously violated when short-term catastrophic events caused by farming practices (e.g., tillage or mowing) or weather (e.g., heavy snowfall, hail, or flooding) occur. Nests that were expected to hatch before or were initiated after the event would likely have different hatch rates from those subject to the event. In most studies samples are not large enough to estimate overall nest success by partitioning the sample into subsamples affected by or not affected by the event.

Some of the advantages of the modified Mayfield method are: (1) it provides survival rates that are less biased than apparent rates for samples of clutches that are active when found, (2) the calculations are not as complicated as the maximum likelihood method, (3) it is robust in regard to moderate variation in daily survival rates, and (4) estimated standard errors are available for setting confidence intervals and testing for statistical differences in survival rates.

**Essential Data**

For each nest, only species identification, age when found, and ultimate fate are needed to estimate nest success rates by this method (Johnson and Klett 1985). With two exceptions the same nest records are used to calculate both the Mayfield and shortcut estimator of nest success. When the shortcut method is used all nests of unknown fate and all nests that were damaged by investigators or were abandoned because of investigator disturbance are excluded. These nests are used in the Mayfield calculations if the known component of exposure is > 0.

**Calculations**

The same assumptions used for the Mayfield method apply to the shortcut method.
We denote by h the mean age at which hatching occurs (Appendix Table B-1), and
by f the mean age of clutches when found. If all the nests are found at age
f, then apparent nest success (Eq. 1) is the percentage of nests that survived
the interval h-f and the daily survival rate for that interval is the h-f root
of apparent nest success. **(Equation 4)**:

The shortcut estimator of nest success (P_{3}) is then calculated
from Eq. 2. In this special case (all of the nests are found at age f), s provides
the maximum likelihood estimate of the daily survival rate (Bart and Robson
1982). The result is only approximate when actual ages vary and f is the mean
age when found.

Detailed instructions for computing nest success by the shortcut method are presented in Appendix Table B-3.

**Advantages and Limitations**

The shortcut estimator requires minimal information for each nest and is easy to compute. The method is useful for quick estimates and is recommended for preliminary analyses or to check the calculations of the Mayfield estimator. Because the shortcut estimator is an approximation of the Mayfield and maximum likelihood estimator and is not useful for statistical evaluation, we recommend that the last two methods be used whenever possible.

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