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Apparent nest success is expressed as the proportion of observed nesting attempts in which eggs hatch, P = N_s/(N_s + N_u), where N_s = number of successful clutches and N_u = number of unsuccessful clutches.

In Table 1, Appendix B:

1. there were 26 total breeding pairs;
2. nest fate was unknown or nests were not found for 5 of the 26 pairs;
3. a nest was found for each of 21 pairs, 1 of these pairs was also known to renest;
4. the resulting sample size was 22 nests, 14 of which were successful, 8 of which were not (some nests were protected by predator exclosures);
5. apparent nest success P = 14/(14 + 8) = 0.636 or 64%
   (Note that the sample included a clutch of sterile eggs for Appam Lake pair no. 3. It excluded White Lake nos. 2 and 3 and Goose Lake no. 4 because no nests were found, even though observation of chicks at White Lake no. 3 indicated there had been a successful nest.)

Calculating Mayfield nest success

Mayfield success is expressed as the estimated proportion of nests in which eggs hatch, P = (1 - N_u /E)^h, where N_u = number of unsuccessful clutches, E = exposure-days (total), and h = clutch age at hatching (35 days for piping plovers).

In Table 1, Appendix B:

1. the sample includes all nests (total 24, including 2 of unknown fate);
2. exposure-days is the interval from nest discovery until hatching or date of nest loss (losses occurring between visits are assumed to occur at the midpoint between visits, e.g., E = 10.5 days for a nest found 7 June, incubated when visited 14 June, and destroyed when visited 21 June; the exposure period for a nest of unknown fate ends on the last date that a viable nest was observed);
3. from table example, N_u = 8 nests, E = 405.0 days total (includes renest for White Lake pair no. 9), h=35 days;
4. P = (1 - 8/405.0)³⁵ = 0.497 or 50%.
   (This result may not seem remarkably lower than the 64% apparent success estimate derived from the same database, but this is a moderate level of nest success for piping plovers, and disparity between Mayfield and apparent estimates increases as true success decreases. Note: a practical reference with additional examples based on duck nests is Klett et al. 1986.)
5. if needed, the Mayfield estimate of daily nest survival, s, is the hth root of P or, more simply, s = 1 - N_u/E. In this example, s = 1 - 8/405.0 = 0.980

When fates of nests are independent, apparent nest success can be treated as a binomial proportion with the standard error SE(P) = ([P-P²]/N)^1/2 where N is the total number of nests used to calculate P. The assumption of independence is most likely to be violated when nests belong to ≥2 clearly identifiable groups (e.g., nests inside and outside predator exclosures), because fates of nests in the same group are likely to be correlated.

A 95% confidence interval is approximated by P ± 2(SE), and a 90% interval by P ± 1.65(SE). Thus, a standard error and confidence limits can be derived in the preceding example where P = 0.636 and N = 22: SE = ([0.636-0.6362]/22)^1/2 = 0.103; 95% confidence limits are 0.636 ± 2(0.103) = 0.430 to 0.842, or 43% to 84%.

Note: to construct confidence intervals from standard errors, N should be reasonably large (>20 at typical rates of nest success).

Standard errors for Mayfield estimates of nest success are calculated similarly as long as each exposure-day is independent and equally likely to result in the destruction of a nest. Thus, the equation is the same as for apparent nest success except that P becomes the Mayfield estimate of daily nest survival (s) and N becomes the total number of exposure-days (E) used to estimate P (detailed support is in Johnson 1979).

Thus, from preceding calculations of Mayfield nest success with P = 0.497, s = 0.980, and E = 405.0: SE(s) = ([s - s²]/E)^1/2 = ([0.980 - 0.980²]/405.0)^1/2 = 0.007. An approximate 95% confidence interval for s is 0.980 ± 2(0.007) = 0.966 to 0.994. A 95% confidence interval for the corresponding Mayfield success estimate (P) is calculated by raising confidence limits for s to the power of h (again, h = 35 for plovers): 0.966³⁵ to 0.994³⁵ = 0.298 to 0.810, or 30 to 81%.

Calculating fledging rate and pair success

Fledging rate is an estimate of the mean number of flighted juveniles produced per breeding pair, based on total numbers of pairs and 18-20 day old chicks. Pair success can be defined as either the proportion of breeding pairs that have nests with hatched eggs, or that produce fledged (18-20 day old) young; the latter is used here.

In Table 1, Appendix B:

1. there were 26 breeding pairs and all are included in the sample;
2. even though nests or young were not observed for 3 breeding pairs (White Lake nos. 2 and 3 and Goose L. no. 4), they still are included because each pair defended a breeding territory for several weeks; a nest may have been attempted but could have been overlooked, or was destroyed before being discovered;
3. 22 fledgling plovers were produced by 26 pairs of plovers, so the fledging rate was 22/26 = 0.85/pair.
4. pair success for this example was 10/26 = 0.384 or 38%, based on 10 pairs successfully producing ≥1 18- to 20-day-old young each.