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Piping Plovers and Least Terns of
the Great Plains and Nearby

A draft protocol for assessing piping plover reproductive success
on Great Plains alkali lakes

Appendix C -- Examples: calculating apparent and Mayfield nest success estimates;
calculating fledging rate and pair success (based on data in Table 1, Appendix B).

Calculating apparent nest success

Apparent nest success is expressed as the proportion of observed nesting attempts in which eggs hatch, P = Ns/(Ns + Nu), where Ns = number of successful clutches and Nu = 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 - Nu /E)h, where Nu = 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, Nu = 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 - Nu/E. In this example, s = 1 - 8/405.0 = 0.980
Standard errors and confidence intervals for nest success estimates

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.

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