MAYFIELD LOGISTIC REGRESSION: A PRACTICAL APPROACH FOR ANALYSIS OF NEST SURVIVAL

Hazler, Kirsten R

ABSTRACT.-

Mayfield logistic regression is a method for analyzing nest-survival data that extends the traditional Mayfield estimator by incorporating explanatory variables (e.g. habitat structure, seasonal effects, or experimental treatments) in a logistic-regression analysis framework. Although Aebischer (1999) previously showed that logistic regression can be used to fit Mayfield models, few ornithologists have put that finding into practice. My purpose here is to reintroduce this underused method of nest-survival analysis, to compare its performance to that of a dedicated survival-analysis program (MARK), and to provide a practical guide for its use. Like the traditional Mayfield method, Mayfield logistic regression accounts for the number of “exposure days” for each nest and allows for uncertain fates (censoring), thus avoiding the bias introduced by typical applications of logistic regression. Mayfield logistic regression should be widely applicable when nests are found at various stages in the nesting cycle and multiple explanatory variables influencing nest survival are of interest. Received 4 February 2003, accepted 28 March 2004.

RESUMEN.-La regresión logística de Mayfield es un método para analizar datos de supervivencia de nidos que extiende el estimador tradicional de Mayfield mediante la incorporación de variables explicativas (e.g. estructura de hábitat, efectos estacionales o tratamientos experimentales) en el esquema de un análisis de regresión logística. Aunque Aebischer (1999) mostró previamente que la regresión logística puede ser usada para ajustar los modelos de Mayfield, pocos ornitólogos han puesto en práctica este hallazgo. Mi propuesta aquí es reintroducir este método poco usado en los análisis de supervivencia de nidos, para comparar su rendimiento con el de un programa de análisis de supervivencia (MARK) y para brindar una guía práctica para su uso. Como el método tradicional de Mayfield, la regresión logística de Mayfield considera el número de días de exposición para cada nido y admite situaciones inciertas (censura), evitando así el sesgo introducido por las aplicaciones típicas de la regresión logística. La regresión logística de Mayfield puede aplicarse ampliamente cuando los nidos se encuentran en varias etapas en el ciclo de nidificación y cuando hay interés en la influencia de múltiples variables explicativas sobre la supervivencia de los nidos.

ACCURATE ESTIMATION OF nest-survival rates is a critical component, and often the primary focus, of many ornithological studies. In many cases, researchers are interested not only in actual survival rates, but also in how various factors (e.g. habitat structure, seasonal effects, or experimental treatments) affect the probability of nest success. Here, I review methods currently in use to address those goals and reintroduce a method which I have named “Mayfield logistic regression.” I include analyses of two sample data sets to illustrate the method’s ease of application and to compare its performance to that of a dedicated survival-analysis software package. Finally, I enumerate the assumptions of the method and address possible violations.

REVIEW OF EXISTING METHODS

The Mayfield method (Mayfield 1961, 1975) is commonly used to estimate daily survival rates of nests. Prior to Mayfield’s introduction of the method, most studies reported nest success simply as the ratio of successful nests to total nests found. As Mayfield pointed out, that is not problematic if all nests are monitored from the initial stage of building and followed through termination. However, in most field studies, nests are found at different stages in the nesting cycle, and nests that fail early are least likely to be monitored. Therefore, the observed ratio of successful to total nests is biased high and overestimates true nest success. To eliminate the bias, Mayfield’s method accounts for number of “exposure days” (i.e. number of days during which a nest is under observation – from the time it is found until it fails, fledges, or is censored). For a group of nests, the daily failure rate, r, is F/E – where F is total number of nest failures and E is number of exposure days – summed over all nests. Daily survival rate, S, is simply 1 – r, and probability of a nest surviving for a nesting cycle of d days is S^sup d^. In practice, nests are usually not monitored every day, and the true number of exposure days is unknown. Mayfield (1961) assumed that nests lost during an interval of several days failed at the midpoint of that interval. He did not specifically address how to calculate exposure days for nests believed to have fledged during an interval or for those with uncertain fates. Researchers calculate number of exposure days for such nests in slightly different ways (see review by Manolis et al. 2000), which can have some effect on the Mayfield estimator.

Johnson (1979) presented the Mayfield model in terms of a likelihood function. The likelihood is based on binomial probabilities of nest success or failure. Survival is modeled in discrete time, with integer numbers of days, and nests are checked at intervals of f days. If t = 1, the model is straightforward. Probability of a nest surviving the 1-day interval is simply S, and probability of failure is 1 – S, where S is the daily survival rate. If t > 1, Mayfield’s midpoint assumption is used for failed nests. Probability of surviving the t-day interval is S^sup t^, whereas probability of failing during the interval is modeled as S^sup 0.5t-1^ (1 – S).

Johnson (1979) also presented a more appropriate model, which assumes that the time of nest failure is known only to have occurred within a certain interval, rather than making Mayfield’s midpoint assumption. In that model, as before, the probability of a nest surviving an interval of t days is S^sup t^. In contrast, the probability that the nest fails during the interval is simply 1 – S^sup t^. Johnson demonstrated that the traditional Mayfield method performs very well if intervals between nest checks are small (1-6 days). For modest data sets, he recommended continued use of the Mayfield method (with a slight adjustment if intervals are long) because of its robustness, combined with its ease of calculation as compared with the theoretically correct model, which requires iterative computer routines for estimation. Hensler and Nichols (1981) and Bart and Robson (1982) have also presented evaluations of, and extensions to, the Mayfield model. Pollock and Cornelius (1988) and Natarajan and McCulloch (1999) also proposed some alternatives. Yet the Mayfield method in its original form continues to be probably the most widely used estimator of nest survival.

The Mayfield method assumes that nests have a common probability of survival. However, the goal of many nesting studies is not simply to calculate a nest-survival rate for a particular species in a particular area, but to assess which factors potentially affect survival rates, causing certain nests to be more prone to failure than others. For some studies, the Mayfield estimate may suffice. For example, to determine how a certain silvicultural treatment affects nesting success, one can calculate separate Mayfield estimates for nests in plots subjected to treatments versus nests in control plots and compare them statistically with the confidence intervals presented by Johnson (1979). That approach requires enough nests in each treatment to calculate robust estimates and strongly restricts the number of potential factors that can be examined.

If a researcher wishes to consider multiple predictors of nest success or explanatory variables on a continuous scale (or both), a method allowing for individual covariates is required. Logistic regression, with nest success or failure as the binomial response variable, is frequently employed to examine how various habitat characteristics affect probability of nest success (e.g. Hoover and Brittingham 1998, Bisson and Stutchbury 2000, Moorman et al. 2002). Logistic regression is a powerful tool if nests are found at the beginning of the nesting cycle and if nest fates are certain. However, as discussed above, that is usually not the case, so the typical logistic-regression analysis suffers from the same bias that initially prompted development of the Mayfield method. Moreover, nests with uncertain fates must be excluded from analysis, which can also introduce bias, in addition to reducing sample size.

Natarajan and McCulloch (1999) proposed a binomial, daily survival-rate model specifically designed to deal with heterogeneous survival rates among nests. It is a flexible approach that can incorporate nest-specific explanatory variables, time-varying covariates, and “pure heterogeneity” (random effects). However, to my knowledge, that approach has not been applied to any field data besides the authors’ own example. Presumably, ornithologists have not adopted the method because it presents complex likelihood equations, because software designed to implement it is not available, and because it was published in a statistical rather than an ornithological journal.

Recently, a nest-survival module was made available in MARK (White and Burnham 1999), a general-purpose survivalanalysis program designed for wildlife biologists. The nest-survival module extends Johnson’s (1979) binomial model to incorporate explanatory variables. Strengths of the program include the ability to relax the assumption of constant survival probability across the breeding season, readily accessible estimates of standardized parameter coefficients and survival, and automated ranking of candidate models by Akaike’s Information Criterion (AIC; Akaike 1973) for those who have embraced the information-theoretic paradigm of data analysis (Anderson et al. 2000). However, the program cannot import data from a standard spreadsheet file, necessitating the additional step of converting data structure to a very specific text-based format. More importantly, the process of model construction is fairly unwieldy and can be daunting to those who do not have a full understanding of design matrices. The program does not generate P-values for traditional frequentist hypothesis testing, which may be viewed by some as a disadvantage. Finally, estimation routines frequently fail to converge if covariates are not standardized, so estimates of unstandardized coefficients are not readily available.

Manolis et al. (2000) recommended the use of Cox (proportional hazards) regression, which allows for individual covariates, incorporates number of exposure days, and can handle uncertain nest fates. Unlike the discrete-time binomial models discussed above, Cox regression models survival in continuous time and assumes that the exact failure time is known. It relaxes the Mayfield assumption of a constant survival rate over time; instead, it assumes an underlying baseline hazard function that does not need to be specified. However, because the analysis is predicated on this unspecified hazard function, it is difficult to obtain an actual estimate of daily survival, which is a primary focus of most nestsurvival studies. Furthermore, Cox regression requires the specification of a well-defined time origin (Allison 1995, Williams et al. 2002), and it is not clear whether that should be based on the age of the nest (frequently difficult to ascertain) or the day of the nesting season. Different results may be obtained depending on which definition of time origin is used (J. Gannon pers. comm.). For reasonable parameter estimates, there must be enough nests “at risk” at any given time; that can be a problem if one is analyzing a sparse data set. Cox regression is widely used in the social and medical sciences (Allison 1995) and has also been applied to analysis of survival in wildlife populations (Williams et al. 2002). Given large data sets and complicated analysis requirements (e.g. time-varying covariates; Allison 1995), Cox regression holds great promise for estimating effects of various explanatory variables on nest survival. However, with more limited data sets, a simpler, less data-intensive method is preferred.

MAYFIELD LOGISTIC REGRESSION

Aebischer (1999:528) demonstrated that “the duality between Mayfield models and binomial models means that Mayfield models can be fitted by logistic regression” and thus easily incorporate individual covariates. Most ornithologists, however, are apparently unaware of the practical application of that approach. I have coined the term “Mayfield logistic regression” for the method, which essentially approximates the analysis used in program MARK and belongs in the class of discrete-time binomial models. Analysis can be conducted using standard statistical software, models are easy to construct, models generally can be fit with either raw or standardized covariates without convergence problems, and results may be readily interpreted using either traditional hypothesis tests or the information-theoretic framework, which is rapidly gaining popularity (Anderson et al. 2000).

In typical nest studies to date, S is the estimated probability of a nest surviving the entire nesting cycle or, in some cases, the probability of surviving to some predefined cutoff stage (e.g. beginning of the potential fledging period). Each nest represents a single binomial “trial,” where the outcome is success or failure. In contrast, for Mayfield logistic regression, S is a daily survival rate. Each nest represents multiple binomial trials, depending on the number of exposure days. As a simple example, a nest that was observed for five days and failed on the fifth day is considered to have undergone five binomial trials, with four successes followed by a failure.

Like the traditional Mayfield method, Mayfield logistic regression requires a means of calculating number of exposure days or binomial trials, given that nests are not visited daily and that some have uncertain fates (Manolis et al. 2000). For a nest that survives to fledging or is censored (uncertain fate), I assume that the last active date is the last exposure day, whereas for a nest that fails, I use the mid-point between the last check and the last active date (rounded up to the nearest day). That is equivalent to the “Last Active-A” approach of Manolis et al. (2000) for calculating exposure days for the Mayfield estimate. For general use with Mayfield estimation, Manolis et al. (2000) recommend their “Last Active-B” approach, which uses the midpoint for successful as well as failed nests, and the last active date only for nests with unknown fate. However, I prefer the “Last Active-A” approach because, based on the simulation results from Manolis et al. (2000), it appears to be less biased when true mortality rates are not constant over the nesting cycle. Program MARK also requires that observations for successful or uncertain nests be truncated on the last active date.

I have implemented Mayfield logistic regression using PROC LOGISTIC in SAS (SAS Institute, Gary, North Carolina), but the options described here are most likely available in other statistical software packages. In prior nest studies using logistic regression, single-trial syntax was used (i.e. 1 nest = 1 trial). For Mayfield logistic regression, I use the events/trials syntax, where the event is success (0) or failure (1) over the observation interval (i.e. not necessarily the ultimate fate of the nest if nests are censored at the end of the interval), and number of trials is number of exposure days. Note that events/ trials is not entered as a proportion; instead, the actual numbers of events and trials are entered. Because a nest cannot fail more than once, the number of events in this case will always be either 0 or 1. Note too that because I am actually modeling nest failure, signs of all coefficients and the intercept need to be reversed to interpret their effects on survival.

METHODS AND RESULTS

SAMPLE DATA SET 1: EFFECTS OF HABITAT STRUCTURE

To illustrate use of the method, I present analysis of a data set that is treated more thoroughly elsewhere (Hazler et al. unpubl. data). In this example, I considered effects of three stand-level variables – midstory height (MIDHT), snag basal area (SNAGBA), and vertical density or shrub cover (VERTDENS)-on daily survival rates of Acadian Flycatcher (Empidonax virescens) nests (n = 163). Because I was interested in stand-level effects, in theory I could have used the Mayfield estimator to calculate survival rates for each stand and then regressed those estimates on the attributes of the stands. However, in several of the stands, I had an inadequate number of nests to obtain reliable Mayfield estimates. Instead, I assigned to each nest the attributes of the stand in which it was found and used Mayfield logistic regression. (The reader may argue that this is a case of pseudoreplication; I address that issue later. For simplicity in presenting this example, I assume here that nests represent independent samples.)

A portion of the input data set is shown in Table 1. The first column is a nest identification code for reference only. The next three columns are necessary for calculation of observed exposure days. Numbers in those columns are dates converted to integers, starting with 1 for the first egg date of the season. (Julian dates could also be used.) DAYFOUND is the first day on which the nest was observed at risk (i.e. eliminating any days prior to the first egg). LASTACTIVE is the last day on which the nest was known to be active. For nests that were censored or believed to have fledged, LASTCHECK is equated to LASTACTIVE, even if there was a subsequent check revealing that the nest was empty. For nests that were known to fail in the last interval, LASTCHECK is the day the nest was found empty or destroyed. OBSDAYS is the number of observed exposure days. For nests that survived the observation period, OBSDAYS = LASTACTIVE – DAYFOUND. For nests that failed, OBSDAYS = (LASTCHECK + LASTACTIVE)/2 – DAYFOUND. That value is rounded up to the nearest day. FAIL takes the value O if the nest survived the interval, or 1 if it failed. Explanatory variables MIDHT, SNAGBA, and VERTDENS could be substituted with any number of covariates, including class variables.

To calculate model parameters using SAS, I used the code presented in Appendix 1. It should be clear from this example that it is a straightforward procedure to format the data for analysis, and to construct the desired model or models. The next important question is: How well does the method perform? I compared output for the same data set using MARK with output from the Mayfield logistic regression in SAS. I ran the same model, but standardized the variables prior to analysis (standard normal transformation with mean = 0 and standard deviation = 1), because MARK was unable to generate estimates with unstandardized variables. Estimates for the coefficients were within 0.2% of each other (Table 2). In addition, the estimate of daily survival generated by the model with all (standardized) covariates set to 1 was 0.97921 (MARK) versus 0.97917 (SAS), a negligible difference. For a 29-day nesting cycle, daily survival rates translate to an overall probability of success of 0.54 for both methods.

SAMPLE DATA SET 2: ANNUAL AND SEASONAL EFFECTS

As a second example for comparison of Mayfield logistic regression with MARK, I used a set of Acadian Flycatcher nests (n = 177) monitored in a single 56-ha plot from 2001 to 2003 (Hazier et al. unpubl. data). I was interested in differences in survival rates among years as well as seasonal differences. I treated YEAR as a fixed class variable. To account for seasonal effects, I used the continuous variable MIDPT, which is the midpoint between the first and last observation days for each nest. I also included an interaction between YEAR and MIDPT. Again, it was necessary to standardize the continuous variable for comparison with MARK.

The SAS code for this example is shown in Appendix 2. Older versions of SAS do not allow class variables, so dummy variables must be coded into the data set by the user. The current version of SAS automatically constructs dummy variables when a class variable is declared. However, the user may wish to specify a parameterization other than the default, and that can be done with the “param =” option (SAS Institute 2000). Refer to Hosmer and Lemeshow (1989) or Allison (1999) for a discussion of dummy variable coding.

For this example, SAS and MARK generally produced parameter estimates that were within

DISCUSSION

ASSUMPTIONS OF MAYFIELD LOGISTIC REGRESSION

Mayfield logistic regression, like any other analytical method, carries certain assumptions. It is important to understand the assumptions of the method and to be aware of possible violations that may affect results. I address the assumptions below, but a detailed analysis of how violations of the assumptions will affect results is beyond the scope of this paper.

Known number of exposure days.-Like the traditional Mayfield estimator, Mayfield logistic regression assumes that number of exposure days is known exactly. Of course, this assumption is violated; the researcher must estimate the number of exposure days. Despite that fact, Johnson (1979) showed that Mayfield’s (1961) estimator performs well when intervals between nest checks are short. The same property appears to hold for Mayfield logistic regression as well, as evidenced by the above comparison of results between SAS and MARK output. Thus, although this assumption is obviously violated, it appears to have a negligible effect on the results. I stress, however, that intervals between checks must be short; the longer the intervals, the more biased the estimates will be. I suggest intervals of 2-3 days. (Disturbance to nests from frequent checks can be minimized by observing nesting behavior [e.g. incubating or feeding] from a distance on some checks to determine whether nests are active.)

Uninformative censoring. – Nests with unknown fates are censored observations. Random censoring occurs if a nest can no longer be monitored for some reason that is unrelated to the status of the nest; this is uninformative censoring and does not cause problems for survival estimation. Another type of censoring occurs when nests survive to the potential fledging interval (Manolis et al. 2000). After a nest reaches that stage, its ultimate fate is often ambiguous. If nests at that stage are collectively more or less likely to fail than other nests, censoring is informative and can result in biased survival estimates. Williams et al. (2002) recommend that all exposure days after the first day of the potential fledging interval should be excluded from calculation of the Mayfield estimator. However, the simulation results of Manolis et al. (2000) indicate that such an “early termination” approach actually results in greater bias in the Mayfield estimator than a “last active” approach. I expect that this finding extends to Mayfield logistic regression as well, but more research with simulated data sets will be required to determine how best to deal with informative censoring and uncertain nest fates.

Constant survival. – Mayfield logistic regression assumes a constant survival rate over time. If the stage in the nesting cycle is important, one can stratify the data by using an indicator variable for nest stage (using an appropriate repeated-measures model to accommodate multiple observations per nest). With the second sample data set above, I showed how seasonal effects can be modeled. However, the method still assumes that survival probability is constant over the entire period of observation for any given nest. If a more thorough investigation of time-varying covariates is desired, Cox regression may be preferable to Mayfield logistic regression.

Homogeneity. – Mayfield logistic regression assumes that all nests have a common daily survival rate, constrained only by the explanatory variables included in the model. However, there will be other, unobserved factors affecting nest survival; that unobserved heterogeneity, if severe, can lead to “heterogeneity shrinkage,” which is the attenuation of coefficients toward 0 (Allison 1999). Logit models are more susceptible to this problem than linear models because they lack an explicit error term, and it is thus recommended that all relevant covariates be included in logit models, even if they are not of particular interest to the researcher (Allison 1999).

Mayfield logistic regression also assumes that nests represent independent samples. However, lack of independence arises when there is unobserved heterogeneity operating at the level of groups (rather than individual nests), such that survival probabilities of nests in one group differ from survival probabilities in another group because of some factor(s) not considered in the model. That type of heterogeneity, also called overdispersion (Burnham and Anderson 1998, Allison 1999), is an often overlooked or ignored aspect of nest-survival studies using individual covariates. The standard protocol for nest-survival studies is to establish several research plots in which nests are located and monitored (Martin and Geupel 1993). Thereafter, if a logisticregression approach is used to examine effects of various explanatory variables, nests are typically treated as independent observations. However, one could argue that nests from the same stand or study plot are not independent, and that this is a case of pseudoreplication (Hurlburt 1984). For example, all nests within a stand may be within the territory of a single wide-ranging nest predator. Additionally, many nests within a stand represent repeated nesting attempts of particular pairs of birds. If observations are not independent, parameter estimates will still be correct, but standard errors will be underestimated because the effective sample size will be smaller than the actual number of nests (Allison 1999, Anderson and Burnham 2002). That, in turn, leads to inappropriate inferences drawn from traditional P-values or AIC rankings. Because of its importance, I will go into some detail on the diagnosis of and remedy for overdispersion.

Overdispersion of stand- or plot-level data can be diagnosed by testing the model for goodness-of-fit (Burnham and Anderson 1998, Allison 1999). If the model does not fit the data well, it could be due either to an incorrectly specified model or to overdispersion. If one is reasonably sure that the model is correctly specified but finds that the model does not fit, the data are probably overdispersed. The statistic used to test for goodness-of-fit is c, the variance inflation factor (also called the heterogeneity factor). It is estimated as the deviance divided by the degrees of freedom or as the Pearson chi-square divided by the degrees of freedom, both of which have a chi-square distribution for grouped data. (Deviance is a likelihood ratio statistic comparing the fitted model with the saturated model [the model having a separate parameter for each observation]: D = -2[log L^sub f^ log L^sub s^], where L^sub f^ and L^sub s^ are the likelihoods of the fitted and saturated models, respectively.) If c > I (even if it is not “significant”), it should be used to adjust the standard errors of the estimates as described below. If c

The AGGREGATE option in PROC LOGISTIC groups the data on the basis of levels of explanatory variables (Allison 1999). If the option SCALE = 1 or SCALE = NONE is specified, no adjustment is made, but LOGISTIC generates the deviance and Pearson statistics needed to calculate c. If c > 1, one can choose to adjust the standard errors (and, by extension, the P-values) – using the Pearson statistic by specifying SCALE = P (Appendix 3), or using the deviance by specifying SCALE = D.

When multiple candidate models are evaluated within an information-theoretic framework, c must be used to calculate QAIC (quasi-AIC), which is used in place of AIC to determine model ranking (Burnham and Anderson 1998, Anderson et al. 2000, Anderson and Burnham 2002). In that case, c should be calculated from the global (most highly parameterized) model and then used to adjust all other models. That can be done by specifying the appropriate numeric value for SCALE. Note, however, that the square root of c should be used because SAS squares the scale factor.

For individual-level data, the variance inflation factor as calculated above is not chi-square distributed and is known to be biased high (Allison 1999). In the sample data set presented above, that was not a problem because I used stand-level data and all observations within a stand were grouped together by the AGGREGATE option. However, there are many situations in which one has true individual-level data, such as measurements of microsite variables around the nest, or a combination of individual- and stand-level data. There are several ways to proceed in such a case.

When modeling effects of nest-level variables, the simplest approach is to calculate c for the most highly parameterized stand-level model, not including any nest-level variables. If no stand-level variables have been measured, the user can specify a null (intercept only) model with a user-defined grouping variable (Appendix 4). Value of c calculated from the stand-level model should then be used to adjust the model(s), including nest-level variables using the SCALE = option. That, in essence, is what the documentation for MARK recommends (Cooch and White 2001), though MARK does not provide any mechanism for calculating c for nest-survival data.

Another approach is to estimate the logit model with generalized estimating equations (GEE; Allison 1999, Hosmer and Lemeshow 2000). That repeated-measures approach assumes that observations are independent between groups but correlated within groups. Sample SAS code to do this using PROC GENMOD is available in Allison (1999). Unfortunately, the method is not likelihoodbased, making model selection problematic. It also suffers from the problem of heterogeneity shrinkage.

Recent statistical developments make it possible to explicitly model hierarchical structure at multiple levels (Bryk and Raudenbush 1992, Goldstein 1995, Allison 1999, Snijders and Bosker 1999). For example, a researcher may want to model the combined effects of landscape-, stand-, and microsite-scale variables on nest survival. There are specialized software packages available for multi-level modeling (see Snijders and Bosker 1999 for a brief overview), and hierarchical logit models can also be implemented in SAS with PROC NLMIXED (for the simplest model specifications) or the macro GLIMMIX (for more complex models). Hierarchical model specification and estimation are fairly complicated, and consultation with a statistician is strongly recommended for most ornithologists. In addition, there has been some criticism of numerical estimation procedures for hierarchical logit models, and certain data structures may yield biased results (Allison 1999, Snijders and Bosker 1999). Nonetheless, hierarchical modeling holds great promise for future work in nest survival and other ecological studies at multiple spatial scales.

CONCLUSION

Mayfield logistic regression combines the strengths of two widely used methods into a single approach. It extends the traditional Mayfield estimator to incorporate individual covariates in a logistic-regression analysis framework. Like the traditional Mayfield method, this approach accounts for number of exposure days for each nest and allows for uncertain fates (censoring), thus avoiding the bias introduced by typical applications of logistic regression. The binomial model is intuitively more appealing than Cox regression in that a daily survival rate is readily calculated after the model parameters have been estimated. It also more readily accomodates sparse data and does not require a specified time origin. The method is simple to implement in a standard statistical software package and generates results comparable to those generated by a dedicated survival-analysis package. Furthermore, the method can deal with overdispersion and is readily adaptable to a hierarchical modeling approach. I expect Mayfield logistic regression to be widely applicable when nests are found at various stages in the nesting cycle and multiple explanatory variables influencing nest survival are of interest.

ACKNOWLEDGMENTS

The Cooper Lab provided a stimulating environment in which to develop these ideas, and I acknowledge especially discussions with J. Cannon (slave to Cox regression) and B. Mattsson (MARK’s pet) as we all struggled with various nest-survival data sets. J. T. Peterson supplied much-needed, patient statistical advice and introduced me to the concept of hierarchical modeling. D. J. Twedt initially encouraged me to think seriously about the problem of overdispersion. The first sample data set was collected by A. Amacher and me while we were graduate students in the Zoology Department at North Carolina State University, working on a project supported by Westvaco Corporation and others, under the direction of R. A. Lancia. I collected the second sample data set as part of my doctoral dissertation research at the University of Georgia, working on a project supported by the U.S. Geological Survey, under the direction of R. J. Cooper in collaboration with D. J. Twedt. Comments from B. Mattsson, R. J. Cooper, J. T. Peterson, M. T. Murphy, and D. H. Johnson improved the manuscript. Financial and academic support during the preparation of this paper were provided by the Warnell School of Forest Resources at the University of Georgia.

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Associate Editor: M. T. Murphy

KIRSTEN R. HAZLER1

Warnell School of Forest Resources, University of Georgia, Athens, Georgia 30602, USA

1 E-mail: krh5938@forestry.uga.edu

Copyright American Ornithologists’ Union Jul 2004

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