Ecol. Appl. 15:281-293.

Ecol. Appl. 15:281-293.
Ecological Applications, 15(1), 2005, pp. 281–293
q 2005 by the Ecological Society of America
A CLUTCH AND BROOD SURVIVAL MODEL THAT DISCRIMINATES
RANDOM AND CORRELATED MORTALITY
BARRY D. SMITH,1,3 W. SEAN BOYD,1
AND
MATTHEW R. EVANS2,4
1Canadian
Wildlife Service, Environment Canada, Pacific Wildlife Research Centre, 5421 Robertson Road,
Delta, British Columbia V4K 3N2 Canada
2Centre for Wildlife Ecology, Department of Biological Sciences, Simon Fraser University,
Burnaby, British Columbia V5A 1S6 Canada
Abstract. Quantitative conservation methodologies such as Population Viability Analysis (PVA) require reliable estimates of life history parameters such as breeding success.
The utility of such metrics for egg-laying species is complicated by the fact that the mortality
of eggs and juveniles can occur both randomly and independently over time, or catastrophically, as in the sudden loss of a clutch or brood. Not knowing the nature of mortality
caused by either or both of abiotic (e.g., weather) and biotic (e.g., predation) events limits
our ability to confidently assess a population’s demography and sustainability, or rank
competing hypotheses. To address this deficiency, we describe a statistical model that
estimates egg and juvenile survival rates continuously from laying to fledging based on
periodic observations of individual clutches and broods. Adjunct data on environmental or
predation threats can be included in the model as covariate series potentially affecting
juvenile survival. Our model can statistically characterize mortality between the extremes
of random and catastrophic mortality and can determine if unwitnessed mortalities occurred
independently or were correlated (i.e., overdispersed, where catastrophe is extreme overdispersion). Overdispersion is estimated as a parameter of the beta-binomial probability
distribution of survival outcomes, which differs from its treatment in Program MARK
where overdispersion is an a posteriori diagnostic referred to as ĉ. We used data for the
sea duck Barrow’s Goldeneye to illustrate our model. Specifically, we contribute to the
argument that a larger brood confers a fitness advantage to a tending hen by concluding
that brood size on hatch day is positively correlated with a juvenile’s probability of surviving
to fledge.
Key words: beta-binomial; breeding success; brood amalgamation; catastrophe; clutch parasitism; Mayfield method; mortality; overdispersion; Program MARK; survival.
INTRODUCTION
One of the key methodologies for assessing a population’s sustainability over time is population viability
analysis (PVA; Beissinger and McCullough 2002, Morris and Doak 2002). Effective use of analyses such as
PVA requires that an analyst have confidence in the
life history parameter estimates and their uncertainties
that enter such models. However, expressions of uncertainty often tacitly assume that survival estimates
arise from a simple binomial process where individuals
independently either live or die, and whose rate may
or may not change over time. The three most wellknown statistical tools for estimating survival rates for
bird clutches and broods are the Kaplan-Meier productmoment survival estimator (Kaplan and Meier 1958),
the Mayfield method (Mayfield 1961, 1975), and Program MARK (White and Burnham 1999; available onManuscript received 22 October 2003; revised 20 April 2004;
accepted 27 April 2004; final version received 18 May 2004.
Corresponding Editor: T. R. Simons.
3 E-mail: [email protected]
4 Present address: Biology Department, Mount Allison
University, Sackville, New Brunswick E4L 1G7 Canada.
line).5 The Mayfield method for nest success has found
wide use in bird demographics over the last four decades, and some authors have modified or refined the
Mayfield method to adapt it to their particular data
(Johnson 1979, Johnson and Shaffer 1990, Grand and
Flint 1997, Stanley 2000, Manly and Schmutz 2001,
Dinsmore et al. 2002). The Kaplan-Meier productmoment survival estimator has found broad generic
applicability in survival analysis and hypothesis testing
in a variety of fields from medicine to demography.
However, like the Mayfield method, it assumes that the
deaths of individuals are random and follow a binomial
probability distribution.
A well-known contemporary analytical tool for population demographers is Program MARK (White and
Burnham 1999). Program MARK offers a suite of options for survival estimation and modeling using observational or capture–mark–recapture (CMR) data that
includes a nest survival model (Dinsmore et al. 2002)
that has evolved from the Mayfield method. The principal contribution of Program MARK is its capacity
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^http://www.cnr.colostate.edu/;gwhite/mark/mark.htm&
282
BARRY D. SMITH ET AL.
for robust and realistic, though potentially highly parameterized, survival models, and its ability to empirically deal with overdispersion, i.e., the tendency for
individual mortality events to be correlated. Program
MARK exploits the contemporary availability of powerful computers to undertake data analyses that were
impractical in the recent past. Perhaps more important,
it has implemented contemporary theory for model
ranking based on the information-theoretic approach to
model selection and interpretation (Burnham and Anderson 2002). Thus it has the ability to estimate survival rates and their uncertainty for direct use in demographic population models or for hypothesis ranking
among competing models.
Despite the robustness of analytical tools such as
Program MARK, there remain many circumstances
where specific hypotheses or particular data structures
are not well suited to the suite of statistical options
available in the literature. One key deficiency concerns
a lack of robustness in accommodating the distributional characteristics of clutch and brood mortalities.
In particular, it has been recognized by demographers
that a survival rate is not a generic metric, but integrates
an individual’s success at avoiding mortalities due to
random biotic (e.g., predation) and abiotic (e.g., weather) events (Morris and Doak 2002). Such predation or
weather events are not likely to affect all eggs in a
clutch, or all juveniles in a brood, independently. For
example, a predator may attack more than one juvenile
in a brood of ducklings, or a violent weather event may
destroy an entire brood. Overall survivorship of eggs
and juveniles will represent an individual’s success at
enduring all of these threats.
The model we present here addresses two limitations
of the Mayfield, Kaplan-Meier, and Program MARK
methodologies. None of the above models deals explicitly with overdispersion during the parameter estimation phase of model fitting (though Program
MARK deals with overdispersion as an a posteriori
correction). Likewise, none accommodates the reality
that an individual’s survival likely results from enduring a mixture of random (independent) and correlated
(overdispersed) mortality processes. Specifically, our
model offers two advantages for modeling breeding
success from laying to fledging. First, survivorship is
statistically partitioned into random and correlated
mortality profiles. Thus the assumption that mortality
events be statistically independent, i.e., binomially distributed, is relaxed. The overdispersed partition may
range from partial to full (catastrophic). This nonindependence of mortality events is accommodated by
use of the beta-binomial probability distribution for
model prediction error (Mood et al. 1985, McCullagh
and Nelder 1989). Second, survivorship estimates can
be measured from laying through hatching, then from
hatching to fledging without the need to observe hatching.
Ecological Applications
Vol. 15, No. 1
A potential constraint of our approach is that survivorship curves are parameterized to follow a Weibull
probability distribution (Walpole et al. 1998). As such,
our model trades off the advantages of our parsimonious approach against robustness in survivorship
curves offered by distribution-free models such as
those offered by Program MARK. However, our more
flexible error structure can reduce the need for flexibility in the survivorship model. Goodness-of-fit
(GOF) assessments can be used to judge any consequences of this trade-off. Our model incorporates the
information-theoretic features of model ranking and
GOF testing (Burnham and Anderson 2002) that would
be familiar to users of Program MARK and are key to
parsimonious model selection, hypothesis ranking, and
adjudication of the quality of a model’s fit to data.
Researchers can judge the utility of the clutch and
brood survivorship model we describe here for their
scientific inquiries by addressing the following features
of their hypotheses and data. If (a) your purpose is (1)
to estimate clutch and/or brood survival rates, their
uncertainty and distributional (random or correlated)
characteristics for use in a demographic or simulation
model, or (2) to rank models or test hypotheses concerning the survival rate of eggs in a clutch or juveniles
in a brood (i.e., investigate the effects of predators,
weather, pesticides, etc.), and (b) you have data on
steady or declining clutch and/or brood sizes periodically over time, clutch and/or brood age, and optionally
a brood covariate series (e.g., weather, or a stage or
condition variable), and (c) you are comfortable with
assuming almost synchronous hatching of all eggs in
a clutch, specifying a laying age and/or a fledging age,
assuming negligible measurement error, and assigning
all eggs or juveniles (precocial or altricial) observed
to a family, then: you can estimate clutch and/or brood
survival rates and their uncertainty, have survival rates
vary with age or time, relate survival to a covariate
data series, and partition mortality into its random and
correlated components.
Our model was motivated in part by demographic
questions concerning the breeding success of the sea
duck Barrow’s Goldeneye (Bucephala islandica) in the
central interior (Chilcotin-Cariboo) region of British
Columbia, Canada. Our particular interest in Barrow’s
Goldeneye in this region stems from the unique grassland and fragmented forest mosaic habitat near Riske
Creek, British Columbia (see Plate 1). This habitat is
rare and unique in British Columbia and is geographically isolated from similar habitat to the east, particularly in Canada’s prairie provinces. Decades of forestry and fire suppression have resulted in this unique
habitat being further diminished by timber harvesting
and forest encroachment upon the grassland.
Conservation concerns for the Chilcotin-Cariboo
population of Barrow’s Goldeneye initially arose due
to their being secondary cavity nesters that lay 4–15
eggs (Godfrey 1986), primarily in cavities excavated
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CLUTCH AND BROOD SURVIVORSHIP
283
PLATE. 1. Aerial view portraying landscape characteristics of the Chilcotin-Cariboo region near Riske Creek, British
Columbia (B.C.), Canada. Photo credit: W. S. Boyd.
by Pileated Woodpeckers (Dryocopus pileatus; Evans
et al. 2002). Barrow’s Goldeneye tend to choose cavities roughly 12 m above the ground and in aspen or
fir trees within ø100 m of a small, shallow pond (Evans
2003). Their choice of such cavities helps minimize
egg predation by black bears and small mammals
(Evans et al. 2002). Hatching of all eggs in a clutch
occurs synchronously, with the hatched young undergoing a coordinated freefall from their cavity and then
being led to an adjacent pond by the hen. The territoriality of Barrow’s Goldeneye usually results in each
small pond accommodating a single brood, with larger
ponds sometimes accommodating multiple, but isolated, broods (Savard 1982, 1984). Brood rearing occurs
on ponds shallow enough for the young to dive for
invertebrate prey (Evans 2003). While on or around the
pond the young are vulnerable to avian and mammalian
predators and harsh weather events such as heavy rain
or hailstorms.
The key scientific queries concern the potential loss
of riparian areas as a source of cavities due to forestry,
the possibility that climate change would alter the productivity (invertebrate biomass) of the ponds for foraging juveniles, and that a changing landscape from
forest encroachment would increase predation threats,
particularly from avian predators, on juveniles (Evans
2003). Consequently, over the past two decades Barrow’s Goldeneye has attracted research attention from
both conservation and behavioral scientists. Conservation questions addressed, for example, whether the
use of nest boxes would increase clutch survivorship
by providing greater protection from predation, resulting in more and larger clutches (Savard 1988, Evans
et al. 2002). Similarly, behavioral ecologists ques-
tioned the evolutionary advantage of the high prevalence of conspecific clutch parasitism (Eadie and Fryxell 1992, Eadie and Lyon 1998, Eadie et al. 1998,
Lyon and Eadie 2000) and brood amalgamation (Savard
1987) in Barrow’s Goldeneye and related species.
Much of the scientific argument concerning the evolutionary consequences of these behaviors has relied
on theoretical models (Johnstone 2000, Broom and
Ruxton 2002a, b, Öst et al. 2003) and genetic sampling
and interpretation (Andersson and Åhlund 2000, Lyon
and Eadie 2000). Given this backdrop, we applied our
clutch and brood survivorship model to observations
of known clutches and broods made in 1995, and 1997
to 2000, at Riske Creek to empirically address two
hypotheses related to Barrow’s Goldeneye conservation. Hypothesis I: Is there is a different probability of
surviving to fledge for juvenile Barrow’s Goldeneye
hatched in larger vs. smaller broods? Hypothesis II:
Does the foraging quality of a brood-rearing pond (as
measured by invertebrate biomass) affect the probability that a juvenile in a brood using that pond will
fledge?
SURVIVORSHIP MODEL
Our model was developed on the premise that the
survival rate of eggs in a clutch, or juveniles in a brood,
can vary with age (a), and in the case of broods (b),
in relation to abiotic and biotic covariates. A full description of the model is presented in the Appendix,
whereas here we describe the model conceptually and
with only sufficient detail and parameter definitions to
enable a reader to qualitatively interpret the results we
present for Barrow’s Goldeneye.
BARRY D. SMITH ET AL.
284
FIG. 1. Example plots of (a) probability density functions
(pdf) and (b) their associated survivorship attenuation functions for random (R) and correlated (C) mortality processes
and both additively combined (R and C). The functions for
R and C portray situations with an initial period when young
juveniles experience a high rate of correlated mortality (perhaps catastrophic losses of broods) that diminishes with time
as the rate of random mortality increases, then diminishes
with age. The parameter values for this example are: aB,R 5
0.001, bB,R 5 2.5, aB,C 5 0.1, bB,C 5 0.9, cB 5 0.7, and fB 5
0.05.
We developed our model using the Weibull probability density function (pdf) as a tractable and flexible
model of survivorship probabilities over time (Walpole
et al. 1998). In its simplest formulation it represents a
constant survival rate with an exponential distribution
of survivorship,
v(a; a, b) 5 abab21e2aab
(1)
(a . 0; b . 0) with its attenuation, or survivorship,
function (1 2 cumulative probability function) A(a)
being described by
A(a) 5 e2aab.
(2)
When the shape parameter b 5 1, survivorship is constant at the instantaneous rate a. Values for b ± 1
introduce age dependence in survivorship.
A key feature of our model is its ability to partition
survivorship into random (R) and correlated (C) components. To achieve such a model we chose to construct
a new pdf as a contagious mixture of two Weibull distributions representing the random and correlated components of mortality independently for both clutches
(or nests, N) and broods (B). The two scenarios of
random (R) and correlated (C) mortalities are additive
(Fig. 1) for both clutches and broods such that
A•(a) 5 (1 2 v(a))A•,R(a) 1 v(a)A•,C(a)
(3)
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where the subscript ‘‘dot’’ can represent exclusively
either clutches (N) or broods (B), and v(a) represents
the proportion of a clutch or brood vulnerable to a
correlated mortality process at age a.
The survivorship function for both clutches and
broods (Eq. 3) must be bounded in time. By defining
a 5 0 to correspond to the age that a clutch hatches,
increasingly negative ages apply to increasing younger
clutches, while positive ages apply to broods. We therefore define a negative number of days (I), corresponding to the age all clutches in the dataset are initiated.
Likewise, for broods we define a positive number of
days corresponding to the age (D) beyond which the
disappearance of a juvenile from a brood might be due
to fledging rather than mortality. Consequently, the age
range for clutches is a 5 I to 0, while that for broods
is a 5 0 to D. The proportion of clutches and broods,
respectively, vulnerable to a correlated mortality process at age a 5 I and a 5 0, respectively, are defined
by cN and cB, and diminish with age at instantaneous
rates fN and fB.
One goal of our model was to allow both the random
and correlated survivorship profiles for broods to be
functions of external factors, our so-called brood covariates. We identified two potential brood covariates
directly associated with basic data collection: expected
brood size on hatch day (EN,b[a 5 0]) and the day of
the year that hatching occurred, t. We refer to these as
intrinsic brood covariates. Additionally, one or several
adjunct brood covariates may have also been measured.
Our model allows these covariates and their coefficients to operate on the Weibull parameters aB,• and bB,•
(yielding a9B,•,b and b9B,•,b) to modify the shape of the
survivorship function (Eq. 3), where the subscript
‘‘dots’’ can represent either random (R) or correlated
(C) mortality. Note that any covariates operating upon
bB,• introduce age dependence in survivorship.
A key model assumption is no, or more practically,
negligible measurement error. That is, we assume that
counts of the number of eggs in a clutch or juveniles
in a brood are accurate. Therefore all data records (r,
r 5 1 to R) for each clutch or brood must exhibit a
steady or declining number of individuals over time.
As such, our model error structure presumes that deviates from predicted survivals arise from actual stochastic outcomes. Further, we consider the basic sampling or observational unit to be a clutch or brood
followed through time, with their eggs and juveniles,
respectively, being considered elements of the sample.
Survivorship estimates are therefore inherently weighted by clutch or brood size. We also make the point
here that our implementation of the model treats individuals alive on hatch day as juveniles in a brood.
Readers should also recognize that our definition of
a clutch or brood is robust in the sense that an analyst
is free to create the analyst’s own definition of a clutch
or brood. For example, an analyst might choose to define a brood before and after an amalgamation event
February 2005
CLUTCH AND BROOD SURVIVORSHIP
285
FIG. 2. Examples of plausible probability
mass distributions of survivorship outcomes,
p•[s•(a 1 i)], for n•(a) 5 10, m•,•(a) 5 0.6, and n•
5 0. (a) No overdispersion, u2•,• (a) 5 0, generates
a binomial distribution of survivorship outcomes; (b) partial overdispersion, u2•,• (a) 5 0.2,
generates a greater spread of possible survivorship outcomes according to a beta-binomial
distribution; while (c) full overdispersion, u2•,•(a)
5 1.0, generates a beta-binomial distribution
with only two possible outcomes, either all n•(a)
individuals survive or die. Plot (d) portrays an
example in which 70% of the mortalities follow
a random mortality process (C), while 30% of
mortalities (c• 5 0.3; f• 5 0) are correlated (C)
and fully overdispersed as per plot (c).
as two separate broods, and perhaps accompany the
analysis with a categorical brood covariate that indicates whether or not the brood resulted from an amalgamation. Likewise, an analyst may choose to define
a brood that has moved between two ponds, as two
separate broods.
To address the deficiency of traditional models to
accommodate the nonindependence of mortality events
we chose the beta-binomial probability mass function
(pmf) to model prediction error. The advantage of the
beta-binomial pmf is that its definition includes a parameter, u2•,•(a), that explicitly accommodates overdispersed (i.e., correlated) outcomes when u2•,• (a) . 0. Here
the subscript dots represent the four survivorship scenarios N, R; N, C; B, R; and B, C. If u2•,•(a) 5 0 there
is no overdispersion and the distribution limits to the
binomial pmf; thus by our definition u2•,R( a) 5 0 always.
If, in the extreme, u2•,•(a) 5 1, the beta-binomial distribution is fully overdispersed such that the n•(a) individuals in a clutch or brood either all survive or none
survive; by our definition a catastrophic outcome at a
survival rate of m•,•(a 1 i) over the time interval i. Note
that we have made u2•,•(a) a function of age,
u2•,C(a) 5 u2•,C(0)e2n • a
(4)
to accommodate the probable scenario that the degree
of correlated mortality (C) is likely to diminish with
age, especially for juveniles in a brood. Our survivorship model predicts a probability, p•[s•(a 1 i)], of observing s•(a 1 i) of n•(a) individuals surviving a time
interval i. For illustration, we draw attention to the
graphic examples (Fig. 2) illustrating random and overdispersed survivorship outcomes.
Recognizing that even a determined observer is unlikely to witness many clutches hatching, we realized
that some of the robustness of our model would rest
with its ability to accept data for clutches and broods
lacking observations of the number of eggs or juveniles
alive on hatch day. Our model was therefore constructed to calculate the expected number of eggs hatched
(EN,b[0]) during the clutch to brood transition (NzB) as
a probabilistic mixture of RzR, RzC, CzR, and CzC survivorships.
Once the probabilities of observing any outcome s•(a
1 i) have been defined, we can calculate the negative
ln-likelihood of each possible outcome for each data
record r using
lFr[s•(a1i)] 5 22ln[p•[s•(a 1 i)]]
(5)
where Fr[s•(a 1 i)] 5 1 if the outcome s•(a 1 i) for
prediction m•,•(a 1 i) was observed, else Fr[s•(a 1 i)]
5 0. We include the factor 2 to make Eq. 5 equivalent
to the G statistic for evaluation using likelihood ratio
tests (Burnham and Anderson 2002). The likelihood of
the model is therefore,
L[s• (a 1 i )] 5
O O
R
n • (a)
r51 s • (a1i)50
l Fr[s•(a1i)] 3 Fr [s• (a 1 i )]
(6)
where R is the number of observations of Fr[s•(a 1 i)]
5 1. The maximum-likelihood parameter estimates and
their covariance matrix are obtained when L is minimized (LMIN). The quality of model fit (GOF) was diagnosed by parametric bootstraps, which also yielded
confidence limits for the derived survival estimates and
an a posteriori estimate of overdispersion ĉ (White and
Burnham 1999).
HYPOTHESES, DATA PREPARATION, UTILE METRICS
Our purpose is to report on two hypotheses concerning survivorship to fledging of Barrow’s Goldeneye juveniles, primarily to illustrate our model. However, our results have implications both for Barrow’s
Goldeneye conservation, and our understanding of the
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BARRY D. SMITH ET AL.
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FIG. 3. Typical (a) observed, (b) predicted,
and (c) simulated profiles of the observed number of eggs (brood ages a , 0) in a clutch and
juveniles (brood ages a $ 0) in a brood. Clutches and broods for each tending hen are connected by gray lines. The observed data (a) are
those for the years 1995 and 1997–1999. These
data were analyzed to investigate Hypothesis I
(R 5 1090). It is evident that many more broods
were observed than clutches. Note that the observed data are truncated at a brood age of 56
days corresponding to the analyst’s choice of D
5 56 for age at fledging. The model predictions
(b) are those provided by Model 1 of Table 1.
Likewise, the simulated data (c) are one realization using the maximum-likelihood estimated
parameters of Model 1 of Table 1.
fitness implications of the reproductive behaviors of
clutch parasitism and brood amalgamation. Hypothesis
I concerns possible differences in the probability of
surviving to fledge among juveniles reared in broods
of different sizes, as measured or inferred on the day
the eggs hatched (hatch day). Hypothesis II concerns
possible differences in the probability of surviving to
fledge among juveniles reared on ponds with differing
productivities, as measured by estimates of invertebrate
biomass (Evans 2003). Invertebrate biomass (milligrams per sample) was estimated from benthic core
samples and pelagic activity traps collected among 20
ponds in 1995 and 1997 to 1999 a priori qualitatively
judged to be of low, medium, and high invertebrate
productivity (Evans 2003). An estimated interannual
correlation of 93% among ponds supported that this
measure had merit as a reliable index of pond productivity. Invertebrate biomass varied by roughly an order
of magnitude among the ponds sampled, all of which
were observed to support Barrow’s Goldeneye broods
in at least one of the years sampled.
We had available for analysis a set of observations
of the number of eggs in a clutch and juveniles in a
brood for individually followed families (Fig. 3a). Off-
spring associated with an adult tending hen, identified
by her unique nasal disc pairing, allowed each egg or
juvenile observed to be assigned to a specific hen.
However, clutches may have been parasitized, so we
generally did not know if a family was composed of
eggs from more than one hen. Typically broods were
observed and counted every 2–5 days, but sometimes
more or less frequently. Clutches were observed much
less frequently than broods. The calendar date ( t) of all
observations was recorded and used to calculate clutch
and brood ages. If clutches were not observed at, or
just before, hatch, as was typically the case, calendar
hatch date was usually inferred from the observed stage
of juvenile development when broods were first observed on a pond (Gollop and Marshall 1954). Our
analyzed dataset included egg counts only for dates on
or after the date the maximum number of eggs in a
cavity was observed. Our dataset did not include broods
that we knew underwent brood amalgamation or for
which hatch date, and therefore clutch and brood age,
could not be confidently calculated. Further, observations of clutches outside the age range I # a, a 1 i #
D were excluded from our dataset. Within the subset
of data that qualified for analysis (Fig. 3a), a few fam-
February 2005
CLUTCH AND BROOD SURVIVORSHIP
ilies were first followed as clutches, while most were
not followed until they were first seen as broods on a
pond. We chose I 5 240 days and D 5 56 days for
the analyses we present. We also clarify that for Barrow’s Goldeneye I refers to the age the tending hen
began to incubate her full clutch in order to assure
synchronous hatching. Egg laying for any hen will have
taken place over several days. Fewer data records qualified for investigating Hypothesis II (R 5 659) than
for Hypothesis I (R 5 1090) since Hypothesis I could
use data from families on ponds for which there was
no estimate of pond productivity.
For an accepted model fit, we consider three metrics
to be of special interest to many analysts, and appear
in our results. One is the probability, at age a, that a
juvenile will fledge at age D. For hatch day (i.e., a 5
0), AB,•(0) and AB(0) represent survivorships to age a
with their corresponding vulnerabilities to random [1
2 v(0)] or correlated mortalities [v(0)],
p[fledge(0, D )]
5
[1 2 v (0)]AB,R (0)m B,R (D ) 1 v (0)AB,C (0)m B,C (D )
.
AB (0)
(7)
This metric has particular utility for expressing the relative effect of model covariates on a juvenile’s propensity to fledge.
A second metric is expected brood size on hatch day,
EN,b[0]. This metric provides an estimate of the number
of juveniles alive in brood b on hatch day when there
is at least one observation of the number of eggs alive
prior to hatch. In this study we use EN,b[0] as an intrinsic
covariate to investigate Hypothesis I. It has particular
value in that it mitigates an observer’s inability to count
the number of juveniles in a nest on hatch day. It is
worth noting that for some interpretations EN,b[0] might
be considered a better metric than an actual count of
juveniles on hatch day if the analyst’s purpose is to
infer a hen’s intended initial brood size; i.e., analyses
drawing fitness interpretations; however, the two metrics will tend to be very highly correlated.
Lastly, we present a measure of dispersion more intuitive than u2•,•, specifically
EIU 5 1 1 u2•,•(a) 3 (n•(a) 2 1).
(8)
This metric calculates the ‘‘effective independent unit’’
(EIU), a statistical measure of the number of individual
eggs or juveniles that tend to associate as a single mortality event such that the hypothetical outcomes of such
mortality events would follow a binomial distribution.
An EIU value of, say 2.3, for juveniles might be interpreted to mean that a predator tends to take, on average,
2.3 juveniles per mortality event. This metric has proven
informative in other sampling applications where individual birds within a flock do not associate independently
(Iverson et al. 2004). Conversely, when u2•,• . 0 the ‘‘ef-
287
fective independent sample size’’ (EISS) for a clutch or
brood observation is reduced from n•(a) to
EISS 5
n• (a)
.
1 1 u 2• , • (a) 3 [n• (a) 2 1]
(9)
When interpreting our results we entertained both
the information-theoretic and hypothesis-testing paradigms for adjudicating our two key hypotheses and
their alternatives (Anderson et al. 2000, Burnham and
Anderson 2002). We first used AICC to assess the
weight of evidence in our data for each hypothesis
(model ranking). We then used likelihood ratio tests to
execute probabilistic comparisons among paired competing models differing only in a single main effect.
RESULTS
Competitive model trials to investigate Hypotheses
I and II using our data from all ponds produced a distinct ranking of models (Tables 1 and 2). The highest
ranked models for both hypotheses narrowly passed
parametrically bootstrapped GOF diagnostics of model
adequacy (p 6 1 SE 5 0.03 6 0.02 for Hypothesis I;
p 6 1 SE 5 0.06 6 0.02 for Hypothesis II). More
satisfying values for p could have been obtained had
we chosen to remove a few outlier data points that
contributed disproportionately to model deviance
(LMIN). However, we had confidence that our relatively
large number of data records (R) effectively neutralized any bias from these outliers. Our choice not to
censor outliers resulted also in bootstrapped estimates
of ĉ 6 1 SE slightly greater than unity, at 1.08 6 0.04
and 1.06 6 0.05 for the best ranked models (Model 1)
for Hypotheses I and II, respectively. The quality of
the fit for Model 1 concerning Hypothesis I can be
visualized in the survivorship predictions (Fig. 3b) and
by comparing the observed data (Fig. 3a) with a simulated realization (Fig. 3c) generated using the maximum-likelihood parameter estimates for Model 1.
With respect to Hypothesis I, the second highest
ranked model, Model 2 (ignoring Model 1 with function u2B,C(a) for the moment), strongly supports a parametrically and statistically strong relationship between
the probability, on hatch day, that a juvenile will fledge
at age D 5 56 days, p[fledge (0, D)], and expected
brood size on hatch day, EN,b[0]. Model 2 is an ø500
times more probable fit to our data than its direct competitor, Model 6 (Pair A in Table 1, Fig. 4), lacking
EN,b[0] as a covariate. A likelihood ratio test favored
Model 2 (p[Model 2 [ Model 6] 5 0.0004, DLMIN 5
20.53, df 5 4). Model 2 also identifies strong yeareffects, with the effect of EN,b[0] varying among years
to the extent that little effect is evident in 1997, while
in other years there is a distinct tendency for p[fledge
(0, D)] to increase as EN,b[0] increases. Model 2, with
year-effects, is ø104 times a more probable fit to our
data than its competitor, Model 7, that lacks yeareffects (Pair B in Table 1). A likelihood ratio test sig-
Ecological Applications
Vol. 15, No. 1
BARRY D. SMITH ET AL.
288
TABLE 1. Models ranked by increasing AICc, and associated statistics for Hypothesis I: Are there differences in the probability
of surviving to fledge among juveniles reared in broods of different sizes as measured or inferred on their hatch day?
Model
Pair
1
2
3
4
5
6
7
8
9
10
C
A, B, C
A
B
Model description
NULL
NULL
NULL
NULL
NULL
NULL
NULL
NULL
NULL
NULL
1 YEARR 1 EN,b[0]RC with u2B,C(a)
1 YEARR 1 EN,b[0]RC
1 YEARR 1 AGER 1 EN,b[0]RC
1 YEARR with u2B,C (a)
1 YEARR 1 AGER
1 YEARR
1 EN,b[0]RC
1 AGER 1 EN,b[0]RC
1 AGER
(EC, JRC)
DAICc
K
w
LMIN
0.00
14.40
16.26
21.02
24.10
26.91
32.64
34.29
36.86
42.30
14
13
14
10
10
9
10
11
7
6
0.99
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1522.67
1539.09
1538.93
1551.72
1554.80
1559.62
1563.34
1562.99
1573.57
1581.02
Notes: The baseline NULL (EC, JRC) model identifies a constant mortality rate with a correlated mortality (C) process for
both eggs (E) and juveniles (J) and, in the case of juveniles, also statistically identifies a random mortality process (R). The
following symbols represent those covariates challenged with explaining our clutch and brood data: YEAR, among-year
differences in juvenile survivorship; AGE, age-dependent differences in juvenile survivorship; EN,b[0], juvenile survivorship
differences with expected brood size on hatch day; PP, juvenile survivorship differences among ponds with different invertebrate productivities. The symbol u2B,C (a) indicates that the degree of correlated mortality among juveniles can diminish with
brood age. ‘‘Pair’’ identifies, using shared characters, paired rank comparisons referred to in the text; blank cells in this
column indicate that the model was not paired with another model; K represents the number of estimated parameters; w
represents AICc weights; and LMIN is the model deviance. Models are described by their ln-linear additive covariates that
operate on the Weibull parameters a and b.
Hypothesis I model fit summaries and sampling statistics: number of clutches plus broods, 117; number of eggs, 990;
number of juveniles, 7707; number of clutch predictions made and evaluated, 45; number of brood predictions made and
evaluated, 894. The bootstrapped estimates of ĉ 6 1 SE for the null and best models are 1.07 6 0.04 and 1.08 6 0.04,
respectively, indicating minimal overdispersion. The best model (AICC 5 1550.73) passed the parametric bootstrap diagnostic
for GOF.
nificantly favors Model 2 (p[Model 2 [ Model 7] ,
0.0001, DLMIN 5 24.25, df 5 3).
Competitive model trials to investigate Hypothesis
II using our data from those fewer ponds for which we
had covariate data on pond productivity also produced
a distinct ranking of models (Table 2). As for the original dataset used to investigate Hypothesis I, Model 3
investigating Hypothesis II also strongly supported a
positive relationship between p[fledge (0, D)] and
EN,b[0], again with year-effects (Fig. 5a), though the
statistical strength of the relationship is weaker due to
the smaller dataset. Indeed, Model 3 excluded pond
productivity as a covariate, indicating insufficient statistical support for the hypothesis that, among the
ponds sampled, p[fledge (0, D)] is influenced by pond
productivity. The direct competitor of Model 3, Model
5 (Pair D in Table 2), was approximately five times
poorer at explaining our data than was Model 3. Model
7, which included pond productivity, but not EN,b[0],
as a covariate, ranked poorly as a putative model to
explain our data, though there is a slight tendency for
the p[fledge (0, D)] to increase with pond productivity
in years other than 1997 (Fig. 5b).
The best ranked models investigating Hypotheses I
and II include the function u2B,C(a) (Eq. 4) with nB . 0,
indicating that the degree of correlated mortality among
juveniles (and the EIU, Fig. 6a) diminished with brood
age as, incidentally, did their vulnerability to correlated
mortality (Fig. 6b). The models that included nB . 0
were ø1300 and 14 times more probable than their
competitors with nB 5 0, for Hypotheses I (Pair C in
Table 1) and II (Pair E in Table 2), respectively. Like-
TABLE 2. Models ranked by increasing AICc, and associated statistics for Hypothesis II: Are there differences in the
probability of surviving to fledge among juveniles reared on ponds with different invertebrate productivities as measured
by estimates of invertebrate biomass on selected ponds?
Model
1
2
3
4
5
6
7
8
Pair
E
D, E
D
Model description
NULL
NULL
NULL
NULL
NULL
NULL
NULL
NULL
1 YEARR
1 YEARR
1 YEARR
1 YEARR
1 YEARR
1 YEARR
1 YEARR
(EC, JRC)
1
1
1
1
1
1
1
EN,b[0]RC with u2B,C(a)
PPR 1 EN,b[0]RC with u2B,C(a)
EN,b[0]RC
AGER 1 EN,b[0]RC
PPR 1 EN,b[0]RC
PPR
PPR with u2B,C(a)
DAICc
K
w
LMIN
0.00
3.50
5.29
7.13
8.52
20.87
21.40
24.47
14
16
13
14
15
11
12
6
0.77
0.13
0.06
0.02
0.01
0.00
0.00
0.00
849.05
848.53
856.35
856.18
855.56
875.95
874.47
889.59
Notes: Definitions are as in Table 1. Hypothesis II model fit summaries and sampling statistics: number of clutches plus
broods, 61; number of eggs, 574; number of juveniles, 5408; number of clutch predictions made and evaluated, 39; number
of brood predictions made and evaluated, 536. The bootstrapped estimates of ĉ 6 1 SE for the null and best models are 1.05
6 0.05 and 1.06 6 0.05, respectively, indicating minimal overdisperison. The best model (AICc 5 877.14) passed the
parametric bootstrap diagnostic for GOF.
CLUTCH AND BROOD SURVIVORSHIP
February 2005
289
FIG. 4. The probability (with 95% confidence intervals), on hatch day, that a juvenile
Barrow’s Goldeneye will fledge at D 5 56 days,
p[fledge (0, D)], as a function of expected brood
size on hatch day, EN,b[0]. The values portrayed
are those reported by the highest ranked model
(Model 1) of those used to investigate Hypothesis I (Table 1). Year effects are clearly evident.
There is no evidence that p[fledge (0, D)] is
influenced by EN,b[0] in 1997, whereas in other
years there is a clear tendency for juveniles
hatched into larger broods to have an increased
p[fledge (0, D)].
lihood ratio tests affirmed the statistical contribution
of nB . 0 to model fit (Hypothesis I: p[nB 5 0] ,
0.0001, DLMIN 5 16.41, df 5 1; Hypothesis II: p[nB 5
0] 5 0.007, DLMIN 5 7.30, df 5 1). This was anticipated
since juveniles would be expected to behave more independently of their siblings as they aged, thereby lessening group vulnerability to predation or weather
threats. The inclusion of u2B,C(a) in all competitive model pairs significantly improved the fit of these models
but did not change the relative ranking of models based
on the covariates of age, year, EN,b[0], or pond productivity.
For neither Hypotheses I nor II was there statistical
evidence of an age-effect on juvenile survivorship independent of any putative covariates, i.e., no support
for either bB,R ± 1 or bB,C ± 1. Nevertheless, our highest
ranked models for both hypotheses (Model 1) included
intrinsic brood-effect parameters operating on bB,R and
bB,C, respectively, such that b9B,R,b ± 1 and b9B,R,b ± 1.
Thus an effect of EN,b[0] was to change daily survivorship with age among broods. The tendency was for
young broods with higher values for EN,b[0] to experience higher survivorships early in life (Fig. 7), which
eventually resulted in a higher overall p[fledge (0, D)]
for those broods.
Finally, Model 1 concerning Hypothesis II afforded
us an opportunity to look for a relationship between
EN,b[0] and pond productivity for those clutches and
broods for which we had adjunct data on pond productivity. However, we found no evidence that the
EN,b[0] for Barrow’s Goldeneye hens using particular
ponds may be determined in part by the pond’s productivity. Such evidence would support an hypothesis
that hens obtain some nutrition for egg production exogenously once having arrived on the breeding
grounds.
DISCUSSION
Our clutch and brood survivorship model successfully evaluated two key hypotheses concerning the
breeding success of Barrow’s Goldeneye in British Columbia. We confidently concluded that the probability
FIG. 5. (a) The probability (with 95% confidence intervals), on hatch day, that a juvenile Barrow’s Goldeneye will
fledge at D 5 56 days, p[fledge (0, D)], as a function of
expected brood size on hatch day, EN,b[0]. The values portrayed are those reported by the highest ranked model (Model
2) of those models used to investigate Hypothesis II that
include pond productivity as a covariate (Table 2). This result
is similar to that portrayed in Fig. 4, which is based on a
larger sample size. (b) The p[fledge (0, D)] (with 95% confidence intervals) as a function of standard deviates of pond
productivity measured as mean invertebrate biomass per
pond-year (milligrams per sample). The values portrayed are
those reported by Model 7 of those used to investigate Hypothesis II. The results indicate both parametrically and statistically weak evidence for the p[fledge (0, D)] to be higher
on the more productive ponds. The inadequacy of this relationship is emphasized by the very low rank of this model
when compared to models including EN,b[0] as a covariate.
290
BARRY D. SMITH ET AL.
Ecological Applications
Vol. 15, No. 1
FIG. 6. (a) The predicted proportion of juveniles alive at the plotted brood age that are
vulnerable to a correlated mortality process.
This proportion will diminish with brood age
when fB $ 0, as in this result for Model 1, challenging Hypothesis I. (b) The effective independent unit (EIU) vs. brood age. EIU will diminish with nB(a) as well as with brood age
when nB $ 0 as in this result for Model 1, challenging Hypothesis I. The scatter within and
among years for both panels (a) and (b) arises
from differences among broods in their expected brood size on hatch day, EN,b[0] and, for
panel (b), also from differences in the number
of juveniles alive at brood age a, nB(a).
that a juvenile would survive to fledge was positively
related to brood size on hatch day in some years, but
found little support for the argument that pond productivity affected juvenile survival. More generally, we
think this demonstration of our model introduces an-
FIG. 7. Daily survivorship (with 95% confidence intervals) of juveniles during the first day after hatch, and according to year, vs. expected brood size on hatch day, EN,b[0],
for Model 1, challenging Hypothesis I.
other robust analytical tool for investigating environmental effects (e.g., pesticides, predation, habitat alterations, weather, etc.) on the reproductive success of
birds, or for providing high-quality parameter estimates
and a measure of their uncertainty for inclusion in population viability (PVA) or similar analyses. With respect to similar analyses, we have used our model successfully on a previously published dataset of our colleagues (Gill et al. 2000, 2003) to challenge the null
hypothesis that pesticides do not affect the reproductive
success of American Robins (Turdus migratorius) nesting in fruit orchards of the Okanagan Valley, British
Columbia. As we expected, we found no detectable
effect of pesticides on reproductive success in accordance with the authors’ original interpretations using
the Mayfield method (Mayfield 1961, 1975) and Program MARK’s nest survival model (White and Burnham 1999, Dinsmore et al. 2002). The reason for our
expectation arises from our recognition that overdispersion in a dataset acts to reduce the effective independent sample size (EISS, Eq. 9) and thus appropriately decreases the power to falsely detect a significant
effect. That is, our model reduces the probability of
February 2005
CLUTCH AND BROOD SURVIVORSHIP
making a Type II error (Walpole et al. 1998) when
survivorship outcomes are not independent. A corollary to this benefit of our model is that analyses that
do not explicitly account for overdispersion run a higher risk of falsely detecting statistical correlations,
which can ultimately lead to fictitious interpretations
of cause and effect.
Readers may have perceived that our model is not
limited in application to demographic analyses of bird
reproduction, but can be applied to any species where
an interpretation of its reproductive life history is analogous to that of birds, e.g., egg-laying reptiles. Indeed,
when there is no need to model the clutch-to-brood
transition, our model can be applied to any species
where an integer number of offspring in a brood can
be accurately counted over time, there is a desire to
explicitly account for overdispersion, and the model’s
caveats and assumptions stated in the Introduction are
acceptable to the analyst.
We illustrated our model using data on Barrow’s
Goldeneye clutch and brood survivorship to challenge
two hypotheses. (Incidentally, in preliminary analyses
we found no support for the hypothesis that juvenile
survivorship was influenced by hatch day of the year,
t). Our modeling results clearly supported that a juvenile’s probability of surviving to fledge at 56 days
increased with its expected brood size on hatch day in
some years. This finding supports the life history argument that conspecific clutch parasitism has a fitness
advantage for the juveniles (Eadie and Lyon 1998, Eadie et al. 1998, Lyon and Eadie 2000) and perhaps also
for the recipient hen (Eadie and Lumsden 1985, Eadie
et al. 1988). The juveniles of both the tending hen, and
the hen that deposited her eggs in that tending hen’s
nest, are conferred a survivorship advantage by having
their offspring as members of larger broods. However,
this interpretation must be tempered by the realization
that the tending hen is probably not indifferent to the
parentage of the brood she is tending. There is evidence
in Common Eiders (Somateria mollissima) that a tending hen, or her ducklings, may act to preferentially
increase their fitness over that of the other ducklings
in amalgamated broods (Öst and Bäck 2003), a socalled ‘‘selfish herd’’ behavior (Hamilton 1971, Eadie
et al. 1988). We point out that we did not have information on which, if any, of the broods in our analysis
were formed through clutch parasitism, but this seems
certain to be true for the largest of broods (i.e., those
with brood sizes on hatch day of 20–25 juveniles
[Evans et al. 2002; J.-P. Savard, personal communication]). Likewise, we did not follow the survivorship
of broods that were observed to increase in size by
brood amalgamation. However, our interpretations of
a higher probability of surviving to fledge in larger
broods endorses the fitness value of brood amalgamation (Savard 1987).
A conservation interpretation of increased juvenile
survival in larger broods is that increasing the size of
291
broods in a region, such as the Riske Creek region of
our study, appears to be a conservation option if survival to fledge is considered to limit population growth.
Thus our results add another question to conservation
planning. That is, what is the trade-off between providing nest boxes to increase the number of Barrow’s
Goldeneye nesting opportunities in underutilized
ponds, vs. increasing the survivorship of offspring in
currently used ponds? The answer is not evident with
our current knowledge. However, Barrow’s Goldeneye
have invested in the life history fitness option of relinquishing offspring to the care of another, perhaps
more established or closely related (Andersson and Åhlund 2000, Lyon and Eadie 2000) hen. This suggests
that this option might be preferable to a hen raising
her own offspring in a more risky habitat, perhaps despite nesting opportunities provided by artificial nest
boxes. Though nest boxes have proven successful,
large-mammal (e.g., bears) and small-mammal (e.g.,
squirrels) predation can defeat their efficacy (Evans et
al. 2002), perhaps more so in less preferred habitat.
However, our study supplements the findings of Evans
et al. (2002), which demonstrate a significantly increased clutch size for nest boxes over natural cavities.
Notwithstanding unconsidered factors, our results imply that such increases in clutch size can disproportionately increase the expected number of juveniles
fledged.
Had our data supported a positive relationship between pond productivity and the probability of juveniles surviving to fledge, we would have been able to
provide guidance as to which ponds would have the
highest priority for nest boxes. Unfortunately, we found
no convincing evidence of such a relationship, possibly
because there was insufficient contrast in pond productivity, with no pond having a productivity below a
critical threshold affecting juvenile survival. Supporting this interpretation of adequate productivity, we also
found no evidence of a relationship between expected
brood size on hatch day and pond productivity, given
that it has recently been established that Barrow’s Goldeneye hens from the Riske Creek region acquire the
vast majority of their nutrition for egg development
locally (Hobson et al. 2004). Our failure to detect such
a relationship must be interpreted with the understanding that only ponds that supported at least one brood
were included for consideration in this analysis. Clearly
ponds depauperate of prey biomass would be poor
choices for brood rearing. More positively, there appears to be a considerable range of pond productivities
that support successful rearing of Barrow’s Goldeneye
broods.
We conclude by emphasizing the key contributions
of our model for advancing our understanding of the
dynamics of reproduction in birds and perhaps other
egg-laying species. Principally, we provide a method
and model application for measuring and statistically
evaluating survivorship during the critical life history
Ecological Applications
Vol. 15, No. 1
BARRY D. SMITH ET AL.
292
phase of egg-laying to fledging. We particularly want
to emphasize two elements of our modeling approach.
First, we demonstrate the utility of our model for statistically discriminating between random and correlated mortality events. We think this is a key advance that
reinforces the need for demographic models, including
population viability models, to strive for realism concerning survivorship dynamics. Second, our emphasis
on overdispersion (correlated mortality) reinforces that
mortality events are unlikely to be random events, particularly in young broods, and indeed may be fully
correlated, i.e., catastrophic. We implore investigators
to recognize this potential feature of brood survivorship
when they draw statistical inferences from their similar
data. To that end we have also introduced the concept
of the effective independent sample size (EISS, Eq. 9;
see Hypotheses, data preparation, utile metrics), which
we trust will motivate readers to take heed of the potential for nonindependence of individual mortalities.
Finally, despite the benefits of our statistical modeling approach for the hypotheses challenged here,
there potentially remain with our model the same subtle
suite of biases that can also plague studies relying on
the more traditional Mayfield (Mayfield 1961, 1975)
and Kaplan-Meier (Kaplan and Meier 1958), or the
more contemporary Program MARK (White and Burnham 1999) methodologies. Since we can only draw
statistical interpretations from the data we collected,
clutches or broods that failed before they were witnessed by an observer introduce interpretive biases to
which a researcher must be astute. We consider such
biases in our particular study to be minimal because of
the dutiful nature of data collection and the easily observed brood rearing by Barrow’s Goldeneye hens. Our
most overt bias is our compulsory selection only of
ponds supporting broods for challenging Hypothesis II.
As with all modeling interpretations, our ultimate conclusions are conditional upon the constraints that determined what data were collected and the circumstances under which they were collected.
ACKNOWLEDGMENTS
We thank David Green and Brent Gurd of the Centre for
Wildlife Ecology at Simon Fraser University for constructive
reviews prior to submission, and Steve Dinsmore and an
anonymous reviewer for astute and helpful formal reviews
upon submission. This work was motivated and influenced
in part by the team of authors responsible for the development, implementation, and wise use of Program MARK, specifically, David Anderson, Ken Burnham, Gary White, and
Evan Cooch. The model and manuscript were improved in
content and organization thanks to feedback (after oral presentations) from participants at the North American Sea Duck
Conference and Workshop, 6–10 November 2002, Victoria,
British Columbia, Canada, and the Third North American
Duck Symposium, 5–9 November 2003, Sacramento, California, USA. We particularly thank John Simpson (University
of Guelph, Ontario, Canada) for beta-testing our model application as part of his M.Sc. thesis and providing valuable
feedback for improving the model’s implementation.
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APPENDIX
A detailed mathematical description of the clutch and brood survivorship model associated with the study is available in
ESA’s Electronic Data Archive: Ecological Archives A015-008-A1.
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