U.S. Department of the Interior

U.S. Geological Survey

Open-File Report 2020-1027

Some Approaches to Accounting for Incidental Carcass 

Discoveries in Non-Monitored Years using the Evidence of 

Absence Model
Cover: Wind turbines in Gorgonio Pass Wind Resource Area, California, March 20, 

2019. (Photograph by Manuela Huso, U.S. Geological Survey.)
Some Approaches to Accounting for Incidental 

Carcass Discoveries in Non-Monitored Years using 

the Evidence of Absence Model

by Daniel Dalthorp, Paul Rabie, Manuela Huso, and Andrew Tredennick

Open-File Report 2020-1027

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

DAVID BERNHARDT, Secretary

U.S. Geological Survey

James F. Reilly II, Director

U.S. Geological Survey, Reston, Virginia: 2020

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Suggested citation:

Dalthorp, Daniel, Rabie, Paul, Huso, Manuela, and Tredennick, Andrew, 2020, Some approaches to 

accounting for incidental carcass discoveries in non-monitored years using the Evidence of Absence 

model: U.S. Geological Survey Open-File Report 2020-1027, 24 p., https://doi.org/10.3133/ofr20201027. 

ISSN 2330-1258 (online)
iii

Contents

Executive Summary....................................................................................................................................... 1

Three Methods to Account for Incidentals.............................................................................................. 1

Evaluation of the Methods ..................................................................................................................... 2

Results................................................................................................................................................... 2

Introduction.................................................................................................................................................... 3

Three Approaches for Accounting for Incidental Carcasses .......................................................................... 4

Exclude Incidentals (exI): M* ∼ [ M ∣ gs, xs ] ................................................................................... 4

Truncated Prior (prI)............................................................................................................................ 4

Include Incidentals (inI): M* ∼ � M ∣

∣ g'

, xs + xi �............................................................................. 4

Simulation Study............................................................................................................................................ 5

Results........................................................................................................................................................... 5

Tables.......................................................................................................................................................... 12

Conclusions ................................................................................................................................................. 20

Limitations ................................................................................................................................................... 21

Frequencies, Not Magnitudes .............................................................................................................. 21

Sensitivity and Specificity..................................................................................................................... 21

Alternatives .......................................................................................................................................... 22

References .................................................................................................................................................. 22

Figures

Figure 1. Convergence of estimator coverages when detection probability for incidental carcasses (gi) 

is 0.05 and the user-defined credibility level (α) is 0.1. ................................................................................ 7

Figure 2. Estimator coverages for nominal 90 percent CIs. ................................................................... 8

Figure 3. Estimator coverages for nominal 80 percent CIs................................................................... 10

Figure 4. Estimator coverages for nominal 50 percent CIs. ................................................................. 11

Tables

Table 1. M* ∣ {xs, xi} for all three estimators, with α = 0.5, gi = 0.005............................................ 12

Table 2. M* ∣ {xs, xi} for all three estimators, with α = 0.5, gi = 0.01. .............................................. 13

Table 3. M* ∣ {xs, xi} for all three estimators, with α = 0.5, gi = 0.05. .............................................. 14

Table 4. M* ∣ {xs, xi} for all three estimators, with α = 0.1, gi = 0.005. ............................................ 16

Table 5. M* ∣ {xs, xi} for all three estimators, with α = 0.1, gi = 0.01. .............................................. 18

Table 6. M* ∣ {xs, xi} for all three estimators, with α = 0.1, gi = 0.05. .............................................. 19

Abbreviations

EoA evidence of absence

exI Estimator 1 (“exclude incidentals”)

inI Estimator 3 (“include incidentals”)

ITP incidental take permit

PCM post-construction monitoring

prI Estimator 2 (“truncated prior”)
iv

Variables and Parameters

αα user-defined credibility level (1 − αα)

 CI credible interval (Bayesian analog of a confidence interval)

ggss detection probability for formal PCM search process

ggii detection probability for incidental carcasses

λλ true annual mortality rate (for example, bats per year), Poisson distributed

λλ

^ estimated mortality rate

MM true number of fatalities

MM∗ estimated minimum number of fatalities such that Pr(MM ≤ MM∗) ≥ 1 − αα

xxss number of carcasses found during formal PCM searches

xxii number of incidental carcasses found
1 

Some Approaches to Accounting for Incidental 

Carcass Discoveries in Non-Monitored Years using 

the Evidence of Absence Model

by Daniel Dalthorp1, Paul Rabie2, Manuela Huso1, and Andrew Tredennick2

Executive Summary

We evaluate three approaches to accounting for incidental carcasses when estimating an 

upper bound on total mortality (MM) as MM∗ using the Evidence of Absence model (EoA; Dalthorp 

and others, 2017) to assess compliance with an Incidental Take Permit (ITP) (Dalthorp & Huso, 

2015) under a monitoring protocol that includes formal, dedicated carcass surveys that achieve 

an overall detection probability of ggss = 0.15 in the first year, followed by 4 years with no formal 

monitoring but with carcasses potentially discovered incidentally by operations and maintenance 

crews in their normal course of activity or otherwise discovered outside the formal searches. We 

refer to carcasses discovered incidentally as “incidentals” and define xxii as the count of 

incidentals. Similarly, we define xxss as the number of carcasses found during the formal searches 

conducted the first year. 

Three Methods to Account for Incidentals

The difficulty with incidentals is that they cannot simply be included in the carcass count 

that is used to calculate MM∗ because the detection probability for incidentals is typically not 

known. This presents a statistical puzzle in that we know that the incidentals provide information 

about mortality rates, but it is difficult to quantify that information in the calculation of MM∗. We 

explore three possibilities: 

1. exI: exclude incidentals and calculate MM∗ based solely on carcasses found in the formal

search process (with detection probability ggss);

2. prI: incorporate the information provided by incidentals into a truncated, informed prior

distribution to be used in EoA;

3. inI: include incidentals in the carcass count and calculate MM∗ based on all carcasses found

(xxss + xxii), whether in the formal search process (with detection probability ggss or as

incidentals (with detection probability ggii).

The first method (exI) is the default for EoA and is statistically valid. Carcasses

discovered outside the monitored period or outside the search plots are accounted for by the 

spatial and temporal search coverage components in the EoA model. Although valid, the method 

loses statistical power as the number of unmonitored years accumulates. 

1U.S. Geological Survey, Corvallis, Oregon 

2Western Ecosystems Technology, Inc. (WEST), Cheyenne, Wyoming 
2 

The second method (prI) makes use of some of the information in the incidentals 

without requiring an estimate or assumption about detection probability for incidental carcasses 

(ggii) . It is statistically valid and improves slightly on the statistical power of the first method. 

Also, because it does not require an estimate or assumption about ggii , it avoids the uncertainties 

involved in assuming a fixed value for ggii and the expense of estimating ggii if no fixed 

assumption is made about ggii. 

The third method (inI) is an attempt to take maximal advantage of the information 

implicit in the incidental finds. To do so requires enhancing the information by either assuming a 

fixed value for ggii or by adequately estimating it. If an assumed value for ggii is badly mistaken 

(for example, assuming ggii = 0.05 when it is really 0.005 or vice versa), there is substantial risk 

that MM∗ will be grossly inaccurate. However, if it is known that ggii is small (for example, ggii <

0.01), and ggii is assumed to be small, then misspecification of ggii has little effect on the accuracy 

of inI coverage probabilities. Because information in the incidentals is more fully incorporated 

into estimates, discovery of incidentals in non-monitored years can have a strong impact on MM∗, 

providing greater opportunity to detect high fatality rates that went unnoticed in the first year of 

monitoring but increasing the probability of an MM∗ that is much higher than the true MM. 

Evaluation of the Methods

EoA was designed to calculate the upper bounds (MM∗) of 100(1 − αα)% credibility 

intervals for mortality (MM), which can be interpreted as the maximum number of fatalities that is 

compatible with the data. A total mortality that exceeds MMαα

∗ can be effectively ruled out with 

100(1 − αα)% credibility. The natural test for EoA-type models is on the accuracy of MMαα

∗ , that is, 

whether the probability that MM exceeds MM∗ is close to αα. We call the true probability that MMαα

∗ ≥

MM the “coverage probability” or simply “coverage” and call 1 − αα the nominal or target 

coverage. We conducted simulations to test the coverage probabilities for the three methods 

under a wide variety of scenarios under a general framework of: 

• formal PCM searches conducted in the first year, with detection probability of gs = 0.15; 

• 4 years without formal monitoring

o actual detection probabilities for incidentals ranging from gi = 0.001 to 0.05 

o assumed detection probabilities (inI only) ranging from 0.001 to 0.05 

• actual mortality rates ranging from λ = 0.5/year to 10/year; 

• α values of 0.1, 0.2, and 0.5. 

Results 

We found that coverages for all three estimators (assuming ggii known in inI) converged 

to the target 1 − αα as λλ increased, with rate of convergence dependent on αα. Estimators exI and 

prI had similar coverage in almost all cases, with prI outperforming exI in most of the few 

cases where they differed. Both estimators over-covered (implying a tendency for MM∗ to be too 

large) when the annual take (λλ) was small and credibility level (1 − αα) was large. As λλ increased, 

both estimators converged on nominal coverage at close to the same rate when αα was 0.5. 

When ggii was known, estimator inI had more accurate coverage probabilities than did 

exI and prI in most scenarios. In particular, it had much less of a tendency to overestimate when 

λλ was small and ggii was large. This is because inI directly and efficiently incorporates the 

information from the incidentals into the estimates. However, its advantage appeared to largely 
3 

evaporate when desired credibility was 50 percent, in which case inI represented only a 

marginal improvement over exI or prI. 

The relatively good performance of estimator inI in many of the simulation scenarios 

depends on ggii being known, which is usually an unrealistic assumption. When ggii is misspecified 

or poorly estimated, inI can be strongly biased. Because the process of detecting incidentals is 

difficult to predict, designing and conducting trials to accurately estimate ggii would require 

careful thought. In theory, ggii could be estimated using simplified binomial search trials that do 

not require follow-up searches to monitor the fates of carcasses after initial placement. This 

would involve placement of trial carcasses throughout the monitoring period and using the 

fraction of those carcasses that were discovered incidentally as an estimate of ggii (after 

accounting for search coverage).

Introduction

Wind energy facility operators typically conduct post-construction fatality monitoring 

(PCM) studies at wind energy facilities to evaluate the direct impacts to birds and bats, which 

may collide with wind turbines. Counts of carcasses are analyzed with data from experimental 

trials designed to evaluate detection bias (imperfect searcher efficiency, carcass removal, 

unsearched areas, etc.) to arrive at an estimate of total mortality and mortality rate at a site. When 

the goal is to provide assurance of compliance with an incidental take permit of a rare species 

(for which carcass counts may be low or even zero), the Evidence of Absence (EoA; Dalthorp 

and others, 2017; Dalthorp and Huso, 2015) estimator is the most appropriate statistical estimator 

available. In this context, EoA is not designed to give a point estimate of how many rare 

individuals were killed, but to calculate MM∗ and credible intervals, which are the Bayesian analog 

to the confidence intervals of classical statistics. In practice, MM∗ is often used to determine 

whether projects are in compliance with their incidental take permits (ITPs) and to determine the 

mitigation debits accruing to a permit holder. In mathematical terms, MMαα

∗ is defined as the 

minimum number of fatalities such that Pr(MM ≤ MM∗ ∣ XX = xx; gg = gg�) ≥ 1 − αα for the user’s 

choice of credibility level, 1 − αα where MM is the true mortality of the rare species, xx is the 

number of carcasses found in a search process for which the detection probability has been 

estimated as gg� (Dalthorp and others, 2017). In lay terms, EoA is saying you have a 1 − αα chance 

that true mortality was ≤M * given your search process and what was discovered. We refer to this 

value as MM∗ throughout, including subscripts as needed to differentiate among estimators or to 

distinguish credibility levels.

Finding incidental carcasses (defined as bird or bat carcasses found outside of prescribed 

PCM searches) at a wind energy facility has been the source of discussion regarding how they 

should be treated in the context of EoA. Observations of incidental carcasses are not data in the 

sense that they were not observed as part of the data collection process that is modeled in EoA. 

Indeed, the EoA framework is designed to account, through gg�, for carcasses that likely exist but 

were not found during PCM searches, which would include incidental carcasses. However, when 

the purpose of monitoring is to evaluate compliance with the Endangered Species Act and 

associated ITP and to determine mitigation debits for permit holders, these incidental finds can 

represent valuable information.

The question we address here is: How can incidental carcasses be treated in the context of 

EoA while maintaining the strong statistical properties of the estimator? In this report, we focus 

exclusively on one specific monitoring plan in which PCM monitoring is conducted in the first 

year with an overall detection probability of ggss = 0.15, followed by 4 years in which no formal 
4 

PCM is conducted, but the potential exists for bats to be discovered incidentally by workers at 

the site. We describe three approaches to accounting for incidental carcasses when estimating MM∗

in the EoA framework to assess compliance with an ITP (Dalthorp and Huso, 2015). We 

describe a probability-based simulation we conducted to assess the three approaches for handling 

incidentals in terms of coverage probabilities; that is, the accuracy of the level of credibility 

given in statements that can be made using EoA like “Having searched for Indiana bats with an 

overall detection probability of gg and having found none, we can assert with 50 percent

credibility that no more than five were taken in this period.” We provide results and discussion 

of results and conclude with recommendations on approaches to interpreting incidental finds in a 

management context where the objective is to be reasonably sure—with 100(1 − αα)%

credibility—that permitted take is not being exceeded. 

Three Approaches for Accounting for Incidental Carcasses

Exclude Incidentals (exI): MM∗ ∼ [MM ∣ ggss, xxss]

This is the standard EoA calculation of MM∗. The modeling framework of EoA already 

accounts for carcasses that are missed due to unsearched areas, unsearched periods of time, 

carcass removal, or imperfect searcher efficiency, and that the exI approach (“exclude 

incidentals”) is well-supported by statistical theory. An incidental carcass represents one of those 

missed and can be thought of as accounted for in the estimator. EoA acknowledges that some 

carcasses may be missed in the searches, and an incidental represents serendipitous discovery of 

one that was missed.

Truncated Prior (prI) 

The prI estimator makes use of the information that total mortality must be greater than 

or equal to the number of incidentals found by truncating the lower bound of EoA’s prior 

distribution to exclude the possibility that MM < xxii. Instead of using an integrated binomial 

reference prior (Pr(MM = mm) ∝ √mm + 1 − √mm) that EoA uses as a default, prI uses a prior that 

has probability 0 for MM < xxii and Pr(MM = mm) ∝ 1/√mm (a Jeffreys prior) for mm ≥ xxii. When the 

number of incidentals (xxii) does not exceed the number of carcasses found in formal searches 

(xxss), the prI estimator is virtually indistinguishable from the traditional exI. The differences 

between exI and prI are significant only when xxii > xxss and xxss is small. This situation would be 

expected to be rare except when the detection probability for incidentals is not small in relation 

to ggss. 

Include Incidentals (inI): MM∗ ∼ [MM ∣ gg′, xxss + xxii]

Including incidentals in the carcass survey count requires accounting for the detection 

probability of the incidentals, which we define as carcasses that are discovered outside of 

standardized search plots or in non-monitored years. In either case, ggss in the model should be 

slightly adjusted and replaced with a gg′ that includes a component that reflects detection 

probability of incidentals. If the detection probability for survey carcasses (ggss) is substantially 

higher than detection probability for incidentals (ggii), this approach would be fairly robust to 

misspecification of ggii, and ggii could be assumed to be some fixed value. However, as the nonmonitored years accumulate, the overall proportion of carcasses found in the carcass surveys 
5 

decreases (ggss in year 1, ggss/2 for carcasses through the first two years, etc.), and the sensitivity to 

misspecification of ggii increases. Our simulations include evaluation of effect of misspecification 

of ggii. 

Simulation Study

We assess (via simulation) how the estimators perform under various realistic scenarios 

that are rooted in probability and evaluate their performance in terms of coverage. Particularly, 

we adopt the basic framework of annual mortalities as MM ∼ Poisson(λλ), 1 year of PCM 

monitoring at ggss = 0.15 followed by 1–4 non-monitored years, during which the probability of 

finding incidentals is >0. Scenarios include all combinations of λλ ∈ {0.5, 1, 1.5 , . . . , 10} and 

detection probabilities for incidental carcasses ggii ∈ {0.001, 0.0025, 0.005, 0.01, 0.025, 0.05}. In 

addition, estimator inI is based on an assumed ggii taken from the same set as the actual ggii. Thus, 

for example, in some scenarios inI assumed ggii is correct and in others ggii is mis-specified. The 

other estimators make no assumptions about ggii. In the first year, when the carcass surveys are 

conducted, the detection probability for the carcass surveys is assumed to be ggss = 0.15, and 

carcasses that are not found in the searches are available for detection as incidentals. The 

probability of incidental detections in the first year is likely to be lower than in later years 

because most of the would-be incidental discoveries are made by the search teams instead. We 

assume that the detection for incidentals in the first year is ggii/3. The denominator is arbitrary, 

but the estimator is only weakly dependent on that assumption unless ggii is not small in relation 

to ggss. For inI, we then take the assumed gg to be gg′ = ggss + (1 − ggss)ggii/3 for the first year and 

(ggss + (1 − ggss)ggii/3 + (yy − 1)ggii)/yy for cumulative totals over yy years.

We evaluate the estimators in terms of coverage probabilities of MMαα

∗ for MM with αα ∈

{0.1, 0.2, 0.5}, which reflects the intent and common implementation of EoA as a tool for 

determining compliance. In addition, extensive tables show MM∗ values that may be useful in 

evaluating mitigation obligations. This general framework, which includes models for discovery 

of incidentals, allows for evaluation of the estimators rooted in probability theory. 

Results

Under most of the conditions that we tested, the estimators performed similarly. The 

similarities began to break down as the detection probability for incidentals (ggii) increased to 

0.05, and the distinctions were most pronounced when αα = 0.1. Figure 1 shows coverage 

probabilities for the three estimators when ggii = 0.05, yy = 5, and αα = 0.1 for λλ from 0.5 to λλ = 

60 per year, a mortality rate that is unlikely to be seen in practice but is useful for seeing 

convergence of the estimators to nominal coverage. 

A perfect estimator would cover MM exactly 100(1 − αα)% of the time, but exact coverage 

is rarely possible for discrete distributions, especially when counts are small. Estimator 

overcoverage (above the blue target) means that MM∗ exceeds MM more frequently than expected, 

which means the estimator tends to overestimate for that scenario. Similarly, coverage below the 

target indicates a tendency to underestimate. Lines represent coverage probabilities of MMαα

∗ for 

cumulative mortality through yy years. The degree of overestimation or underestimation in 

particular cases can be seen in tables 1–6. 

Additionally, a ggii of 0.05 is probably an upper bound on what might occur for bats, and 

αα = 0.1 is more stringent than the more commonly used αα = 0.5 or 0.2. Thus, figure 1 represents 

the scenarios with the most extreme divergence among the estimators and serves to highlight the 
6 

qualitative differences in how the estimators make use of the information in the incidentals. 

Quantitative differences among the estimators depend on specific combinations of ggii and αα, as 

illustrated in figures 2–4. 

In all cases, coverage probabilities for exI and prI converged to the target coverage as λλ

increased. Although virtually identical over most of the range of λλ, exI and prI diverge from 

each other in figure 1 when λλ was between, roughly, 8 and 15. Under these conditions, prI gave 

somewhat higher coverages than exI, reflecting a tendency to give higher estimates. This gives 

the prI model slightly greater power than exI to detect large λλ’s in the non-monitored years.

When mortality rates are not very high (λλ < 7 in the figure)—which is likely in 

practice—estimators exI and prI have a strong tendency to overestimate the total mortality over 

5 years, with coverage probability of 100 percent when ggii = 0.05 and αα = 0.1. The problem is 

that the information content in the formal carcass surveys in year 1 is progressively diluted as the 

non-monitored years accrue, and the overall detection probability for carcasses arriving over the 

course of 5 years decreases to 3 percent. After accounting for the uncertainty inherent in such a 

small gg, it becomes difficult to rule out mortality of MM > λλ ⋅ 5 when λλ is not large.

The inI estimator can remedy some of the tendency for exI and prI to overestimate 

under the conditions in figure 1 . With ggii = 0.05 the information content in the incidentals is 

potentially high, but harvesting that information efficiently requires knowing or reliably 

estimating ggii. If ggii is known, inI converges to nominal coverage more rapidly than the other 

two estimators and has substantially less of a tendency toward over-estimation. However, this 

requires that detection probability for incidentals is known or reliably estimated. 

In figures 2–4, coverage probabilities for MMαα

∗ are shown for the three estimators for 0.5 ≤

λλ ≤ 10 and ggii ∈ {0.001, 0.0025, 0.005, 0.01, 0.02, 0.05}. The black and green lines represent 

coverages for exI and prI, resp. The solid red lines represent coverages for inI when the value 

assumed for ggii in inI is correct; broken red lines are for inI when ggii is misspecified. The 

topmost red line is for an assumed ggii of 0.001, and successively lower lines are for ggii = 0.0025, 

0.005, 0.01, 0.02, 0.05, respectively. The horizontal, light blue line represents the nominal, target 

coverage (1 − αα). In many panels, the coverages for one or more of the estimators are nearly 

identical for most, if not all, of the values of λλ. As a result, sometimes one or more lines appear 

to be missing. This is because they are entirely coincident with a line that was drawn later. The 

lines were drawn in order: exI first (black), inI next (red), and prI last (green).
7 

Figure 1. Convergence of estimator coverages when detection probability for incidental carcasses (gi) is 

0.05 and the user-defined credibility level (αα) is 0.1. [Horizontal blue line indicates target coverage.] 

In all cases, estimator exI is independent of the detection probability for incidentals (ggii), 

so the black lines are identical within each column in the panels. When ggii is small (for example,

ggii ≤ 0.01), discovery of incidentals is not likely, so the prior in prI rarely needs adjustment. As 

a result, in the small ggii scenarios, prI and exI are virtually indistinguishable. This is reflected in 

the figures by a virtual absence of a black line for exI, which becomes hidden by the green line 

for prI. Additionally, whenever xxii ≤ xxss the adjustment to the prior in prI has trivial impact on 

MM∗ because the posterior distribution for MM will automatically assign probability 0 to values of 

MM that are <xxss, so the incidentals do not provide information that affects the estimation. In other 

words, truncating the prior when the posterior would perform the same (or a possibly more 

pronounced) truncation has virtually no effect. 

Coverages (nominal 90 percent) for exI and prI are largely indistinguishable (fig. 2) in 

the simulation scenarios except when ggii is relatively large, in which case prI had a greater 

tendency to overestimate than did exI. When λλ > 9, exI had a slight tendency to underestimate. 

In these scenarios where ggii is correctly specified for inI (solid red lines), convergence to 

nominal coverage is faster than it is for the other two estimators, especially when ggii is larger 

(bottom rows of figures). This means that when αα = 0.1 and λλ is small, inI shows much less of 

a tendency to overestimate than do exI and prI. This is because, by assigning a detection 

probability to incidentals and incorporating them into the estimator, inI is able to attain a larger 

gg value than either exI or prI, thereby tending to give smaller MM∗ values, especially when ggii

and λλ are large, in which case the incidentals carry more information, which inI is able to 

leverage into greater accuracy (fig. 2).
8 

Figure 2. Estimator coverages for nominal 90 percent CIs. [Red lines are for Estimator 3 (iiiiii) when

detection probability for incidental carcasses (ggii) is specified as 0.001, 0.0025, 0.005, 0.01, 0.02, or 0.05 

(top to bottom lines). Solid lines are for ggii correctly specified and broken lines for detection probability for 

incidental carcasses ggii misspecified (dashed for assumed ggii ≤ 0.01 and dotted for ggii ≥ 0.02). Blue 

lines indicate the target coverage.] 
9 

When ggii is misspecified, the performance of inI is notably less impressive than when ggii

is correctly assumed. For example, when true ggii is 0.01 (third row of panels in fig. 2) and ggii is 

assumed to be >0.01 (dotted red lines below the solid line), inI has a fairly strong tendency to 

underestimate, and when true ggii = 0.01 and ggii is wrongly assumed to be 0.001 or 0.0025, the 

performance advantage of inI over exI and prI vanishes, as reflected in the dashed red lines in 

the fourth row of panels (fig. 2).

In general, with αα = 0.1, if true ggii is small (ggii ≤ 0.01, top four rows of panels) and ggii is 

assumed to be small (dashed or solid red lines), then inI performs similarly to exI and prI but 

tends to have lower estimates where they differ.

When true ggii = 0.05 but is assumed to be small, inI tends to overestimate, matching or 

even exceeding the tendency of prI to overestimate when λλ is very large (λλ > 7 or 8), and ggii is 

assumed to be very small (ggii ≤ 0.005).

Qualitatively, the same kinds of patterns repeat when αα = 0.2 or αα = 0.5, but the 

nuances differ.
10 

Figure 3. Estimator coverages for nominal 80 percent CIs. [Red lines are for iiiiii when the detection 

probability for incidental carcasses (ggii) is specified as 0.001, 0.0025, 0.005, 0.01, 0.02, or 0.05 (top to 

bottom lines). Solid lines are for ggii correctly specified and broken lines for ggii misspecified (dashed for 

assumed ggii ≤ 0.01 and dotted for ggii ≥ 0.02). Blue lines indicate the target coverage.] 
11 

Figure 4. Estimator coverages for nominal 50 percent CIs. [Red lines are for iiiiii when incidental 

carcasses (ggii) is specified as 0.001, 0.0025, 0.005, 0.01, 0.02, or 0.05 (top to bottom lines). Solid lines are 

for ggii correctly specified, and broken lines for ggii misspecified (dashed for assumed ggii ≤ 0.01 and dotted 

for ggii ≥ 0.02). Blue lines indicate the target coverage.] 

When αα = 0.5 (fig. 4), overall performances of exI and prI are virtually 

indistinguishable. When ggii is correctly specified, inI outperforms both exI and prI, but the 

differences are slight except when ggii = 0.05 and λλ is small.

As when αα = 0.1 or 0.2, misspecification of ggii degrades the performance of estimator 

inI. If true ggii ≤ 0.01 and ggii is likewise assumed to be small, the overall performance of inI is 
12 

comparable to that of exI or prI. However, when ggii is not small (≥ 0.02) but is assumed to be 

small (dashed lines in bottom two rows of panels in fig. 4), inI has a strong tendency to 

overestimate, and when true ggii ≤ 0.01 but is assumed to be not small (ggii ≥ 0.02, dotted red 

lines in top four rows of panels in fig. 4), inI has a strong tendency to underestimate.

Tables

Mortality estimates for all three estimators (MMeeee

∗ , MMeeee∗ , and MMinI

∗ ) as a function of 

specific numbers of carcasses found in regular PCM searches (xxss), incidental carcass discoveries 

(xxii), detection probabilities (ggii), and credibility levels (1 − αα) are listed in the following series 

of tables. Also included in the tables are the probabilities that each given combination of xxss and 

xxii occur, depending on the actual annual fatality rate (λλ = 2, 5, or 10) and number of monitored 

years (1 or 5). Tables with all combinations of ggii ∈ {0.005, 0.01, 0.05} and αα ∈ {0.5, 0.1} are 

presented below. 

The tables are dense with information, and little interpretation is provided. Instead, they 

are provided to allow readers to investigate probabilities and magnitudes of potential estimation 

errors. A discussion of Table 1 provides guidance on interpretation. 

Table 1 shows the MM0.5

∗ values for all three estimators when ggii = 0.005 as a function of xxss

and xxii and the number of years of data. It is assumed that ggii is correctly specified in inI. The 

probabilities of each combination of xxss and xxii are shown for each λλ ∈ {2, 5, 10}. 

Table 1. MM∗ ∣ {xxss, xxii} for all three estimators, with αα = 0.5, ggii = 0.005. 

1 year 5 years

xxss xxii MMeeee

∗ MMpppp∗ MMiiiiii

∗

PPPP(xxss, xxii ∣ λλ,ggii = 00. 000000)

MMeeee

∗ MMpppp∗ MMiiiiii

∗

PPPP(xxss, xxii ∣ λλ,ggii = 00. 000000)

λλ = 22 λλ = λλ = 11 λλ = 22 λλ = λλ = 11

0 0 1 1 1 0.7405 0.4687 0.2195 7 7 6 0.7110 0.4234 0.1798

1 0 8 8 8 0.2208 0.3541 0.3296 39 39 34 0.2123 0.3210 0.2689

2 0 14 14 14 0.0325 0.1300 0.2483 72 72 63 0.0311 0.1183 0.2043

3 0 21 21 21 0.0032 0.0330 0.1249 105 105 92 0.0030 0.0297 0.1024

4 0 27 27 27 0.0002 0.0061 0.0457 139 139 121 0.0002 0.0056 0.0373

5 0 34 34 34 0.0000 0.0009 0.0143 172 172 151 0.0000 0.0008 0.0119

6 0 41 41 40 0.0000 0.0001 0.0035 205 205 180 0.0000 0.0001 0.0028

7 0 47 47 47 0.0000 0.0000 0.0007 239 239 209 0.0000 0.0000 0.0006

8 0 54 54 54 0.0000 0.0000 0.0001 272 272 238 0.0000 0.0000 0.0001

0 1 1 3 8 0.0022 0.0036 0.0029 7 10 34 0.0309 0.0466 0.0382

1 1 8 7 14 0.0006 0.0023 0.0047 39 39 63 0.0088 0.0337 0.0586

2 1 14 14 21 0.0001 0.0009 0.0032 72 72 92 0.0014 0.0119 0.0426

3 1 21 21 27 0.0000 0.0002 0.0017 105 105 121 0.0002 0.0034 0.0218

4 1 27 27 34 0.0000 0.0000 0.0007 139 139 151 0.0000 0.0006 0.0082

5 1 34 34 40 0.0000 0.0000 0.0002 172 172 180 0.0000 0.0001 0.0024

6 1 41 41 47 0.0000 0.0000 0.0000 205 205 209 0.0000 0.0000 0.0006

7 1 47 47 54 0.0000 0.0000 0.0000 239 239 238 0.0000 0.0000 0.0001

0 2 1 4 14 0.0000 0.0000 0.0000 7 13 63 0.0007 0.0023 0.0042

1 2 8 8 21 0.0000 0.0000 0.0000 39 39 92 0.0002 0.0017 0.0064

2 2 14 14 27 0.0000 0.0000 0.0000 72 72 121 0.0000 0.0006 0.0044

3 2 21 21 34 0.0000 0.0000 0.0000 105 105 151 0.0000 0.0001 0.0023

4 2 27 27 40 0.0000 0.0000 0.0000 139 139 180 0.0000 0.0000 0.0009

5 2 34 34 47 0.0000 0.0000 0.0000 172 172 209 0.0000 0.0000 0.0002

0 3 1 5 21 0.0000 0.0000 0.0000 7 14 92 0.0000 0.0001 0.0003
13 

1 3 8 9 27 0.0000 0.0000 0.0000 39 39 121 0.0000 0.0001 0.0004

2 3 14 14 34 0.0000 0.0000 0.0000 72 72 151 0.0000 0.0000 0.0003

3 3 21 21 40 0.0000 0.0000 0.0000 105 105 180 0.0000 0.0000 0.0001

For example, when xxss = 2 and xxii = 0, MM∗ = 14 for all three estimators for 1-year data, 

while MM∗ = 72 for exI and prI and MM∗ = 63 for inI for 5-year data. The estimates are higher 

for 5-year data because they use a smaller realized, cumulative gg to account for mortality in the 

non-monitored years 2–4. The probability of that occurrence (xxss = 2 and xxii = 0 through 5 

years) is 0.0311 when λλ = 2 and 0.1183 when λλ = 5 (Table 1). 

When λλ ≤ 10, MM∗ is the same for all three estimators after the first year over 98 percent

of the time (top 6 rows of left panel in table). The cases where they differ are rare. For example, 

when xxss = 6 and xxii = 0, MMeeee

∗ = MMpppp∗ = 41 and MMiiiiii

∗ = 40; the probability of that result is 

<0.00005 when λλ = 2, 0.00011 when λλ = 5, and 0.0035 when λλ = 10 (Table 1).

When the year of formal monitoring is followed by four non-monitored years, MMeeee

∗ =

MMpppp

∗ in all cases where either xxss > 0 or xxii = 0. These cases account for over 95 percent of the 

total. In other words, exI and prI would be in agreement more than 95 percent of the time when 

λλ ≤ 10 for 5-year data and over 99 percent of the time for 1-year data (assuming ggii = 0.005 and 

αα = 0.5, as in table 2). The only practical distinction between MMeeee

∗ and MMpppp∗ for 5-year data is 

when xxss = 0 and xxii = 1, in which case MMeeee

∗ = 7 and MMpppp∗ = 10. This situation would occur 3–

5 percent of the time for λλ = 2, 5 or 10. All other cases where MMeeee

∗ and MMpppp∗ differ occur with 

frequency less than 1 percent combined (Table 1).

When no incidentals are found over the course of 5 years, MMiiiiii

∗ is consistently less than 

MMeeee

∗ and MMpppp∗ , and 95.7 percent of time inI will have a smaller MM∗ than will prI or exI when 

λλ = 2. When λλ is much greater than 2, the probability of finding incidentals increases 

significantly, as does the probability that MMiiiiii

∗ > MMpppp

∗ . For example, when λλ = 5, the 

probability that MMiiiiii

∗ > MMpppp

∗ is 10.2 percent, and when λλ increases to 10, Pr(MMiiiiii

∗ > MMpppp

∗ )

rises to 19 percent (table 2).

Tables with all combinations of ggii ∈ {0.005, 0.01, 0.05} and αα ∈ {0.5, 0.1} are presented 

below. 

Table 2. MM∗ ∣ {xxss, xxii} for all three estimators, with αα = 0.5, ggii = 0.01. 

1 year 5 years

xxss xxii MMeeee

∗ MMpppp∗ MMiiiiii

∗ PPPP(xxss, xxii ∣ λλ,ggii = 00. 00 ) MMeeee

∗ MMpppp∗ MMiiiiii

∗ PPPP(xxss, xxii ∣ λλ,ggii = 00. 00 )

λλ = 22 λλ = λλ = 11 λλ = 22 λλ = λλ = 11

0 0 1 1 1 0.7359 0.4661 0.2169 7 7 5 0.6794 0.3814 0.1444

1 0 8 8 7 0.2223 0.3491 0.3267 39 39 30 0.2055 0.2868 0.2192

2 0 14 14 14 0.0331 0.1308 0.2431 72 72 56 0.0305 0.1077 0.1625

3 0 21 21 20 0.0030 0.0327 0.1215 105 105 82 0.0027 0.0269 0.0803

4 0 27 27 27 0.0002 0.0061 0.0455 139 139 108 0.0002 0.0050 0.0303

5 0 34 34 34 0.0000 0.0007 0.0139 172 172 134 0.0000 0.0006 0.0090

6 0 41 41 40 0.0000 0.0001 0.0033 205 205 160 0.0000 0.0001 0.0024

7 0 47 47 47 0.0000 0.0000 0.0008 239 239 186 0.0000 0.0000 0.0006

8 0 54 54 53 0.0000 0.0000 0.0001 272 272 211 0.0000 0.0000 0.0001
14 

0 1 1 3 7 0.0040 0.0066 0.0068 7 10 30 0.0581 0.0819 0.0641

1 1 8 7 14 0.0013 0.0054 0.0092 39 39 56 0.0172 0.0609 0.0947

2 1 14 14 20 0.0002 0.0018 0.0069 72 72 82 0.0027 0.0223 0.0709

3 1 21 21 27 0.0000 0.0005 0.0031 105 105 108 0.0003 0.0058 0.0357

4 1 27 27 34 0.0000 0.0001 0.0013 139 139 134 0.0000 0.0011 0.0130

5 1 34 34 40 0.0000 0.0000 0.0005 172 172 160 0.0000 0.0001 0.0045

6 1 41 41 47 0.0000 0.0000 0.0001 205 205 186 0.0000 0.0000 0.0008

7 1 47 47 53 0.0000 0.0000 0.0000 239 239 211 0.0000 0.0000 0.0002

0 2 1 4 14 0.0000 0.0001 0.0001 7 13 56 0.0024 0.0087 0.0133

1 2 8 8 20 0.0000 0.0000 0.0001 39 39 82 0.0008 0.0064 0.0193

2 2 14 14 27 0.0000 0.0000 0.0001 72 72 108 0.0001 0.0024 0.0144

3 2 21 21 34 0.0000 0.0000 0.0000 105 105 134 0.0000 0.0005 0.0074

4 2 27 27 40 0.0000 0.0000 0.0000 139 139 160 0.0000 0.0001 0.0031

5 2 34 34 47 0.0000 0.0000 0.0000 172 172 186 0.0000 0.0000 0.0008

6 2 41 41 53 0.0000 0.0000 0.0000 205 205 211 0.0000 0.0000 0.0002

7 2 47 47 60 0.0000 0.0000 0.0000 239 239 237 0.0000 0.0000 0.0001

0 3 1 5 20 0.0000 0.0000 0.0000 7 14 82 0.0001 0.0006 0.0019

1 3 8 9 27 0.0000 0.0000 0.0000 39 39 108 0.0000 0.0004 0.0025

2 3 14 14 34 0.0000 0.0000 0.0000 72 72 134 0.0000 0.0002 0.0021

3 3 21 21 40 0.0000 0.0000 0.0000 105 105 160 0.0000 0.0000 0.0010

4 3 27 27 47 0.0000 0.0000 0.0000 139 139 186 0.0000 0.0000 0.0004

5 3 34 34 53 0.0000 0.0000 0.0000 172 172 211 0.0000 0.0000 0.0001

0 4 1 7 27 0.0000 0.0000 0.0000 7 16 108 0.0000 0.0000 0.0001

1 4 8 9 34 0.0000 0.0000 0.0000 39 40 134 0.0000 0.0000 0.0004

2 4 14 14 40 0.0000 0.0000 0.0000 72 72 160 0.0000 0.0000 0.0002

3 4 21 21 47 0.0000 0.0000 0.0000 105 105 186 0.0000 0.0000 0.0001

Table 3. MM∗ ∣ {xxss, xxii} for all three estimators, with αα = 0.5, ggii = 0.05. 

1 year 5 years

xxss xxii MMeeee

∗ MMpppp∗ MMiiiiii

∗  PPPP(xxss, xxii ∣ λλ,ggii = 00. 0000) MMeeee

∗ MMpppp∗ MMiiiiii

∗  PPPP(xxss, xxii ∣ λλ,ggii = 00. 0000)

λλ = 22 λλ = λλ = 11 λλ = 22 λλ = λλ = 11

0 0 1 1 1 0.7213 0.4413 0.1941 7 7 3 0.4835 0.1633 0.0256

1 0 8 8 7 0.2156 0.3312 0.2888 39 39 16 0.1438 0.1215 0.0378

2 0 14 14 13 0.0315 0.1224 0.2179 72 72 30 0.0211 0.0458 0.0292

3 0 21 21 19 0.0038 0.0304 0.1101 105 105 43 0.0026 0.0117 0.0144

4 0 27 27 25 0.0002 0.0060 0.0405 139 139 57 0.0001 0.0022 0.0053

5 0 34 34 31 0.0000 0.0008 0.0120 172 172 71 0.0000 0.0003 0.0017

6 0 41 41 37 0.0000 0.0001 0.0030 205 205 84 0.0000 0.0001 0.0004

7 0 47 47 43 0.0000 0.0000 0.0006 239 239 98 0.0000 0.0000 0.0000

8 0 54 54 49 0.0000 0.0000 0.0002 272 272 112 0.0000 0.0000 0.0000

0 1 1 3 7 0.0204 0.0311 0.0273 7 10 16 0.2064 0.1728 0.0563
15 

1 1 8 7 13 0.0059 0.0226 0.0420 39 39 30 0.0628 0.1293 0.0845

2 1 14 14 19 0.0008 0.0087 0.0303 72 72 43 0.0089 0.0480 0.0623

3 1 21 21 25 0.0001 0.0022 0.0157 105 105 57 0.0010 0.0119 0.0323

4 1 27 27 31 0.0000 0.0004 0.0058 139 139 71 0.0000 0.0025 0.0118

5 1 34 34 37 0.0000 0.0001 0.0020 172 172 84 0.0000 0.0003 0.0036

6 1 41 41 43 0.0000 0.0000 0.0005 205 205 98 0.0000 0.0000 0.0010

7 1 47 47 49 0.0000 0.0000 0.0001 239 239 112 0.0000 0.0000 0.0003

8 1 54 54 56 0.0000 0.0000 0.0000 272 272 126 0.0000 0.0000 0.0001

0 2 1 4 13 0.0003 0.0012 0.0020 7 13 30 0.0446 0.0935 0.0606

1 2 8 8 19 0.0001 0.0009 0.0028 39 39 43 0.0129 0.0704 0.0891

2 2 14 14 25 0.0000 0.0003 0.0022 72 72 57 0.0020 0.0254 0.0685

3 2 21 21 31 0.0000 0.0001 0.0011 105 105 71 0.0002 0.0060 0.0347

4 2 27 27 37 0.0000 0.0000 0.0003 139 139 84 0.0000 0.0012 0.0123

5 2 34 34 43 0.0000 0.0000 0.0001 172 172 98 0.0000 0.0002 0.0039

6 2 41 41 49 0.0000 0.0000 0.0000 205 205 112 0.0000 0.0000 0.0010

7 2 47 47 56 0.0000 0.0000 0.0000 239 239 126 0.0000 0.0000 0.0002

8 2 54 54 62 0.0000 0.0000 0.0000 272 272 139 0.0000 0.0000 0.0001

0 3 1 5 19 0.0000 0.0000 0.0001 7 14 43 0.0066 0.0336 0.0426

1 3 8 9 25 0.0000 0.0000 0.0002 39 39 57 0.0018 0.0251 0.0660

2 3 14 14 31 0.0000 0.0000 0.0001 72 72 71 0.0003 0.0090 0.0478

3 3 21 21 37 0.0000 0.0000 0.0001 105 105 84 0.0000 0.0022 0.0243

4 3 27 27 43 0.0000 0.0000 0.0000 139 139 98 0.0000 0.0004 0.0094

5 3 34 34 49 0.0000 0.0000 0.0000 172 172 112 0.0000 0.0000 0.0026

6 3 41 41 56 0.0000 0.0000 0.0000 205 205 126 0.0000 0.0000 0.0006

7 3 47 47 62 0.0000 0.0000 0.0000 239 239 139 0.0000 0.0000 0.0002

0 4 1 7 25 0.0000 0.0000 0.0000 7 16 57 0.0009 0.0083 0.0234

1 4 8 9 31 0.0000 0.0000 0.0000 39 40 71 0.0003 0.0068 0.0342

2 4 14 14 37 0.0000 0.0000 0.0000 72 72 84 0.0000 0.0026 0.0257

3 4 21 21 43 0.0000 0.0000 0.0000 105 105 98 0.0000 0.0006 0.0130

4 4 27 27 49 0.0000 0.0000 0.0000 139 139 112 0.0000 0.0002 0.0050

5 4 34 34 56 0.0000 0.0000 0.0000 172 172 126 0.0000 0.0000 0.0014

6 4 41 41 62 0.0000 0.0000 0.0000 205 205 139 0.0000 0.0000 0.0003

7 4 47 47 68 0.0000 0.0000 0.0000 239 239 153 0.0000 0.0000 0.0001

0 5 1 8 31 0.0000 0.0000 0.0000 7 18 71 0.0000 0.0018 0.0102

1 5 8 10 37 0.0000 0.0000 0.0000 39 40 84 0.0000 0.0014 0.0147

2 5 14 15 43 0.0000 0.0000 0.0000 72 72 98 0.0000 0.0005 0.0115

3 5 21 21 49 0.0000 0.0000 0.0000 105 105 112 0.0000 0.0002 0.0053

4 5 27 27 56 0.0000 0.0000 0.0000 139 139 126 0.0000 0.0000 0.0022

5 5 34 34 62 0.0000 0.0000 0.0000 172 172 139 0.0000 0.0000 0.0005

6 5 41 41 68 0.0000 0.0000 0.0000 205 205 153 0.0000 0.0000 0.0002

0 6 1 9 37 0.0000 0.0000 0.0000 7 19 84 0.0000 0.0003 0.0035

1 6 8 11 43 0.0000 0.0000 0.0000 39 41 98 0.0000 0.0002 0.0053

2 6 14 15 49 0.0000 0.0000 0.0000 72 72 112 0.0000 0.0002 0.0039
16 

3 6 21 21 56 0.0000 0.0000 0.0000 105 105 126 0.0000 0.0000 0.0021

4 6 27 27 62 0.0000 0.0000 0.0000 139 139 139 0.0000 0.0000 0.0006

5 6 34 34 68 0.0000 0.0000 0.0000 172 172 153 0.0000 0.0000 0.0002

0 7 1 10 43 0.0000 0.0000 0.0000 7 20 98 0.0000 0.0001 0.0010

1 7 8 12 49 0.0000 0.0000 0.0000 39 41 112 0.0000 0.0000 0.0017

2 7 14 16 56 0.0000 0.0000 0.0000 72 72 126 0.0000 0.0000 0.0012

3 7 21 21 62 0.0000 0.0000 0.0000 105 105 139 0.0000 0.0000 0.0006

4 7 27 27 68 0.0000 0.0000 0.0000 139 139 153 0.0000 0.0000 0.0002

5 7 34 34 74 0.0000 0.0000 0.0000 172 172 167 0.0000 0.0000 0.0001

0 8 1 11 49 0.0000 0.0000 0.0000 7 22 112 0.0000 0.0000 0.0004

1 8 8 13 56 0.0000 0.0000 0.0000 39 42 126 0.0000 0.0000 0.0003

2 8 14 16 62 0.0000 0.0000 0.0000 72 72 139 0.0000 0.0000 0.0004

3 8 21 21 68 0.0000 0.0000 0.0000 105 105 153 0.0000 0.0000 0.0001

0 9 1 12 56 0.0000 0.0000 0.0000 7 23 126 0.0000 0.0000 0.0001

1 9 8 14 62 0.0000 0.0000 0.0000 39 43 139 0.0000 0.0000 0.0002

3 9 21 22 74 0.0000 0.0000 0.0000 105 105 167 0.0000 0.0000 0.0001

Table 4. MM∗ ∣ {xxss, xxii} for all three estimators, with αα = 0.1, ggii = 0.005. 

1 year 5 years

xxss xxii MMeeee

∗ MMpppp∗ MMiiiiii

∗ PPPP(xxss, xxii ∣ λλ,ggii = 00. 000000) MMeeee

∗ MMpppp∗ MMiiiiii

∗ PPPP(xxss, xxii ∣ λλ,ggii = 00. 000000)

λλ = 22 λλ = λλ = 11 λλ = 22 λλ = λλ = 11

0 0 8 8 8 0.7396 0.4689 0.2202 44 44 38 0.7103 0.4244 0.1808

1 0 20 20 19 0.2212 0.3517 0.3271 103 103 90 0.2129 0.3179 0.2686

2 0 29 29 29 0.0326 0.1317 0.2490 152 152 133 0.0315 0.1193 0.2041

3 0 38 38 38 0.0033 0.0330 0.1239 199 199 174 0.0032 0.0299 0.1016

4 0 47 47 47 0.0002 0.0065 0.0467 243 243 212 0.0002 0.0060 0.0386

5 0 55 55 55 0.0000 0.0010 0.0148 286 286 250 0.0000 0.0009 0.0123

6 0 64 64 63 0.0000 0.0001 0.0036 328 328 287 0.0000 0.0001 0.0029

7 0 72 72 71 0.0000 0.0000 0.0008 369 369 323 0.0000 0.0000 0.0006

8 0 80 80 79 0.0000 0.0000 0.0001 410 410 359 0.0000 0.0000 0.0001

0 1 8 10 19 0.0023 0.0035 0.0032 44 48 90 0.0307 0.0455 0.0380

1 1 20 19 29 0.0006 0.0024 0.0048 103 103 133 0.0086 0.0344 0.0569

2 1 29 29 38 0.0001 0.0008 0.0033 152 152 174 0.0012 0.0126 0.0434

3 1 38 38 47 0.0000 0.0002 0.0017 199 198 212 0.0001 0.0030 0.0216

4 1 47 47 55 0.0000 0.0001 0.0005 243 243 250 0.0000 0.0006 0.0077

5 1 55 55 63 0.0000 0.0000 0.0002 286 286 287 0.0000 0.0001 0.0024

6 1 64 64 71 0.0000 0.0000 0.0001 328 328 323 0.0000 0.0000 0.0007

7 1 72 72 79 0.0000 0.0000 0.0000 369 369 359 0.0000 0.0000 0.0002

0 2 8 12 29 0.0000 0.0000 0.0000 44 52 133 0.0008 0.0025 0.0043

1 2 20 20 38 0.0000 0.0000 0.0000 103 103 174 0.0002 0.0019 0.0059

2 2 29 29 47 0.0000 0.0000 0.0000 152 152 212 0.0000 0.0006 0.0047
17 

3 2 38 38 55 0.0000 0.0000 0.0000 199 198 250 0.0000 0.0002 0.0023

4 2 47 47 63 0.0000 0.0000 0.0000 243 243 287 0.0000 0.0000 0.0008

5 2 55 55 71 0.0000 0.0000 0.0000 286 286 323 0.0000 0.0000 0.0003

6 2 64 64 79 0.0000 0.0000 0.0000 328 328 359 0.0000 0.0000 0.0001

0 3 8 13 38 0.0000 0.0000 0.0000 44 54 174 0.0000 0.0000 0.0003

1 3 20 20 47 0.0000 0.0000 0.0000 103 103 212 0.0000 0.0000 0.0005

2 3 29 29 55 0.0000 0.0000 0.0000 152 152 250 0.0000 0.0000 0.0003

3 3 38 38 63 0.0000 0.0000 0.0000 199 198 287 0.0000 0.0000 0.0002
18 

Table 5. MM∗ ∣ {xxss, xxii} for all three estimators, with αα = 0.1, ggii = 0.01. 

1 year 5 years

xxss xxii MMeeee

∗ MMpppp∗ MMiiiiii

∗  PPPP(xxss, xxii ∣ λλ,ggii = 00. 00 ) MMeeee

∗ MMpppp∗ MMiiiiii

∗  PPPP(xxss, xxii ∣ λλ,ggii = 00. 00 )

λλ = 22 λλ = λλ = 11 λλ = 22 λλ = λλ = 11

0 0 8 8 8 0.7379 0.4640 0.2176 44 44 34 0.6813 0.3786 0.1457

1 0 20 20 19 0.2202 0.3503 0.3270 103 103 80 0.2033 0.2872 0.2196

2 0 29 29 29 0.0324 0.1319 0.2437 152 152 118 0.0302 0.1084 0.1637

3 0 38 38 38 0.0033 0.0326 0.1201 199 199 154 0.0030 0.0267 0.0809

4 0 47 47 46 0.0003 0.0063 0.0456 243 243 188 0.0003 0.0051 0.0306

5 0 55 55 54 0.0000 0.0010 0.0137 286 286 222 0.0000 0.0008 0.0092

6 0 64 64 63 0.0000 0.0001 0.0038 328 328 255 0.0000 0.0001 0.0025

7 0 72 72 71 0.0000 0.0000 0.0008 369 369 287 0.0000 0.0000 0.0005

8 0 80 80 79 0.0000 0.0000 0.0001 410 410 319 0.0000 0.0000 0.0001

0 1 8 10 19 0.0043 0.0067 0.0060 44 48 80 0.0584 0.0825 0.0626

1 1 20 19 29 0.0014 0.0046 0.0094 103 103 118 0.0175 0.0611 0.0933

2 1 29 29 38 0.0001 0.0019 0.0067 152 152 154 0.0023 0.0228 0.0699

3 1 38 38 46 0.0000 0.0005 0.0035 199 198 188 0.0003 0.0058 0.0345

4 1 47 47 54 0.0000 0.0001 0.0013 243 243 222 0.0000 0.0011 0.0130

5 1 55 55 63 0.0000 0.0000 0.0004 286 286 255 0.0000 0.0001 0.0041

6 1 64 64 71 0.0000 0.0000 0.0000 328 328 287 0.0000 0.0000 0.0011

7 1 72 72 79 0.0000 0.0000 0.0000 369 369 319 0.0000 0.0000 0.0002

0 2 8 12 29 0.0000 0.0000 0.0001 44 52 118 0.0025 0.0089 0.0135

1 2 20 20 38 0.0000 0.0000 0.0001 103 103 154 0.0008 0.0063 0.0204

2 2 29 29 46 0.0000 0.0000 0.0002 152 152 188 0.0001 0.0025 0.0148

3 2 38 38 54 0.0000 0.0000 0.0001 199 198 222 0.0000 0.0006 0.0073

4 2 47 47 63 0.0000 0.0000 0.0000 243 243 255 0.0000 0.0001 0.0030

5 2 55 55 71 0.0000 0.0000 0.0000 286 286 287 0.0000 0.0000 0.0007

6 2 64 64 79 0.0000 0.0000 0.0000 328 328 319 0.0000 0.0000 0.0002

7 2 72 72 86 0.0000 0.0000 0.0000 369 369 350 0.0000 0.0000 0.0001

0 3 8 13 38 0.0000 0.0000 0.0000 44 54 154 0.0001 0.0007 0.0018

1 3 20 20 46 0.0000 0.0000 0.0000 103 103 188 0.0000 0.0004 0.0028

2 3 29 29 54 0.0000 0.0000 0.0000 152 152 222 0.0000 0.0001 0.0021

3 3 38 38 63 0.0000 0.0000 0.0000 199 198 255 0.0000 0.0001 0.0008

4 3 47 47 71 0.0000 0.0000 0.0000 243 243 287 0.0000 0.0000 0.0003

5 3 55 55 79 0.0000 0.0000 0.0000 286 286 319 0.0000 0.0000 0.0001

0 4 8 15 46 0.0000 0.0000 0.0000 44 56 188 0.0000 0.0000 0.0002

1 4 20 21 54 0.0000 0.0000 0.0000 103 104 222 0.0000 0.0000 0.0003

2 4 29 29 63 0.0000 0.0000 0.0000 152 152 255 0.0000 0.0000 0.0002

3 4 38 38 71 0.0000 0.0000 0.0000 199 198 287 0.0000 0.0000 0.0001
19 

Table 6. MM∗ ∣ {xxss, xxii} for all three estimators, with αα = 0.1, ggii = 0.05. 

1 year 5 years

xxss xxii MMeeee

∗ MMpppp∗ MMiiiiii

∗  PPPP(xxss, xxii ∣ λλ,ggii = 00. 0000) MMeeee

∗ MMpppp∗ MMiiiiii

∗  PPPP(xxss, xxii ∣ λλ,ggii = 00. 0000)

λλ = 22 λλ = λλ = 11 λλ = 22 λλ = λλ = 11

0 0 8 8 7 0.7181 0.4422 0.1915 44 44 17 0.4808 0.1629 0.0266

1 0 20 20 18 0.2170 0.3298 0.2916 103 103 42 0.1471 0.1224 0.0405

2 0 29 29 27 0.0326 0.1218 0.2176 152 152 62 0.0218 0.0446 0.0283

3 0 38 38 35 0.0034 0.0305 0.1093 199 199 81 0.0023 0.0113 0.0150

4 0 47 47 43 0.0002 0.0058 0.0413 243 243 99 0.0002 0.0021 0.0059

5 0 55 55 50 0.0000 0.0008 0.0120 286 286 117 0.0000 0.0002 0.0017

6 0 64 64 58 0.0000 0.0001 0.0031 328 328 134 0.0000 0.0000 0.0004

7 0 72 72 66 0.0000 0.0000 0.0007 369 369 151 0.0000 0.0000 0.0001

8 0 80 80 73 0.0000 0.0000 0.0001 410 410 168 0.0000 0.0000 0.0000

0 1 8 10 18 0.0211 0.0319 0.0267 44 48 42 0.2066 0.1748 0.0549

1 1 20 19 27 0.0061 0.0229 0.0416 103 103 62 0.0609 0.1284 0.0844

2 1 29 29 35 0.0009 0.0086 0.0316 152 152 81 0.0095 0.0478 0.0633

3 1 38 38 43 0.0001 0.0024 0.0156 199 198 99 0.0010 0.0120 0.0315

4 1 47 47 50 0.0000 0.0005 0.0058 243 243 117 0.0000 0.0022 0.0114

5 1 55 55 58 0.0000 0.0001 0.0018 286 286 134 0.0000 0.0003 0.0036

6 1 64 64 66 0.0000 0.0000 0.0004 328 328 151 0.0000 0.0000 0.0011

7 1 72 72 73 0.0000 0.0000 0.0001 369 369 168 0.0000 0.0000 0.0003

0 2 8 12 27 0.0003 0.0011 0.0021 44 52 62 0.0450 0.0932 0.0600

1 2 20 20 35 0.0001 0.0010 0.0026 103 103 81 0.0132 0.0702 0.0909

2 2 29 29 43 0.0000 0.0003 0.0021 152 152 99 0.0019 0.0257 0.0686

3 2 38 38 50 0.0000 0.0000 0.0011 199 198 117 0.0001 0.0064 0.0331

4 2 47 47 58 0.0000 0.0000 0.0004 243 243 134 0.0000 0.0013 0.0123

5 2 55 55 66 0.0000 0.0000 0.0002 286 286 151 0.0000 0.0002 0.0036

6 2 64 64 73 0.0000 0.0000 0.0000 328 328 168 0.0000 0.0000 0.0009

7 2 72 72 80 0.0000 0.0000 0.0000 369 369 184 0.0000 0.0000 0.0003

0 3 8 13 35 0.0000 0.0000 0.0001 44 54 81 0.0064 0.0330 0.0420

1 3 20 20 43 0.0000 0.0000 0.0002 103 103 99 0.0018 0.0245 0.0628

2 3 29 29 50 0.0000 0.0000 0.0002 152 152 117 0.0003 0.0096 0.0487

3 3 38 38 58 0.0000 0.0000 0.0000 199 198 134 0.0000 0.0024 0.0248

4 3 47 47 66 0.0000 0.0000 0.0000 243 243 151 0.0000 0.0005 0.0094

5 3 55 55 73 0.0000 0.0000 0.0000 286 286 168 0.0000 0.0001 0.0027

6 3 64 64 80 0.0000 0.0000 0.0000 328 328 184 0.0000 0.0000 0.0006

7 3 72 72 87 0.0000 0.0000 0.0000 369 369 201 0.0000 0.0000 0.0001

0 4 8 15 43 0.0000 0.0000 0.0000 44 56 99 0.0008 0.0089 0.0222

1 4 20 21 50 0.0000 0.0000 0.0000 103 104 117 0.0002 0.0065 0.0350

2 4 29 29 58 0.0000 0.0000 0.0000 152 152 134 0.0000 0.0025 0.0253

3 4 38 38 66 0.0000 0.0000 0.0000 199 198 151 0.0000 0.0007 0.0136

4 4 47 47 73 0.0000 0.0000 0.0000 243 243 168 0.0000 0.0001 0.0049
20 

5 4 55 55 80 0.0000 0.0000 0.0000 286 286 184 0.0000 0.0000 0.0015

6 4 64 64 87 0.0000 0.0000 0.0000 328 328 201 0.0000 0.0000 0.0004

7 4 72 72 95 0.0000 0.0000 0.0000 369 369 217 0.0000 0.0000 0.0001

0 5 8 16 50 0.0000 0.0000 0.0000 44 58 117 0.0000 0.0019 0.0096

1 5 20 22 58 0.0000 0.0000 0.0000 103 104 134 0.0000 0.0014 0.0152

2 5 29 30 66 0.0000 0.0000 0.0000 152 152 151 0.0000 0.0005 0.0116

3 5 38 38 73 0.0000 0.0000 0.0000 199 198 168 0.0000 0.0001 0.0054

4 5 47 47 80 0.0000 0.0000 0.0000 243 243 184 0.0000 0.0000 0.0023

5 5 55 55 87 0.0000 0.0000 0.0000 286 286 201 0.0000 0.0000 0.0006

6 5 64 64 95 0.0000 0.0000 0.0000 328 328 217 0.0000 0.0000 0.0001

0 6 8 17 58 0.0000 0.0000 0.0000 44 60 134 0.0000 0.0004 0.0036

1 6 20 23 66 0.0000 0.0000 0.0000 103 104 151 0.0000 0.0002 0.0051

2 6 29 30 73 0.0000 0.0000 0.0000 152 152 168 0.0000 0.0001 0.0040

3 6 38 38 80 0.0000 0.0000 0.0000 199 198 184 0.0000 0.0000 0.0020

4 6 47 47 87 0.0000 0.0000 0.0000 243 243 201 0.0000 0.0000 0.0009

5 6 55 55 95 0.0000 0.0000 0.0000 286 286 217 0.0000 0.0000 0.0002

6 6 64 64 102 0.0000 0.0000 0.0000 328 328 233 0.0000 0.0000 0.0001

0 7 8 18 66 0.0000 0.0000 0.0000 44 62 151 0.0000 0.0001 0.0012

1 7 20 23 73 0.0000 0.0000 0.0000 103 105 168 0.0000 0.0000 0.0017

2 7 29 30 80 0.0000 0.0000 0.0000 152 152 184 0.0000 0.0000 0.0013

3 7 38 38 87 0.0000 0.0000 0.0000 199 198 201 0.0000 0.0000 0.0006

4 7 47 47 95 0.0000 0.0000 0.0000 243 243 217 0.0000 0.0000 0.0003

0 8 8 19 73 0.0000 0.0000 0.0000 44 63 168 0.0000 0.0000 0.0003

1 8 20 24 80 0.0000 0.0000 0.0000 103 105 184 0.0000 0.0000 0.0005

2 8 29 31 87 0.0000 0.0000 0.0000 152 153 201 0.0000 0.0000 0.0003

3 8 38 39 95 0.0000 0.0000 0.0000 199 198 217 0.0000 0.0000 0.0001

4 8 47 47 102 0.0000 0.0000 0.0000 243 243 233 0.0000 0.0000 0.0001

0 9 8 21 80 0.0000 0.0000 0.0000 44 65 184 0.0000 0.0000 0.0001

1 9 20 25 87 0.0000 0.0000 0.0000 103 106 201 0.0000 0.0000 0.0002

2 9 29 31 95 0.0000 0.0000 0.0000 152 153 217 0.0000 0.0000 0.0001

Conclusions

All three estimators are statistically valid in the sense that they are rooted in probability 

theory and do not rely on wildly implausible assumptions. The exI (exclude incidentals) and prI

(truncated prior) estimators produce virtually identical results, except in the somewhat unusual 

situation where xxss = 0 and xxii > 0, in which case MMpppp

∗ tended to be greater than MMeeee∗ . Both 

tended to overestimate when λλ was small and credibility level (1 − αα) was large, and both 

converged on nominal coverage at close to the same rate when αα was 0.5. inI, which includes 

incidentals in the observed carcass count and uses an assumed or estimated ggii, had similar or 

much better coverage than the other two when the detection probability in unmonitored years 

was correctly estimated (or assumed), but had poorer coverage properties when badly 

misspecified. However, if ggii was “small” (≤ 1%) and was assumed small, the coverage 

properties were still better than or comparable to coverages for exI and prI even when ggii was 
21 

misspecified (as, say, ggii assumed to be 1% when it is really 0.1%). These small ggii values may 

not be applicable for eagles or other large raptors, which would require a separate analysis.

The probability of detection (gg) in EoA accounts for carcasses that occur outside of the 

study either in time (non-monitoring years) or space (outside of searched plots), so eeee (exclude 

incidentals) in EoA is valid. Although valid, the statistical power of eeee rapidly declines as an 

estimate of gg is diluted through non-monitored years. Additionally, if incidentals are found but 

are excluded from estimation of total mortality (MM∗), it is possible for the estimate of mortality to 

be less than the number of carcasses actually observed. Information available in incidental 

carcass discoveries can be incorporated into EoA by truncating the lower bound of the prior 

distribution for MM (pppp ) to ensure that MM∗ ≥ xx and to improve on the statistical power of EoA. 

If the detection probability for incidentals (ggii) can be estimated, incorporating ggii into the overall 

estimation of gg and including the incidentals in the carcass count used for calculating MM∗ would 

provide a more complete accounting of the information available in incidentals. This would have 

the effect of increasing the overall gg value, leading to lower mortality estimates in most cases 

and partially correcting for the tendency of eeee and pppp and to overestimate mortality when λλ

is small. However, in the unusual event that λλ is small and incidentals are found in nonmonitored years, iiiiii would substantially overestimate MM. If ggii is badly misspecified or poorly 

estimated, MM∗ may be strongly biased. For larger species like eagles or other large raptors, 

accurate estimation of ggii may be more crucial than it is for bats.

Limitations

Frequencies, Not Magnitudes

This modest simulation study is limited to exploration of basic statistical properties of 

three approaches to accounting for incidentals under some simplified scenarios. The primary 

focus is the coverage probabilities for estimating MM and the frequencies with which the 

estimators over- or under-estimate MM∗. The magnitude of over-estimations is discussed only 

briefly, but the extensive results tables show MM∗ values in a wide array of scenarios. These MM∗

values should not be interpreted in isolation but only in conjunction with the probabilities with 

which they occur.

Sensitivity and Specificity

The frequencies with which high fatality rates are not detected (false negatives) and low 

fatality rates are mistakenly interpreted as high (false positives) are discussed only briefly. In a 

general framework of 1 year of monitoring followed by 4 non-monitored years, there is limited 

statistical power to distinguish clearly between high and low fatality rates. The difficulty is 

compounded by the high degree of volatility that is inherent in small count data. In addition, the 

simulations are based on a simplifying assumption that the true number of annual fatalities varies 

as a Poisson random variable with constant mean, but in practice the numbers of annual fatalities 

are likely to be more variable than Poisson or have rates that vary significantly from year to year. 

This would affect the coverage probabilities, increase the frequencies of very high MM∗’s, and 

increase the probabilities that high fatality rates would pass unnoticed in the non-monitored 

years.
22 

Alternatives

The analysis is strictly limited to exI, prI, and inI in the context of 1 year of monitoring 

at gg = 0.15 followed by 4 non-monitored years, which was the scope of the original question 

posed to us. Alternatives such as different numbers of monitored and non-monitored years, 

different ggss values, other ways of accounting for incidentals are beyond the scope of the study. 

References

Dalthorp, Daniel, and Huso, Manuela, 2015, A framework for decision points to trigger adaptive 

management actions in long-term incidental take permits: U.S. Geological Survey Open-File 

Report 2015-1227, 88 p., http://dx.doi.org/10.3133/ofr20151227. 

Dalthorp, Daniel, Huso, Manuela, and Dail, David, 2017, Evidence of absence (v2.0) software 

user guide: U.S. Geological Survey Data Series 1055, 109 p., https://doi.org/10.3133/ds1055. 
Publishing support provided by the U.S. Geological Survey

Science Publishing Network, Tacoma Publishing Service Center

For more information concerning the research in this report, contact the

Director, Western Fisheries Research Center

U.S. Geological Survey

6505 NE 65th Street

Seattle, Washington 98115-5016

https://www.usgs.gov/centers/wfrc
Dalthorp and othrs—Accounting for Incidental Carcass Discoveries in Non-Monitored Years using the Evidence of Absence Model—Open-File Report 2020-1027

ISSN 2331-1258 (online)

https://doi.org/10.3133/ofr20201027