ESTIMATION OF ANIMAL VITAL RATES WITH KNOWN FATE STUDIES

ALL MARKED ANIMALS DETECTED

KNOWN FATE STUDIESSample of n animals followed through time and fate can be

determined Radio telemetry studies Nest success

BINOMIAL SURVIVAL MODEL

Follow n subjects, and observe y survivors f(y|n,s) = ( ) sy(1-s)n-y

L(s|n,y) = sy(1-s)n-y s=y/n; var (s)=s(1-s)/n

Fate of individual is independentAll detected, and fates are knownNo censoring (e.g., no failure of radio)

ny

MULE DEER EXAMPLENumber Released

Alive Dead

Treatment 57 19 38Control 59 21 38

Treatment Controls 19/57 = 0.333 21/59=0.356Var(s) 0.333(1-0.333)/57=0.003899 0.356(1-0.356)/

59=0.00388695%CI 0.211 - 0.458 0.234 - 0.478

2 =0.058 P>=0.81

example from White and Garrott (1990:209-210) in which 120 mule deer fawns in Colorado were equipped with radio transmitters and followed through winter. Sixty-one fawns were on study area near an oil shale development (“treatment”) and 59 were from areas removed from human activity

CONTINUOUS SURVIVAL METHODS(NON-PARAMETRIC APPROACH):KAPLAN-MEIER METHOD

S(t) = ( ) = (1 - )

S(t) = Probability of surviving t time units from the

beginning of the studyd = No. of deaths recorded at time jn = No. of animals alive and at risk at time jt = time units since the beginning of the study

nj – dj

nj

djnj

t

i=1

t

i=1

EXAMPLERADIO-TAGGED BLACK DUCKSWeek 1 2 3 4 5 6 7 8Number alive at start 48 47 45 39 3

428

25 24

Number dying 1 2 2 5 4 3 1 0Number alive at end 47 45 39 34 2

825

24 24

Number censored 0 0 4 0 2 0 0 0S1 = 47/48 = 0.979S2 = 45/47 = 0.957S3 = 39/41 = 0.951 (note: only 41 because 4 were censored)S4 = 34/39 = 0.872S5 = 28/32 = 0.875 (note: only 32 because 2 were censored)S6 = 25/28 = 0.893S7 = 24/25 = 0.960S8 = 24/24 = 1.000

KM ESTIMATOR

Censoring, e.g., transmitter failure But censoring should be independent of survival Keep to a minimum (e.g., predator effect on radios)

Staggered entry: e.g., animals leave study area (but return)

DESIGN CONSIDERATIONS

Capture n animals How many? Use binomial model for sample allocationMust be able to record fates (alive or dead)

at the end of each interval Trade off: study area must be small enough to permit

frequent surveys- but too small may lead to more censoring…

Animals not encountered should be censored, and if later resighted should be considered as a new staggered entry

Try to prevent censoring Censoring must be random and independent of fate

NEST STUDIES AND THE MAYFIELD METHOD

Hatching rate (prop nest success)Many nests encountered late in nesting phasePositive bias in survival (eg., dsr=.99)

“Early” nests have more survival days (s1=.9930=.74, s29=.992=.98)

Chance of failure related to N of daysNeed to adjust survival ratesBasic idea: consider number of days of

exposure, rather than number of nests

HATCHING SUCCESS-BIAS

STUDY DESIGN• Nests marked or uniquely identifiable• Periodically monitored to determine status• Censoring and staggered entry are possible• Record monitoring history for each individual:

date, time, status

MAYFIELD’S ESTIMATORdsr: daily survival rate

dsr = 1 – d / exposure

S = (dsr )t

S: probability of survival for study period

EXPOSURE

Nest No. 1 May 8 May 15 May

Exposure days

1 1 1 1 14 (2*7)2 1 0 3.5 (.5*7)3 1 1 0 10.5 (1*7+.5*7)

Total 28

Survival histories and exposure via the Mayfield method of three hypothetical nests (1-active nest, 0-nest destroyed)

DSR AND SURVIVAL

dsr = 1 – ( d/exposure ) = 1 - 2/28 = 0.9286

var(dsr ) = {(28-2)x2 / (28)3 = 0.0023688

S = dsr34 = 0.928634 = 0.0806

95% confidence interval: 0.002 – 2.240

ASSUMPTIONS• Random sampling• Rates constant (Accommodate through

stratification)• Visits recorded• Pr(s) not influenced by observer• Pr(visit) independent of Pr(survival)

MARK MLE DSRMLE in Mark no need for midpoint

assumption

For details of nesting model in Mark see:

http://www.auburn.edu/~grandjb/wildpop/lectures/lect_04.pdfhttp://www.phidot.org/software/mark/docs/book/pdf/chap17.pdf

NEST SURVIVAL MODEL IN MARKDaily nest survival model Function of nest-, group-, and time-specific

explanatory variables (Dinsmore et al. 2002). Allows visitation intervals to vary Requires no assumptions about when

nest losses occur.Uses encounter histories of individual

nests Likelihood-based procedures Values for time-specific explanatory

variables, such as age, date, and precipitation, are allowed to vary daily.

)(

)(

0

0

1jj

j

jjj

x

x

e

es

INPUT FOR MARK1. day the nest was found2. last day the nest was checked

when alive3. last day the nest was checked4. fate of the nest (0 =

successful, 1 = depredated) 5. number (frequency) of nests

that had each history (usually 1)

nest survival group=1; 1 59 62 1 3; 1 48 48 0 1; 1 37 37 0 1; 1 22 26 1 1; 1 22 24 1 1; 1 12 17 1 1; 1 27 32 1 1; 1 32 32 0 1; 1 45 51 1 1; 1 26 32 1 1;

DESIGN ISSUES: NEST SUCCESSCan predict n of samples (nests)

needed

Trade off between more nests and more visits

Fewer visits & more nests = increased precision Fewer visits = less information on stage transitions and

fledging

WHAT YOU SHOULD KNOW

Assumptions of the models• Random sampling, Rates constant, Visits recorded, Pr(s) not influenced by observer, Pr(visit) independent of

Pr(survival)

Bias associated with hatching rate Many nests encountered late in nesting phase Positive bias in survival

“Early” nests have more survival days, chance of failure related to N of days

Use and limitations of censoring and staggered entry

censoring should be independent of survival and kept to a minimum Animals not encountered should be censored, and if later resighted should be considered as a new staggered entry