ESTIMATION OF ANIMAL VITAL RATES WITH KNOWN FATE STUDIES ALL MARKED ANIMALS DETECTED

KNOWN FATE STUDIES Sample 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) = ( ) s y (1-s) n-y L (s|n,y) = s y (1-s) n-y ŝ=y/n; var ̂ (ŝ)=ŝ(1-ŝ)/n  Fate of individual is independent  All detected, and fates are known  No censoring (e.g., no failure of radio) nyny

MULE DEER EXAMPLE Number Released AliveDead Treatment Control TreatmentControl ŝ 19/57 = /59=0.356 Vâr(ŝ) 0.333( )/57= ( )/59= %CI  2 =0.058 P>=0.81 example from White and Garrott (1990: ) 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 study d = No. of deaths recorded at time j n = No. of animals alive and at risk at time j t = time units since the beginning of the study n j – d j n j djnjdjnj t i=1 t i=1

EXAMPLE RADIO-TAGGED BLACK DUCKS Week Number alive at start Number dying Number alive at end Number censored Ŝ 1 = 47/48 = Ŝ 2 = 45/47 = Ŝ 3 = 39/41 = (note: only 41 because 4 were censored) Ŝ 4 = 34/39 = Ŝ 5 = 28/32 = (note: only 32 because 2 were censored) Ŝ 6 = 25/28 = Ŝ 7 = 24/25 = Ŝ 8 = 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 allocation Must 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 phase  Positive bias in survival (eg., dsr=.99)  “Early” nests have more survival days (s 1 = =.74, s 29 =.99 2 =.98)  Chance of failure related to N of days Need to adjust survival rates Basic 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 ESTIMATOR dsr: daily survival rate dsr ̂ = 1 – d / exposure S ̂ = (dsr ̂ ) t S: probability of survival for study period

EXPOSURE Nest No.1 May8 May15 MayExposure days (2*7) (.5*7) (1*7+.5*7) Total28 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 = var(dsr ̂ ) = {(28-2)x2 / (28) 3 = S ̂ = dsr ̂ 34 = = % confidence interval: – 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 DSR MLE in Mark no need for midpoint assumption For details of nesting model in Mark see:

NEST SURVIVAL MODEL IN MARK Daily 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.

INPUT FOR MARK 1.day the nest was found 2.last day the nest was checked when alive 3.last day the nest was checked 4.fate of the nest (0 = successful, 1 = depredated) 5.number (frequency) of nests that had each history (usually 1) nest survival group=1; ; ; ; ; ; ; ; ; ; ;

DESIGN ISSUES: NEST SUCCESS Can 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