estimation of animal vital rates with known fate studies all marked animals detected
DESCRIPTION
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) nynyTRANSCRIPT
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.
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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