Advances since Program MARK Parameter Estimation from Marked Animals. PLive captures PLive resightings PDead recoveries

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1 Program MARK Parameter Estimation from Marked Animals Gary C. White Dept. of Fishery and Wildlife Colorado State University Fort Collins, Colorado, USA Advances since 985 P More complex time/space models P Covariates more realistic biology P New models innovative data collection Capability Year 2 Encounter Techniques P Standard tags, leg bands, or neck collars P Radios (implants) PPIT tags P Camera traps P DNA samples 3 Types of Encounter History Data PLive captures PLive resightings PDead recoveries 4

2 Estimable Parameters PSurvival P Movement P Emigration/immigration P Reproduction/recruitment P Population size P Rate of population change 5 Models in MARK P Live encounters Cormack, Jolly, Seber P Dead encounters Band or ring recovery P Joint live and dead Burnham, Barker, Lindberg P Known fate radio-tracking P Closed capture-recapture likelihood only, including individual heterogeneity P Robust design Pollock, Kendall P Multistrata design Brownie, Nichols, with dead recoveries Barker P Jolly-Seber POPAN Pradel Link-Barker (7 parameterizations) P Nest success P Occupancy estimation, with robust design 6 Additional Features P Multiple attribute groups P Time, group and individual covariates P Unequal time intervals P Model management P AIC model selection P Quasi-likelihood P Variance Components P Model Averaging Knowledge Required of the User P Windows 95/98/NT/2000/ME/XP P Structure of models P Design matrix and link functions P AIC, model selection P Maximum likelihood estimation L Sharp knives cut people 7 8

3 Data are Required to Gain Reliable Knowledge A statistician s primary function is to prevent, or at least impede research. 9 Computer Requirements P Windows 95/98/NT 4.0/ 2000/ME/XP P CPU speed to compute estimates in reasonable amount of time P$28 Mb RAM 0 Program Architecture P Interface written in Computer Associates Visual Objects P Numerical analysis in 32 bit FORTRAN 95 code P Communication by files Vocabulary of MARK P Encounter histories P Parameter Index Matrices (PIM) P Design Matrix 2

4 Basic Data Encounter Histories P LDLDLD... < Dead recoveries < Joint live and dead < Known fate P LLLLLLL... < Live recaptures < Closed capture-recapture P Dead recovery matrices P Known fate Likelihood Function Maximum Likelihood Estimation Likelihood = Pr(Enc. Hist.) Observed log e (Likelihood) = Num. Observed X log e [Pr(Enc. Hist.)] 3 4 Pr(Enc. History) P How are encounter histories translated into probabilities that are a function of the parameters of interest? Releases Dead Encounters S - S Live Dead r - r Reported Not Reported Encounter History (LD) 0 S ( - S)r 0 ( - S)( - r) 5 6

5 Seber 970 Brownie et al.: f i = ( - S i )r i (-S )r S (-S 2 )r 2 S S 2 (-S 3 )r 3 S S 2 S 3 (-S 4 )r 4 (-S 2 )r 2 S 2 (-S 3 )r 3 S 2 S 3 (-S 4 )r 4 (-S 3 )r 3 S 3 (-S 4 )r 4 (-S 4 )r 4 7 Brownie et al. 985 Model f S f 2 S S 2 f 3 S S 2 S 3 f 4 f 2 S 2 f 3 S 2 S 3 f 4 f 3 S 3 f 4 f 4 8 Dead Encounters 4 encounter occasions LDLDLDLD S - survival prob. r -conditional reporting probability 9 Dead Encounters S S 2 S 3 ( - S 4 )r J

6 Dead Encounters S 2 S 3 ( - S 4 )r 4 Dead Encounters ( - S 4 )r J 2 22 Dead Encounters ( - S 4 )( - r 4 ) + S 4 23 Dead Encounters Model notation: {S(t) r(t)} S, S 2,..., S t ; r, r 2,..., r t. 24

7 Dead Encounters Model notation: {S(.) r(t)} S = S 2 =... = S t ; r, r 2,..., r t. 25 Dead Encounters Model notation: {S(t) r(.)} S, S 2,..., S t ; r = r 2 =... = r t. 26 Dead Encounters Model notation: {S(.) r(.)} S = S 2 =... = S t ; r = r 2 =... = r t. 27 Dead Encounters Model notation: {S(g*t) r(g*t)} S, S 2,..., S t, S 2, S 22,..., S 2t ; r 2, r 3,..., r t, r 22, r 23,..., r 2t. 28

8 Dead Encounters Model notation: {S(g) r(g)} S., S 2.,..., S g. ; r., r 2.,..., r g.. 29 Live Encounters (CJS) Releases N - N Encounter History (LL) p Seen Np Live - p Not Seen Dead or Emigrated 0 N( - p) 0 - N 30 Live Encounters (CJS) 4 encounter occasions LLLL N - apparent survival prob. p - capture prob. 3 Live Encounters (CJS) 0 N p 2 N 2 ( - p 3 )N 3 p 4 32

9 Live Encounters (CJS) 00 N 2 ( - p 3 )N 3 p 4 Live Encounters (CJS) 0 N p 2 N 2 p 3 [N 3 ( - p 4 ) + ( - N 3 )] Live Encounters (CJS) 00 N p 2 [( - N 2 ) + N 2 ( - p 3 ){N 3 ( - p 4 ) + ( - N 3 )}] 35 Live Encounters (CJS) Model notation: {N(t) p(t)} N, N 2,..., N t - ; p 2, p 3,..., p t. 36

10 Live Encounters (CJS) Model notation: {N(.) p(.)} N = N 2 =... = N t - ; p 2 = p 3 =... = p t. 37 Live Encounters (CJS) Model notation: {N(t) p(.)} N, N 2,..., N t - ; p 2 = p 3 =... = p t. 38 Live Encounters (CJS) Model notation: {N(g*t) p(g*t)} N, N 2,..., N t -, N 2, N 22,..., N 2t - ; p 2, p 3,..., p t ; p 22, p 23,..., p 2t. 39 Live Encounters (CJS) Model notation: {N(g) p(g)} N., N 2.,..., N g. ; p., p 2.,..., p g.. 40

11 Extensions to CJS Extensions allow additional parameters to be estimated. P Multi-strata P Jolly-Seber (Pradel, POPAN, Link-Barker) P Robust design P Incorporate dead encounters 4 Multi-Strata Model P Allows transitions between categories, e.g.: < Life stages < Areas (e.g., wintering grounds) < Breeding status 42 Jolly-Seber Model PEstimates of N and p P Leading Zeros used to estimate: < Lambda (8 = N t+ /N t = N + f ) < Recruitment (f = 8 - N) < Seniority (() < Population size (N t ) 43 Robust Design PMultiple trapping occasions close together where no population change is assumed PEstimates of: < Temporary emigration < Population size (N) 44

12 Releases - S Joint Encounters Both live and dead encounters Encounter History (LDL) Live S - Fp Dead Fp r - r Seen Not Seen Reported Not Reported 45 0 SFp 00 S( - Fp) 0 ( - S)r 00 ( - S)( - r) Joint Encounters 4 encounter occasions LDLDLDLD S - survival prob. r - reporting prob. p - capture prob. F - fidelity prob. (N = SF) 46 Joint Encounters 000 S F 2 p 2 S 2 F 3 p 3 S 3 F 4 p 4 ( - S 4 )r 4 Joint Encounters Model notation: {S(t) p(t) r(t) F(t)} S, S 2,..., S t ; p 2, p 3,..., p t ; r, r 2,..., r t ; F 2, F 3,..., F t

13 Joint Encounters 0000 S F 2 p 2 S 2 F 3 p 3 S 3 [F 4 ( - p 4 ) + ( - F 4 )] ( - S 4 )r 4 Barker Live-Dead Model P Extension of joint live and dead recoveries model that allows resightings during intervals P Estimation of survival and fidelity to study area Robust Design Barker Model PBarker model extended to include robust design (also see Lindberg et al. 200) PEstimate temporary and permanent emigration, survival, and population size. Robust Design Barker Model P For wildlife applications, this general model could incorporate data from DNA, radios, harvest, and casual resightings of marked animals P Estimates of survival, temporary and permanent emigration, and population size 5 52

14 Multi-strata with Live and Dead Encounters PStrata-specific live encounters, dead encounters ignore strata PEstimation of transition probabilities, true survival, strataspecific live encounter probability, and dead encounter probability 53 Joint Encounters: Estimation of Radio Effects P Group : regular live/dead encounters < PGroup 2: known fate data entered as joint live/dead encounter history < 0000 < p = < r = PCompare S-hat and F-hat for 2 groups 54 Closed Captures Closed Captures Unmarked p - p c Seen - c p Not Seen - p Encounter History (LL) Seen pc Not Seen Seen Not Seen 0 p( - c) 0 ( - p)p 00 ( - p)( - p) 4 encounter occasions LLLL p - initial capture prob. c - recapture prob. N - population size 55 56

15 Closed Captures Closed Captures PN = M t+ /[ - (-p )(-p 2 )...(-p 4 )] p c 2 c 3 c Closed Captures Closed Captures 0 p c 2 ( - c 3 )c 4 00 ( - p )p 2 ( - c 3 )c

16 Closed Captures Closed Captures 000 ( - p )p 2 ( - c 3 )( - c 4 ) 0000 ( - p )( - p 2 )( - p 3 )( - p 4 ) 6 62 Closed Captures Model notation: {p(t) c(t) N(.)} p, p 2,..., p t ; c 2, c 3,..., c t ; N. 63 Closed Captures with Heterogeneity Pp = Bp a + ( - B)p b where B is probability animal is from mixture distribution a which has p a initial capture probability PSimilarly for remaining p i and c i 64

17 Closed Captures with Heterogeneity Model notation with 2 mixtures (a, b): {B p a (t) p b (t) c a (t) c b (t) N(.)} B; p a, p a2,..., p at ; p b, p b2,..., p bt ; c a2, c a3,..., c at ; c b2, c b3,..., c bt ; N. 65 Known Fate Radio-tracking Data Encounter History (LD) S Live 0 S Releases - S Dead ( - S) 66 Known Fate Known Fate 4 encounter occasions LDLDLDLD S - survival prob. 000 S S 2 S 3 ( - S 4 ) 67 68

18 Known Fate Known Fate 0000 S 2 S 3 ( - S 4 ) 0000 S S 2 ( - S 4 ) Known Fate Model notation: {S(t)} S, S 2,..., S t. 7 Known Fate Equivalent to Kaplan Meier PKM estimate is product of survival estimates for fixed time intervals P Staggered entry < < P Right censoring or missing intervals <

19 Advantages of Using MARK Known Fate over Kaplan-Meier PAlternative models PCovariates PModel selection PModel averaging PVariance components 73 Model Building PIM ú Design Matrix ($) ú Link Function ú Real Parameters 74 Parameter Index Matrix PIM Time-specific apparent survival Parameter Index Matrix PIM Time-specific capture probabilities º 2 º 3 º 4 º 5 p 2 p 3 p 4 p 5 2 º 3 º 4 º 5 p 3 p 4 p 5 3 º 4 º 5 p 4 p 5 4 º 5 p5 ±

20 Parameter Index Matrix PIM Age-specific apparent survival 77 Parameter Index Matrix PIM Age- and time-specific apparent survival, 2 age classes M M 5 M 6 M 7 º 2 º 3 º 4 º 5 M 2 M 6 M 7 2 º 3 º 4 º 5 M 3 M 7 3 º 4 º 5 M 4 4 º 5 78 ± Parameter Index Matrix PIM 2 Groups, Survival Different, Timespecific Model {S(g*t)} Parameter Index Matrix PIM 2 Groups, Survival Same, Time- specific Model {S(t)} Group Group

21 Parameter Index Matrix PIM 2 Groups, Survival Different, Model {S(g)} 8 Parameter Index Matrix PIM 2 Groups, Survival Same, Model {S(.)} Group 82 Group 2 PIM Manipulation PTime-specific (current PIM or all) PAge-specific (current PIM or all) PConstant (current PIM or all) PDiagonal, Row, Column changes PChange a singe cell PExchange or copy whole PIMs PPaste or copy to clipboard PRenumber (with or without overlap) PIM Chart PVisually assess PIMs PDrag boxes to copy PIMs PRight click to select menu choices 83 84

22 Design Matrix PAllows additional constraints not possible using PIMs < Additive models with parallelism between groups, ages PCovariates, including individual covariates Individual Covariates PIncluded in Encounter Histories File PAttached to animal s encounter history PIMs for Design Matrix Examples 5 intervals (6 occasions), 3 groups 87 Group Group Group Model {theta(g*t)} logit(theta) Year Variation in theta by time and by group 88 g g2 g3

23 Link Functions PSin PLogit PLog-Log PComplementary Log-Log PLog PIdentity PMultinomial Logit PCumulative Logit 89 Design Matrix Identity Link Unbounded Real Estimate = X$ 90 Design Matrix Sin Link Bounded 0- Real Estimate = [sin(x$) + ]/2 9 Design Matrix Logit Link Bounded 0- Real Estimate = exp(x$)/[ + exp(x$)] = /[ + exp(-x$)] 92

24 2.5 2 Logit vs. Real Parameters Logit Scale Males Females Year Real Scale Males Females Year 93 Group Group 2 Group logit(theta) Model {theta(.)} Year No variation in theta by time or groups 94 Model {theta(g)} Group Group 2 Group Model {theta(t)} logit(theta) Year Variation in theta by groups g g2 g logit(theta) Year Variation in theta by time 95 96

25 Model {theta(g + t)} Group Group 2 Group Model {theta(g*t)} logit(theta) Year Variation in theta by time and by group Additive model with no interactions g g2 g logit(theta) Year Variation in theta by time and by group g g2 g Model {theta(t)} Group Group 2 Group Model {theta(g + T)} logit(theta) Year Time trend in theta logit(theta) Year Time trend in theta by groups Additive model with no interactions g g2 g

26 Design Matrix Manipulation Model {theta(g*t)} PAdd and delete columns PProducts of columns or copy a column logit(theta) Year Time trend in theta by groups Additive model with interactions g g2 g3 PIntercept or trend covariates PIdentity matrix P Time or group indicator variables PCovariates or individual covariate names P Functions of columns and/or individual covariates PCopy values or rotate column up/down PPaste values from clipboard PRetrieve columns from previous models 0 02 Numerical Estimation PSeparate FORTRAN code PRuns as a separate process in a different thread PCan be used separately from interface program Numerical Methods P Encounter histories parsed to compute probability as function of parameters P Numerical optimization of likelihood with quasi-newton algorithm P Singular value decomposition of information matrix to obtain VC matrix and rank 03 04

27 Features of Results Browser P Retrieve PIMs and Design Matrix of models to construct a new model P Retrieve full output P Retrieve parameter estimates P Retrieve variance-covariance matrix POutput to a NotePad window, to Excel, to the clipboard, or printed directly PGraphics 05 Hypothesis Tests PLikelihood ratio tests PAnalysis of Deviance (ANODEV) 06 Goodness-of-fit Procedures 07 Advanced Analyses PModel averaging PVariance components estimation PQuasi-likelihood to correct for overdispersion PBootstrap procedure 08

28 Model Selection Older Approaches PForward selection PBackward selection PBest R 2 PAll possible models 09 Model Selection Information-Theoretic Approach PAIC PAkaike s Information Criterion PAIC = -2log( ($)) + 2K PMARK ranks models based on their AIC values 0 Model Selection Burnham and Anderson (998) PAIC = -2log( ($)) + 2K PAIC c = -2log( ($)) + 2K + 2 K(K + )/(n - K -) P)AIC c = AIC c - minaic c Pw i = exp(-) i /2)/3exp(-) j /2) Selecting Best Model P Estimated precision too good P95% confidence interval coverage <95% 2

29 Model Averaging Model Averaging PIncorporate model selection uncertainty into parameter variances PObtain proper confidence interval coverage 3 4 Model Averaging w i ' exp(&) i /2) R j exp(&) j /2) j' var ˆ ' R j w j j ' var(ˆ2 ˆ j M j ) % ˆ2 j & ˆ2 2 2 San Luis Valley Mallard Data, from Brownie et al. 985, page 92 Survival Parameter (S) Adults Parameter Model Weight Estimate Standard Error {S(a*t) f(a*t)} {S(a) f(a*t)} Weighted Average Unconditional SE % CI for Wgt. Ave. Est. (logit trans.) is to Percent of Variation Attributable to Model Variation is 35.3% 5 6

30 Variance Components PSampling Variance, e.g., Standard Errors PProcess Variance, e.g., Variance across time PSum is Total Variance Process Variance F 2 PNeeded for valid Population Viability Analysis PForecasting population trends 7 8 Variance Components Variance Components var( ˆ S) F2 n % n 2j n i' var(ŝ ˆ i S i ) ˆF 2 ' n n & j i ' S i & S 2 n n & j i ' Ŝ i & Ŝ 2 & n n & j i ' Var(Ŝ ˆ i S i ) 9 20

31 Variance Components PMean PLinear Trend PUser-defined linear function Beta-hat SE(Beta-hat) Example S-hat SE(S-hat) S-tilde SE(S-tilde) RMSE(S-tilde) Estimate of F 2 = with 95% CI ( to ) 2 22 Example Continued Program MARK Assumptions P Animal fates are independent P Binomial variation P Violations lead to overdispersion 23 24

32 Quasi-likelihood Estimation PCorrection for overdispersion PVariance inflation factor PQAIC = -2log( ($))/ + 2K PQAIC c = -2log( ($))/ + 2K + 2 K(K + )/(n - K - ) Var( ˆθ) SE(ˆθ) ' P ĉ Goodness of Fit PEstimate of over dispersion, c PRejection of global model as fitting the data Goodness of Fit Logistic Regression PDeviance/degrees of freedom PHosmer-Lemeshow test 27 Goodness-of-fit Testing Decide if c> for global model PProgram RELEASE PPrograms ESTIMATE/BROWNIE PDeviance -2log( model) - -2log( saturated) PBootstrap procedure 28

33 Bootstrap Procedure PCompute estimates from data PCompute parametric bootstraps from estimates PDetermine likelihood of observed deviance PEstimate as observed deviance divided by mean of bootstrap deviances (or Pearson chi-square) 29 Goodness of Fit Bootstrap Approach Percent Relative Bias EBV=2 Releases= Bootstrap RELEASE 5:0.5 5:0.8 0:0.5 0:0.8 5:0.5 5:0.8 Occasions:Survival 30 Program Documentation PHelp file PWWW < Gentle introduction Evan Cooch < Theory description David Anderson Ken Burnham PVarious other publications 3 Program Availability PSingle 3Mb zipped setup file PNine.4Mb zipped setup disks PUpdate zip file of executable files ö Includes help file and examples 32