Bayesian Time-Stratified-Petersen Estimators for Abundance

Transcription

1 Bayesian Time-Stratified-Petersen Estimators for Abundance Simon Bonner 1 Carl James Schwarz 2 1 Department of Statistics University of Kentucky Lexington, KY 2 Department of Statistics and Actuarial Science Simon Fraser University Burnaby, BC, Canada stat.sfu.ca 1 / 24

2 Objectives Estimate outgoing salmon smolt abundance Estimate total number of outgoing juvenile salmon. Separate estimates for hatchery and wild fish. Estimate characteristics such as percentiles of run timing. 2 / 24

3 Sampling Protocol 3 / 24

4 Sampling Protocol Sample of fish is marked, transported above trap, and released. Marked fish are recaptured to estimate capture efficiency. Unmarked fish are captured along with marked fish. 4 / 24

5 Notation Statistics: n i number of fish marked and released in week i. m ij number of marked fish released in week i and recovered in week j. u i number of unmarked fish captured in week i Subdivided into AD (adipose clipped) and NC (not clipped) fish. Parameters: p i recapture rate in week i. θ ij prob of movement from i i + j 1. U i total outgoing population in week i. U = U i grand total outgoing population. Two cases Diagonal case: m ij = 0 when j > i. Non-diagonal case: m ij can be > 0 when j > i. 5 / 24

6 Diagonal case: sample data: JC 2003 Chinook Marked Fish Unmarked Fish JW D n i m ii AD NC AD+NC log(flow) ,135 4, , ,452 10, , ,199 2, , ,427 30,542 39, , ,243 13,337 17, , ,412 26,706 35, , ,703 26,831 34, Total 50,489 2,459 41, , ,995 25% of hatchery fish are AD-fin clipped. 6 / 24

7 Diagonal case: Problems/opportunities in real data Problems: Sparse data - few marked fish released/recaptured, e.g. jw 39. Missing data - no marking done and/or no unmarked captured, e.g. jw 9. Anomalous data - fish disappear after release, e.g. jw 41. Opportuntities Reduce handling of endangered species by skipping marking in some weeks. Reduce costs by skipping some weeks. Change allocation of effort among weeks. 7 / 24

8 Diagonal case: Current methods Complete Pooling Separate weekly estimates Simple Petersen (Weekly) Stratified-Petersen Û = ( )( ) u i n i weeks ( weeks ) m ii weeks Û i = u i n i m ii ; Û = Û i weeks 4.2 (SE.081) million 16 (SE 3.7) million 5 (SE.21) million (ex JW 41) Implicit assumption Allow catchability of homogeneity to vary by of catchability over week. 43 weeks. Pool over problems. Poor estimates for with sparse, problematic or no data. 8 / 24

9 Diagonal Case: Alternatives - catchability Pooling weeks that have similar catchability: Arbitrary. How to estimate se properly to account for pooling? How to deal with missing weeks? 9 / 24

10 Diagonal Case: Alternatives - Abundance Draw a curve through the log(u): Arbitrary. How to estimate se properly to account for interpolations? 10 / 24

11 Diagonal Case: Proposed methodology Hierarchical and Spline model: Fit a spline curve to the log(u i ) as the underlying trend with variability about trend line. Hierarchical model for p i around a common mean and/or covariate adjusted value. Note that smoothing both p and log(u) does NOT work well. 11 / 24

12 Diagonal Case: Technical details (a) Likelihood - standard stratified Petersen estimator. L = s i=s [( ni m ii ) p m ii i (1 p i ) n i m ii ] [( Ui u i ) p u i i (1 p i ) U i u i ] (b) Hierarchical model on logit(p i ) [Could use covariates as well including spline on logit(p)] logit(p i ) Normal(µ p, σ 2 p) Imputes missing p i using range of possible capture rates from hierarchical model. Borrows information from other weeks where data is present or not sparse. 12 / 24

13 Diagonal Case: Technical details (c) Spline model (with error ) for log(u i ) using B-spline basis functions and knots every 4th week and smooth changes in coefficients. log(u i ) = K+q k=1 b k B k (j) + ε logu i b k+1 = b k + (b k b k 1 ) + ε spline k Spline interpolates reasonable values for abundance based on variation of other weeks around smooth curve. Allow for slow changes in general shape of spline. We also allow for pre-specified shocks ( jumps ) in the spline (e.g. arrival of hatchery fish). (d) Sensible priors everywhere. 13 / 24

14 Diagonal Case: Advantages/Disadvantages of methodology Advantages: Borrows information from other weeks for estimating catchability and weekly run size. Provides weekly (and total) estimates. Able to estimate run timing. Automatic model selection Sparse data - fits very smooth curves. Small effective number of parameters. Rich data - overwhelms the smooth curves. Large effective number of parameters. Odd data easily handled by setting it to missing. Disadvantages: Not amenable to hand computations. Do other weeks provide useful local information? 14 / 24

15 Diagonal Case: Results - JC 2003 Chinook - Abundance Allowed for 2 jumps in spline when hatchery fish arrive. Pooled Petersen: 4.2 (SE.081) million fish. Stratified-Petersen 5.0 (SE.21 ) million fish (ex j.w. 41) Spline method 5.3 (SE.18 ) million fish 15 / 24

16 Diagonal Case: Results - JC 2003 Chinook - Catchability Notice range of catchability in jw 9, 41, etc. For additional structure in p use covariates such as log-flow or fit a spline. 16 / 24

17 Diagonal Case: Results - JC 2003 Chinook - Wild vs. hatchery 25% of hatchery fish are AD-fin clipped. Add an additional binomial component to the likelihood. Add separate splines for wild and hatchery fish. 17 / 24

18 Non-Diagaonal Case: Conne River, NL, Canada Date n i Subsequent recoveries u i Total / 24

19 Non-diagonal case - modifications to model Add a component to likelihood for movement among strata: (a) Likelihood stratified Petersen estimator (Darroch) L(p, U n, m, u) = ( ) n i J s (θ m ii, m i,i+1, m i,i+2,... i+j 1 p i+j 1 ) m ij i=1 T k=1 j=1 ( 1 θ i+j 1 p i+j 1 ) ni m i ( Uk u k ) p u k k (1 p k) U k u k 19 / 24

20 Non-diagonal case - modifications to model (b) Non-parametric smoother for movement based on continuation ratios. ( logit θ i,i+k 1 max l=k θ i,i+l ) ( N µ (NP) k, σ (NP)2) 20 / 24

21 Non-Diagonal Case: Results - Five models for p 21 / 24

22 Non-Diagonal Case: Results - Five models for log(u i ) 22 / 24

23 Non-Diagonal Case: Results - Comparison of estimates Model for 95% CI Effective U, p Û interval width parameters Simple, Pooled 80 ( 69, 92) Simple, Hier 90 ( 73, 111) Hier, Hier 85 ( 70, 103) Cubic, Hier 79 ( 67, 92) Spline, Hier 78 ( 66, 92) (a) Virtually no loss in precision with very sparse data compared to simple pooling methods. (b) Automatic model selection with small number of effective parameters. 23 / 24

24 Summary - BTSPAS Advantages: Hierarchical model borrows information from other weeks. Spline forces estimates of abundance to follow smoothish curve. Easy to interpolate for strata with missing or odd data. Automatic (internal) model selection depending on data richness. Allows for flexibility in sampling design not available under earlier methods. Disadvantages: Careful of interpolations before and after last sampling. Key assumption that patterns visible = patterns hidden. Available as BTSPAS package in R with complete examples. Biometrics (online early) DOI: /j x 24 / 24