Bayesian Time-Stratified-Petersen estimators for abundance. Sampling Protocol. Simon J. Bonner (UBC) Carl James Schwarz (SFU)

Transcription

1 Bayesian Time-Stratified-Petersen estimators for abundance Sampling Protocol Simon J. Bonner (UBC) Carl James Schwarz (SFU) Simple-Petersen or Stratified-Petersen methods are often used to estimate number of outgoing smolt or returning salmon. These methods are inadequate to deal with heterogeneity in catchability among strata and with missing data from strata caused by crew illness, high water flow, or other causes. We propose a Bayesian spline-based methodology to estimate abundance and run-timing which provides several compelling advantages over the more traditional estimators. The hierarchical model for capture probabilities and the spline model for the general shape of the run curve, allow information to be shared among stratra within a Bayesian framework and allows great flexibility to deal with missing data. It is self-calibrating- for strata with poor data, extensive pooling across strata take place but with strata with rich data, the information for a particular stratum takes precedence. The methodology automatically adjust measures of precision for heterogeneity in catchability among strata (which is ignored in the simple-petersen) and shares information from neighbouring strata (unlike the Stratified-Petersen). Examples from estimating the number of outgoing number of salmon smolt in the Trinity River, CA will be present - rotary screw-traps are used to capture fish - a sample of fish is marked and transported above trap and released - a portion of the marked fish are recaptured to estimate capture efficiency - unmarked fish are captured along with marked fish 1 2

2 Notation Data: - batch marks change weekly - unit of analysis is the (julian) week - 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 (may include the n i ). This can often be (partially) subdivided in hatchery and wild fish. Hatchery Chinook are 25% adfin clipped; Hatchery Steelhead are 100% ad-fin clipped. - Current program is basically diagonal recoveries, i.e. m ij = 0for j>i. (but see Bonner and Schwarz, 2010) Parameters: - p i - recapture rate in week i - U i - total outgoing population in week i. - U =! U i - grand total outgoing population weeks 3 Sample JC 2003 Chinook Data Marked Fish Unmarked Fish AD JW D n i m ii m i,i+1 m i,i+2 AD* NC +NC f ,135 4, , ,452 10, , ,199 2, , ,427 30,542 39, , ,243 13,337 17, , ,646 6,282 7, , ,412 26,706 35, , ,703 26,831 34, , ,651 11,309 14, , ,677 3, , ,343 1, Total 50,489 2, , , ,995 * 25% of hatchery fish have AD-fin clip 4

3 Objectives and Problems Objectives: - estimate total number of outgoing juvenile salmon - separate estimates for hatchery and wild fish - estimate characteristics such as percentiles of run timing Problems: - sparse data (e.g. few marked fish released/recaptured) - missing data (no marking done and/or no unmarked captured) - anomalous data (e.g. fish disappear after release) Population Estimation Key concept (of mark-recapture): - recapture of marked fish provide estimate of screw-trap capture efficiency (e.g. the screw-trap is capturing 5% of the fish that pass the location) - use the estimated recapture rate to expand the number of unmarked fish captured. Methods: Complete Pooling Simple Petersen " # $ Û =! weeks " # $ u i % " & ' # $! weeks! weeks m ii % & ' n i % & '!" Separate weekly estimates (Weekly) Stratified- Petersen Û i = u in i m ii ;Û =! weeks Û i JC 2003 Chinook Estimates 4.2 (SE.081) million 16 (SE 3.7 ) million 5 (SE.21 ) million (ex JW 41) 5 6

4 Population Estimation Example JC 2003 Chinook Julian Week n i m ii * u i Û i ,616?? 10 1, , , , ,557 23, , ?? , , , , , ,580 1,274, , , , , , , ,534 12,414, , , ,734 Pooling 50,489 2, , million * Adjusted for less than 7 days sampling. Complete Pooling Best possible precision BUT Unable to estimate run timing easily Handle missing marking weeks. Unable to deal with weeks missing capture of unmarked fish Implicitly assumes homogeneous capture. Est small bias - whew SE large bias(!) Odd data has little effect Population Estimation!" Separate weekly estimates Comparison Weekly estimates may have poor precision but overall estimate has acceptable precision Estimate run-timing No estimate if don t mark in a week No estimate if don t recapture in a week Allows for heterogeneous capture across weeks, but assumes homogeneity within weeks Odd data could lead to highly biased weekly estimates 7 8

5 Population Estimation alternatives? (Brief Introduction to) Splines Spline flexible curve with no long-term pattern; based on local behavior Try pooling weeks that are similar (partially stratified) - arbitrary - how to estimate se properly? - dealing with missing weeks esp. with no unmarked fish Spline and Hierarchical model a) fit a smooth curve to the Ûi (spline) as the underlying trend but allow variability about trend line b) allow p i to vary around a common mean (hierarchical model) 9 Elements of spline fit - knots where the pieces of the spline meet - degree of spline (linear, quadratic, or cubic) 10

6 Knots - many knots = wiggly fit - fewer knots = smoother fit (Brief Introduction to) Splines Linear spline fit: (Brief Introduction to) Splines - rough rule of thumb, use n/4 knots Degree of spline - linear = joined line segments not differentiable at knots - quadratic/cubic = first/second derivatives are smooth Fitting splines = ordinary least squares (or variants thereof) - e.g. linear spline fit has design (X) matrix of the form: 1, x, (x-knot1) +, (x-knot2) +, where (arg) + is 0 if arg < 0 [1,] [2,] [3,] [4,] [5,] [6,] [7,] [8,]

7 (Brief Introduction to) Splines Quadratic spline fit. Design matrix has: 1, x, x 2, (x-knot1) + 2, (x-knot2) + 2, (Brief Introduction to) Splines Quadratic spline fit with fewer knots (every 8 th point) (less wiggly ) You can also control wiggliness by enforcing penalties on derivatives, or penalties on changes in coefficients

8 (Brief Introduction to) Hierarchical Models Recall that in each of 40 weeks we have a Petersen experiment there is a separate p i for each week In a pooled-petersen experiment, all p i are assumed to be equal (unrealistic) In a stratified-petersen experiment, all p i allowed to unequal and no information from other weeks is used in week of interest (too general) Hierarchical model: p i drawn from a common distribution with mean µ p and sd! p - if! p small, then sharing of information across weeks - if! p large, then each p i is separate from each other Proposed Spline & Hierarchical Method (a) Likelihood standard stratified Petersen estimator (b) Hierarchical model on logit(p i ) [Could use covariates as well including spline on logit(p)] (c) Spline model (with error ) for log(u i ) using B-spline basis functions and knots every 4 th week and smooth changes in coefficients (d) Sensible priors everywhere 15 16

9 Proposed Spline & Hierarchical Method Advantages: - borrows information from other weeks for estimating catchability and weekly run size. - gives weekly (and total) estimates - estimate run timing - if missing marking week, uses range of capture rates seen in other weeks to impute range of possible capture rates for weeks with no marking done - if missing unmarked fish in a week, uses spline to interpolate reasonable value for outgoing total based on variation of other weeks around smooth curve - automatically adjusts for amount of heterogeneity in capture-rates across weeks. If small variation, estimates have precision similar to pooled-petersen. If larger variation, estimates have realistic standard errors - odd data easily handled (simply set to missing) Disadvantage: - not amenable to hand computations - difficulty to fit Bayesian methods useful - computer programs are complex JC 2003 Chinook - Spline Model - Allowed for 2 jumps when hatchery fish arrived. Pooled Petersen: 4.2 (SE.081) million fish. Stratified-Petersen 5 (SE.21 ) million fish (ex j.w. 41) Spline est: 5.3 (SE.18 ) million fish 17 18

10 Capture Efficiency Estimation JC 2003 Chinook - Hierarchical Model Separating Wild and Hatchery JC 2003 CH YOY Fish Above results are a mixture of wild and hatchery fish. 25% of hatchery fish have adipose fin clip; hatchery YoY fish released starting in jw 22. In jw 40, hatchery 1 released; no further wild fish in stream (?) Marked Fish Unmarked Fish AD JW D n i m ii m i,i+1 m i,i+2 AD* NC +NC f ,135 4, , ,452 10, , ,199 2, , ,427 30,542 39, , ,243 13,337 17, , ,646 6,282 7, Notice range of catchability in j.w. 9, 41, etc - Additional structure in p use covariates such as log-flow or spline Need to expand ad-clipped fish by factor of

11 Separating Wild and Hatchery JC 2003 CH YOY Fish Population Estimation Separating Wild and Hatchery YOY CH Fish JC Run Timing W.YOY (millions) H.YOY (millions) Pooled Petersen* 0.74 (SE.02) 1.3 (SE.03) Stratified-Petersen 0.71 (SE.04) 2.3 (SE.17) Spline 0.87 (SD.11) 2.3 (SD.12) *SE Not adjusted for interpolation of ad-clipped. Note that after jw 40, hatchery fish are classified as age 1+ Wild 0% 10% 30% 50% 70% 90% 100% Mean SD Hatchery Mean SD

12 Evaluation of Trinity River Restoration Program Evaluation of Trinity River Restoration Program Estimates of log(abundance) available at JC, WC, and PT (not all sites in all years Estimates of log(abundance) available using flow-based method (i.e. look at fraction of flow that is sampled by rotary screw trap) Need to align series and extrapolate

13 Evaluation of Trinity River Restoration Program Summary - borrows information from other weeks o spline forces estimates to follow smoothish curve o capture rates come from common distribution - estimates available at weekly and total level with realistic se - estimates available for wild vs hatchery groups with realistic se - run timing estimates available - easy to interpolate for weeks with missing/odd data - able to add covariates (e.g. log-flow for p) (not shown) - model fitting complex no hand computations - careful of interpolations before and after last sampling - assumption that patterns visible = patterns hidden - Allows some flexibility in sampling, e.g. every second week - BTSPAS package in R has complete examples 25 - Bonner and Schwarz (2010a; 2010b) for more details - Phase2/TrinityReport-Final%20Phase2.pdf for full details. 26