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# SAMPL ANIM POPNS ST 506

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This 292 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 506 at North Carolina State University taught by K. Pollock in Fall. Since its upload, it has received 22 views. For similar materials see /class/223941/st-506-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15

Lecture 20 Logistic Regression Methods Methodology Shriner et al2002 Example Occupancy Methods Methodology Examples I want to acknowledge Jim Nichols for some of the occupancy slides Draft Proposal Due next Tuesday Logistic Linear Regression YO or 1 now unlike before Where Y continuous or discrete variable Binary Regression Model and can be Viewed as a generalised linear regression Many applications for ecologists Useful in study of species habitat relationships in ecology Where 1 represents species present and 0 represents species absent Although sometimes there are detection questions that complicate this Logistic Linear Regression PltK1gt7ri PltK0gt1 7r uYlX1X2Xp7rl VarYlX1X2Xp7ril 7ri Y is a binary variable and so this work related to binomial distributions in terms of mean and variance structure Logistic Linear Regression More complex now in that the probability that Y1 is related to the X variablesThis gives What we call a link function In this case logit link logirlt7rigtLogeltl 7 gt 139 280 81X182X2 8po Logit link makes it a generalised linear regression Simple Logistic Regression ExampleLastLeCture P visitation I l I I l 39 j O 20 40 60 80 Lea F height cm Figure 91 1 Relationship betmreen leaf height and wasp visitation for the C t lniw i7 Ious plant Darlingtonia califm nica Each point represents a different plant in a pnj39iij ulation in the Siskiyou Niountains of southern Oregon Ellison and Gotelii unpul quot shed data The xaxis is the height of the leaf 1 continuous predictor Variable i The y axis is the probability of wasp visitation Although this is a continuous Vill39i39v abie the actual data are discrete because a plant is either visited 1 or not 0 Logistic regression fits an S shaped logistic curve to these data Logistic Iegnsrsl39 39 sion is used here because the response variable is discrete so the relationship has my upper and lower asymptote The model is t using the logit transformation Equu tionh938 The bestd t pawnmeters using maximum likelihood by iterative fitting are E0 2 7293 and 1 01 15 The P V iue for a test of the null hypothesis that 3 0 is 0002 suggesti g that the probability of Vvasp Visitation increases xvith increasing leaf height Interesting Example Ramsey and Sohafer Famous Case of Pioneers in California stranded in the winter in the Sierra Nevada Mountains Well known case because there were suggestions of cannibalism Perhaps the results are no surprise to the women in the Class 45 Chapter 20 Logistic Regression for Binary Response Variables Display 201 Sex age and survival status of Donner Party members 15 years or over Name Antoine B reen Mary Breen Patnck B urger Charles Demon John Dolan Patiick Donner Elizabeth Donner George Donner Jacob Donner Tamsen Eddy Eleanor Eddy William Elliot Milton Fosdick Jay Fosdiek Sarah Foster S arah F0 ster William Graves Eleanor Graves Elizabeth Graves Franklin Graves Mary Graves William Halloran Luke Hardkoo Mr Herron i liam Noah J ames Keseberg Lewis Keseberg Phillipine McCuteheon Amanda McCutcheon William Murphy John Muiphy Lavina Pike Ham et Pike William Reed James Reed Margaret Reinhardt J o seph Shoemaker Samuel S mith James S n yder John Sp tzer Augustus Stanton Charles Trubode J B Williams Bay is Williams Eliza we male female male male male male female male male female female male male male female female male female female male female male male male male male male female female male male female female male male female male male male male male male male male female Survival no yes yes no no no no no no no no yes no no yes yes yes yes no no yes yes no no yes yes yes yes yes yes no no yes no yes yes no no no no no no yes no yes Donner Party Survival Plots Age and Gender logit r 1633 0078age1597f39em 105 0037 0753 A plot of the tted model is shown in Display 2010 The conclusions of the analyses in Display 206 and a statement of the summary is provided in Section 2011 Display 2010 Fit of logistic regression model for Donner Party Survival with age and fem 1 Female as explanatory variables 10Ao agoi 2 O 08 Survival 0 1 Females and 06 A Females I Estimated 0 M0100 Probability 04 000 88 0 0 a 0000 20 30 40 50 60 Age years Logistic Linear RegressionzBird Occurrence with Habitat Example Shriner Simons and Farnsworth 2002 A GISbased Habitat Model for Woodthrush in Great Smokey Mountain National Park Chapter 47 In Predicting Species Occurrences Issues of Scale and Accuracy JM Scott et al eds Island Press Logistic regression Model Y is 1 present and 0 absent Absent really means rare or hard to detectbe careful here X s habitat variables Southern Appalachian Study Sites Great Smok Mountains ational Park Pisgah and Cherokee National Forests Part of lar est rotected orested andscape In the eastern United tates International recognized re ugIa of temperate forest biodiverSIty High floral diversity Largest remaining stands of primary forest in the eastern United States Over 100 breeding bird species 75 Neotroplcal migrants Kentucky Tennessee C hatt anooga Georgia Atlanta Virginia I Roanoke Virginia Knoxville Charlotte South CaroHna 100km HUI North CaroHna WinstonSalem Great Smoky Mountains National Park National Forests Wilderness Areas Modeling Approach Logis ric regression no r Regular Mul riple Regression Model inpu r is presenceabsence which is The mos r commonly available Type of dis rribu rion da ra Minimal informa rion los r if majori ry of coun rs are 039s and 139s Nice probabili ry of Occurrence maps can be drawn Woodthrush Example Methods 3743 10 min poin r coun rs E along Trails 250500 m apar39T where all birds of all ll species coun red from 199699 Her39e focus on wood rhr39ush only De rec rion probabili ry es rima rion perhaps an issue Occupancy Models HABITAT MODEL X VARIABLES Slope ElevaTion AspecT Geology 24 DisTurbance HisTory 5 VegeTaTion Type 14 Landform Index RelaTive Slope PosiTion Topographic Convergence Index Topographic RelaTive MoisTure Index ShannonWiener Index of Topographic Complexi ry Woodthrush ExamplezMethods Included all the X s listed before plus squared topographic variables plus interactions between topographic variables Class variables were vegetation types geology types and disturbance classes Backward Elimination and Forward Selection used to aid in model selection Also maX Concordance used Model Cross Validation 1833 points in model development set 1910 points in model validation data set Random Proc Logistic in SAS Woodthrush ExamplezResults Tables 471 and 472 give resultsSigni cant vars were Elevation and its squared term Topographic complexity index ne scale landform involving elevationslope and aspect at the point and its squared term Landform index site protection ie cove slope or ridge and landform index interaction with elevation Geology Classes Disturbance Classes Vegetation VariablesVery Costly sampling at each point but were not important 534 PREDICTING SPECIES OCCURRENCES TABLE 471 Result of logistic regression model selection for wood thrush presenceabsence using model selection data Standard chi39squarf Variablea DF Parameter estimate error Wald Pr INTERCPT 1 123752 192466 49268 00264 ELEV 1 90385 19063 224803 lt 00001 ELEV2 1 lt 0001 lt 0001 433424 lt 00001 WI 1 28090 09866 81065 00044 SW12 1 00231 00075 95938 00020 Lli 1 08238 03433 57578 00164 ELLFI 1 12637 04531 77668 00053 GEOL 21 404717 00065 DISTURB 4 103836 00344 Concordance Concordant 779 percent Discordant 217 Tied 04 percent 356040 pairs Variable abbreviations are as follows ELEV elevation ELEV2 elevation squared SWI Shannon Wiener Index of Topographic Complexity SW12 SWI squared LFl Landform index ELLFI elevation LFl GEOL bedrock type DISTURB disturbance history type Disturbance History Matters Greatly Realise this disturbance was around lOOyrs back TABLE 472 Parameter estimates fer the disturbance histery type class variable Estimates are relative 113 the undisturbed type Type Parameter estimate Selective cut 013222 Light commercial cut D365 Heavy cut O3603 Settlement eree 041522 Probability map 3 draining WM 1quot ml II nal m il mm m Eliu Fiji l5 Pmbabil y 0f dangling EWGDU Thrush quot 39 028 quot DEE 5 51 LEI mi I Survey MEIER Woodthrush ExamplezConclusions Vegetation sampling Each point had veg data collected 10m radius circle Species composition 0o cover of multiple layers of forest Ted Simons notes that their results are not uncommon When compared to other studies It is not clear if the scale is too ne Probably so Woodthrush ExamplezConclusions Validation seems fairly good 86 of observed points classi ed correctly if one uses prediction probability of 03 and above to classify the point as present However not clear Why they chose this value Woodthrush ExamplezConclusions Very large study so estimates have reasonably good precision This will not always be the caseNeed much larger sample sizes in logistic regression than regular multiple regressionThis is especially so here because so many class variables Look at RSE SEEstimate of some of the key variables For example Elevation RSE 191 904 21 LFl RSE 034 082 41 20 often used as an adhoc indicator of good precision in wildlife ecology studies Woodthrush ExamplezConclusions Points were along trails and thus not random but that is probably not very relevant We are conditioning on the X s obtained anyway Spatial autocorrelation ignored Points 250500m apart but Simons suggested 1000m should have been usedSigni cance levels over stated Effective sample size less than actual sample size Probably not a large problem See Lichstein papers on Simons website for more detail Woodthrush Example Conclusions Use of Counts vs Presence Absence in Regression That are advantages of standard multiple regression vs logistic multiple regression There are pro and cons to eachThink about What they are Logistic Linear Regression SAS Procs Proc LOGISTIC Proc CATMOD Proc PROBIT Woodthrush Example Conclusions Absent does not truly mean absent but only rare here This leads to the next topic on occupancy Occupancy Methods Generalised Logistic regressions to allow for detection probability being less than 1 for a species That is apparent absence may not be true absence if species is hard to detect Truly absent What you want Present but undetected possible Ecological Metrics Different Choices for Different Situations Individual Species Metrics ST 506 Population Metrics Presence Absence or Occupancy Metric This Lectures Focus Community Metrics ST 506 Species Richness Species Diversity Species Abundance Curves One Common Theme in the Recent Research on all Metr39ics is the Need To Account For39 Detection Probability Individual Species Metrics Population Metrics Abundance Relative Abundance Indices Absolute Popn Size or Popn Density Demographic Parameters Survival Rates Recruitment Rates Movement Emigration and Immigration Sometimes We Want a Metric that Covers Larger Spatial Scales and also One that May Be Easier to Measure than these Common Popn Metrics Patch Occupancy Rate Metric The Problem Conduct presenceabsence detectionnon detection surveys for a particular species of interest Estimate what fraction of sites or area is occupied b a speCIes W when the speCIes is not always detec ed WI h certainty even when present ie plt 1 We can also relate to important covariates as in regular logistic regression Naive Occupancy rate estimates Shriner et al paper may be biased low because some sites where the species was not detected are occupied In other words apparent absence of a species from a site may ust be a failure to detect the s eCIes his could be be m ecause indIVIduals are hard to etect or because the speCIes is very rare or both Also can we detect changes in the occupancy rate temporally or spatially Patch Occupancy Metric Motivation Some Reasons why The Information may be Needed Extensive Monitoring Programs Geographic Range Changes climate change habitat fragmentation pollution Meta Population Processes Habitat Selection Invasive Species Spread Occupancy Estimation Reference MacKenzie D I Nichols J D Royle J A Pollock K H Bailey L L and Hines J E 2005 Occupancy Es rima rion and Modeling Inferring Pa r rerns and Dynamics of Species Occurrence Elsevier San Diego USA Make Sure You all Run Ou r and Buy If So I ge r Some Royal ries Model Parameters gal probability site i is occupied pl probability of detecting the species in site i at timej given species is present Both parameters could be functions of covariates Patch Occupancy Metric Key Design Issue is Replication to Estimate Detection Probability Replication is crucial if we are to separate occupancy from detection probability This will cost moreThink of the Shriner study for example Where there were 3743 points There are several types of replication possible Usual method is Temporal replication several repeat visits to sample units Within a relatively short period of time eg a breeding season Spa l39ial replica l39ion randomly select a subsample of sites Within each sample unit Observer replica l39ion have several observers go to each site independently Data Summary Detection Histories A detection history hi for each Visited site or sample unit 139 1 denotes detection 0 denotes nondetection Example detection history hi 1 O O 1 Denotes 4 Visits to site Detection at Visits 1 and 4 Data Summary Detection Histories 4 Site UJN Visits to S sites S is known Detection History 1101 1001 1100 0000 Note detections histories with all Os allowed as part of data Different to capturerecapture 1011 A Probabilistic Model Very Similar to CaptureRecapture Models in Concepts Used Sites that are occupied For example Prdetection history 1001 Pr h 1001 llz39 1711 1pzrz1 pzr3pi4i Model Key Issue Apparent vs True Absence Sites Where the species was not detected at all These sites may or may not be occupied Prdetection history 0000 Pr 11k 0000 First Term Site is Occupied but Species Escapes De39l39ec39l39ion Second Term Site is Unoccupied Model The Likelihood Function The combination of these statements forms the model likelihood S MaXim g hlh2hs HPrUzI be obta i1 Remember it may be more compeX as covariates likely modelled VarCov matrix estimated using inverse of Fisher Information or parametric bootstrap arameters can Parameters cannot be sitespeci c Without covariates 3 Visit Example All Cells If 3 Visits then there are 8 capture histories with expected numbers 111 110 101 011 100 010 001 000 anmm NWp1p21p3 pr11 p2p3 Nw1plp2p3 pr11 p21 p3 NW139p1p21p3 Nwlple p pg NW11901 p21p31 w We can estimate the parameters l p1 p2 p3 0r submodels using ML methods Model The Likelihood Function The combination of these statements forms the model likelihood S ugg h1h2hs 1 1pm i1 Maximum likelihood estimates of parameters can be obtained VarCov matrix estimated using inverse of Fisher Information or parametric bootstrap However parameters cannot be sitespeci c Without additional information covariates Model Assumptions The detection process is independent at each site No heterogeneity that cannot be explained by covariates Sites are closed to changes in occupancy state between sampling occasions That is sample in a short time Window Model Covariates Sitespeci c model wandor p eg habitat type patch size patch isolation Surveyspeci c model p eg local environmental conditions water temperature Use link function such as logistic 630 31 32 pij 1e 0 1xi 2y1j Model Software Windowsbased software Program PRESENCE J Hines ampD MacKenzie is available at the Patuxent Software Site Program MARK G White Fit both prede ned and custom models with or Without covariates Provide maximum likelihood estimates of parameters and associated standard errors Assess model t Simple Example Anurans at Maryland Wetlands Droege and Laehman Frogwatch USA NWFUSGS Volunteers surveyed sites for 3minute periods after sundown on multiple nights S 29 wetland sites Piedmont and Coastal plain 27 Feb 30 May 2000 Covariates Sites habitat pond lake or swamp marsh wet meadow Sampling occasion air temperature Example Anurans at Maryland Wetlands Droege and Lachman American toad Bufo americanus Detections at 10 0f 29 sites Naive 1029O34 Example Anurans at Maryland Wetlands B americanus Model AAIC I whabptmp 000 050 013 Jptmp 042 049 014 1habp 049 049 012 kJp 070 049 013 Naive Example Anurans at Maryland Wetlands B americanus NoTice ThaT The na39I39ve occupancy esTimaTes have a serious negaTive bias 034 vs 05 NoTice ThaT choice of model has liTTle effecT on The esTimaTe or iTs SE NoTice ThaT 529 is a very small no of siTes To moniTorll Thus The SEs are quiTe large under all The differenT models The largesT RelaTive SE is abouT 28 quot0 O14O49 Also This is why The AIC is having Trouble disTinguishing beTween The models Clearly This esTimaTe will be of mosT use an ecologisT if compared To esTimaTes in differenT years in a moniToring program To deTecT Trends in occupancy Model assumptions Closure Surveys are independent No unmodeled heterogeneity Closure What if Occupancy changes during the survey Is a season de ned appropriately The max length of time that is reasonable will depend on the species Lack of independence Surveys are not independent if the outcome of survey A is dependent upon the outcome of survey B Usually parameter estimates may be OK but standard errors too small Unmodeled heterogeneity In occupancy probabilities parameter estimates should still be valid as average values across the sites surveyed In detection probabilities occupancy will be underestimated similar to capture recapture studies covariates may account for some sources of variation Another Example Giant Weta Study An ancient giant insect Order Orthoptera now endangered in NZ due to introduction of small rodents P116 in Occupancy Book Therefore the need to look at occupancy within a reserve and how it is affected by habitat browse ys non browse by cattle and goats S72 sites which are 3m radius plots searched between 3 and 5 times in March 2004 Other Examples Giant Weta Analysis Occupancy covariate browse or no browse Detection probability covariates day and observer Best Model using AIC is VBrowse pday observer Other Examples Giant Weta Analysis Occupancy Estimation Naiveq 049 SE 006 t7Browse 077 17No Browse 050 19 063 SE 008 Detection Probability Estimation 13 varied widely from 010 069 That is why the occupancy estimate of 063 is so much higher than the naive occupancy estimate of 049 Design Issues for Occupancy Studies What is a season How to de ne a sampling unit Selecting sampling units Repeat surveys More units vs more surveys What is a Season A season is a period of time during which it is reasonable to assume occupancy is static or changes occur completely at random Depends very much on the target species and study objective How to de ne a sampling unit Should be assessed on a casebycase basis Large enough to have a reasonable probability of occupancy but not so large that any measure may be meaningless Size matters How to de ne a sampling unit Is there a natural de nition Pond At What scale do you want to measure occupancy Is the species territorial What density does it occur at What is the size of it s home range Selecting sampling units How units are selected determines how results can be generalized Each sampling unit within the population should have a nonzero probability of being selected If units are selected such that occupancy is different than for the population of interest estimates may be biased eg surveying only at historic sites Repeat Surveys Repeat surveys do not necessarily imply repeat visits Discrete visits Multiple surveys Within single visit Single observer conducting multiple surveys Multiple observers each conducting a single survey Multiple survey plots Within a larger sampling unit Allocation of effort For a standard design there is an optimal number of repeat surveys per unit The optimal number depends upon values for l and p Does not depend upon number of units or total number of surveys Reasonably robust to effect of cost Number of Surveys L5 L6 L7 L8 L9 26 L3 L4 16 L2 15 OJ 18 20 23 34 16 10 17 14 13 11 10 OJ L2 L3 L4 L5 L6 L7 L8 L9 HOW many units Once the number of repeat surveys has been determined how many units to survey can be determined from the variance equation A 11 Varw 1 W 19 Kp1pK 1 pi 11p1lt How many units Example if I z 7 any p z 04 should use 5 surveys per unit p 092 To achieve a SE of 004 use S183 based on Equation 61 in the reference book and on the previous equation General recommendations on allocating effort When detection probability is gtO5 at least 3 surveys per unit More surveys will be required when p is lower For rare species survey more units less intensively Increasing spatial replication with insuf cient repeat surveys may not be worthwhile General recommendations on allocating sampling effort Example if l m 04and p z 03 011 o 007 Surveying 80 units 5 times gives a decrease of 36 Surveying 200 units twice gives SEN SE With only 2 surveys per unit would require 500 units to achieve same level of precision or increase total effort 250 Nonstandard designs Repeatedly surveying a subset of units and elsewhere only once does not generally provide a more ef cient design Surveying a unit repeatedly until rst detection up to a maximum may provide a more ef cient design but may be less robust Note the optimal maXimum number is higher than values given in the previous table The standard design Where each site visited the same no of times is probably best Final Comments on Design Designing studies tends to be an iterative affair Simulation and pilot studies can provide useful information on how designs and eld methods are likely to perform Program GENPRES Hines is available at the Patuxent Software Site Multiple Seasons Changes in Occupancy There are multiple Visits Within each of several seasons Fig 71frorn Mackenzie et al 2005 attached Within a season we have closure While between seasons there can be local extinction 8 and also local recolonisation y Fig 71 and 72 attached This is similar in concept to the robust design used for capturerecaptureCh 19 Text and for looking at changes in communitiesCh 20 text Conclusions A generalised logistic regression analysis which always for accounting for uncertain detection There is a cost to replication There is a value to replication unless the detection probability is close to 1 There is good software available to compute the estimates LECTURE 10 CLOSED CAPTURERECAPTURE REMOVAL MODELS CATCH EFFORT MODELS SELECTIVE REMOVALS Some Review of Last Lecture MARK RABBIT EXAMPLE Model Selection We tted Mt and MO Rabbit Analyses for 506 Delta AICc Model Model AICc AICc Weight Likelihood Par Deviance 536099 000 100000 10000 30000 5548 469416 6668 000000 00000 20000 74299 Note How Mt so much favoured Lets look at each output MARK RABBIT EXAMPLE Model Output Mt Rabbit Analyses for 506 Real Function Parameters of Mt 95 Con dence Interval Parameter Estimate Standard Error Lower Upper 1p 05200943 01359513 02714729 07591460 2p 00866824 00309946 00422018 01697371 3N 16150918 40414581 11580591 29141774 Note estimate of N we get from the Lincoln Peterson MARK INSECT EXAMPLE Data format N0tepad File Capture history No animal gp1M gp2F Note at end of line 1063 76 0125 37 11 1419 P picivorus Male and Female Data Delta AICC Model Model AICc AICc Weight Likelihood Par Deviance LPCombl318987 000 086021 10000 40000 12354 LPBoth 1315353 363 013979 01625 60000 11891 P picivorus Male and Female Data LP Both Sexes LP Comb Sexes Parameter Estimate SE Estimate SE 1p Ml 037 0077 035 0049 2p M2 018 0044 019 0030 3p Fl 034 0063 035 0049 4p F2 020 0041 019 0030 5N M 21086 4054 21268 2909 6N F 27653 4590 27538 3645 Note 1amp3 and 2amp4Lets simplify by putting them equal in the combined analyses Gains in precision The Estimates from Best Model P Pioivorus Male and Female Data Parameter 11p M 21p F 3N M 4N F LP Sexes Combined Estimate 03524099 01946450 21268334 27538463 St Error 00492986 00303306 29093079 36456268 Taxi cab Data Part of Input File Carothers 1973 Scheme A taxicab data true population 420 occasions10 groups1 1 0001000001 1 1000100000 1 0000010000 1 00010010001 0000000001 1 11010100001 0010000000 1 8 01010000101 There are 283 recordsone for each taxi cabs history note the format used for comments also this one does not use summary format like the rabbit and insect data OUlIgtUJN Taxi Cab Example PIMS M Mt PIMS Structure p 12345678910 c 2345678910 N 11 Note In the full model c PIMS 11121314 15 16171819 with N PIM 20 M0 PIMS Structure p 1111111111 c 111111111 N2 Mb PIMS Structure p 1111111111 c 222222222 N 3 Summary of Some Output From CAPTURE Inside MARK MO 368 with standard error 144896 underestimates when there is heterogeneity Model Mh Suggested for use here from old Model selection Procedure J acknife 471 with standard error 3632 Chao 407 with standard error 2742 From MARK Directly with PIMS Model MO 368 with standard error 144896 underestimates when there is heterogeneity Can also t MO and Mt with PHVIS Can also t Finite Mixture approach for heterogeneity but did not work too well here Nhat 463 with huge SE of 27388 using the Closed Captures and then with Heterogeneity procedure Will not insist you know this one in detail SummaryClosed Capturerecapture Models Summary Closed CaptureRecapture Design Issues Precision Issues Need adequate capture probabilities and numbers of samples to estimate standard errors that are small enough ie RSE 20 is good level to strive for Look at Tables in Otis et al 1978Note that good model selection requires much larger capture probs than just estimation under one assumed correct model Full Simulation Studyideal Simpler Approximation Use Expected Values for guesses of What the data might be like and do analysis on that data using MARK or CAPTURE The precision SEs you get is a fairly good estimate as to What you would get if you ran real data With those parameter values Use of Expected Value Method for Precision Evaluation Example N500 Lincoln Petersen Study Equal capture probability case N500 p05 for both periods then n1250 n2250 m2125 Mt Estimate 4985 SE 222 RSE 4 Summary Closed CaptureRecapture Design Issues Minimise Model Bias Satisfy Assumptions 1 Closure Short studiesno mortality no recruitment no immigration or emigration Check With telemetry sometimes 2 Equal Catchability Heterogeneityoften hard to avoid unless one can use different methods of capture in each sample Which is not usually feasible Rerandomise trap locations each time Collect covariate data for Huggins method or to stratify on Trap Response often hard to avoid unless one can use different methods of capture in each sample Which is not usually feasible Time Variationtry to eliminate so that simpler models can be used 3 No Tag Loss Obviously avoid check out in pilot studies Use double tagging method to estimate tag loss if it is a problem Use of Expected Value Method for Model Bias Evaluation Example N500 Lincoln Petersen Study Where there is heterogeneity with two groups Average p05 Group 1 250 animals p 09 Group 2 250 animals p 01 n1250 n2250 m2205 Estimate 304 Bias 196 Equal capture probability case N500 p05 then n1250 n2250 m2125 Estimate 498 Bias O Use of Expected Value Method for Model Bias Evaluation Effect of Heterogeneity on LP Estimator 2 Groups N5 00 p 09 01 Bias 196 p 08 02 Bias 133 p 07 03 Bias 70 p 06 04 Bias 21 p 05 05 Bias 0 DENSITY ESTIMATION WITH CLOSED CAPTURERECAPTURE 143 Nested Grids To account for edge effects Trapping Webs Uses Distance Ideas Nested Grids Fig 142 X X X X X X X X d K X X X X X X X X X X X X X 1 X X X X X X X X X X X X X X X X X X X X X X X FIGURE 142 Nested trapping grids a 4 X 4 grid b 6 X 6 grid c 8 X 8 grid d 10 X 10 grid After Otis et 111 1978 Trapping Web Fig 143 FIGURE 143 Schematic diagram of a trapping web with 16 lines each of total lungtlx AT wit 20 traph per line after Anderson 39I ul 1983 Traps are equally spaced along each line Points equidislanl be tween traps are denoted by b with 1 representing the Center 0ft e we and lb located just beyond the last trap Captures in lhe eleventh ring of traps are assigned to he annulus A which has urea ha 7 70 After Anderson et MI 1983 REMIWAL DATA Nonselective Selective Removals Removals 144 or Changeinrati0145 N onselective Removals Equal Sampling Effort Unequal Sampling Effort Controlled studies Uncontrolled Studies Small mammal trap grids Commercial sheries Electro shing in small streams Hunter species with control through complete check stations REMOVAL MODELS CATCH EFFORT MODELS Closed Closed Closed Population with Constant Effort 1 Two Sample Case Example Maecolaspis aVida a beetle removal data from rst two sets of sweeps of a net Each set is four sweeps Menhinick1963 Ecology 44 617621 2 2 2 N n1 2 135 135 230890 nl n2 135 76 59 132111 112135 76 59 204370 n1 135 135 Closed Population with Constant Effort Example Maecolaspis aVida a beetle removal data from rst two sets of sweeps of a net Each set is four sweepsMenhinick1963 Ecology 44 617621 minim n2 n1 n2 4 1352762211 594 SEN 4281 Closed Population with Constant Effort 1 Two Sample Case Assumptions Closure Capture probability p is constant over animals Capture probability constant over time Failure Sometimes if p is low even if the assumptions are valid the method will fail if 111 112 or 111 lt 112 MARK Trick to Get this Model Output Input File 10 135 01 76 Running Model Fix e0 MARK Output Removal Example MehinickzMARK Estimate SE Lower Upper 1p 045 00793 030 060 20 0 Fixed 3N 3027 3982 2516 4181 Note parameter 2c set to 0 Closed Population with Constant Effort 2 Multiple Samples better precision M closed population N except for removals constant p all animals all samples m n1 n2 nk removals Approaches Regression Approach First Then Use of Mb and Mbh Time Popn Size Observed Expected removals removals l N r11 Np 2 N39ni 1 12 N39n1p i N39Xi n1 N39XDP Xi cumulative catch r1i catch Estimation a Regression Model E Np pxi intercept slope catch cumulative catch 5 515 Estimation b Use Program CAPTURE Mb Maximum Likelihood Removal by marking is statistically equivalent to removal physically Could use Model Mb 39 39 TURE Mbh Maximum Likelihood can extend this removal model to Mbh that is there is gf capture probabilities over animals However each Wintcgapture probability over samples 7 r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r J USE of CAPTURE on the Web Example of the Removal Models Mb Mbh task read population zippen 7 Whitefish data 2526151312135 Task population estimate zippen Line 2 7is the no of occassions Line 3 Entries are the no of removals on each occasion For model With heterogeneity Mbh replace zippen With removal Example White sh Coregonus clupeaformz s Ricker 1958 150 A small lake on an island in Lake Nipigon Ontario was shed by gill nets in an identical manner for 7 successive weeks the same sizes of nets positions and lengths of sets were repeated each week For White sh of fork length 1314 inches 3335 cm the weekly catches and and cumulative catches are given in the table below Week i 1 2 3 4 5 6 7 1 Catch ni 25 26 15 13 12 13 5 catch Xi Cumultive O 25 51 66 79 91 104 30 Catch lO 20 4O 6O 80 Cumulative catch 100 120 Plot of catch ni versus cumulative catch Xi for a population of White sh data from Ricker 1958 WHITE FISH Physiea1 Removal Ricker 1958 p 150 1 2 3 4 5 6 7 Catch 25 26 15 13 12 13 5 Cumulative 0 25 51 66 79 91 104 catch Number removed 109 See attached output for details Mb N 138 SEN147 M A A bh N 138 SEN147 NO HETEROGENEITY DETECTED IN THIS EXAMPLE A The regression estimate was N 1 Mehinick 10 Period Removal Data Removal Data for k10 were 726344323l23l7181113 Regression Approach 0 Use of MARK or CAPTURE is better Mb Mbh not shown You could run it yourself in CAPTURE if you want 0 Regression Approach 5 I 2 K r 5 p lt 0 o N o IOO 200 300 400 N Z NUMBER RE MOVED PREVIOUSLY FIG 1 Progressive decrease in numbers of M aecolaspf avida Say caught in successive sweeping period In a 225 m2 area of a Sericea lespedeza stand I Mehinick 10 Period Removal Data Model Mb in MARK 0 xed at 0 Parameter Estimate SE Lower Upper 12p 01971 0017 01652 02334 20 3N 3641 1199 3466 3952 Closed Population Multiple Samples with Variable Effect gag n1 n2 nk removals f1 f2 fk effort in each sample Parameters N initial population size pi 1 3 z kfi when k is small T 1 Probability of capture Catch coef cient Two parameter Model N k Model Assumptions Closed population except removals 2 All animals all samples k constant 3 The relationship for pi holds ie fish caught independently 4 Information on effort and catch complete fro commercial sheries Estimation We can generalize earlier Regular Model Time Popn Size Observed Expected l N n1 Nkf1 2 N n1 n2 N n1k f2 l N39 Xi ni N Eni f1 Nk 39 kXi Linear Relationship Example Common Crab Fishery Example Blue Crab Callinectes sapz39dus Fischler 1965 I The total catches per week for 12 weeks from a commercialsize male crab population are given in following Table I Three different kinds of gear were used trotlines crab trawls and shrimp trawls I From the ratios of total catch over total effort for each gear the relative fishing powers of the 3 gears were 100 142 and 106 I For example one crab trawl day was equivalent to 142 trotlines operating for 1 day I These powers were then used to convert the weekly fishing efforts of crab and shrimp trawls into standard units of trotlinesday Example Blue Crab Callinectes sapz39dus continued I The assumption of a closed population seems reasonable because i Natural mortality was negligible ii The catches ni Table have been corrected for recruitment from precommercial to commercial size so that the population under study was the population of commercialsize male crabs at the beginning of the experiment iii Only the male population is considered as tagging studies indicated that commercialsize females were emigrating out of the area while the shery was operating Catcheffort data for a population of commercialsize male crabs from Fischler 1965 Table 11 1 Hi Yi Xi week 1b effort nifi 1b 1 33 541 194 1729 0 2 47 326 248 1908 33 541 3 36 460 243 1500 80 867 4 33157 301 1102 117 327 5 29 207 357 818 150 484 6 33 125 352 941 179 691 7 14191 269 528 212 816 8 9 503 244 389 227 007 9 13115 256 512 236 510 10 13 663 248 551 249 625 11 10 865 234 464 263 288 12 9 887 227 436 274 153 200 I l 50 Catch per unit effort 100 50 50 100 150 200 250 300 350 Cumulative catch Plot of catch per unit effort nifi versus cumulative catch Xi for a population of male crabs redrawn from Fischler 1965 A Regression N 330 300 95 CI 299600 373600 Another Example Gould W R and Pollock K H 1997 Catch effort maximum likelihood estimation of population parameters Canadian Journal of Fisheries and Aquatic Science 54 890897 I would like you to check this for your self in the text TABLE 1411 Catch Effort and Temperature Data for a Commercially Harvested Lobster Population Period Catch 11 Effort 12 Temperature ti 1 60400 33664 79 2 49500 27743 77 3 28200 1 7254 63 4 20700 14764 35 5 11900 11190 31 6 15600 16263 29 7 13200 14757 31 3 25400 32922 3 256 9 29900 4551 9 3 4 10 32500 43523 36 11 24700 37478 40 12 27600 43 367 59 13 22200 37960 61 At Port Maitland Nova Scotia Canada 1950 1951 reanalysis of data after Paloheimo 1963 cited in Gould and Paloheimo 1997b bEffort is in thousands of trap hauls e out This value was missing so we used the average of the two adjoining periods TABLE 1413 Comparison of Parameter Estimates Standard Errors for Three Models for a Commercially Harvested Lobster Population N o 31 2 Effort plus temperature model pf 549974 394 0030 011 47780 0182 0004 002 Effort model pf 472270 329 0037 21840 0108 00036 Constant probability model p 716860 289 84200 0172 1 AI Port Maitland Nova Scotia Canada 1950 1951 reanalysis of data after Paloheimo 1963 cited in Gould and Paloheimo 1997b SELECTIVE REMOVAL OR CHANGEIN RATIO MODELS Different ages or sexes are removed selectively so the proportions of the ages or sexes change after the removal Deer with a controlled hunt is a good example Removals selectively made towards males in harvest Removals harvest are known Also need an unbiased estimate of the proportion of males both before and alter the hunt Estimate Removal Iiializ lizrii l39riz Time 2 Sample Estimate Motivation for Estimation Approach 2 1 N1 2 N21 2 N11 7i1 N1p1r11 p2 N2 N17i N17i Rearrangement gives N1 Iii F1172 71 172 Theoretical Result N r11 r1172 1 p1 p2 Estimation A r rf9 11 12 Nl A A 171 172 N1 2 This is Equation 1440 p327 Theoretical Variance for N1 Nil6171131 N 22 V0413 p1 p2 2 VaFUVI Note the importance of the difference p1 p2 to the variance There must be a change in ratio or the method wont work at all Selective Removal Assumptions 1 Closed Population except for the removals Both types of animals are equally catchable or sightable in each of the two samples before and after the removals The number of removals is exactly known Remington Farms Deer Herd Example Pre Hunt Post Hunt 31 00963 92 00381 t r11 56 Antlered r21 54 Antlerless r1 110 56 110x00381 00963 00381 2 890 total deer N11 2 86ant1ered deer 890x0963 A721 2 804ant1er1ess deer N1 Note BOX 5 next gives the SEs and a summary of the example in a slightly different notation r Box 5 Using the CIR method to estimate pop ulation size of an ered and antlerless deer Antlered actype and antleriess39 1 ytype deer were observed during 54 prehunt and 52 posthunt road counts on Remington Fanns Maryland Con ner et a 1986 One hundred and twenty xtype and 1126 ytype animals were observed before 56 antlered deer Rx and S4 an erless deer Ry were removed by hunters during a 1week season After the season 43 x type and 1086 ytype deer were observed Therefore 102 x11 1201246 00963 132 32112 431129 00381 NA Rx quot39 R182V081 2 56 110o0381o0963 00381 51309109532 890 575 149 51 1353 00963890 86 53913 14 In this example both 1 and 3 types were re moved Because the ytype annexless animals were probably more observable than the xtype an imals 1 was probably gt11 and the population estimates were likely to be biased high Validity of Assumptions in the Example 1 Closed Population except for the removals OK Fairly short time period perhaps some migration Both types of animals are equally catchable or sightable in each of the two samples before and after the removalsAnt1erless more sightable than antlered deer and this probably means the estimates are biased high The number of removals is exactly knownOK as controlled hunt Lecture 13 Sampling Design Issues Strati ed Random Sampling Comment on the BACI Homework and Critiques Brief General Comments on Jim Gilliam s Lecture last Thursday Important Basic Sampling Designs Stratified Random Sampling Maj or Focus Special Issues on Sampling of Animals Detection Probability Models Methods of Estimating Detection Probability Later Lectures 9 Will consider a few of the BACI Designs you found This should help us see the breadth and complexity of their use 9 I will consider a few of the papers that you critiqued that bring out important points I was very happy with the way you handled the critiques and they were of a very high standard 9 I still have a few more to finish grading by Thursday Jim Gilliam Lecture Discussion 9 He uses a mix of Experiments and Observational Studies and suggests iterating between the two Nick Haddad made a very similar point 9 My opinion is that a strong overall research strategy will arise if it includes and integrates Mathematical Modelling of Popn or Community Dynamics EXperiments often on a component of the overall puzzle Strong Inference due to replication randomisation and control Often on a small spatial and temporal scale Observational Studies on larger more realistic spatial and temporal scales Weaker inference due to possible confounding effects will lead to the need to iterate back to the other approaches to understand and clarify confounded effects Jim Gilliam Lecture Discussion O Observational Studies He focused a lot on principal components analysis as a way of reducing dimensionality 91 will explore and illustrate this at some length later in the semester I will talk about Principal Components Analysis and then using Principle Components in a multiple regression model All Sampling Design Topics Basic Sampling Designs Simple Random Sampling Systematic Random Sampling Strati ed Random Sampling A Brief Review Simple Random Sampling We began by considering simple random sampling Without replacement which is the simplest probability sampling method It is analogous to using a completely random design in experimental design It is most useful if our sampling units are homogeneous Now Consider the Simplest Possible Problem Estimate the Population Mean u Simple Random Sampling Estimation of the Population Mean Represent the Population y1 y2 yN Population Mean Finite Population Parameter N 2 Yr 1 N Represnt the Sample by y1 y2 yn Sample Mean Estimate of the Parameter 2371 l y I is an estimate of u Simple Random Sampling Standard Error of Sample Mean Standard Result SE of 2 Finite Population Result SE of 2 NN n Note The second term is called the nite population correction factor Simple Random Sampling Another ProblemEstimate Population Total Population Total Parameter This is a very important new parameter in nite populations N 1 Population Total Estimate f W SE N SEW ltNltN ngt n Simple Random Sampling Sample size needed for desired precision p 6465 Example 2 n39n1nN 1quot 20m 196 N 910 CV 0486r 01 2 n 196x0486 Q91 010 n392L283 191910 smaller than 91 Systematic Random Sampling Systematic random sampling often gives a better spatial coverage than simple random sampling Here is an example Think of sampling along a transect of length 100 meters Where you start at a random point in rst 10 m 7 meters from Excel and then every 10th meterThe systematic random sample will be 7172737475767778797 1 also chose a completely random sample of n10 using Excel 18203359638590919296 Notice clumping along transect random does not mean uniform NOTE Only major danger of systematic random sampling would be if there is some cyclical pattern of response along the transect This does sometimes happen Strati ed Random Sampling The population is divided up into L homogeneous strata The stratum sizes are N1 N2 NL Within each stratum a simple random sample of size n1 n2 nL is taken It is important to realize that the sampling is independent in the different strata Note Analogous to randomised complete block design in experimental design Focus of Todays Lecture Strati ed Random Sampling Why Stratify Strata of Interest Increase Ef ciency of Overall Population Estimators Strati ed Random Sampling How to Stratify and How Many Strata Pick Homogeneous Strata Usually Use 510 Strata Strati ed Random Sampling Purposes of Strati cation Strata of Interest Increase Ef ciency of Overall Population Estimators How to Stratify and How Many Strata Pick Homogeneous Strata to increase ef ciency Usually Use 510 Strata If too many then the sample size within strata is too low Strati ed Random Sampling Estimation Methods Individual stratum means totals and variances Over all population means and totals Estimation of Overall Population Mean u I yst ZZZ1 This is a weighted average of the individual stratum means Here WI 2 N l N Estimation of Overall Population Mean n 7 ZLIVVJ C Varm ZleWfVarm 2 l n l Estimation of Overall Population Mean u Example Williams et al 2002 P66 N1 90N2 100N3 400N4 320N910 721 221712 221723 221714 20n83 l 4 W1 010W2 011W 2044W4 20352W 1 1 71 2050 72 1500 73 3000 74 2100 7 010205 0111 0443 0 03 521 2 2425 Estimation of Overall Population Mean u Example Williams et al 2002 I Valm Wm Ni nz39 N n Val2 N1 9ON2 100N3 400 N4 320N 910 ml 21n2 21n3 21n4 20n83 W1 2 O10W2 O11W3 O44W4 035 512 2160522 90 532 90542 40 Varyst O11Check yourself if you like Example based on Helicopter Survey Stratified Random Sample Very Widely Used Mule Deer Helicopter Example Kufeld et a1 1980 Journal of Wildlife Management 44 632639 0 8 strata of different sizes based on different regions in different habitats 0 Sampling Unit is a plot Where a complete count of mule deer is made 0 Very good example of use of the standard sampling methodology Individual stratum estimates and then overall population estimates sum of the stratum estimates Reasonable precision of estimates 0 No adjustment for detectability Assumes all animals seen in each plot so there is likely a negative bias on the estimates Table 1 Esdmalcs of the wintering mule deer population bascd on a suati cd random samplc of clgm swam on me uncompangr Plamau Colorado 1977 79 from Kufcld cl a1 1980 Stratum Paramster Year 1 2 3 4 5 6 7 8 Total Area kmz All 133 285 262 317 150 207 103 231 1688 No quadrats Total All 206 440 405 489 232 319 159 357 2607 Samp1q All 12 35 23 38 13 25 9 38 193 Deer popuiation 1977 189 289 581 1361 428 2386 724 5440 11401 estimates V 1978 275 1307 352 3861 892 4772 901 5524 17884 39 1979 155 1446 669 2458 339 5589 247 6182 17085 90 CL 1977 39 22205 1978 14042 12951 1979 Example based on Helicopter Survey Detection Probability Issues There is no adjustment for detectability Assumes all animals seen in each plot This is likely untrue There is likely a negative bias on the population estimates We now take a short detour into the Whole issue of the estimation of detection probability because it clearly is so central to design in Wildlife and sheries studies ST 506 will involve study of this topic in far more detail Components of Sampling Design Standard Finite Sampling Methods Need to understand some of the key concepts of nite population sampling Absolutely crucial for any graduate student in ecology However often cannot apply Without modi cation to animal sampling Special Issues on Sampling of Animals Plants don t move Animals do and are often hard to detect We have to adjust animal counts to obtain good inferences This requires detection probability be estimated A very difficult and complex task Covered in detail in ST 506 but it has to be mentioned here as it can be so central to your design Again absolutely crucial that you be aware of this issue OVERVIEW OF METHODS OF ASSESSING POPULATIONS Direct methods of monitoring Census Method Count all animals in the population a usually unrealistic practice Sampling Methods Count animals in sampling units or areas Absolute Abundance Relative Abundance Estimate popn size by adjusting Use incomplete count as for unseen animals and only an indeX of population size part of area being sampled OUR MAIN FOCUS CHEAPER BUT PROBLEMS CENSUS Usually it is impossible to count all the members of a wildlife population for practical reasons Exceptions are small localized populations of highly Visible and valuable endangered species Examples are the California Condor the Puerto Rican Parrot popn size in wild is only around 40 birds ABSOLUTE ABUNDANCE ESTIMATION Not all area sampled and not all animals seen N Ca C count of animals seen a fraction of area sampled relates to nN in s r sampling A 3 estimate of the fraction of animals seen or caught Note there are many ways to estimate 3 For example capture recapture line transects etc ABSOLUTE ABUNDANCE ESTIMATION Example Aerial survey of caribou in Alaska C count of caribou seen 551 a fraction of area sampled 01 say n150 N1500 plots 3 estimate of the fraction of animals seen or caught05 Note there are many ways to estimate 3 For example two independent observers in the planeWe will discuss details of this later NzCa 5510lX05 551x 10 x 2 211020 ABSOLUTE ABUNDANCE ESTIMATION CAPTURE METHODS CaptureRecapture Removal and CatchEffort ChangeinRatio or Selective Removal COUNT METHODS Line Transects Variable Circular Plots Double Sampling eg ground aerial Counts from multiple observers with mapping All can be viewed as methods to estimate 3 in NzCa Complete Counts on a Subsample of Plots Example Aerial Survey with Complete Ground Counts on a Subsample oFPlots Description Key Assumption The complete count is a truly complete accurate count on the same plot at the same time as the aerial count Key Questions 1 Is it possible to do an accurate complete count in aerial survey applications it is often impossible to do a complete ground count even in a Few plots 2 Fit is possible is it possible at reasonable expense For a reasonable size sample Otherwise 3 based on a very small subsample will be very imprecise Complete Counts on a Subsample of Plots Example Aerial Survey with Complete Ground Counts on a Subsample oFPlots Numerical Example Thompson 1992 and p 251 252 Williams et al Aerial Survey alone n 20 plots out of N 100 plots in whole area 240 moose detected 100x240 20 detection probability 2 1200 moose estimated if ignore a O2 100 Ground Survey on Subsample of 5 plots E air count 7 0 2 ground count Estimation of Population Size N12901500 8 080 01 Ni C 240 1500 a I 02x08 Complete Counts on a Subsample of Plots Example Arctic Breeding Bird Survey with some plots searched twice All plots standard search Some plots more thorough search problem is itthey are really complete counts Another MethodWe Will return to Aerial Surveys CAPTURERECAPTURE MODELS LINCOLNPETERSEN MODEL N Population size n1 No of marked animals in the population n2 Sample size m2 No of marked animals in the sample Sample Population m 2112 n1N N nlnzmz CHAPMAN S MODIFICATION TO REDUCE BIAS m1 lLl n2 l N l NCZW4 m21 This estimator is approximately unbiased Unbiased If you repeated the study many times and took the average N it would be equal to N PETERSEN MODEL ASSUMPTIONS 1 Closure 2 Equal Catchability 3 Zero Mark Loss Marking is de nitive Now returning to Aerial Surveys Multiple Observers Apply the LincolnPetersen mode n1 seen by rst observer 112 seen by second observer m2 seen by both observers m2 I91 quot2 Multiple Observers Two independent observers Key Assumptions 1 No matching errors 2 Closed population 3 The observers are really independent Multiple Observers Two independent observers Examples aerial surveys Two observers in same plane One observer 1n plane one on ground we gs Examples Bird point counts Two observers at the same points Multiple Observers Two independent observers Example Henny and Burnham 1971 JWM Osprey nests 0c 2 1 ie total area is sampled there is one air observer and one ground observer Data n1 51 seen by air observer n2 63 seen by ground observer m2 41 seen by both observers Two independent observers 39 A quot11n211 0 1 52x64 1 7824 42 SEZ7 311 A n1 51 T 065 am N 7824 A 112 63 T 081 gmqu N Modeling Availability amp Perception Detection Processes Two Processes Availability animals have to be available to be detectedIn many animal populations not all animals are availableeg dugongunderwater birds don t sing Perception even if animals are available then they still have to be detected This is also uncertain a dugong on the surface may still be missed MODELING OVERALL DETECTION PROBABILITY The probability of detection is made up of a probability that the area is sampled plus an availability process and a detection process P animal gr PHIL ea gamipu quot Estimating Availability amp Perception Detection Processes Availability Very hard to estimatespecial studies For example one can use the robust capturerecapture design Often assume all animals are available and do nothing which may not be good depending on the application Perception The methods I mentioned earlier can be used to estimate this quantity and often it assumed to be the total detection probability Some References on Detection Probability Buckland ST Anderson D R Burnham K P Laake J L Borchers D L and Thomas L 2001 Introduction to Distance Sampling Oxford University Press Oxford Pollock K H Marsh H Bailey L L Farnsworth G L Simons T L and Alldredge M W 2004 Separating components of detection probability in abundance estimation An overview with diverseex l I W L Thom on Ed quotampling Rare and Elusive Species Concepts Design fulation Parameters Island Press Washington D C Bailey L L T R Simons parameters for plethodonf Journal of Wildlife Man39 Pollock K H Marsh H Y 006 Modeling availability and perception processes for strip and line tr niectst an application to dugong aerial surveys Journal of Wildlife Management 70 255262 nig 39 e tectability capture design Simons TR MW Alldredge K H Pollock and J M Wettroth 2007 Experimental analysis of the auditory detection process on avian point counts The Auk 1243 986 999 Strati ed Random Sampling Total Sample Size and Sample Allocation Issues for Strata Strati ed Random Sampling Sample I Allocation Rules to Obtain Better Precision Equal Allocation allocate the sample size equally in all the strata Proportional Allocation allocate proportional to the size of the stratavery Widely used Optimal allocate proportional to size and stratum variances and inversely proportional to costs in the different strata Strati ed Random Sampling Sample I Allocation Rules to Obtain Better Precision Equal Allocation allocate the sample size equally in all the strata ONot usually sensible unless all strata are equal size in terms of overall estimates precision 9 However maybe good if you want to compare stratum means as your primary focus Strati ed Random Sampling Sample Allocation Rules to Obtain Better Precision Proportional Allocation allocate proportional to the size of the stratavery Widely used Commonly usedtakes account of stratum sizes being different nl nNlNnWl Strati ed Random Sampling Sample Allocation Rules to Obtain Better Precision Proportional Allocation nl nNiNnWi Example with 4 strata n1 2 83x010 n2 2 83x011 n3 2 83x044 n4 2 83x035 n1 2 83 n2 2 91 n3 2 365 n4 2 2905 Note the weights given earlier in lecture Use rounded values n1 28112 29113 237114 29 for total 11 83 Strati ed Random Sampling Sample Allocation Rules to Obtain Better Precision Optimal Allocation O Takes account of stratum sizes different variances and different costs of sampling in different strata see p 67 of Williams et a1 2002 reference 9 Optimal Allocation is not used as much as proportional allocation but it can result in a gain in precision if the costs and variances are known or well estimated from a prior study or a pilot survey Thursday Ted Simons Lecture Simons TR MW Alldredge K H Pollock and J M Wettroth2007Experimental analysis of the auditory detection process on avian point counts The Auk 1243 986999 ST 506 Homework Set 10 2008 Due Tuesday November 4 2008 Q1 Program MARK provides AIC model comparisons for an alligator population at Lake Ellis Simon North Carolina subject to capturerecapture from 1976 to 1979 by Manley Fuller a former Ph D student of Dr Phil Doerr There was just one capture period each year Four models are tted using varying assumptions about constancy of survival and capture probabilities over time Then the parameter estimates and SEs for each model are also given Here we just consider survival and capture probability estimation based on the recaptures Model Delta AICC AICc Parameters l p 18583 000 l pt 18854 271 4 lt p 18921 337 4 I t pt 19062 478 5 Real Function Parameters of Phit pt PIM 95 Con dence Interval Parameter Estimate Standard Error Lower Upper 1Phi 07352941 01455371 03908088 09232408 2Phi 08175824 01550904 03686232 09717563 3Phi 05927490 23270495 02021374E10 10000000 4p 07200000 01595594 03527872 09238422 5p 05358423 01208847 03081002 07495562 6p 05927490 23270494 02021374E10 10000000 Real Function Parameters of Phit p PIM 95 Con dence Interval Parameter Estimate Standard Error Lower Upper 1Phi 07855523 01498374 03905479 09544207 2Phi 07539437 01169131 04711652 09133294 3Phi 06175588 01316335 03513308 08280107 4p 05972312 00977566 04006899 07668259 Real Function Parameters of Phi pt PIM 95 Con dence Interval Parameter Estimate Standard Error Lower Upper 1Phi 07763123 00945402 05441921 09098149 2p 06972303 01528817 03577103 09049608 3p 05560607 01060756 03504571 07441041 4p 04653489 01287048 02399885 07058034 Real Function Parameters of Phi p PIM 95 Con dence Interval Parameter Estimate Standard Error Lower Upper 1Phi 07257951 00830339 05388314 08570676 2p 05827875 00970619 03897915 07533655 a Based on the information given you need to decide which model to use I would then like you to write a brief report on what you have learned from this study about survival and capture rates and the assumptions behind the sampling method and model b If there were 20 51 41 and 25 total animals captured each year marked and unmarked use the above estimates to obtain a population size estimate in each year Don t worry about getting SEs for the estimates Q2 Input the following two arti cial data sets into MARK and then fit the CJS model phit pt and then the restricted version phi p for each data set Phoenixl 1 1 1 5 18 101 130 110 202 100 150 PhoeniX2 1 11 32 101 130 110 148 100 690 Comment on the important points for the models within a data set and then compare the estimates and precisions between data sets What is going on here Q3 I strongly advise you to play around with the dipper data yourself so that you see how to manipulate PIMS and use predefined model sets for multiple groups using the recaptures option in MARK If you do create a new test data set by copying the one you have so that you do not lose Gary s example analyses Lecture 18 CORMACK JOLLY SEBER MODEL Survival and Capture Probability Estimation Ch 17 Exam 2 Next Week Open Population Models More than two samples necessary Additions Births or Immigrants not separable Deletions Deaths or Emigrants not separable These models allow estimation of apparent survival rates and recruitment as well as population sizes FULL J OLLYSEBER MODEL PARAMETERS AND UNOBSERVED RANDOM VARIABLES Population Sizes N1 N2 NH Nk Survival Rates 19 2 9 quot k2 9 k1 Capture Probabilities p1 ppm PM pk Recruitment Numbers B1 B2 Bk2 Bk1 Marked Population Sizes M1 M2 V Mk1 Mk J OLLYSEBER MODEL This model makes the following assumptions 1 Every animal present in the population at the time of i th sample i l 2 k has the same probability of capture pi 2 Every marked animal present in the population immediately after the i th sample has the same probability of survival i until the i lth sampling time i l 2 kl 3 Marks are not lost or overlooked 4 All samples are instantaneous and each release is made immediately after the sample 5 All emigration is permanent 6 Fates of animals are independent FULL J OLLYSEBER MODEL A A Po ulation Sizes p N19N29Nk1Nk s 1R t A A A urvlva a es 19 2 9 quot k2 9 k1 Capture Probabilities p1 1321Ak1 Pk Marked Population Size M1 M2 MM Mk A Recruitment Numbers B1 A32 Bk 2Bk1 OPEN CAPTURE RECAPTURE MODELS FOR ESTIMATING DEMOGRAPHIC PARAMETERS Components of Data Recaptures this component is Where we estimate survival and capture probabilities You can see that from the intuitive estimators First Captures and Recaptures allows estimation of population sizes and recruitment parameters and also survival and capture probabilities You also can see that from the intuitive estimators OPEN CAPTURE RECAPTURE MODELS FOR ESTIMATING DEMOGRAPHIC PARAMETERS We do not discuss the likelihoods much in this class but There are three components of the Full Likelihood for the JollySeber Model LL1First CapturesL2Losses on CaptureL3Recaptures L1First Captures allows estimation of population sizes and recruitment parameters and also includes survival and capture probabilities L2Losses on Capture important that we allow for this but these parameters are not of biological interest L3Recaptures this is the reduced CormackJollySeber likelihood and this component is where we estimate survival and capture probabilities CORMACKJOLLYSEBER MODELCh 17 Survival Rates r31 r32 13H k1 Capture Probabilities pl 132 lm pk Marked Population Sizes M1 MWMk WE JUST FOLLOW THE MARKED ANIMALS AND DON T USE I EESTO TOTAL RATIOS TO GET POPN SIZE OR BIRTH CORMACKJOLLY SEBER MODEL Comprehensive Survival Modeling Extension to Stage Structured Models for survival Within a stage and movement between stages SURVIVAL MODELING EUROPEAN DIPPER EXAMPLE Illustrated by European Dipper data taken from Lebreton at al 1992 using SURGE MARK better now and I reanalysed in that programThere are seven years with one recapture event per year There were data for the two sexes separately but they were found to be similar so I combined them for ease of presentation Five models were compared using AIC Akaike Information Criteria and we show details of our analysis for those models Dipper Combined Sex Analysis Delta AICC Model AICC AICc Weight Par Phif p PIM 666160 000 089445 3 Phi p PIM 670866 471 008505 2 Phit p PIM 673998 784 001776 7 Phi pt PIM 678748 1259 000165 7 Phit pt PIM 679588 1343 000109 11 Best Model Model Details First we will look at the full CJ S Model Next Look at the Best Model Where survival differs between ood and non ood years and Where p is constant over years 1Phi 2Phi 3Phi 4Phi 5Phi 6Phi 7p 8p 9p 10p 11p 12p Dipper Full C S Model Phit pt PIMOutput Parameter Estimate 07181818 04346708 04781705 06261177 05985335 07284299 06962027 09230769 09130435 09007892 09324138 07284328 Standard Error Lower 01555470 03610409 00688290 03075047 00597091 03643839 00592656 05048461 00560517 04855434 00000000 07284299 01657637 03302969 00728778 06161497 00581758 07140650 00538330 07360176 00458025 07684926 00000000 07284328 Last phi and p and not separately estimable Upper 09199575 05710588 05942685 07333741 07019412 07284299 09141508 09889758 09778505 09672856 09828579 07284328 EUROPEAN DIPPER EXAMPLE SUMMARY Very nice example which shows the value of the AIC procedure Very simple model is adequate only 2 survival parameters estimated and 1 capture probability Estimates are much more precise than for the full CJ S model Meadow Vole Example from Text P436439 in Text There are 6 sampling occasions and 2 sexes The approach will be to look at the full CJ S for both sexes first Then we will show you can do better using AIC on a set of submodels An additive model on survival across sexes combined with a constant capture probability turns out to be best TABLE 178 Model Selection Statistics for Different Models of Time and SexSpecific Variation in Capture and Survival Probabilities of Meadow Voles Model Parametersquot Deviance AAICC 4324 p 7 749 000 cm 1 6 782 123 65t 375 8 742 133 cpsJ pg 14 617 140 Pt pg 7 769 198 255 pf 10 725 382 1254 39 11 705 394 54 95 12 686 414 cm 1954 14 649 460 cpsd pg 11 71 4 477 lt91 1951 10 738 512 cps4 p54 17 591 525 cm pt 9 762 540 cpst pp 14 684 817 34 pg 1 15 667 850 cps 7954 9 850 1422 cp p54 8 879 1505 Psr pSH 7 946 1967 cps pt 6 971 2013 cp 135 6 976 2067 cp pt 5 1004 2144 cps p 3 1082 2515 cps pg 4 1066 2562 P 195 3 1098 2674 ltp p 2 1120 2690 At Patuxent Wildlife Research Center 1981 see data in Tables 175 and 176 bParameter numbers computed in program MARK White and Bumham 1 999 438 Chapter 17 Estimating Survival Movement and Other State Transitions TABLE 177 Parameter Estimates under the General TwoSex CJS model pm pm for Meadow Voles Studied at Patuxent Wildlife Research Center Laurel Maryland 1981 Capture probability Survival probability F 1 M 1 F Capture Sampling ema e fa e emale Male period dates aorta SE D aortas a 88180 1 02741 J J 089 0052 086 0052 2 81 85 088 0055 090 0039 078 0066 058 0066 8 8 29 9 2 090 0057 082 0071 068 0066 071 0072 4 103 107 096 0037 091 0059 069 0069 059 0009 5 1031 114 100 J 083 0069 J t 6 124 128 J J J J See data in Tables 175 and 176 bParameter not estimable under CIS model Standard error not estimated Full Model Summary Capture probabilities high and fairly constant over time and sexes Survival probabilities vary over time and seem higher for females Lets see if we can get a simpler model to use Multiple Group Model Notation st each seX and time has a distinct parameter st there is a constant additive effect of seX at each time see next panels s there is a seX effect but no time variation t there is time variation but no effect of seX there is no time variation and no effect of seX Multiple Group Model Notation stModel Logit g ij 9 ai 8 i 2 sex j time period 5 Model Logit 17 7 a tModel Logit 4157 7 Model Logit 4157 7 Multiple Group Model Notation Model s t there is a constant additive effect of seX at each time Legit m as 0 With s 2 Female or 1 Male Female LOgif m 3 Male Logit Ua j TABLE 178 Model Selection Statistics for Different Models of Time and SexSpecific Variation in Capture and Survival Probabilities of Meadow Voles Model Parametersquot Deviance AAICC 4324 p 7 749 000 cm 1 6 782 123 65t 375 8 742 133 cpsJ pg 14 617 140 Pt pg 7 769 198 255 pf 10 725 382 1254 39 11 705 394 54 95 12 686 414 cm 1954 14 649 460 cpsd pg 11 71 4 477 lt91 1951 10 738 512 cps4 p54 17 591 525 cm pt 9 762 540 cpst pp 14 684 817 34 pg 1 15 667 850 cps 7954 9 850 1422 cp p54 8 879 1505 Psr pSH 7 946 1967 cps pt 6 971 2013 cp 135 6 976 2067 cp pt 5 1004 2144 cps p 3 1082 2515 cps pg 4 1066 2562 P 195 3 1098 2674 ltp p 2 1120 2690 At Patuxent Wildlife Research Center 1981 see data in Tables 175 and 176 bParameter numbers computed in program MARK White and Bumham 1 999 Additive Survival Constant Capture Model Much simpler summary than the full model Survival estimates plotted in next gure Capture probability estimate is constant at 090 SEQ 002 10 391 Males a 08 55 E 5 Lu 067 52 U a J TB 04 2 Z 1 w 02T 00 r 1 F 17 i 1 2 3 4 5 Time Period FIGURE 172 Estimated monthly survival probabilities and 95 confidence intervals from model cpsH p for male and female meadow voles at Patuxent Wildlife Research Center 1981 Mark Demonstration Dipper Two groups make it a bit more complex Open Existing File to see Gary White analyses already there Open New Analysis Non estimable parameters in the full CJ S model Use of Prede ned Models approach to save time it takes to use the PIMS Use of PIMS to account for the ood vs non ood years models which don t t into that Mark Demonstration Dipper PIMS Structures There are 4 PIMS now Male and Female Survival Male and Female Capture probabilities Mark Demonstration Dipper PIMS Structures There are 4 PIMS now Male and Female Survival Male and Female Capture probabilities Mark Demonstration Dipper PIMS Structures Survival Full Model Males 123456 23456 3456 456 56 6 Females 789101112 89101112 9101112 101112 1112 12 Mark Demonstration Dipper PIMS Structures Survival Best Model Males 122111 22111 2111 111 11 1 Females 122111 22111 2111 111 11 1 Mark Demonstration Dipper PIMS Structures Capture Best Model Males 333333 33333 3333 333 33 3 Females 333333 33333 3333 333 33 3 CORMACKJOLLY SEBER MODEL Extension to Stage Structured Models for survival Within a stage and movement between stages Open CR Models for Multiple States Survival and Movement Modeling for Multiple Sites or States Arnason Schwarz Nichols Brownie Pollock State transitions are now uncertain unlike age where an animal automatically moves to the next age Estimate Statespeci c transition probabilities Statespeci c survival probabilities Statespeci c movement probabilities Uses Metapopulation modeling Breeding proportions and costs of reproduction Testing conditionrelated survival weight classes PARAMETER NOTATION I39S i transition probability ie The probability that an animal alive and in state r at time i is alive and in state s at time i l S capture resighting probability for an animal in state s at 1 time 1 5 x at TNW I I J A gt A gt Capture gtAA History 151 p 1AAP2A A gt B gt Capture gt AB History A p 31923 A gt A gt No Capture gt A0 History A gt B gt No Capture gtA0 History AB B 1 1AAp 1 p2 LocationA Probability AA 41924 AB 1533195 A0 1 41924 13195 LocationB Probability BB 15133195 BA 15131195 B0 1 1513319 1MP CAPTURE HISTORY FORMAT Now our capture histories generalize to include state as well as caught or not Next is an example with 5 periods and two possible states EXAMPLE Some of many Poss1ble Capture Histories Five Periods and Two States A and B 0 means not detected in either state AAOAA ABAOA AOBAO A0000 BBA00 BAA00 000BA 0AOB0 MULTISTATE MODELS Can Use of SURVIV MS SURVIV and MARK on two examples to illustrate this methodology Microtus Mass Class Model A B C D Nichols et al 1992 Ecology Canada Geese going to three different wintering grounds ABC Mid Atlantic Chesapeake Carolinas Hestbeck et all99l Ecology MICROT US MASS CLASS EXAMPLE Small eld mouse studied over 4 periods months Interested in weight transitions survival and movement from stage to stage between periods A lt 22 g Juveniles B 22 33 g Subadults C 34 45 g Small adults D gt 45 g Large adults MICROT US MASS CLASS EXAMPLE TRANSITION Survival MovementPROBABILITIES WS Transition EstimateSt Error AB 087005 B B 051006 B C 024006 C C 066005 C D 008008 D C 022022 D D 056006 Not all transitions are shown as some do not occur The animals lost weight WINTERING CANADA GEESE Markresight of neck collared migratory geese in the Winter 198384 to 198586 3 different Winters Where estimates possible A Mid Atlantic Region NY NJ DE PA B Chesapeake Bay Region MD VA C Carolinas Region NC SC All birds breed in Canada and come south for Winter only transition probabilities between years estimated and then survival and movement probabilities estimated SEPARA T 1 ON OF S UR VI VAL AND MO VEMEN T irs SirWirs If movement is at end of interval 39 Stays in r S r z Survives rs w Inrat i 1 Dies Moves to s WIN T ERIN G CANADA GEESE SURVIVAL PROBS SHOWN HERE Sr Location EstimateSE A 058 0015 B 066 0016 C 049 0019 WIN TERIN G CANADA GEESE MOVEMENT PROBS ONLY SHOWN HERE W Transition EstimateSE AA 0710016 A B 0290016 A C 0000001 B A 0100006 B B 0890007 B C 0020002 C A 0070010 C B 0370024 C C 0560025 Piobability of movement given survived and they assumed the anlmals moved at the end of 1nterva1They add to 1 1n each set WIN TERING CANADA GEESE MOVEMENT PROBABILITIES Birds in the Carolinas are much less likely to return there the next Winter 056 than for the other areas Whereas 07 l returning to Midatlantic And 089 of returning to Chesapeake Most faithful birds Conclusions and Complex Applications urvival and Movement Applications Metapopn modeling in different atches of fragmented hab1tat D1s ance between patches as a covar1ate Stages can be life stages microtus example Combine tag returns amp recaptures to evaluate value of marme reserves Use of radio telemetry to get more detailed info on movement j RESEARCH ON SEPARA T 1 ON OF S UR VI VAL AND M O VEMEN T PARAMETERS Recent work has focused on how to separate transition probabilities into survival and movement probabilities more generallyThis is not easy to accomplish realistically Work has been joint with my former Ph D student MiJeom J oe Reference is Joe and Pollock 2002 Journal of Applied Stat1sticsProceedings of the EURING conference MarkzMSSURVIV Example I will open the example and show you the basic structures SEPARA TION 0F SUR VIVAL AND M0 VEMEN T Additional Slides on Our researchNot covered in Class SEPARA T I ON OF S UR VI VAL AND M O VEMEN T t xed We began by pretending that we knew the movement time t between the two sampling periods i and il We then have rs r t rs s l t i Si 9 51 Where we have 5 is the survival rate from i to il given the animal is alive at i and the animal is in state r for the whole interval Wi is the probability of moving from state r to state s at it for an animal alive in state r at it Note that W1 21 W1 in a two stage or patch system SEPARA T I ON OF S UR VI VAL AND M O VEMEN T t xed In previous work the animals were assumed to move at the beginning or the end of the interval so that t0 rs s rs i Sz39Wi ort1 bis Sim SEPARA T 1 ON OF S UR VI VAL AND MO VEMEN T t random variable We continue by assuming the movement time t between the two sampling periods i and i1 comes from a known distribution ft We then have 1 a J Serwal famt 0 An important special case is where ft is uniform which corresponds to the movement time being randomly distributed in the intervalFor this case we can obtain an explicit form for the integral SEPARA T 1 ON OF S UR VI VAL AND MO VEMEN T t uniform random variable We continue by assuming ft is uniform We then have 1 fs wig J SZ SSHdt O 39rs WirSSiSSir S1S11nSir Sz39s 1 and I39S WiI39SSiS 1 SEPARA T 1 ON OF S UR VI VAL AND MO VEMEN T t uniform random variable Simulations When the movement rates were high High correlations between estimates Some failure to converge High relative standard errors When the movement rates were low Estimates are much better behaved Currently planning to apply to some butter y data of Haddad SEPARA T 1 ON OF S UR VI VAL AND MO VEMEN T t another random variable We continued by assuming ft is beta with known parameters In practice we would have to estimate the distribution from telemetered animals Lecture 17 JOLLYSEBER OPEN MODEL Survival and Capture Probability Estimation Ch 17 Population Size and Recruitment Estimation Ch 18 Read Wildlife Monograph Ch 4 The old fashioned approach but I think you will find it useful Part of the class today will follow this material Note Reading 171 and 173 then later 181t0183 Open Population Models More than two samples necessary Additions Births or Immigrants not separable Deletions Deaths or Emigrants not separable These models allow estimation of apparent survival rates and recruitment as well as population sizes FULL J OLLYSEBER MODEL PARAMETERS AND UNOBSERVED RANDOM VARIABLES Population Sizes N1 N2 NH Nk Survival Rates 19 2 9 quot k2 9 k1 Capture Probabilities p1 ppm PM pk Recruitment Numbers B1 B2 Bk2 Bk1 Marked Population Sizes M1 M2 V Mk1 Mk PARAMETER ESTIMATION Can obtain MLE s numerically using a program like JOLLY or MARK Allows parameter constraints goodnessof fit tests to be easily computed 39 First I shall present explicit ML estimators with their intuition to motivate the model better This is the old fashioned approach I will not expect you to calculate estimators this way 39 To start with we shall express estimators in terms of the marked population sizes Later we shall show how to estimate these quantities CAPTURE BECAPTURE EXPERIMENTSmPOUOCk et al 19 Table 41 Notation for the JollySaber model described in detail in Chapter 4 of this monograph PARAMETERS M the number of marked animals in the population at the time the ith sample is taken 139 1 k M1 5 01 Ni the total number of animals in the population at the time the ith sample is taken 139 1 k B the total number of new animals entering the population between the ith and i 1th sample and still in the population at the time i 1th sample is taken i 1 k u 1 35 the survival probability for all animals between the ith and 139 1th sample i 1 k 1 pi the capture probability for all animals in the ith sample 1 1 k STATISTICS m the number of marked animals captured in the ith sample 139 1 k ul the number of unmarked animals captured in the ith sample 1 1 k nI ml ui the total number of animals captured in the itb sample 139 1 k R the number of the ni that are released after the ith sample 139 1 k 1 This may not be all of the n due to losses on capture as discussed in the text n the number of the Rt animals released at i that are captured again i 1 k h 1 21 the number of animals captured before i not captured at i and captured again later i 2 k ll J OLLYSEBER MODEL This model makes the following assumptions 1 Every animal present in the population at the time of i th sample i l 2 k has the same probability of capture pi 2 Every marked animal present in the population immediately after the i th sample has the same probability of survival i until the i lth sampling time i l 2 kl 3 Marks are not lost or overlooked 4 All samples are instantaneous and each release is made immediately after the sample 5 All emigration is permanent 6 Fates of animals are independent Parameter Estimation J ollySeber Model First will cover intuitive approach This is an open population version of Mt Later I Will break up into components for more formal inference Estimation of Capture Probability An estimate is obviously the fraction of animals captured at time i so that i 2 mi Mi 2 ni Ni 44 Note Only p2 p3 pk1 are estimable p1 and pk are not estimable Population Size Estimation minizMiNi Similar to the LincolnPetersen model solve the equation and obtain N1 niMimi Ni ni i Note Miwill be estimated later 41 Note Only N2 N3 Nk1 are estimable Not N1 nor Nk SURVIVAL RATE ESTIMATION Mi mi R is the number of marked animals alive in the population just after time i M11 is the number of the above animals alive at time i 1 Therefore an intuitive estimator of survival is i1 Note M i1and Mi have to be estimated and that equation will be given later Note bl 2 K2 are estimable K1 is not estimable SURVIVAL RATE ESTIMATION We really should refer to the parameter as apparent survival Q SiFi 1 apparent survival SI true survival E delity prob did not emigrate Note We cannot get true survival from capturerecapture data alone Add telemetry to estimate F or assume that Fl in particular applications Estimation of Birth Numbers An equation for estimation of birth numbers is 31 2 Nil A1NiniRi 43 The rst term of the equation represents the total number of animals at time i 1 and the second term represents those animals at time i 1 that were present at time i Estimation of Marked Population Size The derivation is given on the next page M mi RiZiri 45 Note M2 M3 M k1 are estimable The Mi are crucial to the estimation of all the other parameters in this intuitive approach M1 0 N1 is not estimable Mk is not estimable ZiMi39mi Proportion of marked animals not seen at i that are seen again rR 1 1 Proportion of animals seen at i that are seen again ZiMi39mi 11 R which gives M1 2 mi R1Ziri Note I wont expect you to use these equations I just want you to understand roughly Why the method works FULL J OLLYSEBER MODEL A A Po ulation Sizes p N19N29Nk1Nk s 1R t A A A urvlva a es 19 2 9 quot k2 9 k1 Capture Probabilities p1 1321Ak1 Pk Marked Population Size M1 M2 MM Mk A Recruitment Numbers B1 A32 Bk 2Bk1 ASSUMPTIONS and MODEL ROBUSTNESS Heterogeneity of capture probabilities Serious negative bias of population size estimators Mild negative bias on survival estimators ie Survival estimators robust to heterogeneity Trap Response of Capture probabilities Trap Happy Negative bias on population size estimators Little effect on survival estimators Trap Shy Positive bias on population size estimators Little effect on survival estimators ASSUMPTIONS and MODEL ROBUSTNESS Heterogeneity of survival probabilities Complex but if survival rates vary over animals in a consistent way from sample to sample there is often a positive bias in survival rates Interactions between heterogeneity of capture and survival probabilities can cause positive or negative biases in survival rates Assumptions continued 39 Mark Induced Mortality Positive bias on population size estimators especially when capture probability is small Severe negative bias on survival estimators Tag Loss Positive bias on population size estimators especially when capture probability is small Severe negative bias on survival estimators 39 Emigration Confounded not separable from mortality We should call survival apparent survival Temporary emigration causes problems that Will be discussed later EXAMPLE Gray squirrel study in an English oak forest Twoyear study with monthly sampling Baited traps with com Toe clipping used for marking Natural boundaries suggest that survival and recruitment estimators are not confounded with migration processes Data Table 43 Estimators Table 44 Old approach now we would use MARK so we can t a variety of sub models but it still illustrates the key points High capture probabilities except for August 1973 to December 1973 Therefore the precision of estimators is generally very good Table 43 Capture recapture statistics for a gray squirrel population at Afice Holt Forest Research Station Surrey En gland November 1972 September 1974 Period Date nia n2 Ri n 2 1 Nov 1972 46 46 43 2 Dec 1972 16 12 46 4 1 1 3 Jan 1973 48 42 18 48 3 4 Feb 1973 46 42 46 15 9 5 Nlar 1973 51 46 50 46 8 6 Apr 1973 37 37 37 35 17 7 Vlay 1973 41 11 41 4O 11 8 Nlay Jun 1973 42 39 42 37 12 9 Jun 1973 47 13 47 40 6 10 Jul 1973 31 26 31 26 2O 11 Aug 1973 8 7 8 8 39 12 Sep 1973 2 2 2 2 45 13 Oct 1973 1 0 1 1 47 14 NOV 1973 4 3 4 3 45 15 Dec 1973 9 8 9 8 10 16 Jan 197 1 19 17 18 17 31 17 Feb 1974 19 14 19 18 34 18 IVIar 1974 27 20 27 24 32 19 Apr 1974 36 36 36 32 20 20 May 1974 45 34 44 33 18 21 Jul 1974 71 46 73 15 5 22 Aug 1974 22 20 22 2 O 23 Sep 1974 3 2 2 3 Notation is explained in Table 41 Table 44 JollySaber estimates and approximate standard errors3 for a gray squirrel pepulation at Alice Holt Forest Research Station Surrey England November 1972 September 1974 A A A Period Date Sub SE lt15 SE 8 SE 1 Nov 1972 094 0037 Dec 1972 471 039 096 0030 63 077 3 Jan 1973 513 070 100 0004 45 127 4 Feb 1973 560 119 099 0023 51 153 5 Mar 1973 605 151 094 0041 00 106 6 Apr 1973 549 123 095 0038 00 000 7 May 1973 523 060 100 0030 39 122 8 May Jun 1973 565 206 090 0052 37 145 9 Jun 1973 546 157 092 0067 87 330 10 Jul 1973 589 459 084 0066 22 660 11 Aug 1973 518 599 100 0000 00 599 12 Sep 1973 13 Oct 1973 14 Nov 1973 15 Dec 1973 583 920 093 0115 10 657 16 Jan 1974 553 430 098 0068 131 825 17 Feb 1974 664 814 100 0071 68 1014 18 Mar 1974 745 791 093 0067 00 628 19 Apr 1974 584 213 099 0071 182 422 20 May 1974 760 612 100 0168 339 886 21 Jul 1974 1103 1810 021 0048 00 223 22 Aug 1974 219 000 23 Sep 1974 a 52207 and SEquot 15 include only sampling variation or quoterror of estimationquot 8 6 was obtained using the full variance estimator of Jolly 1965 5 Notation explained in Table 41 Gray Squirrel Example Conclusions Population Sizes Slight negative bias possible due to heterogeneity and trap happy response Survival Rates Should be very accurate and precise Values are very close to one except the last one This may be due to losses of young animals but it is dif cult to tell because the study ends here Recruitment Numbers There is only recruitment in April and May of 1974 In 1973 there was no recruitment It was a bad year for squirrels to reproduce poor acorn crop FULL J OLLYSEBER MODEL A A Po ulation Sizes p N19N29Nk1Nk s 1R t A A A urvlva a es 19 2 9 quot k2 9 k1 Capture Probabilities p1 1321Ak1 Pk Marked Population Size M1 M2 MM Mk A Recruitment Numbers B1 A32 Bk 2Bk1 MODEL RESTRICTIONS AND EXTENSIONS The Full JollySeber version of model has a very large number of parameters Perhaps we could reduce the number of parameters Principle of Parsimony Have as small a number of parameters as realistically possible Advantages l Simplicity 2 Reduce problems of nonestimability 3 Smaller SEs of remaining parameter estimates 4 If too simple will induce bias so Use AIC Criteria that we discussed earlier to strike the right balance MODEL RESTRICTIONS Model A Full Model 19 29m9 k4 p19p29 399pk Model B Constant Survival 1 2ama k 1 Model C Constant Capture p1 p2aquota pk p Model D Constant Survival Constant Capture 1 2ama k 1 p1 ppm pk p MODEL RESTRICTIONS AND EXTENSIONS PROGRAMS Program JOLLY allows one to restrict capture and survival probabilities to be equalModels AB and D We can illustrate with Rose39at39e Tern Data See Ch 533 of Monograph if you like Program JOLLYAGE allows one to include different age classesWe can illustrate with Canada Goose Data See Ch 67 of Monograph if you like These can be run remotely from the Patuxent website similar to how I showed you for CAPTURE Program MARK is a newer program and allows one to restrict capture and survival probabilities to be equalModels AB C and D and any other models you can think of It can also handle the age dependence of the Canada goose example as well Roseate Tern Example JOLLY Brie y presented Adult terns captured and banded on nests on Falkland Island Connecticut Recaptures in subsequent years Study covered 19781984 Summary data in Section 533 Of Monograph We then t J ollySeber Model A Constant Survival Model B Constant Survival and Capture Model D Conclusions on Roseate Tern Data Model B is the best model based on various tests Now if we used MARK would use AIC We display the Model B output in Table 56 of Monograph Table 56 Estimates and approximate standard errors under the JollySaber model A the constant survival model B and the constant survival and capture model 0 for roseate tern data collected by Spendelow 1982 on Falkner Island Connecticut from 1978 to 1984 Model and yea 4i SE 35 SE M SE N SE 13 SE Model Aw JollySeber model i 1978 074 0137 1979 052 0075 025 0068 68 119 544 1418 143 956 1980 090 0114 023 0049 101 113 426 827 200 848 1981 061 0085 036 0054 160 172 581 807 20 454 A 1982 055 0102 037 0056 192 232 372 513 26 236 1983 034 0064 143 217 230 387 x39 066 0033 031 0026 Model B constant survival model 1979 027 0060 64 636 525 1020 192 1148 1980 019 0037 120 880 536 849 144 761 1981 067a 0028 042 0049 146 984 517 494 42 411 1982 036 0042 190 141 376 318 28 281 1983 028 0043 166 166 273 295 171 391 1984 040 0066 135 168 358 509 Model D constant survival and capture model 1979 58 637 466 440 55 334 1980 111 902 360 332 383 435 1981 0 69a 0023 0 31 0 023 166 121 667 55 6 22 36 6 1982 203 151 420 353 6 229 1983 171 167 268 262 242 315 1984 175 227 462 457 1 These estimates pertain to all years in the study because of the assumptions of constant survival andor capture OPEN CAPTURE RECAPTURE MODELS FOR ESTIMATING DEMOGRAPHIC PARAMETERS Components of Data Recaptures this component is Where we estimate survival and capture probabilities You can see that from the intuitive estimators First Captures and Recaptures allows estimation of population sizes and recruitment parameters and also survival and capture probabilities You also can see that from the intuitive estimators OPEN CAPTURE RECAPTURE MODELS FOR ESTIMATING DEMOGRAPHIC PARAMETERS We do not discuss the likelihoods much in this class but There are three components of the Full Likelihood for the JollySeber Model LL1First CapturesL2Losses on CaptureL3Recaptures L1First Captures allows estimation of population sizes and recruitment parameters and also includes survival and capture probabilities L2Losses on Capture important that we allow for this but these parameters are not of biological interest L3Recaptures this is the reduced CormackJollySeber likelihood and this component is where we estimate survival and capture probabilities CORMACKJOLLY SEBER MODEL Comprehensive Survival Modeling Extension to Stage Structured Models for survival Within a stage and movement between stages CORMACKJOLLYSEBER MODELCh 17 Survival Rates 1 2 k2 k1 Capture Probabilities pl 132 9 quot 13191 9 pk WE JUST FOLLOW THE MARKED ANIMALS AND DON T USE IRIIIARRIIEEESTO TOTAL RATIOS TO GET POPN SIZE OR BIRTH CormackJollySeber Model Motivation of Model Structure and Estimation We consider only recaptures and three sampling occasions There are seven capture histories 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 CormackJollySeber Model Motivation of Model Structure and Estimation Expected Values of Cells X111 R1 1p2 2p3 X110 R1 1p21 2173 X101 R1 1139p2 2p3 X100 R11 1p2 2p3 01121 2p3 91139 p2 2p3 X011 R2 2p3 X010 R21 2p3 X001 n0 Chance for recaps yet Note We dont start the cell structure until after first capture and therefore dont have to consider new entries into the popn CormackJollySeber Model Motivation of Model Structure and Estimation Intuitive Estimation Can we separately estimate the components of 2P3 No Only the product of the two CormackJollySeber Model Motivation of Model Structure and Estimation X011E R2 02p3 02173 X011R2 CormackJollySeber Model Motivation of Model Structure and Estimation Expected Values of Cells X111 5 R1 1p2 2p3 X101 5 R1 1139 p2 2p3 X111 X101 5 R1 1 2p3 XIIIX111 X101 5 192 A92 XIIIX111 X101 CormackJollySeber Model Motivation of Model Structure and Estimation Expected Values of Cells X111 5 R1 1p2 2p3 51 X111R1132 2133 Therefore using previous results we can estimate the rst survival rate MARK does it better MLEs Generalisations of the CormackJollySeber Model using MARK Multiple age classes Multiple data sets Parameter constraints to achieve parsimony Model selection Minimize AIC 2 log L 2pars There is penalty for over parameterization SURVIVAL AND CAPTURE PROBABILITY MODELING Continued Can allow for covariates so that if Xi is some covariate like weight or body condition IOgit Z a Xi legit qt 1n 1 1 logit is most common link function used It keeps survival rates between 01 log 05 Xi would not SURVIVAL MODELING EUROPEAN DIPPER EXAMPLE Illustrated by European Dipper data taken from Lebreton at al 1992 using SURGE MARK better now and I re analysed in that programThere are seven years with one recapture event per year There were data for the two sexes separately but they were found to be similar so I combined them for ease of presentation Five models were compared using AIC Akaike Information Criteria and we show details of our analysis for those models Dipper Combined Sex Analysis Delta AICC Model AICC AICc Weight Par Phif p PIM 666160 000 089445 3 Phi p PIM 670866 471 008505 2 Phit p PIM 673998 784 001776 7 Phi pt PIM 678748 1259 000165 7 Phit pt PIM 679588 1343 000109 11 Best Model Model Details First we will look at the full CJ S Model Next Look at the Best Model Where survival differs between ood and non ood years and Where p is constant over years 1Phi 2Phi 3Phi 4Phi 5Phi 6Phi 7p 8p 9p 10p 11p 12p Dipper Full C S Model Phit pt PIMOutput Parameter Estimate 07181818 04346708 04781705 06261177 05985335 07284299 06962027 09230769 09130435 09007892 09324138 07284328 Standard Error Lower 01555470 03610409 00688290 03075047 00597091 03643839 00592656 05048461 00560517 04855434 00000000 07284299 01657637 03302969 00728778 06161497 00581758 07140650 00538330 07360176 00458025 07684926 00000000 07284328 Last phi and p and not separately estimable Upper 09199575 05710588 05942685 07333741 07019412 07284299 09141508 09889758 09778505 09672856 09828579 07284328 ST 506 2008 Some Brief Summarv Notes on F quot quot The Two Sample Removal Model An Example ofa Sampling and F quot quot Problem As an example of a statistical sampling model let us consider a study to monitor freshwater sh in streams we want to estimate population size and detection probability using an electrofishing removal study We denote the data collected n1 fish captured and removed in sample 1 11 fish captured and removed in sample 2 We assume a closed population and equal detection probability of all fish at both time points The parameters are N the population size and pthe probability of detection at each time Note that the assumption of equal catchability is stronger than in the simple capture recapture model we consider a bit later in the semester because we require equal capture probabilities for the two samples nlNgnZN nl NanZnlHZ 132111 n2n1 Nznl f We use the hat symbol to emphasize that this is an estimate rather than the true parameter This is a classic example of the sort of model based estimator we will be using in this class Violation of model assumptions can lead to serious biases in the estimator We also need to be concerned about the sample sizes as they will determine the precision of the estimator The twin themes of model bias and precision will appear repeatedly in what we do in this class Properties of Estimators Bias The difference between the expected value and the parameter Precision Variance and St Error The average squared deviation from the expected value of the estimator Accuracy Mean Squared Error Variance Biasz The average squared deviation from the parameter Methods Ochtmmnn of A M We basically used this method with the removal model described earlier when I was describing the intuitive derivation One expresses what the expected value of some observed value would be equal to in terms of the unknown population parameters Method of Least Squares You should have seen this in a regression class such as ST 511512 Consider tting a straight line to some data You could minimise the sum of squared differences between the tted line and the observed points Minimisez yl a x 2 The resulting estimators of the slope 5 and intercept 0c are well known and called the least squares estimators The can also be shown to be maximum likelihood if the errors are normally distributed Method of Maximum Likelihood Derive the probability distribution of the observed data as a function of the parameters and then view this as a function of the parameters this is the likelihood function Find the values of the parameters which maximise this function These are the maximum likelihood estimators MLEs A very simple example R Retained tag L Lost tag p prob tag retainedl p prob tag lost PRRLRLRRRRL p71 p3 Lp p71 p3 The likelihood is maximised at f xn 710 2 07 The shape of the likelihood will be shown on the whiteboard

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