Line Transect/Point CountLine TransectLine transects and point counts are used widely to count animals.They are variants of the same approach.We will begin with transects which are somewhat simpler to describe.Line TransectyiwwA=areaL=length of transectTypical Layout:Line TransectD= density = N / A = (number counted)/(area covered) = n/2Lw (or = n/kπr2)n= number of animals countedL=length of the transect(k = no. points counted)w = effective width(r = effective radius)Field ExperimentTest these ideas with a known NOld ArboretumPlace birds in a known area (A)Estimate their density withLine transects and point countsField ExerciseApply line transects and point counts to estimating the number of birds in parks and residential areas in MoscowEach person walk at least 2 blocks or (200 m) each in parks and in residential areas or do atleast 4 point counts in each area Field ExerciseMinimum count required: 20 groups of birds in each “habitat”Note: For birds (or other animals) in groups record each group as a single observation. The density estimate will be for groups of birds which you would multiply by average group size to estimate birds per unit area (hectare).Assumptions:1. Animals are randomly and independently distributed over the population area.2. The sighting of one animal is independent of the sighting of another.3.No animal is counted more than once.Assumptions:4. Animals are detected at their intial location prior to disturbance by the observer.5. The response behavior of the population as a whole does not change during the course of the census.6. The animals are homogeneous with regard to their response behavior, regardless of sex, age, etc.Assumptions:7. The probability of an animal being seen, given that it is a right-angle distance y from the line transect path (irrespective of which side of the path it is on), is a simple function g(y) of y, such that g(0)=1 (i.e. probability 1 of seeing an animal on the path is 1.0).Distance Sampling: Key ReferencesSeber, G.A.F. 1973. The Estimation of Animal Abundance. Hafner, NY.Buckland, S.T., D.R. Anderson, K.P. Burnham, J.L. Laake D. L. Borchers and L. Thomas. 2001. Introduction to Distance Sampling: Estimating Abundance of Biological Populations. Oxford University Press, Oxford.Seber (1973)Detection Curve = g(y)g(y)= Prob.(animal seen |animal at y)Observed Detection Function = f(y)f(y)=Prob.(animal at y | animal seen)If set w = Integral of g(y) dyThen f(y) = g(y)/wExamples of f(y)=detections Density EstimateHow do we estimate density?Old approach:Make an assumption about g(y)Derive an estimatorFind parametierscalculate itNew Approach to DensityFind a function which fits f(y) wellThen, assuming that all animals directly on the line (y=0) are detectedg(0) = 1From f(y) = g(y)/wf(0) = g(0)/w = 1/wSo Estimate of w = 1/f(0)Density EstimateD^ = n / 2LwD^ = n f(o) / 2LSo we must find a function f(y) which fits the observed detection distance curve well and then determine f(0)Note: In Lecture Outline notes on web w is symbolized by aDetection CurveWhat is a good model for f(y)?30+ proposed and usedBuckland et al. 2001 criteriaa. Model robustness (flexible)b. Pooling robustnessc. Shape criterion (shoulder)d. Efficiency (small variance)Modelling g(y)2 step process:1. Select a “key function” as a starting point2. A flexible form (a “series expansion” is used to adjust the key function (using 1-2 parameters) to improve fit of model to distance data.Key functionsUniform1/wHalf-normal -y2/2s2eHazard-rate -(y/s)-b1-eSeries ExpansionCosineSimple polynomialHermite polynomialTruncationOften required to find a good model and get a good fit (outliers).Recommend truncating observations beyond distance at which prob. detection falls below 10%.Likelihood Ratio TestUse this to judge requirement for adjustment terms to a key functionAllows evaluating whether addition of m2 terms to m1 already in model significantly improves it.H0: Model w/ m1 adjustments is true modelHa: Model w/ m1+m2 adjustments is trueΧ2 = -2 ln (L1 / L2)where L1 and L2 are maximum likelihood functions for models 1 & 2Sequential ApproachFit a key function, then fit a low order adjustment term.If adjustment improves model fit significantly,then test next order adjustment, etc.Default approach in DISTANCEBuckland et al. recommend α = .15 to increase power.Akaike’s Information CriterionOptimization approachAIC = -2 ln (L) + 2qwhere ln (L) is log-likelihood function evaluated at the max. likelihood estimates of model parameters (q= no. of parameters)Model with lowest AIC is selectedGoodness of fitUseful tool for model selectionCompares no. of detections in each distance interval to expected no. under fitted model.POVCPPaired Observer Varible Circular PlotDeveloped by Kissling and Garton (In Auk, July 2006)Combines distance estimation approach with double observer estimation of probability of detection for objects at center of plot.POVCPTwo observers stand at plot center and independently record every bird and distance as well as any bird movements on a simple plot map.After 8 minute count observers compare maps. [Observers get feedback, i.e. must stay sharp and learn from each other.]At end of day each observer enters their observations into a database which notes birds seen by both or notPOVCPEach observer’s data first analyzed with DISTANCE to determine at what distance detection probability falls below 1.0 (approx. perfect detection distance).Each observer’s detections and misses are analyzed and modelled with logistic regression to estimate each observer’s prob. of detection at y=0.0, plot center (g(0)) w/ covariates(rain, veg type, etc.).POVCPCorrection factors are calculated from θ=1/g(0) for each observer.Each observers count at a point is converted to a density estimate from D=(Σθf(0)n)/2πA single density estimate for each count is then calculated by averaging the 2 observers density estimates at that point incorporating each observer’s effective area and their correction factor.POVCPWe applied this to surveys of beach strands left from timber harvest in SE Alaska in 2001 and 2002.Comparing estimates to surveys analyzed by 4 other standard methods, the estimates were remarkably more precise and showed that other standard methodsare biased low because of birds missed close to the plot center.POVCPDetection probabilities at plot center varied by observers and by bird species from .61 to 1.0 for Hermit Thrush, .83 to .99 for Winter