Mark-recapture models for closed populations

Transcription

1 Mark-recapture models for closed populations A standard technique for estimating the size of a wildlife population uses multiple sampling occasions. The samples by design are spaced close enough in time so that the change in population size during the study is negligible. A population with no changes due to births, deaths, or migration is termed a closed population. During each sample occasion, the individuals in the population face some chance of being caught in the sample. Those animals caught are marked with an individually identifiable mark and released back into the population. The identification of individuals in the samples allows modeling of capture histories which leads to improved estimates of population size. In a typical small mammal study, live traps are baited and laid out in the study area in a large grid. The traps are visited each morning and the animals caught are recorded, marked if necessary, and released. Such a study might last several days or weeks and can provide a reliable estimate of population size during that time if care is taken during design, conduct, and analysis of the study. The data for, say, a study with $ sampling occasions is summarized in the form of frequency counts for capture histories. An animal's capture history can be represented as a string of!'s and "'s, for instance, "!" signifies that a particular animal was captured on the first day, not captured on the second day, and captured on the third day. Each animal in the population is thereby categorized into

2 one of ) possible capture histories, with a resulting frequency count of animals displaying each capture history: Capture day frequency " # $ count probability Capture " " " ] " 1 " history " "! ] # 1 # "! " ] $ 1 $ "!! ] % 1 %! " " ] & 1 &! "! ] ' 1 '!! " ] ( 1 (!!! ] ) 1 ) The sampling scheme induces a multinomial distribution for the frequency counts: ], ],..., ] µ multinomial( >, 1, 1,..., 1 ). " # ) " # ) Here > is the total population size, and 1 4 is the probability that an animal will have capture history 4. The last count, ] ), is the number of animals that did not appear in the sample and is unobserved. The capture history probabilities depend on the probabilities of events which cause animals to end up or not in the samples. The various closed population mark-

3 recapture models represent different constructions for the 1 4 's. For instance, suppose every animal has an equal probability of being captured on any given day, but also that the common capture probability varies in value from day to day. Let ) " œ capture probability on the first day, )# œ capture probability on the second day, ) $ œ capture probability on the third day. The different possible capture histories have the following probabilities: 1 " œ ))) " # $ 1 # œ )")# ") $ 1 $ œ )" ") # ) $ 1 % œ )" " )# ") $ 1 & œ ") " ) #) $ 1 ' œ " )" ) # ") $ 1 ( œ " )" ") # ) $ 1 œ " ) " ) ") ) " # $ Parameter estimation In general a closed population capture recapture study will have < capture occasions, producing 5 œ # < possible capture histories and corresponding categories in the multinomial likelihood. The category probabilities 1", 1#,..., 15 are modeled as functions 1 " œ1" ), 1 # œ1# ),..., 15 œ15 ) of underlying parameters ) œ ) ", )#,..., ) 6, representing different structures for capture probabilities. The essential structural problem in parameter estimation is that the number > of observations in the

4 multinomial distribution (and along with it, the last count ] 5 ) is unknown. The data are the observed counts C", C#,..., C5". The likelihood is a multinomial probability: > C C C C, C,..., C " # 5 " # 5 P ), > œ 1 1 â1. Here " # 5 C5 œ > C" C# âc5", 1 œ"1 1 â1, 5" " # 5" > >x C C C C xc xâc x,,..., œ. " # 5 " # 5 The integer ML trick can be used to find the relationship between the integer ML estimate of > and the other model parameters. Set P ), > > P ), >" "œ >815 "œ!, where 8 œ C" C# âc5" (the number of unique animals sampled), to find that s 8 > œ floor. "s 1 5

5 The computational task is to find the ML estimates of the other parameters. One can substitute the expression for >s into the multinoial log-likelihood and numerically maximize the log-likelihood for the remaining parameters. ML estimates for the two-sample mark recapture model with time-dependent capture probabilities )" and )# can be found symbolically. The model has four possible capture histories ((1 1), (1 0), (0 1), (0 0)) and three parameters: ) ", ) #, and >. Thus, the model is a saturated multinomial model, with 1 " œ )")#, 1 œ ) "), # " # 1 œ ") ), $ " # 1 œ " ) "). % " # ML estimates of )", )#, and > are found by equating 1", 1#, and with their saturated multinomial ML estimates: 1 $ C" > œ )) " #, C# > œ )" ")#, C$ > œ " " # ) ).

6 Solving the equations algebraically gives the ML estimates: )s œ " )s œ # C CC " $ ", C CC " # ", s CC " # CC " $ > œ C ". The estimate for > is recognized as the Lincoln-Peterson estimate. The integer ML estimate for > turns out to be floor C C C C ÎC. " # " $ " Wald intervals The large sample results for ML estimates in ordinary reduced-parameter multinomial models must be modified for these mark-recapture models, because the number of trials > is itself an unknown parameter. The relevant results were proved and reported by Sanathanan (1972 Annals of Mathematical Statistics 43: ). According to Sanathanan's results, the Fisher information matrix for the scaled ML estimates given by s) ), s) ),..., s) ), s> > Î > " " # # 6 6 is in the following form:

7 M ), > œ E + w " Here E is the information matrix per observation for the parameters )", )#,..., ) 6 when > is known, with the element in the 3th row and 7th column given by 5 # ` log P ) E œ " `14 ) `14 ). `) `) 1 ) `) `) 4œ" Also, + is an 6 " column vector in which the 3th element is + œ 3 " `1 ) 1 ) `) k. 5 3 Finally, + 6" is the scalar quantity given by + œ 6" "15 ) 1 ). 5 As >Ä, the joint distribution of the scaled ML parameter estimates approaches a multivariate normal distribution with mean vector! and variance-covariance " matrix Z ), > œ M ), > Þ Asymptotic confidence intervals are formed using as standard errors the square roots of the diagonal elements of the matrix given by s>z s), s>.

8 It is straightforward to work out the Fisher information matrix for the two-sample mark-recapture model with timedependent capture probabilities ), ) : " # " " )! " " )" ") " " " M > œ! ), )# " )# ") #. " " " " )" ") # " ) " ) " ) ") " # " # The asymptotics for mark-recapture models are especially slow, and the resulting coverage rates for the Wald intervals tend to be inaccurate. An essential problem is that the speed of convergence depends on the number 8 of unique animals appearing in the samples, which is usually low in comparison to > in wildlife studies. Profile likelihood A more informative approach is to construct a profile likelihood for >. Using the multinomial log-likelihood function, one fixes the value of > and numerically maximizes the function with respect to the other parameters. One repeats the mazimization for a succession of integer values of >, starting say at its minimum possible value of 8 and proceeding upward through the integers. The resulting graph of the maximized log-likelihood at different values of > is the profile log-likelihood. The range of values of > for which the profile log-likelihood is within 1.92 ( œ " # #; Þ!&, 1 df) of its maximum value is a valid asymptotic 95% confidence interval for >. Simulations

9 have suggested that the profile likelihood confidence intervals, as well as parametric bootstrap confidence intervals, for mark-recapture estimates of population size have much better coverage properties than symmetric Wald intervals (Evans et al Journal of Agricultural, Biological and Environmental Statistics). Models for 1", 1#,..., 15 that are in wide use include: (1) time varying capture probabilities (as in the above example), (2) capture probabilities that vary behaviorally, such as enhanced (trap-happy) or diminished (trap-shy) chance of capture after being captured, (3) capture probabilities that are heterogeneous among individuals, (4) various combinations of the above. Mark recapture estimation for closed populations was unified in a wellknown paper by Otis el al. (1978 Wildlife Monograph); those authors produced and freely distributed a computer program known as CAPTURE that made mark recapture models for closed populations widely accessible in wildlife management. # R program to calculate profile log-likelihood for population size # using data from a two-sample capture recapture survey. # Counts are entered into the vector yy here: capture histories (1 1), # (1 0), (0 1), in that order. yy=c(7,80,7); # Data in example are from: Skalski et al Ecology 64: # Initial patameter values are calculated from multinomial ml estimates. theta10=yy[1]/(yy[1]+yy[3]); theta20=yy[1]/(yy[1]+yy[2]); psi10=log(theta10/(1-theta10)); psi20=log(theta20/(1-theta20));

10 nn=sum(yy); tt0=nn/(1-(1-theta10)*(1-theta20)); # Range of values for profile plot. Adjust "tlo" and "thi" to get a # good plot. tlo=110; thi=370; t.vals=tlo:thi; loglike.vals=t.vals; # ML objective function "negloglike.ml" is negative of log-likelihood; # the optimization routine in R, "optim", is a minimization # routine. The three function arguments are: psi = vector of # parameters (transformed to real line), ts = fixed population size for # profile calculation, ys = vector of frequencies. negloglike.ml=function(psi,ts,ys) { theta1=1/(1+exp(-psi[1])); # Constrains 0 < theta1 < 1. theta2=1/(1+exp(-psi[2])); # Constrains 0 < theta2 < 1. k=length(ys)+1; p=rep(0,k); p[1]=theta1*theta2; p[2]=theta1*(1-theta2); p[3]=(1-theta1)*theta2; p[4]=1-(p[1]+p[2]+p[3]); nn=sum(ys); ys=c(ys,ts-nn); ofn=-(lfactorial(ts)-sum(lfactorial(ys))+sum(ys*log(p))); return(ofn); } for (tt in tlo:thi) { ii=tt-tlo+1; MULTML=optim(par=c(psi10,psi20), negloglike.ml,null,method="nelder-mead",ts=tt,ys=yy); loglike.vals[ii]=-multml$val; } ci.height=max(loglike.vals)-3.843/2; # = 95th chisquare(1) percentile ci.cut=rep(ci.height,length(t.vals)); plot(t.vals,loglike.vals,type="l",xlab="population size",

11 ylab="profile log-likelihood"); points(t.vals,ci.cut,type="l",lty=2); # Calculate Wald confidence interval. that=tt0; theta1=theta10; theta2=theta20; I.mat=matrix(0,3,3); I.mat[1,1]=1/(theta1*(1-theta1)); I.mat[2,2]=1/(theta2*(1-theta2)); I.mat[3,3]=(1-(1-theta1)*(1-theta2))/((1-theta1)*(1-theta2)); I.mat[1,3]=1/(1-theta1); I.mat[3,1]=I.mat[1,3]; I.mat[2,3]=1/(1-theta2); I.mat[3,2]=I.mat[2,3]; Vhat=solve(I.mat); Vhat; me=1.96*sqrt(that*vhat[3,3]); c((that-me),that,(that+me));