Estimation of Time-Specific Survival Rates from Tag-Resighting Samples: A Generalization of the Jolly-Seber Model. C. Brownie 1 and D. S.

Transcription

1 Estimation of Time-Specific Surviva Rates from Tag-Resighting Sampes: A Generaization of the Joy-Seber Mode C. Brownie 1 and D. S. Robson Division of Nutritiona Sciences and Biometrics Unit, Corne University, Ithaca, New York 14853, U. S. A. SUMMARY The resighting rather than recapturing of individuas that were initiay captured, tagged and returned to a popuation offers a means of circumventing the often traumatic and sometimes fata effect of repeated capture and handing of wid animas in a tag-recapture study. The behaviora effect of initia capture and tagging, however, must often be accounted for in the mode. If sighting records are restricted to tagged individuas, with no attempt at estimating a tagged/untagged ratio in the popuation, the mode requirements are simpified to incude ony the modeing of mortaity among tagged individuas and to excude recruitment parameters in the unobserved, untagged portion of the popuation. Short-term capture/ tagging effects of specified duration (i.e., asting for ony one time period) are incorporated into the proposed mode, and their biasing effects thereby eiminated from estimates of time-specific surviva rates. Standard error formuas and tests of the mode are provided in this generaization of the Joy-Seber method of tag-recapture anaysis. 1 Present address: Department of Statistics, Nucear Sciences Center, University of Forida 32611, U. S. A. Key words: Birds; Joy-Seber mode; Sighting records; Tagging effect on surviva; Tag-recapture; Time-specific surviva rates.

2 -2-1. Introduction In many tag-recapture studies, recorded sightings rather than physica recaptures provide the recovery information on animas which have been captured, distinctivey tagged and returned to the popuation. For many species, utiization of sighting rather than recapture records has the obvious practica advantage of greater efficiency. If teescopic or eectronic devices are used for fied identification of the distinct tags, then there is the more important advantage that tagged animas may be monitored in a reativey undisturbed state. Thus the traumatic and even fata effect of repeated capture and handing may be avoided. In addition, the possibiity of biases due to some tagged animas becoming "trap-happy" or "trap-sby 11 may be circumvented by the use of sighting records. Cormack (1964) describes the estimation of time-specific surviva rates from sighting records. The we known Joy-Seber (J-S) method can aso be appied to sighting data, as indicated by Joy (1965). With the correct interpretation, the surviva estimators of J-S and Cormack are equivaent. Both of these modes are based on the often unreaistic assumption that tagging has no effect on behavior. However, initia capture and tagging does disturb the anima so new reeases may initiay have different surviva and sighting probabiities than other extant tagged animas. In this situation the J-S estimators of surviva wi be biased. Robson (1969) considered this probem in the tag-recapture context, and presented a genera probabiity mode which aows for an effect of tagging on surviva of specified but varying duration. Poock (1975) extended this mode to aow for a tagging effect on 11 catchabiity" as we as on

3 -3- surviva. Though potentiay usefu, these modes and the resuts for estimating popuation size are presenty avaiabe ony in a cumbersome notationa format which deters their appication. Cormack (1972), in a more intuitive manner, outined the estimation of surviva rates under Robson's (1969) mode, but his treatment does not provide sufficient detai for impementation of the methods (e.g., variance estimators are not given). We deveop here, for the tag and sighting study, a mode which aows for a short-term (one period) effect of tagging on surviva. Athough based on a different interpretation of the samping process, this mode is equivaent to the simpest generaization of the J-S mode considered by Robson (1969) and Poock (1975). Unike their mode formuation, we describe estimation and testing here in the simpest rather than in the most genera setting. Our objective is to present a generaization of the J-S mode in a readiy impemented format which permits estimation of time-specific surviva rates in the presence of a tagging effect on surviva. To make interpretation easier, we have used notation which is generay simiar to that of Joy (1965). Use of the methods deveoped is iustrated with data from a study on migrating sandpipers. methods to more compex modes is not deat with here. reader is referred to Brownie and Robson (1980). Extension of the The interested 2. The Experimenta. Situation and Notation 2.1 The Experimenta Situation The experimenta situation is that described in Cormack (1964). Animas are referred to as birds for convenience. At reguary spaced time intervas, a known number of birds is captured, tagged and returned to the popuation. These "instantaneous" batch reeases are cosey foowed

4 -4- (or preceded) by a batch sighting in which the number and identity of ony tag~ed birds is recorded. Tags are unique, so that the capture or sighting history of each individua can be foowed separatey. In practice, the sighting operation wi usuay not be instantaneous, but cover a period of time caed the sighting period. The period of surviva, or the period to which surviva rates appy, is the time between the start of one sighting period and the next. be short in reation to the period of surviva. is the compement of mortaity pus emigration. The sighting period shoud In this context, surviva It is not possibe to distinguish between these two sources of "apparent" mortaity. In this simpest situation the correspondence between the tag and sight data and tag-recapture data shoud be cear. Note, however, that in order to appy tag and sight modes to tag-recapture data there must be no "osses on capture". More compex situations may arise because the "recovery" of tagged birds and the "trapping" of new birds for tagging are two separate operations. These situations have no anaog in the usua tag-recapture context. For exampe, in trapping new birds for tagging there may be some recaptures of previousy tagged birds, so that we have records of recaptures as we as sightings of tagged birds. If capture is assumed to affect behavior, there are numerous possibe ways in which such recaptures coud be treated. For simpicity, we assume here that such recaptures do not occur. A second probem arises if the reease of newy tagged birds takes pace during, or just before, the sighting period. In this case the newy tagged birds are more ikey to be sighted uness they immediatey disperse. Such sightings of newy marked birds are to be ignored as indicated in

5 -5- Section 6.1. A third probem arises if the reease of marked birds and the sighting period are separated by a substantia time interva, during which mortaity may occur. The Joy-Seber mode is not appropriate in this case, as discussed in Section Notation and Assumptions The foowing definitions may be appied in the tag-recapture context by repacing the words "sighted" and "resighted" by "recaptured". k the predetermined number of time periods in the study. N. the number of birds tagged and reeased at time i, i =,... ' k - 1. u.. = the number of birds tagged at time j and first sighted J --- at time i, j = 1,. ' k - 1, i = j +, ' k. V. = the number of birds sighted at time j, and next rej sighted at time i, j = 2, ' k -, i = j +,... ' k. n.. = u.. + v.. = the number of birds tagged or sighted at j and next J J J sighted at i, j =2,., k-1, i =j +,, k. m. = the tota number of tagged birds sighted at time i, i=2,,k. V. the tota number of birds subsequenty resighted from the m. birds sighted at time i, i = 2,, k- 1. U. = the tota number of birds utimatey sighted from the N. tagged at time i, i=,..., k-. R. = U. + V. = the tota number of birds utimatey sighted from the N. + m. tagged or sighted at time i, i = 2,, k- 1 Z. = the number of birds tagged before time i that are not sighted at i, but are sighted after i, i = 2,., k- 1 Note that v ij is undefined for j = 1, so et ni= ui and R 1 = u 1 In symbos,

6 -6- u21 = n21 i=2 m. i-1 i-1 i-1 2: u J.. + Iv J.. I nij i = 3,...' k j= j=2 j= v. = J k I vij i=j+ j = 2, 1 ' U. = J k L uij i=j+ j = 1,... ' k - 1 ' and i=2 z. = Z. 1 +U. 1 +V. 1 -m.= Z. 1 +R. 1 -m i = 3,... ' k - 1 Definitions of the subtotas m., U., V., and Z. can be checked by referring to Tabe 1. I Insert Tabe 1 here Two modes are considered. The simpest (Mode 1) is that of Cormack (1964) and assumes that tagging has no effect on surviva. The parameters of this mode are S. the probabiity that a tagged bird aive at time i survives to time i + 1, and p. the probabiity that a tagged bird aive at time i is sighted at time i In the more genera mode (Mode 2) it is assumed that tagging has a oneperiod effect on surviva, so that new reeases have a different surviva rate S~ from that, s., of other extant tagged birds. A basic assumption of the modes is that tagged birds behave independenty of each other. The popuation from which tagged birds are drawn

7 -7- is assumed to be homogeneous with respect to factors which may infuence surviva such as age or sex. Like Cormack (1964) we assume that the process by which birds are captured for tagging does not provide information about popuation size or birth parameters. Thus the quanti ties N. (the number 1 tagged at time i) are treated as known constants rather than as informative random variabes (as in Joy, 1965). not estimabe. Popuation sizes and birth rates are For Mode 2, the data are represented in an array in Tabe 1 in a format which is simiar to that of Tabe 1 in Joy (1965) but, for ater purposes, separatey recording first sightings and resightings. The Mode 1 representation of the data is equivaent to that of Joy (1965) and is iustrated in Tabe 2. Comparison of Tabes 1 and 2 shoud carify the reationship between our notation and that of Joy (1965). further aid the reader who is famiiar with Joy's notation, we ist the foowing simiarities and differences: To Brownie and Robson k N. 1 N. +m. 1 1 m. 1 ui = ni u.. +v.. =n.. 1J 1J.J u = R U. +V. = R z. 1 s. 1 pi n. = s. 1 1 Joy nio (assuming no osses on capture) mi ni n.. ( j = 2,., k-1).j R ~ (i = 2,... ' k-1) zi i pi In making these comparisons, differences in the experimenta situations must

8 -8- be kept in mind. I Insert Tabe 2 here I 3. Mode 2 Mode 2 is a generaization of the Joy-Seber mode which is based on the assumption that capture and tagging affect the surviva probabiity of a bird ony in the period immediatey after reease. Except for those birds which are newy tagged, a other birds have the same probabiity of surviving to the next period. Aso, it is assumed that capture and tagging have no effect on subsequent sighting probabiities. Define S~~ = probabiity a bird tagged and reeased at time i survives ~ to i + 1, i = 1,., k - 1. S. =probabiity a bird tagged before i, and aive at time i, ~ survives to i + 1, i = 2,, k - 1. pi 1 - ~ = probabiity of being sighted at time i for tagged birds which have survived to i, i = 2,.., k p. probabiity a tagged bird aive at time i is sighted at i or ater p. +a.s.p. "j_ ~ ~+ 1 { pk i=2,,k- i=2 Part of the chain of events resuting in sightings for the N 1 birds reeased at time 1 under Mode 2 is indicated in Figure 1 beow.! Insert Figure 1 here Reca that if there is some overap between the reease of tagged birds and the sighting period at time i, then any sightings of these new reeases are ignored (see Section 6.1), and this possibiity is not incuded in Figure 1.

9 -9- foows. The ikeihood of the d<t8. array [ u 0 0, v 0 0 } under M~de? ~ derived :is J J Note that the entries (uo 0 or Vo 0) in any coumn of the data array J J in Tabe correspond to a set of mutuay excusive events. (uo 1 o,u. 2.,,uko) is Mutinomia with parameters +, +, Thus, ( * * * ) No,sop. 1,s.q P 2'...,s.o. 18"+ qk 18k 1Pk L+ - - and for i = 1,..., k- these vector random variabes are mutuay independent. Conditiona on the vaue of m., the vector (v. 1.,v. 2.,,vk.) is aso +, +, Mutinomia with parameters (m.,s.p.,s.o. s. p. 2'..,s.o. s. qk sk pk)' + L+ + + L i=2,,k- Using an appropriate factorization, the distribution function P[[uo.,v.. } ] J J is obtained as a product of conditiona mutinomia distributions, with parameters as indicated in Tabe 3 in a format corresponding to that of the dab array in Tabe 1. I Insert Tabe 3 here The resuting ikeihood function for Mode 2 is L 2 [tu..,v.. }] J J k-1 u. N.-u. TT S ~~ (1 s ~ ) m P p i= k-1 Z. Z.+V. m.-v. ( ) X TT ~ Si 1-Sipi+ i=2 Noting that the Z. are functions of the V., U. and m., we can now identify a minima sufficient statistic for L 2 as

10 -10- The distribution of~ 2 is obtained as a product of conditiona Binomia distributions as ['[~ ~ J k-1 k-1 T P[U./N. X JT f'[m.m.+7.. ]P[V.Im.]. i= i=2 k-1 Z m. Z. V. m. -V. X JT (m i + i ) ( p. I p. ) 1 ( 1- [ p. I p. ] ) 1 f:u.i ) ( S. p. 1 ) ( 1-S. p. 1 ) 1 1 m. 1 \ V. 1 J ]_ = --- [] The foowing maximum ikeihood (ML) estimators are easiy identified from [ 1]: ~1 = u.in., + A S. p. 1 = V. m., + ]_ /':'-.. p.p. = m.(m.+z. ), ]_ ]_ i = 1,... 'k-1 i=2,,k- i=2,,k- and we sove for p., S. and~~ ]_ This gives A p. V. I(Z.+V. ), ]_ ]_ ]_ i=2,,k- A " v. ]_ m. + 1 Z. + +V. + 1 s. =--- ]_ m. V. 1 m. 1+z. 1 ]_ J ' i=2,,k-2 and u. m. 1 z. 1 +v. 1 ~~ = +_ + + ]_ Ni Vi+ mi++zi+, i = 1,... 'k-2 Asymptotic variances of these ML estimators obtained, for exampe, as in Seber (1973), are

11 -11- Var(}\) 2{1-1 1 p. r 1 E (V. ) E (V. +Z. ), ' i=2,,k- and Var(S.) "'i~ Var(s.) = 82{ 1 1_+ 1 ie(v.) E(m.) E(V. 1 ) + E(V i+) - E(mi+ +Zi+)} + ' E(m. + 1 )E(V Z. + 1 ) 8~~2{ 1 1 E(U.) N. E(V. 1 ) + 1 E(m. 1 +z. 1 ) + + i=2,...,k-2 E(V. + 1 ) - E(m z. + 1 ) } + ' i = 1,... 'k-2 E(m. + 1 )E(V Z. + 1 ) E(m. 1 +z. 1 ) + + Non-zero asymptotic covariances are isted in Appendix 1. Estimates of the arge-sampe variances and covariances are obtained by repacing parameters by their ML estimates, e.g., E(V.) is repaced by the observed vaue V., and S. by "' S., etc. 4. Mode 1 and a Test of Mode 1 Versus Mode Mode 1 and Its Reationship to the Joy-Seber Mode In order to determine whether capture and tagging effect the surviva rate in the period immediatey foowing reease, we compare Mode 2 with Mode 1, which is Cormack's (1964) mode and is aso essentiay the J-S mode. Thus, under Mode 1 a tagged animas aive at time i, regardess of capture history, have the same probabiity p. of being sighted at time i. Under Mode 1 it is not necessary to distinguish between first sightings, or u.. 's, and resightings, or v.. 's. However, the u..,v.. notation is used J J J J in order to faciitate comparison with Mode 2. The representation of the data under Mode 1 is iustrated in Tabe 2. Aowing for differences in notation, it can be seen that this is equivaent to the representation in Tabe 1 of Joy (1965).

12 -12- As undermode2, the ikeihood of the data array [u. 1,u.. +v..,j22} ~ ~J ~J is obtained as a product of conditiona mutinomia distributions. These distributions are characterized in Tabe 4 by exhibiting the size parameters and ce probabiities in a format corresponding to that of the data array in Tabe 2. I Insert Tabe 4 here I The ikeihood function under Mode 1 reduces to 1 1 [ u. 1.,u.. +V.., j~2j J ~J where ~ = [2 J Comparison with p. 234 of Joy (1965) shows that if there are no osses on capture (i.e., T =1), then 1 1 is the [ J n.o N.o-n o NiO! / (NiO - nio)! Pi ~ ( 1 - pi) ~ ~ same as 1' except for the factor This factor, modeing the probabiity distribution of nio' does not appear in 11 because we do not treat nio (i.e., Ni) as an informative random variabe. The reader is referred aso to the comments at the end of Section ~.3 of Seber (1965). A minima sufficient statistic for [2] is ~ = [m2,m 3,,~_ 1,!)_, R2,, ~-}, 9..nd the distribution of ~ under Mode 1 is

13 -13- This P<H's to the foowing ML Pstimators: p. A s. :::::: m. Z. (N. +m.) ' m R. R. N.+m. i=2,,k- i=2,,k-2 ~-1 Note that, aowinr, for differences in notation, and assuming no osses A on cu.pture, m. + Z. (N. + m.) /(R.) is the same as Joy's M., and the above esti mators are identica to the J-S estimators. For competeness, the asymptotic variances and covariances of these estimators are incuded in our notation in Appendix 2, equivaent to the corresponding formuas in Joy (1965). 4.2 Test of Mode 1 Versus Mode 2 but we note that they are We can now derive a test of the assumption that capture and tagging have no effect on surviva and sighting rates, against the aternative that capture and tagging affect surviva in the period immediatey foowing reease, i.e., a test of Mode 1 against Mode 2. The test is based on the conditiona distribution under Mode 1 of ~ 2 given ~ denoted by P~ [~2 1~ 1 ], and given by k-1 = r1 (m. )(N.) / (Ni +m.) I I v:- Lf. v. +Lf. i=2 --- [3]

14 -14- Each hynergeometric variabe in [3] can be approximated by a chi-square variabe in the usua w~y, and a contingency chi-square test on one degree of frc'ec1om r'1 b< en.rried out ( r.ee Tabe 5). For i = 2,, k-1, these chi-f,quare statistics are asymptoticay independent and may be added to give a tota chisquare statistic on k-2 degrees of freedom. Tabe 5 Contingency tabes for test of Mode 1 against Mode 2 u. v. N. - U. m.- V. N. m. i=2,,k- N. +m. Rejection of Mode 1 in favor of Mode 2 woud indicate that tagging does affect surviva during the period foowing reease. This test has been suggested by Robson (1969) in the tag-recapture context as a test for initia mortaity of fish due to tagging (see aso Seber, 1973, Tabe 5.10). 5. Tests of Fit to Modes 1 and Non-discriminant Goodness-of-Fit Tests For each of the Modes 1 and 2 the residua distribution, i.e., the conditiona distribution of the data array given the minima sufficient statistic, is used to obtain a goodness-of-fit test of the mode (see aso Brownie and Robson, 1976). The derivations of the residua distributions are straightforward and are omitted for brevity. We need the foowing notation: i=2,,k and j j j = 2,... 'k-1 i=j+,,k

15 -15- Note that m~~. J is obtained by summing from eft to right across the entries in the i th row of Tabe 1 as far as u.., hence m~~ = m J i+,i Mode 1 The residua distribution for Mode 1 is R. zi ) 2 ( * * krr ni+,i'.,~i mi+,i-1',~,i-1 i=2 ( Z. +R. = Z. 1 +m. 1 \ I m m-1~ ni!'-. i+' i+2, i' ' Ki/ For i = 2,, k-2, the corresponding contingency tabes are mi~ i+,i-1 n. 1. mi+ + ' mi~ n. 2. m-1~ i+2,i- + ' i+2, i ~~ ~,i-1 ~i ~i Z. R. Z. +R. each yieding a chi-square statistic on k- i - 1 degrees of freedom. (If pooing is necessary this shoud be done by combining rows eement by eement and reducing the degrees of freedom appropriatey.) The chi-squares corresponding to each tabe are asymptoticay independent and may be added to give a tota chi-square for the goodness-of-fit test of Mode 1.

16 -16- PM "'2 [[ u.., v.. ] IJ2] J J k-2 T( i=2 I u. ) (' v. \ ( z. \ '* *I \ ui+, i' 'uki vi+, i' ' vki} mi+, i-1' '~, i-1) with contingency tabes * m '- vi+, i ui+,i mi+ ~~ * m. + 2 '- 1.. V. + 2,. ui+2,i m. + 2,. * * i =2,..,k-2 mk i-1 vki ~i ~,i ' z. v. u. Z. 1 +m These are used as described for Mode 1, to obtain a goodness-of-fit test to Mode 2. When pooing is necessary, we recommend using these goodness-of-fit tests based on the residua distributions PM [[ u.., v..} 1-<~n ], rather than the conven J J ){J tiona test based on a statistic of the form X 2 = z(o- E) 2 /E, where E is obtained using ML estimates. When ces are pooed in order to cacuate X 2, A and E is based on the unpooed data, the statistic X 2 does not in genera have a centra chi-square distribution under the nu hypothesis that the mode is correct. 5.2 More Genera Modes If the goodness-of-fit tests resut in rejection of Modes 1 and 2, there may be severa reasons for inadequacy of the modes, incuding heterogeneity of

17 -17- the popuation samped. Another possibe reason is that the tagging effect is more extensive than the assumptions of Mode 2 permit. The methods of the preceding sections are easiy extended to form a series of increasingy genera modes. For exampe, a generaization of Mode 2 is obtained under the assumption that tagging affects not ony surviva during the period after reease, but aso the sighting rate at the start of the foowing period. A sti more genera mode assumes that the tagging effect extends beyond surviva and sighting one period after reease to surviva in the second period foowing reease. Estimation and testing procedures for modes refecting these assumptions are easiy obtained using the methods of Sections 2 and 3, and are described in Brownie and Robson (1980). 6. Practica Considerations In this section we consider features of practica importance which are pecuiar to the tag and sight experimenta situation, and have no anaog in the conventiona tag-recapture context. 6.1 Sightings of New Reeases In theory, the reease of newy tagged birds and sightings are assumed to occur simutaneousy at time i, i = 2,, k-1, and we have so far ignored the possibiity that birds tagged and reeased at "time i" may be sighted at "time i" In practice, this is a very rea possibiity, as sighting wi usuay foow after the reease of tagged birds. If the time between reease and sighting is short (as it shoud be), then newy tagged birds may not have dispersed propery and so wi be sighted with higher probabiity than survivors of previous reeases. In this case a different sighting rate (say p.) * wi appy to these new reeases, and the sightings at i of birds reeased at i (denoted by uii) wi not provide information about pi' nor about Si. It i[~ easiy proved that the binomia estimator u.. /N. of p~ is independent of

18 -18- the estimators of a other parameters under Modes 1 and 2. It is therefore useess to record the u.. sightings and we do not incude them in the data array. 6.2 Reease Foowed after an Interva by Sighting For practica reasons, reease and sighting may reguary be separated by a time interva which is substantia reative to the period of surviva. In this case the assumptions concerning surviva and sighting rates must be examined carefuy to determine which modes are appropriate. We define "time i + 1" to be the tirne of the sighting which foows the ith reease of tagged birds, i = 1,... 'k-1. Then the period of surviva to which S. reates is the period between sightings at i and at i + 1. Even if tagging has no effect on surviva or sighting rates Mode 1 wi not be appropriate. This is because the surviva rate S. for the period be tween sightings at i and at i + 1 wi not appy to birds in the i th reease in the much shorter period between their reease and sighting at i + If there is a tagging effect on surviva, but the interva between reease and the foowing sighting is ong enough for this effec~ to wear off, and for birds to disperse propery, then Mode 2 wi be appropriate. If birds do not disperse by the foowing sighting period or if the tagging effect on surviva persists beyond the first sighting period, then more genera modes are needed. 7. An Exampe The methods we have described are appied here to tag and sighting records coected in a study carried out to investigate factors infuencing daiy emigration rates of semi-pamated sandpipers (Caidris pusia), migrating from a staging area on the shore of Sibey Lake, North Dakota. Each day, the reease of newy tagged birds was foowed by a visua survey of the popuation, and

19 -19- si~htings of previousy ta~ged birds were recorded. There was virtuay no mortaity during the 2k-month study period, so daiy "surviva" rates were assumed compementary to the departure or emigration rates. The researcher questioned the vaidity of the J-S mode (Mode ) for these data, as he fet that the trauma experienced during capture and tagging may have resuted in the premature migration of some birds away from the study area on the day foowing reease. He was prepared, however, to assume that surviva (and departure) rates were otherwise unaffected by capture history, hence Mode 2 seemed appropriate. Efficient use of the data was essentia as numbers tagged were not arge, so the comparison of Modes and 2 was aso important. For iustrative purposes, ony a portion of the data set, corresponding to the 30-day period to , is used here. Tabe 6 contains the tagging and sighting records dispayed as in Tabe for Mode 2, with first sightings and resightings recorded separatey in aternating coumns. Records for days 10 to 20 are not dispayed. The compete data set wi be contained in David Lank's Corne University Ph.D. thesis. I Insert Tabe 6 here I The summary statistics used in cacuating the Mode 2 estimates, and A the resuting estimates, are presented in Tabe 7. Note that - S. is the J_ A estimated departure rate on day i for birds tagged before i, and Var(S.) is J_ A A the estimated variance of S. and of - S. As N J_ J_ 4 = N 9 = O, the parameters s4 and S~ are not estimabe. iustrated beow. A A* The cacuation of Si and Si for i = 8 (day 8) is

20 -20- and "' V8 m9 Z9+V9 88 =-- 29/31 X 24/20 X ms V9 m9+z = "'* = U8 m9 Z9+V9 11/20 X 24/20 X ss N8 V9 m9+z = I Insert Tabe 7 here "' The estimate s 8 = iustrates an unappeaing sma-sampe property of the unconstrained ML estimators of surviva for many tag-recapture modes (i.e., the property that the estimates may exceed 1). For this data set, it is ikey that on many days the true surviva rates are cose to 1, hence it is not surprising that many of the estimates (which are not very precise) are greater than 1. Various methods of adjusting these estimates have been suggested, e.g., Buckand (1980), none of which are entirey satisfactory. In the sandpiper study, such adjustments were not attempted since the reative magnitudes of the estimates were of greater interest than the actua vaues. Resuts for the test of Mode 1 versus Mode 2 are contained in Tabe 8. As described in Section 4.2, singe degrees of freedom chi-square statistics are obtained from tabes of the form u. N.- U. N ; e.g.' for i = 10, v. m. -v. m N. +m. 145 yieding a chi-square vaue of 0.37 on 1 degree of freedom. Tabe 8 contains these chi-square vaues with the exception of those corresponding to

21 -21- tabeg where an expected ce frequency of< 5 occurred. This criterion may be unnecessariy stringent to ensure vaidity of the chi-square approximation, but it was used because of the additiona concern that sma ce frequencies woud invaidate the assumption of independence among the individua chi-square statistics. I Insert Tabe 8 here The tota test statistic of on 14 degrees of freedom (obtained by summing entries of Tabe 8) is significant at the 1% eve, indicating that Mode 2 is preferred to Mode. Examining the individua vaues in Tabe 8 shows, however, that ony three of these are "arge" (> 3. 84, say). Thus for many of the tagging occasions there is itte evidence of an immediate effect on surviva. With a few exceptions, the J-S (Mode 1) surviva estimators wi probaby not be seriousy biased, and may be preferred in this situation because of their greater precision. The vaidity of Mode 2 is assessed by means of the goodness-of-fit test described in Section 5.1. A considerabe amount of pooing was necessary to meet the criterion of expected ce frequencies of at east five. For i = 24, the raw tabe , after pooing, gave with a chi-square vaue of 1.72 on 2 degrees of freedom. The tota test statistic gave a vaue of on 55 degrees of freedom, indicating a poor fit to Mode 2. There are severa possibe expanations for the inadequacy of Mode 2. The study area was a staging site for the sandpipers during the course of

22 -22- their southward migration. Birds were arriving at and departing from the study area (staging site) during most of the study period. On any day, for a given bird the departure probabiity may be reated to this unknown sojourn time as we as to current or impending weather conditions and other environmenta factors. Heterogeneity of departure probabiities induced by this type of "age-dependence" coud account for ack of fit to the Mode 2 assumption of ony "date-dependence". The Mode 2 assumption that surviva is independent of capture history, except immediatey after tagging, might therefore be fase. Another possibe reason for departure from the Mode 2 assumptions (hence aso from those of Mode 1) is that tagging may have resuted in out-migration from the immediate study area which was temporary and of varying duration, rather than permanent. Examination of individua contingency tabes (not presented here) showed that ack of fit to Mode 2 was not generay a serious probem. More compex modes woud have required additiona information and woud not have been usefu for these data due to the sma numbers invoved. The Mode 2 surviva estimates were therefore used to obtain information concerning the effect of factors such as weather patterns on departure rates. This is discussed in Lank's thesis. ACKNOWLEDGEMENT We wish to thank David Lank, Department of Bioogy, Box 8238, University Station, Grand Forks, North Dakota 58202, for permission to use his data in the exampe.

23 -23- REFERENCES Brownie, C. and Robson, D. S. (1976). Modes aowing for age-dependent surviva rates for band return data. Biometrics 32, Brownie, c. and Robson, D. S. (1980). Estimation of time-specific surviva rates from tag-resighting sampes: A generaization of the Joy-Seber mode. Paper No. BU-725-M in the Biometrics Unit Mimeograph Series, Corne University, Ithaca, New York, U. S. A. Buckand, S. T. (1980). A modified anaysis of the Joy-Seber capturerecapture mode. Biometrics 36, Cormack, R. M. (1964). Estimates of surviva from the sighting of marked animas. Biometrika 51, Cormack, R. M. (1972). The ogic of capture-recapture estimates. Biometrics 28, Joy, G. M. (1965). Expicit estimates from capture-recapture data with both death and immigration-stochastic mode. Biometrika 52, Poock, K. H. (1975). A K-sampe tag-recapture mode aowing for unequa surviva and catchabiity. Biometrika 62, Robson, D. S. (1969). Mark-recapture methods of popuation estimation. In New Deveopments in Survey Samping, N. 1. Johnson and H. Smith, editors, New York: Wiey. Seber, G. A. F. (1965). A note on the mutipe-recapture census. Biometrika 52, Seber, G. A. F. (1973). Estimation of Anima Abundance and Reated Parameters. London: Griffin.

24 -24- APPENDIX 1 Non-zero Asymptotic Covariances of the Mode 2 Estimators of Surviva and Sighting Rates A A 1 E(m. +Z.) Cov(p.,S.) = p.s.{ E(V.) J. E(m.)E(V.+Z.) J. J. J. }' i = 2,. 'k-2 Cov(S., "' "'* J. S.) J. -1 { 1 1 E(Vi+1)- E(mi++Zi+1)} = s. s. - + ' 1 1 E(V.+) E(m.++Z. 1 ) E(m. 1 )E(V. 1+Z. 1 ) J. J. J.+ J.+ J.+ J.+ i = 2,.. 'k-2 ' c-t A = -s~.. { E(zi+) } Cov(S.,p. 1 ) p J. J.+ J. J E ( V. ) J.+ E(m. J.+ 1+z ) E(m. J.+ 1 )E(V z. J.+ 1 )- i = 1,... 'k-2 i = 2,... 'k-2 ' A cov(si,si+1 ) A i=2,,k-3 i = 1,... 'k-3

25 -25- APPENDIX 2 Asymptotic V::trLmccc~ :me Non-zero Covari::mcc'~~ of the Mode 1 or J-S Estimators of Surviva and Sighting Rates Var(.) = p~~{ }' A Var(S.) 1 1 E(R.) N.+E(m.) E(m.) = s~{-1- _ 1 + ~ + 1 [ 1 _ 1 J 1 E(R.) N.+E(m.) E(R. 1 ) N. 1 +E(m. 1 ) i = 2,... 'k-1 ' i=2,,k-2 i =1,.,k-3 cov(p.+1's.) = -s.p.+1 ~+1{ 1 J. E(R. ) Ni+1+E(mi+1) E(Zi+1) i = 1,... 'k-2 with R. = m. _ 0

26 -26- Tabe 1 Representation of sighting (u) and resighting (v) data for Mode 2 for a study with k = 5 sighting periods Number of Birds Tagged and Reeased Number of Row Birds Sighted N N2 N3 N4 Totas Time 2 u21 m2 3 u v u m" _J 4 UL~ v42 u42 v43 u43 m4 5 u51 v52 u52 v53 u53 v5j+ u54 m5 Co. Totas u v2 u0 V, u v4 U1, L.-J ""

27 -27- N s~~ pyv32 yu2 ) p q~ s3 1 )~ 8 PVU31 ) s3 > 2 ) q~ 83 ) Figure 1

28 -28- Tabe 2 Representation of data under Mode 1 for a study with k = 5 Number of Birds Tagged and Reeased Number of N N2 N N4 Row Birds Sighted 3 Totas Time 2 u21 m2 3 u31 u32 + v32 = n32 m3 4 u41 u42 + v42 = n42 u4 3 + v 4 3 = n4 3 m4 5 u51 u52 + v 52= n52 u53 + v 53 = n53 u54 + v 54 = n54 m5 Co. Totas u = R U2+V2=R2 u3 + v3 = R3 U4+V4=R4

29 -29- Tabe 3 Parameters of conditiona mutinomia distributions for Mode 2, corresponding to the data array in Tabe 1 N m2 N2 m3 N3 ~~ Time 2 81P2 ~~ 3 81q2S2p3 * 82P3 82P3 ~ ~~ 4 8~S2q3S3p4 82q3S3P4 82q3S3P4 * 8 4 S3p~ ~I!. ~~ ~~ Co. totas S}_P2 82P3 82P3 83P4 Sjp!+

30 -30- Tabe 4 Parameters of corresponding conditiona mutinomia distributions N N2 +m2 N3 +m3 N4+m4 Time 2 81P2 3 81~ 8 2P3 82P ~S2q3S3p4 S2q3S3p4 83P4 5 81~S2q3S3q4S4p5 S2q3S3q4 84P5 S3q4S4p5 84P5 Co. Totas 81P2 82P3 83P4 8405

31 e e ~ ~.~...,..._....,.~...---or- -- e Tabe 6 Tag and sighting records for sandpipers, Juy 25 to August 23, 1978, Lake Sibey, North Dakota Time tagged and number tagged Row totas mi ': w +' {i r t~ H w s::.... ':.... ': a! +' 'r'i "' w a ~ oo I w... I CoL totas

32 -32- Tabe 7 Summary statistics and estimates for tag and sighting records in Tabe 6 Summary statistics Parameter estimates A A A A* Day N. u. v. m. z. s. 1- s. var(s.) ~ var(s.) p. J_ J_ J_ J_ J_ J_ J_ o.o o: o.o o.oobo

33 , Tabe 8 Test of Mode 1 versus Mode 2 Individua singe degrees of freedom chi-squares Chi-square Chi-square Chi-square Chi-square i vaue i vaue i vaue i vaue Tota chi-square vaue with 14 degrees of freedom