Reduced-parameter models for analysis of capture-recapture date from one- and two-age class open populations

CLASS OPEN POPULATIONS

A report of research for U.S.F.W.S. Research Work Order, Unit Cooperative Agreement No. 14-16-0009-1522, Work Order #4.

Mimeograph Series No. 1675

Cavell Brownie Department of Statistics North Carolina State University

Management of any animal population requires some knowledge of population size and of survival and reproductive rates, the two parameters that determine changes in population size. Estimation of population size, survival rate and reproductive rate is thus an extremely important cOJllPonent of wildlife management and research programs.

Efficient methods for estimating survival rate from banding and band recovery data were developed by statisticians and published as early as 1970-71. However, these methods saw very little use until the development of computer programs (ESTIMATE and BROWNIE) to compute these estimates and the publication of the associated monograph by Brownie et al. (1978). Now, largely because of this computer software, these methods are widely used by biologists.

models which assume either constant sl:lrvival or constant survival and capture rates. This is presented in two parts. Part 1 relates to the models appropriate when data is available for one age-class only, and Part 2 describes the models for data from young and adults.

NOTATION

The notation used is close to that of Jolly (1982) and Seber's text. The correspondence between this notation and symbols in the first page of output from program JOLLY follows.

Jolly (1982) Program JOLLY NM NU NN=NM+NU NS R

z

P Meaning

# of marked animals in sample i. # of unmarked animals in sample i. Total # caught in sample i.

# released from sample i.

# caught in sample i and later recaptured. # caught before and after but not in

sample 1.

capture probability.

Some additional notation needed is s

R. ~

.

the number of sampling occasions

the number of marked animals captured at i but not released (d

i < mi )

the number of animals first caught at i and released and m.-d. = R.-R. )

~ 1. 1. 1..

t. 1.

ljl

ljlti

₌

ljli =

qi l-p._1. X_s

=

1

the number of time units between samples i and i+1 The constant survival rate per unit of time, so that survival rate between samples i and i+1,

Here I-X.

=

probability an animal alive just after sample i is subsequently

~

recaptured.

MODEL B ALGORITHM

Necessary

input

mi ' ui ' n i ' Ri ' ri ' zi ' t i ' di ' plus "starting values" or initial estimates

R. s

1..

a

a

~ and p. ,

1. i=2, .•. ,s.

We should consider two methods for obtaining these starting values.

OPTION 1:

First compute (in the main program) the Jolly-Seber estimates ~i ' i=1, •.• ,s-2 ,

p. ,

1=2, ••.,s-l •

1.

Then the starting values are

i=2, .•.,s-1

s-l E Pi i=2

s-2

Pass these starting values to the Model B algorithm.

This will usually be the best procedure - except in the case of sparse data sets where problems are encountered in computing the Jolly-Seber estimates

$.,p ..

For such data sets,

it may be possible to get Model B (or D) estimates. In fact, this is one of the reasons for developing the Model B algorithm. The following option would therefore be useful.

OPTION 2: Before computing the Jolly-Seber estimates (or tests), call the Model B algorithm with starting values

~O

,

P~""'P~

supplied by the user in some way (e.g., read in with the data?).

Printed output

Model B estimates ~,Pi' ($ti if desired), standard errors, and correlations or covariances.

Also, Ni,B

i

(M

i?), standard errors, and correlations or covariances.

Goodness of fit test to Model B

Output which need not be printed out

I-X.

_~ i=l, ••• ,s-l

i=2, ••• ,s-1

Computations

First, the iterative procedure is carried out to produce Maximum likelihood (ML) estimates of

$ ,

Xl"",X_s-_l ' From these estimates (and with X 1)'_s= we obtain

i=2, •.••s .

Lastly.

M

_i •

N

i •

B

i are obtained.

Some notation

Recall X = 1 (by definition). s

i=l ••••• s-l. computed recursively.

,f,0. 0 0

Given starting values ~

PZ •...•

Ps. calculate

i=2, ••• ,s • and

computed recursively.

o

working backwards from X

s

=

1 •

o

etc •• to l-X l .

1- 0 = (,f,O)ts-l(l 0 0) X_{s -l} ~ -qs Xs •

Then. ~O

...

is an sxl vector of starting values

o

for the iteration procedure. Xl

~n represents the corresponding vector at the nth iteration, n=1,2,3, ••.

e~ is the ith element of en

~

~n is an sxl vector of first order partial derivatives evaluated using elements in ~n, n=0,1,2, .••

~n is the ith element of ~n i

Hn is an sxs matrix, involving second order partials, evaluated using elements in ~n , n=0,1,2, •.•

vn

=

(Hn)-l is related to the variance-covariance matrix

V~j is the element in the ith row, jth column of Vn •

Explicit formulas for computing

~O

, HO ,

~l

, etc., are given below, but first the basic iteration scheme is outlined.

ITERATION SCHEME

°

Compute ~ HO, using the starting values in

eO

(i.e.,

.

~O

°

°

d

°

0)

us~ng ~ , Xl"'" _{Xs - l ' an P2""'Ps .}

°

°

-1

Compute V = (H )

1st iteration

Compute

~1

= <pI

In matrix notation,

Specifically,

a

1

=

<pI = <pO

s

0

_t:.~

+ r

V..

1 1 1

j=l ~J J

a

1 = 1 0

s

0 _t:.?

Xi - 1 = Xi -1

+

r

Vij i=Z , ••• , s , i

j=l J

Using

a

1 and

~O,

check for convergence (see below). If

1 1

convergence criterion is not satisfied, compute PZ, ••• ,ps '

. 1 .

a

1 b

us~ng e ements ~n _ , y

Use these to compute t:.1 , and HI 1

2nd iteration

2

Compute ~ and check for convergence using

for all i, i=1,2, ••• ,s.

~2

and

~l

•

nth iteration

Continue iterating in this way until convergence criterion is satisfied. That is, at the nth iteration, n=1,2, .••

and convergence is declared if

16~

-

6~-11

< 10-5

If convergence occurs at t e n th h ·1terat1on, _. 6n- l is t e vectorh n-l

of ML estimates and V the estimated variance-covariance matrix for the parameters ~ , Xl, ...,X

s-l . MLe's and variances n-l n-l

for P2' ••• 'P are obtained from 6 , V (see below).

s

-If the convergence criterion is not met at the nth iteration (i.e.,

16~_6~-11 ~

10-5 for any i), then compute

~n

,

~

and go to the (n+l)st iteration.

Formulas for computing ~n

If L represents the log likelihood function under Model B, the elements of ~ are the first order partials oL/08_i • Thus, ~l

=

oL/o~ , ~i

=

oL/oX_i_

1 ' i=2, ••• ,s , given below.

l s-lt'l{Z. m.(l-X.)} ~ = _

r

-l:.::- -2:. _ 1 1

1 ~ i=2 Xi qi Pi

[OMIT

if)

~i = - -I

[qi-l mi-1

- z. l+R. l-r. 1

]

- 1

[qimi _ z ]

5=3 Xi - l P_i-1 1 - 1- 1 - <jJti-l Pi i

qiXi

i=3, ••• ,s-1 m

1

[qS-l ms_1

]

5

~₅ = - - - z +R -r - l-x

Xs - l p5-1 5-1 5-1 5-1 5-1

And so

o

To compute the elements of ~ , in the above expressions for 6. ,

1

, respectively. Similarly,

~1

is

. 0 0 0 0

replace ~,Pi,qi,Xi w1th $ _{,Pi,qi,X i}

1 1 1 1

computed using ~ ,P.,qi'X, (produced after the 1~ iteration).

1 1

Formulas for elements in the Formulas for computing Hn

The elements of H correspond to -o2L/o6i06j , so H is of dimension sxs and is symmetric, i.e., H

ij • Hj i .

diagonal and upper triangle are given below, the rest are obtained by symmetry.

E1e1ents in row 1 are:

1 s-l

H

= -

1:

11 ep2 i-2 2

t. 1

~-2 Xi

[

OMIT

if)

s=3

.

=

-t m (l-q X )

s-2 s-l s-l s-1

2 2

ep _{Ps-1 Xs - 1}

+

ep

Rows 2 to s:

[

OMIT if] s=3

i=3, •••,s-1

H ss

1

=

-2 Xs-1

i=2,3, •••,s-1

H._~,j

=

0 i=2, ••• ,s-2, j>i+1 .

n

Elements of H , n=O,l,2, .•• are eva uate1 d by subst~tut~ng ~· . ~n,x.,p.,q.n n n

~ ~ ~

for the parameters ~,Xi,Pi,qi in the above expressions for H ij

Printing Model 8 estimates ~,

P2' •••

'Ps •

[Question: Do we need to print (~)t1,..•,(~)ts-1 in addition to ~

1]

After convergence, ~ contains the MLe's ~, _{X1""'Xs- 1 and V the} estimated variances-covariance8.

i=l, •.. ,s-l .

Vi +1 ,i+1 = Var(X.)~ i=l, ..• ,s-l •

= Cov(~,

X.

1)

J-From §, V, obtain

i=2, ...,8 •

(Note for

p ,

define

s and the above formula works.)

2 2

t. l(l-q.X.)

~- ~ ~

4>2

Var($)

Var(p )_s

=

2t 1P Var ~ + s- s

ep epts-1 Cov(~,xs-1')

A

is evaluated by substituting Xi for Xi ' qi for q i ' etc., and V

11 for Var ~, etc.

Var(p.) and covariances which must be computed are given below in terms of

1.

the elements of V. and with ~ti written as ~., (Reca11~. = (~)ti)

1. 1.

i=2, .• "s-1 (1)

Var(p )

=

s (2)

+

_1_{tj_1(1-qjXj)_ep - ep -

V1~

_...

+

~

V.._A.

-

q V_{• • '+1}}

i-1 'l'j_1 J 1.,J

(t

P

v.

11

_ <

s-1 s V + 1+1,sl

qi

l

~

1,i+1

~s-1

J

i=2, ... ,s-1 (4)

NOTE: Cov(P.,P

i)

=

Var(p.) , so that (3) with j=i, and (1), should yield

1 1

the same numerical result for a given value of i . Also note that Cov(p.

,p.)

= Cov(P. ,Pi) •

1 J J

i=2, .•• ,s-1

(5 )

Cov(~,p_s) =

- t P

s-1 s

~ V

_u

VIs

---~s-l

Model B Estimates of M

i ' Ni ' Bi ' Variances and Covariances

A

N

_i

= U.

+

M.

1. 1.

d,

7111.'+1

=

~i{mi-p,

_1. 2._m. 711._1.

+

R._1..}

1.

i=2, ••• ,s

i=2, ••• ,s-1

i=2, ••• ,s

i=2, ••• ,s

i=2, ••• ,s-1

(7a)

(7b)

(8)

(9)

(10)

NOTE: Define z =0 , and, as before,

X

=1 , so that (7a) holds

s s

for i=s.

A

To compute variances/covariances for M_i ' N_i '

B

i ' first compute the matrices X and Wdefined below, replacing the parameters ~ , ~i ' qi ' Xi ' etc. by the estimates ~ , $i ' qi ' Xi ' etc.

1 i f k=j

Define a =

kj

j

1T q~1>~

~=k+1

i f k<j

Compute a

Row 1 of the s by (s-1) matrix X

Rows 2 to s of X

o

For example, if s=4,

j=Z, ••• ,s-l

i=2, ...,8

j=l, ••• ,i-Z; i=3, .•. ,s

j=i,i+l, .•• ,s-l; i=Z, ••• ,s-l

(11)

(1Z)

(13)

Q3 X3 tIMZ<PZ q4X4 3 tk_l<Pk~(tk3

0 l:

cP X

_z

cP _{k=Z Xk}

-qZXZMZ Q3X3(1-cPl)MZcPZ Q4X4(1-<Pl)MZ<PZ(tZ3

Xl (I-Xl) _{cPlXIX Z <PIxlx Z}

X =

Q4X4(1-<PZ)M3<P3

0 -Q3 X3M3

X

_z

(l-x_Z) _{<P ZXZX 3}

0 0 -Q4 X4M4

Next, compute W = VX by matrix multiplication, so that W is s by (s-l) • Elements of Ware denoted W

ij in the formulae below.

2t._~-1 ₊ 2 }

+ ,l,M _{W1 i-I} -M-(-:-I-=--""'"") W. . 1

~ i ' i -X_{i -1} ~,~- i=2, ••• ,s (14)

M. t. 1 {] ] - Mit. 1~- M.~ }

+

<P WI, i-I

+

<P VII + I-Xi-I VIi

M. { Mit. 1 M. }

+

]

~- ~

l-X_j_

1 Wj ,i_l

+

$ V1j + I-X i-I Vji

j=i+l, ••• ,s; i=2, ••• ,s-1 (15)

+ t.] -1M. {] t·~-1(I-q.x.)~ ~ V

u

I

<j> qiVl,i+l _<P V

n

- <Pi-I!

M.] { t. 1 (l-q.X.)~- ~ ~

-~}]

V 1j

+ q . V . .

-l-X_j_l ~ J,~+1 <P _$i-l

{

t _{s-1 s V}

P

₊

-!!-

V}

cfl 11 cfl_{s - 1}

M

{t

P

V}

_ j s-1 s V . + ~

1-Xj_1 cfl 1J cfl_{s -1} j=2, .•• ,s (17)

NOTE: The variances and covariances in (1) through (10), and (14) through (17) are used in computing Var(N.) , Var(B.) , and

~ ~

covariances. It is probably not necessary to print out

i=2, •••,8 (18)

u.

u.

~ ~ ~ ~ ~ (~~) ...J... (~~)

Cov(N~ ,N

j ) = Cov(M.,M.) - - Cov p.,M. - Cov p.,H.

... ~ J Pi ~ J P_j J ~

i=2, •..,5-1; j=i+l, .••,5 (19)

[OMIT₅₌₃

if]

U1+1 ~ ~ }

+ ---- Cov(P.,Po+₁)

P1

+

1 ~ ~

1=3"",5-1 (21)

[OMIT_{5=3 or}

if]

4

UO+₁ } <P.U. {-q.<p.toU.

~ (~~) ~ ~ ] J ] ]

+ Cov <P'P_j

+

₁

+ ----

~

P_j

+

₁ Pi ~

<P.U.

~ (~ ~)

- _p. Cov P'+1'P,_{~ J}

J

MODEL D ALGORITHM

Necessary input

As for Model B, with starting values via OPTION 1 or OPTION 2.

OPTION 1:

Starting values are

~O

a Model

B

estimate

~

s

pO

=

_1_

r

A

s-l i=2 Pi

where ~ and P2""'Ps are the MOdel B estimates.

OPTION 2:

Read in values for

~O

, pO .

Printed output

Model 0 estimates ~ and

p ,

($t i a ~. if desired), standard errors,

~

and covariances. Also, Model 0 estimates

N. ,

Hi

(M.

?), standard

~ ~

errors, covariances.

Computations

The iterative procedure produces ML estimates ~ and

p ,

and the (estimated) variance-covariance matrix V (dimension 2 by 2). At each iteration, it is necessary to compute

i=2, ..• ,5-l

q=l-p , and X =1 as before;" s

Yl

-

0

Oi{qYi

R.-rJ

Yi +l = + ~ ~ i+l=2, ..• ,s

Xi

(24)

and

1+1 = 2, ...,s

j=l, .•• ,s-l

Iteration Scheme

k=l, ...,j as on page 14.

Iteration proceeds as for Model B but with the vector of starting

values and 6 (2 xl) and H (2x2) as defined below.

FormuLas for computing 6

At the nth step, ~n is computed by substituting the current

1 ~n n n n h b f 1

va ues ~ , p , Xi ' Y_i into t e a ove ormu as.

(25)

Formulas for computing H (2x2)

+

~~ ~ to_l(l-qXo)i~lto_l~o_l

o_l{Yo+(l-qXo)Wj} (27)

't' i=3 ~ ~ j =2 J J ,~ J J

+}

~

_{ti_l[l-qX i )}

i~l epo~o

i_l[qXjWj-Yo)

't' i=3 j=2 J J, J

(28)

s E

i=2

At the nth step, compute Hn by substituting the current values

(29)

respectively, in formulas (27), (28), (29). the inverse of the symmetric matrix HU •

Printing Model 0 estimates p ,

p

A A

After convergence,

e

contains the Model D estimates p , p and V the variance-covariance matrix.

A

~

That is, ¢

.. e

Var .. V

u

1

P

6 2 Var A .. V 22 = _P

Cov(~,p) .. V I2

Print estimates, standard errors and covariance.

Model 0 estimates of M. , N. , B. , variances and covariances

1 1 1

{

d. }

A A ~

71/'+1 .. p.

7I/.-p -

'IlJ.+R.

1. . 1. ~ m. "'1. 1.. 1.

i=2, ••. ,s

i=2, •••,s-1

i=2, ••. ,s

i=2, •..,s

i=2, ... ,s-l

(30)

(31)

(32)

(33)

(34)

To obtain variances/covariances for

M

Row 1 of X

j=2, . . . .5 or j-l=l . . . .,5-1 (35)

Row 2 of X

(36)

Compute W

=

VX by matrix multiplication, and the variances below which involve the elements of W and V, and S1 ' P

1 defined below.

8-1 S. _I:

t k Q1-1,k-1(1-xk ) 1"'2, ••• ,5-1 Compute _S1 = </l k=i

(37)

a

1=5

5-1

Xi

+

q</li I: _{Q1k Xk}

+

₁ i=2, .••• 5-1 k=1

P. = ₍₃₈₎

~

1 i=8

SjM

_J.

P.M.]

...L.._...L_-V J J V

1-QX_j 12 - 1-QX j 22

i<j, i=2, •.• ,s-1, j=i+1, ••• ,s (40)

[Note that (40) with i=j should give the same numerical result as (39) for the same value of i.]

i=2, ..•,s (41)

The variances and covariances in (39), (40), (41), and in V are used to compute Var(N

i), Var(Bi), and covariances.

i=2, ••• , s

(42)

U.

( A ~) (A ~) ...l.. (~~)

Cov N.,N. = Cov M.,M. - Cov p,M.

~ J ~ J P ~

u.

~ (A A )

- - Cov p,M.

P J

[OMIT_s=3

if)

(OMIT_{s-3 or 4}

if )

i=2, •••,s-1

i=3, .••,s-1

(44)

(45)

TESTING BETWEEN MODELS A, BAND D.

There are several test statistics which could be used. Jolly (1982) presents two statistics [equations (47), (48)J which are

r i

proportions

a- '

i=l, ••• ,s-l. However, the proportions

i

on the also

provide information about fit of the models, and this information is not utilized in Jolly's test statistics. I would like to compute and print separately the components corresponding to these two proportions to see how much difference the second component makes, especially for the data in Jolly's example. For the general user, however, only the sum of the two components need be printed.

Jolly presents two different types of test statistics, one for

comparing B with A (eqn. 47), the other for comparing D with B (eqn. 48). The latter is based on the likelihood ratio; the former is based more directly on a chi-square statistic. The two are equivalent in large samples, and there is no overwhelming statistical argument supporting the use of one type rather than the other. We can either compute the test statistics as Jolly does (but with the second component included), or we can compute a likelihood ratio test statistic in both cases.

Test of Model 0 VS Model 8

Notation

X

iB and

X

iD are the model B and model D estimates of Xi ' respectively. i=l •••• ,s-l •

and are the model B and model D

estimates, respectively. of

Compute

L

₁

=

-2 E

s-l {r

[l-

X

J

i10g I_AiD

i=l e X_iB

i=2 ••••,s-1 .

(47)

(48)

Compute Print out

Print out "Total chisquare ="

"Degrees of freedom =" s-2

Test of ModeL 0 vs ModeL A

Compute two test statistics, print out both.

(i) L₁

=

-Z E

5-1 {r . l o g ·

fR

i (l-X iD)] +

i=l ~ e _{r i} (49)

Print out L l ,

L

Z

(50)

and "Total chi square"

"Degrees of freedom"

=

Zs-5

"Probability"

=

(computed in usual way).

(ii) Compute

[r.-R. (l-x iD) J2

T

=

~ ~

Ii R.X.D(l-X._D)

~ ~ ~

[mi-(mi+Zi)~iDJ2

TZi

=

(mi+Zi)~iD(l-~iD)

i=l, •••• s-l

i=Z, ••• ,s-l

(51)

(52)

Print out individual chi-square values T₁₁, ..• ,T_l,s-1

Also print out

and _{TZZ ' ... •TZ•5-1}

and "Total chi square" "Degrees of freedom" "Probability"

=

Test of Model B vs Model A (Omit if s=3)

Compute two test statistics as for the test of D vs A •

(i) Compute L_l as in (49) but with

X

_iB in place of

X

_iD • Compute L

2 as in (50) but with

P

iB in place of

P

iD •

Print out ~,L2 and (in the usual format) the total chi-square • L_l+L₂, with degrees of

freedom • s-3, and probability.

(11) Compute T

l i as in (51) but with

X

iB in place of

X

iD

.

Compute T

2i as in (52) but with

PiB

in place of

P

iD

.

s-l s-l

Compute T_l •

r

T

l i T2

•

r

T2i

1 2

Comments on Structure of Program

The order in which computations are carried out for models A (i.e., the Jolly-Seber model), Band 0 should depend on the data set to be

analyzed. For "good" data sets, the best way to proceed is to do the computations for model A first, and use the Jolly-Seber or model A estimates $ i ' ~i to get starting values for the model B algorithm. For poor data sets where some summary statistics are zero, it may be better to start with the simplest model (model D) then proceed to B then A. This leads to the follo~ng possibilities:

OPTION 1: (for good data sets--should be the default?)

(i) First compute Jolly-Seber estimates (model A estimates). (ii) Use these to get starting values for the model B algorithm

(see the model B instructions).

(iii) Use model B estimates

$ ,

Pi to get starting values for the model 0 algorithm (see the model 0 instructions).

(iv) Proceed to tests. OPTION 2: (for poor data sets)

(i) Begin with t e mo eh d 1 0 a gorit1 hm uS1ng start1ng va ues· • 1 ~0, p0 read in with the data.

(ii) Use the model D estimates ~ , ~ to get starting values for model B;

o

~ , and

(i.e., ~ from model 0 is passed as the initial value,

o

~ from model D is passed as p., i=2, •.. ,s, to the

1

model B algorithm). This is instead of reading in_.

~O

and p?₁ as initial values for Model B as I had suggested earlier. (iii) Proceed to Jolly-Seber (model A) computations and tests,

With age-dependent models, the time for an animal to mature from the first to the second age-class must be the same as (or simply related to) the period between successive bandings. Thus, as in Pollock (1981), the period between bandings and the time spent in age-class 0, are both assumed to be one year in the models considered here. That is, ti=1 for i=l,... ,s-l, so that for the models with constant survival, <Pi=<pti=<p , i=I,... ,s-l. I cannot think of a useful way to generalize and allow variable ti in the two-age-class models.

Outline of models considered

Model A2 - variable or time-specific survival for adults and young, variable capture rates.

age dependent generalization of Jolly-Seber Model

same structure as Pollock's (1981) model, but with Mi viewed as variables, not fixed parameters.

estimable survival and capture rate parameters are

rates.

age-dependent generalization of JollY's (1982) Model B.

estimable survival and capture rate parameters are 19a , 19Y,

Model D2 - constant survival for adults and young, constant capture rates.

age-dependent generalization of Jolly's (1982) Model D. estimable survival and capture rate parameters are 19a , ~y, p.

Notation: Because of the complexity of formulae below, it seemed less confusing to use superscripts "a" and "y", instead of a prime or "0" or "1" to denote age dependence. The relationship between notation here, that in Pollock (1981), and the JOLLYAGE output is indicated below (but note that I have a question concerning the equivalence of NB(I) and zi).

1

_NN(I)

a

n. n.

1 1

0

_NN'(I)

_{n~ u~}

n,

₌

1 1 1

1

NM(I)

m. m.

1 1

1

NBCI)

? ?

z. Z,

1 1

R~

₁

SCI)

R~₁

R?

S'(I)

R~

1 1

1

R(I)

a

r. r.

1 1

0

_{R' (I)}

_r~

r,

1 1

N~

N(I) N~

1 1

1

+

M~

M(l)

M.

M.

1 1 1

1

PHI(I)

a

qJ. qJ.

1 1

0

PHI' CI)

qJy'

qJ.

l. l.

PC I)

p.

1

a y

and z1'+1

=

z.+r.+r.-m. 1 '_{1 1 1 1+}

3

(where s is the number of samples)

i=l, ••• ,s-l

i=l. ••• ,s-l

u~ = no. of unmarked adults caught at i

u~ = no. of unmarked adults present just before sample i

].

Define

M

l=O , and zs=0 •

Input for Model B2 algorithm

r~_]. s , u.a

].

plus starting values for ~a, mY,

_...

p

_2'····

p s .

OPTION 1:

These starting values can be computed by averaging the

·1 ·0

Pollock (1981) estimates ~. and~. , and from

]. ].

m.

].

z.

m.+R~ ~_{]. ]. a}

r.

1

i=2, ••. ,s-1

OPTION 2: Starting values for ~a, ~y. P2,."'Ps can be obtained from the Model D2 estimates of constant ~a. ~y and P (to be described later).

The iterative procedure is carried out to produce ML estimates

Aa Aa Aa Ay

lP , Xl ' ••• Xs-l ' lP •

Pi =

-!

_Aa

[l-:~_l

-

[l-x~Jl

Xi

From these estimates,

is obtained, i=2, ••• , s .

. . 1 a,O y,O 0 0

G1ven start1ng va ues lP , lP , P2' ••• 'Ps ' calculate

o

0

q.

=

I-p.

1 1 i=2, ••• ,s, and then obtain

a I-x·₁ a starting values for I-Xi '

using the formulas

and l-x~,

1

= lPa[l-qi+l

i=l, •.• ,s-l

where

Iteration is carried out as for Model B in program JOLLY, but with 8, ~, H and V as defined below.

The vector of estimates ~ contains the elements (s+l)xl

....

8

=

(s+l)xl

These are used to re-evaluate 11 (s-l)xl

H (s+l)x(s+l)

and

v

(s+l)x(s+l)

-1

=

H using the formulas below.

Blements of A

-1 s-l 1

= -

t

-a . 2 a

<P 1= Xi

I

Zi q.

1

(1)

(2)

[m

i

q i

_ Z. +

R~

-

r~]

Pi 1 1

1

(3)

i+l=3, ... ,s

(4)

(5)

(6)

(7)

i+l=3, ..• ,s

s-l

(1-;/1')

RY

_1,

H

=-

1

L

1,s+1 rftamY i=l Y

,. ,. xi

7

(9)

H. '+1_1,1

+ 1

[_<IIa a_xi+1

)2

i+1=3, ••• ,s

i=2, ••• ,s-1

(10)

(11)

H.. = 0 i=2, ••. ,s-1

,

i+1<J~.s (12)

1J

y

R.

1

H._1,S+1 = _a1-_y i=2, ...s (13)

If Xi-1

Row s+l

s-l

H

=~

E

s+l, s+l

(uy)l"

<II i=l

(14)

Elements H.. for which i>j are obtained from the above by symmetry, 1J

i.

e.,

H.. =H .. ,

1J J1 i=2, ...

,s,

8

After convergence, the elements of 9 are

and i=2, ...,s .

and VII = var(;a) , V_s+l,s+l = var(;Y) ,

It may not be necessary to print out x~ (or 1-x~) and

~ ~

variances/covariances. However,

p,

and variances/covariances should

~

be calculated (as below) and printed.

1=2, ••• ,s-1

(15)

V. . ]

V + ~ - V

Ii ~a qj i,j+1

2ii,j<s (16)

Cov(Pi

,p

_{s )}

=~

_a

[(1-:;

x~J

[Ps

_~a

_V

_u

+

v::]

x·

~

+~

[PO~:li

+

v::] _

qi [Po

VI '+1!~ +

Vi:;.o]]

a a

~ ~

i=2, ..• ,s-1

9

i=2, ••• ,s-1

_ Vli _ a

Vn]

; ['It

V1,i+l

(l-

q_{i Xi)}

a a

Aa X·1 cp cp

Cov(cp ,Pi) =

_-1

[Vls + Ps Vll] a

cp

i=s

(19)

[qi Vi+l,s+l

-V. 1 a

V1'S+1]

Ay

_-1

l,S+

(l-

qi Xi)

(20)

Cov(cp ,Pi) = _{a a a}

X·₁ cp cp

i=2, ••• ,s-1

d 2 f a Ba.

Mo el

B

estimates 0 M. , N. , 1 1 1

Note that M~ = 0 by definition, M. is actually M~

1 1 1

B~ are not estimable.

1 i=s Also, N~ 1 A M.

=

1 m.+z. 1 1 l-qi

x~

i=2, ••.,s

a u. Aa 1

U_{i -}- -

_p.

1

"a "'a .. N.

=

U. + M.

1 1 1

10

i=2, ••.,s

i=2, ... ,s

X and W defined below.

r

if i=j, j=l, ••• ,s-l

Let lX. , =

~J _{a a}

if i<j, j=2, •.• ,s-1,i=l, ••• ,j qi+l~ •••qj~

Thus, lX • • = (~a

)

J-~

II

j q if i<j

~J _{'k=i+l k}

Elements of

X

(s+l) by (s-l)

RY Y a 1 f j \

-=-q X -~

2 2

x

Y~ 1

Rows

2

to s

R~

~ lX.~,]

'j

X~_~ j=2, .•• ,s-1

a

{a a a

_{Y Y Y}

₁

q'+l X'+l l X i j R_{l q) il-q) ) +} R_{l q)} ~l-q) )

X_{2j =} ,] ,] a

X2 Xl Xl

a

{lMi-j

a a

RY .Yj

qi Xi q. 1 + R. 1)q)

x..

1 = ~- ~- + ~-1

~,~- a a Y

I-X·_~-1 _{Xi-l xi-l}

11

j=2, ••. ,s-1

I

Xi-l Xi-l i=3, ••• ,s-1, j=i, ••• ,s-l

X•. = 0

~J

Row 9+1

j~i-2, i=3, ••• ,s

X l '_{s+ ,J}

j

a = qj+l Xj+l t

i=l

R~ ex ••

~ ~,]

X~_~ j=l, ••• ,s-l

+ _2 WI . 1 +

_....:2~

W. . ]

~a ,~- 1- a ~,~-l

,.. Xi-l

i=2, •••,s

q. x~

M.

M.

M.

,] ,] ~ ~ ~

W. . 1 = ex. 1 . 1 + - WI . 1 +

l-q. a ~- ,J- a J- ₍

l-X~_l]

~,J-~ Xi

ql ,

M. [M.

+-al J. _V

a a 11 ql ql

M.

1

+ W1,i-l +

--(~~---]

VIi

l-X~_l

M.

+ _ ...

,]1....-a

I-x· 1

_r

[

W•. 1

J,~-M.

~

+ -_a VI'_J + ql

M.

~

V.•_~J

J

i<j, i=2, ••• ,s-1, j=i+l, ••• ,s

M.

+-.1 a <P

[

qi

i+I,j-I

a

a

<P <P

M.

+ _ ...

JL...-a

I-x· I

J-V••

V

-

!J.

-HI,j a

<P

(I-q.

_].

x~)

₁

_V

)1

a

Ij

<P

W . I P

=_

S,J- _ - ! W

a a

I,j-I

<P <P

M. [P

VI )

_-.1 -!V +--!

a a 11 a

<P <P <P

i=2, ••• ,s-I,

j=2, •.. ,s.

M.

[P

V.)

_ ...JL...- - ! V • + ...!J.

a a IJ a

I-x· I

_J- <P <P

j=2, ...,s .

Var(N~)

= Var(M.) +

u~

[q. +

u~

Var(p.) - 2COV(P.,Mi))

1 1 Pi ]. Pi 1 1

i=2.",.s

if.l

(~.a :-.a)

(A

A)

1

(A

A)

COV

N., N •

= Cov

M. , M . - -

Cov

p. ,M .

1 J 1 J Pi 1 J

u~ U~

1 -.1 C

(A

A)

+ -

OV

p. ,po

P_j P_j 1 J

13

tf.

-..l

(A

A)

-

Cov

p.,M.

P_j J 1

[

ua )2 2 a (Ua) 2

i+l ( )

qi~

i C (-a - )

+ ---- Var P'+1 - ov ~ ,po

PHI 1 Pi 1

2q,U~~+1

_{1 1 1 ( - a . )}

+ COV ~ ,P'+l

P_i+₁ 1

2~~~~+1

_{1 1 C (. • )}

OV p.,P₁'+1

Pi Pi+1 1

[a4ITs=3

if]

2 [

U~

y y a 1 . . y

+ [H.]₁ V I I + 2H._{s+ ,s+ 1}

~

--_p.

Cov(p.,~

₁ )

1

U

a

I

a i + 1 . · y

- U. q. VI +1 - - - -

COV(P1'+I'~)

1 1 , S P_i+l

i=2, ... ,s-1

~a

Ua

I

~ i -a • i+1 ·a •

- - - - Cov(~ ,p.) + - - - Cov(~ ,P'+l)

Pi 1 Pi+₁ 1

a.a a

ql U.1 _ • U. 11+ • •

I

+ - - - Cov(P·_1,P.) - - - - cOV(P'_l'P,+_l)

Pi 1 1 Pi+1 1 1

l

auT

if]

Cov(B~ B~)

s=3 l ' J

or 4

+ ---- COV(P.,P·+1) +p. 1 1 1 i -1 i s +1 s+1,

1+

~+1

(4

.Y))

- ---- Cov

P'+I'~

PHI 1

i=3, •.. ,s-1

i=2 •...• s-3. j=i+2 •..•• s-l

Model D2

Necessary Iuput

Same as for Model B2. plus starting values for ~a • ~y and p , via one of several options.

OPTION 1.

Compute starting values for ~a • ~y and p by averaging the corresponding Model A2 estimates.

OPTION 2.

Use Model B2 estimates ~a • ~y • and an average of the

p. •

as

~

starting values.

Read in starting values for ~ , ~ and p . COMPUTATIONS

"'a "'y

The iterative procedure produces ML estimates ~ ,~ and

p ,

and the corresponding estimated variance covariance matrix V (dimension 3x3).

At each iteration, it is necessary to compute q

=

1-p

(with

x

a • 1

as

usual), s

s-l [ a ]k-i

[I-X:]

r

~ q

k=i

o.

= (= ~ap if i=s-l)

l.

0

Iteration ScheIDe

i=l, ••• ,s-l

i=l, ••• ,s-l,

i=l, ••• ,s-l

i=s

Iteration proceeds

as

for Models

B, D, B2,

with ~ ,

A ,

H and V

as

defined below.

B1ements

of 8

(3.1).

"'a

~

fl

=

P

"'y ~

s-1

[a a

R.-r. y R.-r.

YYj]

Al =

-.!

I

[a Y

r. + r. + z. - 6. ....!--! + .£ ....!--!

C9a i=1 1 1 1 1 _Xia _C9a _xiy

s-l

[

R~-r~l

A

2 =

-.!

_{pq i=l}

I

R~1 - .-L.2.a X·₁

s-l

[

RY-r~]

A

3 =

-.!

I

R~ - .-L.2. C9y i=1 1 _XiY

[

_-

R~

₊ mi q ] ₊ [_{q f9Y}]2

a a

Xi 1-q Xi

t

t

2

1

6. m.

+ 1 1

q(l-q

X:)

s-l

1 [[6

i

(1-X~)]2 [R~

Y

Ri ]

=

r - - --

_ + L _

i=1 1-x~ q pq X~ f9a X~

1 1 1

s-l

R~(l-x~)

H =_1_

r

1 1

33

(Y)2. 1

_f9

Y

1= Xi

s-l

H

=_-1

_{r -}

1

12 f9a i=1

1-x~

1

m.

67)

+ 1 1

a

1-q X.

1

H21

,H

31 and H32 are obtained from Hji

=

Hij .

Printing out estimates for Model D2

After convergence, the elements of ~ are

Aa

V is the estimated variance-covariance matrix for q) ,

P ,

and Ay

q) . Thus,

Aa

= var q)

( Aa A) V

12

=

cov q) ,p

=

var ~y

(

A Ay)

V

23

=

cov p,q) These estimates, standard errors and covariances (correlations?) should be printed.

Rstiaates of M. , N~ , B~

~ ~ ~

a

m. +

z.

u.

M.

~ 1 Aa _{- !} Aa _iJ~

+ A

= U. =

P

N. =

M.

i=2 •...,s

1

l-q _Xia 1 1 1 1

Aa B.

1

= UAa Aa qA uAa. + RY. ;.Y

i+l - q) 1 ~ ? i=2, ..•,s-l

Corresponding variances and covariances are obtained using the matrix W

=

V X

3xs-l

where Xis defined below.

Row 1 of X

j

a

=

q xJ'+1

I

i=l

(_{9 g>}a)j-i a

I-x·

_~

1

0 .

R~

R! pq O.

M'J

....!...!+ y 0 ...J.+ ~ ~

a ql q i+l y a

X._~

x·

_~ l-q

x·

_~

t

j=l ••••• s-l

Row

2

of

X

j

a X2j = -q Xj+l

I

i=l

(_{9 g>}a) j-i a l-q xi+l

o.

M.

p]

+ 1 1

l-q X~

1

j=l •••.• s-l

Row 3 of X

Matrix W (3 by s-1)

Compute W

=

VX by matrix multiplication. (l-q ~.)

Also. compute P.

=

o. -

~

~ 1 P

(recall 6 =0) s 21 t tel7l1 vanishes for i=l

j=l •..•• s-l

M. [ [q 6.]2 M. VII 2 M. V22 var

(M.)

= _-,,1.,,-- a + _ _1. 1. + P _1.=---==

a q Xi a a i a

1. l-q Xi ~ l-q Xi I-q Xi

[q

6'1

- 2 1. WI . I + 2

P.

W

2 .

1 -

2

P.

a 1- 1. 1.- 1.

~'

,

i=2, ••.,s

[q.:i]

Cov (M.,M.) =

1 J

a

q

x.

M.

J 1

a l-q X.

1.

q 6 i Mi ~a(l_q X~)

1

I

q 6. M.

W

_

J J

l,j-l ~a(l_q X~)

P. M. V12] V + J J

11 1_{-q X·}a

J

P. M. [

+ 1 1_a W_2,j-l l-q

x.

1

q

6. M.

P. M. V

_22]

J J

V

+ J J

a(

a)

12

a

~ l-q

x.

l-q X.

J J

for i<j, i=2, .••,s-1, j=i+l, ••. ,s

Note that in Cov(M.,M) all terms involving 6 will be 0 •

1 S S

i=2, ...,s

i=2, ..• ,s

1 J 1 J P J P

a a U. U. V

22 + 1 .J

p2

1

i=2, ••• ,s-1 j=i+l, ••••s

Var

(B~)

1

=

q

~+1

+

a Ua (1 +

S-i-p

a] +

p q

ql

i

[

U~+1

- qla

_p

~12

V

₂₂

+ ( Ua.)2 V

_{q 1}

₁₁

+

2[U~+1

_p

- qla

U~]

[

_q

u

a.] V + (RY)2 V

₁

₁₂

_i

₃₃

+ 2R~

1

I

a

a

a

1

ql

U'-U'+

_l

[

~

1

]V

_{23 -}

U~_1

q

V

₁₃

i=2, ••• ,s-1

(<J4IT₉₌₃

ifl

_J

a a ..8

1

+

u~[Ui-ql

U

i-

1]

V

q 1 P

12

+ R~

1

i=3, ••• ,s-1

[<J4IT if ] s=3 or 4

[

a a a

1

C9 U.-U'+l

y , ] . I a

+

R

_i [ p

]V

_{23 -}

U

_j q

V

₁₃

+ R~ J

i=2, .•. ,s-3, j=i+2, •.• ,s-1

Aa

Note that formulae for B. , variances and covariances have been

1

included for Models B2 and D2, but are not given in Pollock (1981). Pollock (1981) states that B~ refers to recruitment of adults

1

through tm.igratioD only [page 523 (first paragraph)]. However,

through survival of unmarked young in year i • If the B. on

~

page 20 do not seem to be meaningful quanti ties, . then the program need not compute the B~

,

variances and covariances.

~

Testing between Models D2, B2 and A2. (i) Test of Model B2 versus Model A2

Let _Pi,B2 and be the

model B2 estimates of Pi '

_x·

a

~ and

x~

~ respectively.

Let

Compute

p.

B2

A = ~~!

=-__

Pi,B2 1 A Aa

-Qi,B2 Xi,B2

[ a

r.-R. I-x.

_~ a(_~ Aa_~!B2)]2

RaAa

_·x·

_B2(1 Aa

_-x·

_B2)

~ ~, ~,

[m.-(m.+z.)p. B2]2

=

~ ~ ~ ~.

(m.+z.)p. B2(1-p. B2)_{~ ~~, ~,}

[r~-R~(l-X~

_{~ ~ ~.B2}

)]2

R

y Ay

_{. x·}

_B2

(1

_-x·

Ay _B2)

~~, ~,

i=2, ••• ,s-1

i=l, ... ,s-l

i=2, ••. ,s-1

i=l, ••• ,s-l

a a

Print out individual chi-square values T_{l1, .•• ,T1,s-1}

and

T22 ,···,T2,s-1

y y

T11 ' .•. , T1, s-1

Also print out

s-l s-1 s-1

Ta t a T₂ t T

2i TY t

Y

= T

U = = TU

1 _{i=l i=2} 1

i=1

"Probability" = (computed in usual way)

( ii) Model D2 versus Model A2.

Let and be the

model D2 estimates of p q

_x·

a

1 and x~1 respectively.

Let

Compute

PD2

.. =

---==----Pi,D2

1-

q

D2

X~,D2

[ a

r.-R.

₁ a(₁ I-x·.. a_1,D2)] 2

R

a..

_·x·

&

_D2

(1"&

_-x·

_D2

)

1 1, 1,

[m.-(m.+z.)p. D2]2

=

1 1 1 1,

(m.+z.)p. D2(1-p. D2T_{1 1 1, 1,}

i=2, ••• ,s-1

i=l, ••. ,s-l

i=2, ••. ,s-1

[ y

r.-R.

y(I-x·.. y

D2

)] 2

1 1 1.

R

y ..y

_{. x·}

_D2(l" y

_-x·

_D2)

1. 1., 1.,

i=l, .•. ,s-l

Print out and s-l 1: 1 & &

T11 ,···,T1

,s-1

T22 ,···,T2,s-1

y y

T11 ' ••• , T1,

s-l

s-l 1: 2

26

and s-lE

"Total chi-square"

=

T~ + T₂ + Ti

"Degrees of freedom"

=

3s - 7

"Probability"

=

(iii) Model

D2

versus Model

B2

Compute +

[R~-r~1

log

X~'D21

1 1 e .a

Xi,B2

i=1, •.•ts -1

.

log

~i,D2

+

e

Pi,B2

[

l-

P.

D21]

z. log 1_.1,

1 e Pi,B2

i=2" •• ,s-1

Print out

=

-2Ir~

log

[1-

Xr

'D2] +

[R~-r~l

log

x

r.

D2]

1 e 1 .y 1 1 e .y

-Xi,B2 Xi,B2

i=lt ,."s-1

a a

L11 ,···,L1,s-1

L₂₂,·",1_2,s-1

y Y

L11 ,·",11,s-1

and

ItDegrees of freedom =It s-2 Itprobability =It Checking for Small Expectations.

In carrying out these tests, before computing individual T_ij or L.. values, it will be necessary to check for small expectations as

1J follows:

(i) B2 versus A2

Check for values <2 :

Ra Aa

_R~(l-x~

B2) a a a

. x·

B2 _{Ri-ri ' r .}

1 1. 1 1, 1

RY Ay

_R~(l-x~

_{B2) R~-r~ , r~} . X· B2

1 1, 1 1, 1 1 1

(m. +z._{1 1}

),0.

_1,B2 (m.+z.) (l-,o. B2)_{1 1 1,} m.• z._{1 1}

i=l, ••• ,s-l

i=l, ••• ,s-l

i=2, ••• ,s-1

(ii)

(iii)

D2 versus A2

As for B2 versus A2 above, but replacing B2 estimates in formulae with D2 estimates (Le., _{rep aC1ng Xi,B2}I . Aa with Aa

Xi.D2

,

etc.).

D2 versus B2

Check for values <2 :

R

y

_{. x·}

AY _D2

1 1,

R~(l-X~

_{1 1,}D2)

R~(l-X~

_{1 1,}D2)

28

Ra Aa_{. X· B2}

1 1,

i=l, •..• s-l

R

y_{. X·}AY _B2

1 1.

i=l, ...• s-l

R~(l-X~

_{1 1,}B2)

(m.+z.)p.

B2

1 1 1,

(m.+z.) (1-.0.

1 1 1,B2)

i=2, ••. ,s-1

When expectations <2 are found, it will be necessary to pool before computing the T.. or L .. value, for example, as described

1J 1J

for testing between models

D, B

and A. Alternatively, the component T.. or

L..

could be omitted entirely (and a degree of

1J 1J

freedom subtracted from the degrees of freedom for the total chi-square) •

Goodness of fit tests

(i) Test of fit to Model B2

"Test of fit" chi-square

=

chi-square for B2 versus A2 + chi-square for test of fit

to A2.

df for test of fit to B

=

df for B2 vs A2

+ df for test of fit to A2.

(ii) Test of fit to Model D2

"Test of fit" chi-square

=

chi-square for D2 vs A2 + chi-square for test of fit

to A2.

df for test of fit to D2

=

df for D2 vs A2

+ df for test of fit to A2.

Brownie, C., Anderson, D. R., Burnham, K. P., and Robson, D. S. (1978). Statistical Inference from Band Recovery Data - A Handbook. Resource Publication No. 131. Washington, DC: Fish and Wildlife Service, United States Department of the Interior. Jolly, G. M. (1965). Explicit estimates from capture-recapture data

with both death and immigration--stochastic model. Biometrika 52, 225-247.

Jolly, G. M. (1982). Mark-recapture models with parameters constant in time. Biometrics 38, 301-321.

Pollock, K. H. (1981). Capture-recapture models allowing for age-dependent survival and capture rates. Biometrics 37, 521-529.

Seber, G. A. F. (1965). A note on the multiple-recapture census. Biometrika 52, 249-259.