Mark recapture lecture 2:

• Jolly-Seber• Confidence intervals

Jolly-Seber

• For an OPEN population

• Repeatedly sampled

• Information on when an individual was last marked

Year

LPB

Col

ony

size

Open populationsIndividuals enter or leave the

population between surveys

Survey 1 Survey 2

Catch nt animals

Check if each animal is marked

Total unmarked (ut ) Total marked (mt )

Mark all withcode for this time

period

Release St (equals nt if no handling mortality)

NO YES

Question: What is formula for proportion marked?

Jolly-Seber

Remember Petersen (biased):

N= C MR

Problem: We don’t know how many marked in population (M)

Sample 1: mark 21 animalsSample 2: mark 41 animalsSample 3: mark 46 animals

How many marked at beginning of sample 4?

Not 21+41+46=108, as some will have died or emigrated

A

A

A

Time 1

AB

A

BB

Time 2

ACC

B

A

B

Time 3

Mark 3, but1 of these emigrates

Mark 3 more, but 1 marked animal dies

Mark 2 more, no loss of marked animals

Time 4

Marked animals in sample 4 (m4 ) = 3

+ Marked animals not in sample 4

=Total number of marked animals in population

How many marked animals are alive and present in the population at time 4?

D

DD

DD

Time 4

Marked animals in sample 4 (m4 ) = 3

+ Marked animals not in sample 4

=Total number of marked animals in population

D

6 marked at end of time 4 (S4 )

D

DD

DD

Time 4

Marked animals in sample 4 (m4 ) = 3

+ Marked animals not in sample 4

=Total number of marked animals in population

DDD

Time 5

D D

6 marked at end of time 4 (S4 )

Time 4

Marked animals in sample 4 (m4 ) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

Time 5

6 marked at time 4 (S4 ), recaptured (R4 )=1

D

DD

DD D

DD

D D

Time 4

Marked animals in sample 4 (m4 ) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

Time 5

6 marked at time 4 (S4 ), recaptured (R4 )=1

D

DD

DD

E

D

E

ED

D D

Time 4

Marked animals in sample 4 (m4 ) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

Time 5

E

D

E

ED

Time 6

D

6 marked at time 4 (S4 ), recaptured (R4 )=1+1

D

DD

DD

E

D

E

ED

D D

Marked animals in sample 4 (m4 ) = 3

+ Marked animals not in sample 4 (> 1)

=Total number of marked animals in population

6 marked at time 4 (S4 ), recaptured (R4 )=1+1

Marked animals alive but not found in sample 4

= Recaptures after sample 4 (Z4 =1)x

factor accounting for animals missed or lost from population (S4 / R4 ) = 6/2 = 3

Marked animals in sample 4 (m4 ) = 3

+ Marked animals not in sample 4 (=3)

=Total number of marked animals in population (M4 = 6)

Marked animals alive but not found in sample 4

= Z4 * S4 = 1* 6 = 3R4 2

Mt = mt + Zt * StRt

Biased formula for number of marked animals in population:

Mt = mt + Zt * (St + 1)(Rt + 1)

Unbiased formula for number of marked animals in population:

Jolly-Seber

Remember Petersen (biased):

N= C MR

Rearrange to:

N = M(R/C)

Number marked in population

Proportion markedin sample

Jolly-Seber

Nt = Mt(?)

Number marked in population (t)

Proportion markedin sample t

Jolly-Seber

Nt = Mt(?)

Number marked in population

Proportion markedin sample

mtnt

? = mt +1nt + 1? (unbiased) =

Question:

m5 = 21S5 = 9R5 = 4Z5 = 10n5 = 43

What is N?

Confidence intervals

( )

A range around the estimate of a parameter which

-if repeated –

would include the true value of the parameter a certain percentage of the time

( )( )

( )

( )( )

( )( )

( )( )

( )

( )( )

( )( )

( )

( )

( )

( )

( )

( )95% Confidence interval:

19/20 of these confidence intervals contain the true value

true value

Example of 95% confidence intervals:

501 British Columbians: “Which party would you vote for in the next provincial election?”:

CI = ± 4.5%

Sept 2004 Dec 2004 Feb 2005 Feb 2009

BC Liberals 43% 40% 46% 52%

NDP 37% 43% 40% 36%

Green 10% 8% 10% 12%

Reform 4% 2% 2% --

Difference between confidence interval and variance:

Variance: know distribution of MANY data points around estimate (mean)

Eg. We measured height of 500 British Columbians (1.4 m + 0.2 m)

Confidence interval: only have ONE parameter estimate, have to guess what the distribution of repeated measurements might look like

Eg. We obtained percentage of British Columbians who “disapprove of Campbell’s performance”, and estimated CI (51% + 4.5%)

Is the ratio ofR/C > 0.10?

Is the number of recaptures, R > 50?

Poisson

Normal

BinomialPetersen

Schnabel

Schumacher-Eschmeyer

Jolly-Seber: complex lognormal assumed,See Krebs p 47

Step 1: Make an educated guess as to the distribution (p 22 Krebs)

Y

Y

Step 2: Calculate CI for either R or R/C (as appropriate)

-see formulae in Krebs

Step 3: Insert upper and lower bound for R or R/C into the formula for estimating population size to obtain CI

For example, if CI for R/C is (0.083, 0.177), to calculate CI for N by Petersen:

N=M/ 0.083 (upper bound) N=M/ 0.177 (lower bound)

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