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There are two issues that currently preclude a direct implementation of logical pooling for columns.
First, unlike pooling rows where physical pooling is equivalent to equating the initial tagging probabilities, there is no equivalent rule for recovery probabilities for column pooling. In the current model, the expected values for cells counts are (Schwarz and Taylor) in the 2 x 3 case (i.e. \(s \le t\)):
| Releases | Recovery stratum 1 | Recovery stratum 2 | Recovery stratum 3 |
|---|---|---|---|
| Release stratum 1 | \(p_1 \theta_{11} r_1\) | \(p_1 \theta_{12} r_2\) | \(p_1 \theta_{13} r_3\) |
| Release stratum 2 | \(p_2 \theta_{21} r_1\) | \(p_2 \theta_{22} r_2\) | \(p_2 \theta_{23} r_3\) |
| Unmarked | \(\sum{(1-p_i)\theta_{i1}r_1}\) | \(\sum{(1-p_i)\theta_{i2}r_2}\) | \(\sum{(1-p_i)\theta_{i3}r_3}\) |
where \(p_i\) is the tagging probability in release stratum \(i\); \(r_j\) is the recovery probability in recovery stratum \(j\); and \(\theta_{ij}\) is the probability of moving from release stratum \(i\) to recovery stratum \(j\). In the case of \(s < t\), the recovery probabilities are not separately identifiable because in any column, we can multiply \(\theta_{ij}\) by a constant \(k\) and divide \(r_j\) by \(k\) and get exactly the same expected values. Hence one can always force \(r_i = r_j\) for any (\(i\) and \(j\)) pair by appropriate choice of \(k\) values for each column.
Second, in theory, you can always pool columns and NOT affect the fit. As an analogy, the SPAS model is (to the first order approximation) a regression problem, i.e. given a matrix of recoveries and data
| Stratum | Recovery stratum 1 | Recovery stratum 2 | Recovery stratum 3 |
|---|---|---|---|
| Release stratum 1 | \(m_{11}\) | \(m_{12}\) | \(m_{13}\) |
| Release stratum 2 | \(m_{21}\) | \(m_{22}\) | \(m_{23}\) |
| Unmarked | \(u_1\) | \(u_2\) | \(u_2\) |
You want to find estimates of \(\beta_1\) and \(\beta_2\) such that \[u_1≅\beta_1 m_{11}+\beta_2 m_{21}\] \[u_2≅\beta_1 m_{12}+\beta_2 m_{22}\] \[u_3≅\beta_1 m_{13}+\beta_2 m_{23}\]
Pooling columns 2 and 3 reduces this system of equations to: \[u_1≅\beta_1 m_{11}+\beta_2 m_{21}\] \[u_2+u_3≅\beta_1 (m_{12}+m_{13})+\beta_2 (m_{22} +m_{23})\] which has the identical fit.
Like the regression analogy, pooling columns is like pooling two data points in a regression setting by adding the respective \(X\) and \(Y\) values. You will very similar same regression estimates, particularly if you do a weighted regression to account for the doubling of the variation when you add to points together.
So in theory, pooling columns should have negligible effect on the estimates of the population size. So at the moment, I doubt that it is possible to implement logical pooling of columns and compare column pooling using AIC. It would be possible to implement logical pooling of columns but it solves an uninteresting problem of testing if the product of movement and recovery are identical for all release groups in the two columns which is seldom biologically plausible.
This sample data set was adopted from the Canadian Department of Fisheries and Oceans and represent release and recaptured of female fish in the Lower Shuswap region.
test.data.csv <- textConnection("
160 , 127 , 72 , 82 , 3592
24 , 66 , 13 , 10 , 532
7960 , 9720 , 6264 , 7934 , 0 ")
test.data <- as.matrix(read.csv(test.data.csv, header=FALSE, strip.white=TRUE))
test.data
#> V1 V2 V3 V4 V5
#> [1,] 160 127 72 82 3592
#> [2,] 24 66 13 10 532
#> [3,] 7960 9720 6264 7934 0We now fit two models examining the impact of pooling columns with different types of pooling rows.
library(SPAS)
mod..1 <- SPAS.fit.model(test.data,
model.id="No restrictions",
row.pool.in=1:2, col.pool.in=1:4)
#> Using nlminb to find conditional MLE
#> outer mgc: 25668.84
#> outer mgc: 31226.96
#> outer mgc: 27658.72
#> outer mgc: 8924.818
#> outer mgc: 6133.671
#> outer mgc: 743.5753
#> outer mgc: 27.31259
#> outer mgc: 0.4124965
#> outer mgc: 0.08327987
#> outer mgc: 0.03053576
#> Convergence codes from nlminb 0 relative convergence (4)
#> Finding conditional estimate of N
SPAS.print.model(mod..1)
#> Model Name: No restrictions
#> Date of Fit: 2024-01-25 12:19
#> Version of OPEN SPAS used : SPAS-R 2023-03-31
#>
#> Raw data
#> V1 V2 V3 V4 V5
#> [1,] 160 127 72 82 3592
#> [2,] 24 66 13 10 532
#> [3,] 7960 9720 6264 7934 0
#>
#> Row pooling setup : 1 2
#> Col pooling setup : 1 2 3 4
#> Physical pooling : TRUE
#> Theta pooling : FALSE
#> CJS pooling : FALSE
#>
#>
#> Chapman estimator of population size 273430 (SE 10793 )
#>
#>
#> Raw data AFTER PHYSICAL (but not logical) POOLING
#> pool1 pool2 pool3 pool4 V5
#> pool1 160 127 72 82 3592
#> pool2 24 66 13 10 532
#> 7960 9720 6264 7934 0
#>
#> Condition number of XX' where X= (physically) pooled matrix is 45.64022
#> Condition number of XX' after logical pooling 45.64022
#>
#> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
#>
#> Conditional Log-Likelihood: 285428.4 ; np: 12 ; AICc: -570832.8
#>
#> Code/Message from optimization is: 0 relative convergence (4)
#>
#> Estimates
#> pool1 pool2 pool3 pool4 psi cap.prob exp factor Pop Est
#> pool1 110.8 134.4 86.5 109.4 3592 0.014 72.3 295561
#> pool2 24.0 66.0 13.0 10.0 532 1.000 0.0 645
#> est unmarked 8009.0 9713.0 6250.0 7907.0 0 NA NA 296206
#>
#> SE of above estimates
#> pool1 pool2 pool3 pool4 psi cap.prob exp factor Pop Est
#> pool1 5.4 6.5 4.2 5.3 59.9 0.001 3.5 13978
#> pool2 4.9 8.1 3.6 3.2 23.1 0.000 0.0 0
#> est unmarked NA NA NA NA 0.0 NA NA 13192
#>
#>
#> Chisquare gof cutoff : 0.1
#> Chisquare gof value : 31.95852
#> Chisquare gof df : 2
#> Chisquare gof p : 1.148937e-07mod..2 <- SPAS.fit.model(test.data,
model.id="Pool last two columns",
row.pool.in=c(1,2), col.pool.in=c(1,2,34,34))
#> Using nlminb to find conditional MLE
#> outer mgc: 25647.14
#> outer mgc: 30979.46
#> outer mgc: 26430.02
#> outer mgc: 12526.52
#> outer mgc: 4361.936
#> outer mgc: 373.9823
#> outer mgc: 76.32601
#> outer mgc: 13.19575
#> outer mgc: 0.437045
#> outer mgc: 0.04367686
#> outer mgc: 0.01676388
#> Convergence codes from nlminb 0 relative convergence (4)
#> Finding conditional estimate of N
SPAS.print.model(mod..2)
#> Model Name: Pool last two columns
#> Date of Fit: 2024-01-25 12:19
#> Version of OPEN SPAS used : SPAS-R 2023-03-31
#>
#> Raw data
#> V1 V2 V3 V4 V5
#> [1,] 160 127 72 82 3592
#> [2,] 24 66 13 10 532
#> [3,] 7960 9720 6264 7934 0
#>
#> Row pooling setup : 1 2
#> Col pooling setup : 1 2 34 34
#> Physical pooling : TRUE
#> Theta pooling : FALSE
#> CJS pooling : FALSE
#>
#>
#> Chapman estimator of population size 273430 (SE 10793 )
#>
#>
#> Raw data AFTER PHYSICAL (but not logical) POOLING
#> pool1 pool2 pool34 V5
#> pool1 160 127 154 3592
#> pool2 24 66 23 532
#> 7960 9720 14198 0
#>
#> Condition number of XX' where X= (physically) pooled matrix is 50.91806
#> Condition number of XX' after logical pooling 50.91806
#>
#> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
#>
#> Conditional Log-Likelihood: 295293.6 ; np: 10 ; AICc: -590567.3
#>
#> Code/Message from optimization is: 0 relative convergence (4)
#>
#> Estimates
#> pool1 pool2 pool34 psi cap.prob exp factor Pop Est
#> pool1 110.8 134.4 195.8 3592 0.014 72.3 295561
#> pool2 24.0 66.0 23.0 532 1.000 0.0 645
#> est unmarked 8009.0 9713.0 14156.0 0 NA NA 296206
#>
#> SE of above estimates
#> pool1 pool2 pool34 psi cap.prob exp factor Pop Est
#> pool1 5.4 6.5 9.4 59.9 0.001 3.5 13978
#> pool2 4.9 8.1 4.8 23.1 0.000 0.0 0
#> est unmarked NA NA NA 0.0 NA NA 13192
#>
#>
#> Chisquare gof cutoff : 0.1
#> Chisquare gof value : 31.62033
#> Chisquare gof df : 1
#> Chisquare gof p : 1.874568e-08mod..3 <- SPAS.fit.model(test.data,
model.id="Pool last two columns",
row.pool.in=c(1,2), col.pool.in=c(12,22,34,34))
#> Using nlminb to find conditional MLE
#> outer mgc: 25647.14
#> outer mgc: 30979.46
#> outer mgc: 26430.02
#> outer mgc: 12526.52
#> outer mgc: 4361.936
#> outer mgc: 373.9823
#> outer mgc: 76.32601
#> outer mgc: 13.19575
#> outer mgc: 0.437045
#> outer mgc: 0.04367686
#> outer mgc: 0.01676388
#> Convergence codes from nlminb 0 relative convergence (4)
#> Finding conditional estimate of N
SPAS.print.model(mod..3)
#> Model Name: Pool last two columns
#> Date of Fit: 2024-01-25 12:19
#> Version of OPEN SPAS used : SPAS-R 2023-03-31
#>
#> Raw data
#> V1 V2 V3 V4 V5
#> [1,] 160 127 72 82 3592
#> [2,] 24 66 13 10 532
#> [3,] 7960 9720 6264 7934 0
#>
#> Row pooling setup : 1 2
#> Col pooling setup : 12 22 34 34
#> Physical pooling : TRUE
#> Theta pooling : FALSE
#> CJS pooling : FALSE
#>
#>
#> Chapman estimator of population size 273430 (SE 10793 )
#>
#>
#> Raw data AFTER PHYSICAL (but not logical) POOLING
#> pool12 pool22 pool34 V5
#> pool1 160 127 154 3592
#> pool2 24 66 23 532
#> 7960 9720 14198 0
#>
#> Condition number of XX' where X= (physically) pooled matrix is 50.91806
#> Condition number of XX' after logical pooling 50.91806
#>
#> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
#>
#> Conditional Log-Likelihood: 295293.6 ; np: 10 ; AICc: -590567.3
#>
#> Code/Message from optimization is: 0 relative convergence (4)
#>
#> Estimates
#> pool12 pool22 pool34 psi cap.prob exp factor Pop Est
#> pool1 110.8 134.4 195.8 3592 0.014 72.3 295561
#> pool2 24.0 66.0 23.0 532 1.000 0.0 645
#> est unmarked 8009.0 9713.0 14156.0 0 NA NA 296206
#>
#> SE of above estimates
#> pool12 pool22 pool34 psi cap.prob exp factor Pop Est
#> pool1 5.4 6.5 9.4 59.9 0.001 3.5 13978
#> pool2 4.9 8.1 4.8 23.1 0.000 0.0 0
#> est unmarked NA NA NA 0.0 NA NA 13192
#>
#>
#> Chisquare gof cutoff : 0.1
#> Chisquare gof value : 31.62033
#> Chisquare gof df : 1
#> Chisquare gof p : 1.874568e-08Notice that the population estimate and its standard error is identical to the unpooled case. You cannot compare the AICc values because the data set has changed between the two fits.
#> .id date model.id s.a.pool t.p.pool logL.cond
#> 1 mod..1 2024-01-25 No restrictions 2 4 285428.40
#> 2 mod..2 2024-01-25 Pool last two columns 2 3 295293.64
#> 3 mod..3 2024-01-25 Pool last two columns 2 3 295293.64
#> 4 mod..4 2024-01-25 Pooled Peteren 1 1 60410.94
#> 5 mod..5 2024-01-25 Logical Pooling some rows 6 5 47585.39
#> 6 mod..6 2024-01-25 A single row - Logical Pool 6 5 47580.39
#> 7 mod..7 2024-01-25 Pooled Peteren - Logical Pool 6 1 56584.83
#> 8 mod..8 2024-01-25 Logical Pooling pairs rows 6 5 47584.73
#> np AICc gof.chisq gof.df gof.p Nhat Nhat.se
#> 1 12 -570832.81 32.0 2 0.000 296206 13192
#> 2 10 -590567.29 31.6 1 0.000 296206 13192
#> 3 10 -590567.29 31.6 1 0.000 296206 13192
#> 4 3 -120815.88 0.0 0 NA 70426 4545
#> 5 40 -95090.77 2.8 1 0.096 73440 10424
#> 6 37 -95086.79 13.1 4 0.011 70426 4545
#> 7 13 -113143.67 0.0 0 NA 70426 4545
#> 8 39 -95091.46 3.8 2 0.147 83198 13337mod..3 <- SPAS.fit.model(test.data,
model.id="Physical pooling to single row",
row.pool.in=c(1,1), col.pool.in=1:4)
#> Using nlminb to find conditional MLE
#> outer mgc: 31867.55
#> outer mgc: 31249.72
#> outer mgc: 28928.32
#> outer mgc: 14737.62
#> outer mgc: 2903.05
#> outer mgc: 230.5378
#> outer mgc: 8.523751
#> outer mgc: 0.1178486
#> outer mgc: 4.01421e-05
#> Convergence codes from nlminb 0 both X-convergence and relative convergence (5)
#> Finding conditional estimate of N
SPAS.print.model(mod..3)
#> Model Name: Physical pooling to single row
#> Date of Fit: 2024-01-25 12:19
#> Version of OPEN SPAS used : SPAS-R 2023-03-31
#>
#> Raw data
#> V1 V2 V3 V4 V5
#> [1,] 160 127 72 82 3592
#> [2,] 24 66 13 10 532
#> [3,] 7960 9720 6264 7934 0
#>
#> Row pooling setup : 1 1
#> Col pooling setup : 1 2 3 4
#> Physical pooling : TRUE
#> Theta pooling : FALSE
#> CJS pooling : FALSE
#>
#>
#> Chapman estimator of population size 273430 (SE 10793 )
#>
#>
#> Raw data AFTER PHYSICAL (but not logical) POOLING
#> pool1 pool2 pool3 pool4 V5
#> 1 184 193 85 92 4124
#> 7960 9720 6264 7934 0
#>
#> Condition number of XX' where X= (physically) pooled matrix is 1
#> Condition number of XX' after logical pooling 1
#>
#> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
#>
#> Conditional Log-Likelihood: 287272.7 ; np: 6 ; AICc: -574533.3
#>
#> Code/Message from optimization is: 0 both X-convergence and relative convergence (5)
#>
#> Estimates
#> pool1 pool2 pool3 pool4 psi cap.prob exp factor Pop Est
#> 1 139.1 169.3 108.5 137.1 4124 0.017 57.5 273857
#> est unmarked 8005.0 9744.0 6241.0 7889.0 0 NA NA 273857
#>
#> SE of above estimates
#> pool1 pool2 pool3 pool4 psi cap.prob exp factor Pop Est
#> 1 6.1 7.3 4.8 6 64.2 0.001 2.5 10831
#> est unmarked NA NA NA NA 0.0 NA NA 10831
#>
#>
#> Chisquare gof cutoff : 0.1
#> Chisquare gof value : 38.35232
#> Chisquare gof df : 3
#> Chisquare gof p : 2.380379e-08mod..3a <- SPAS.fit.model(test.data,
model.id="Logical pooling to single row",
row.pool.in=c(1,1), col.pool.in=1:4, row.physical.pool=FALSE)
#> Using nlminb to find conditional MLE
#> outer mgc: 31865.52
#> outer mgc: 31209.17
#> outer mgc: 29588.57
#> outer mgc: 5095.582
#> outer mgc: 3430.827
#> outer mgc: 423.876
#> outer mgc: 603.7936
#> outer mgc: 25.22585
#> outer mgc: 91.84818
#> outer mgc: 0.6298842
#> outer mgc: 0.00497048
#> outer mgc: 3.80732e-07
#> Convergence codes from nlminb 0 relative convergence (4)
#> Finding conditional estimate of N
SPAS.print.model(mod..3a)
#> Model Name: Logical pooling to single row
#> Date of Fit: 2024-01-25 12:19
#> Version of OPEN SPAS used : SPAS-R 2023-03-31
#>
#> Raw data
#> V1 V2 V3 V4 V5
#> [1,] 160 127 72 82 3592
#> [2,] 24 66 13 10 532
#> [3,] 7960 9720 6264 7934 0
#>
#> Row pooling setup : 1 1
#> Col pooling setup : 1 2 3 4
#> Physical pooling : FALSE
#> Theta pooling : FALSE
#> CJS pooling : FALSE
#>
#>
#> Chapman estimator of population size 273430 (SE 10793 )
#>
#>
#> Raw data AFTER PHYSICAL (but not logical) POOLING
#> pool1 pool2 pool3 pool4 V5
#> pool.1 160 127 72 82 3592
#> pool.1 24 66 13 10 532
#> 7960 9720 6264 7934 0
#>
#> Condition number of XX' where X= (physically) pooled matrix is 45.64022
#> Condition number of XX' after logical pooling 1
#>
#> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
#>
#> Conditional Log-Likelihood: 285423.8 ; np: 11 ; AICc: -570825.7
#>
#> Code/Message from optimization is: 0 relative convergence (4)
#>
#> Estimates
#> pool1 pool2 pool3 pool4 psi cap.prob exp factor Pop Est
#> pool.1 121.0 111.4 91.9 122.2 3592 0.017 57.5 236098
#> pool.1 18.1 57.9 16.6 14.9 532 0.017 57.5 37759
#> est unmarked 8005.0 9744.0 6241.0 7889.0 0 NA NA 273857
#>
#> SE of above estimates
#> pool1 pool2 pool3 pool4 psi cap.prob exp factor Pop Est
#> pool.1 6.3 7.5 5.9 6.9 59.9 0.001 2.5 9945
#> pool.1 3.5 6.3 4.3 4.5 23.1 0.001 2.5 1590
#> est unmarked NA NA NA NA 0.0 NA NA 10831
#>
#>
#> Chisquare gof cutoff : 0.1
#> Chisquare gof value : 38.35233
#> Chisquare gof df : 3
#> Chisquare gof p : 2.380369e-08# do physcial complete pooling
mod..4 <- SPAS.fit.model(test.data,
model.id="Physical pooling all rows and last two columns",
row.pool.in=c(1,1), col.pool.in=c(12,12,34,34))
#> Using nlminb to find conditional MLE
#> outer mgc: 31867.53
#> outer mgc: 30927.52
#> outer mgc: 25741.32
#> outer mgc: 14099.8
#> outer mgc: 11055.59
#> outer mgc: 1852.554
#> outer mgc: 165.8634
#> outer mgc: 11.48313
#> outer mgc: 0.2138227
#> outer mgc: 7.423679e-05
#> outer mgc: 2.239631e-11
#> Convergence codes from nlminb 0 both X-convergence and relative convergence (5)
#> Finding conditional estimate of N
SPAS.print.model(mod..4)
#> Model Name: Physical pooling all rows and last two columns
#> Date of Fit: 2024-01-25 12:19
#> Version of OPEN SPAS used : SPAS-R 2023-03-31
#>
#> Raw data
#> V1 V2 V3 V4 V5
#> [1,] 160 127 72 82 3592
#> [2,] 24 66 13 10 532
#> [3,] 7960 9720 6264 7934 0
#>
#> Row pooling setup : 1 1
#> Col pooling setup : 12 12 34 34
#> Physical pooling : TRUE
#> Theta pooling : FALSE
#> CJS pooling : FALSE
#>
#>
#> Chapman estimator of population size 273430 (SE 10793 )
#>
#>
#> Raw data AFTER PHYSICAL (but not logical) POOLING
#> pool12 pool34 V5
#> 1 377 177 4124
#> 17680 14198 0
#>
#> Condition number of XX' where X= (physically) pooled matrix is 1
#> Condition number of XX' after logical pooling 1
#>
#> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
#>
#> Conditional Log-Likelihood: 309568 ; np: 4 ; AICc: -619127.9
#>
#> Code/Message from optimization is: 0 both X-convergence and relative convergence (5)
#>
#> Estimates
#> pool12 pool34 psi cap.prob exp factor Pop Est
#> 1 308.4 245.6 4124 0.017 57.5 273857
#> est unmarked 17749.0 14129.0 0 NA NA 273857
#>
#> SE of above estimates
#> pool12 pool34 psi cap.prob exp factor Pop Est
#> 1 13.2 10.5 64.2 0.001 2.5 10831
#> est unmarked NA NA 0.0 NA NA 10831
#>
#>
#> Chisquare gof cutoff : 0.1
#> Chisquare gof value : 34.97117
#> Chisquare gof df : 1
#> Chisquare gof p : 3.34624e-09# do physcial complete pooling
mod..5 <- SPAS.fit.model(test.data,
model.id="Physical complete pooling",
row.pool.in=c(1,1), col.pool.in=c(1,1,1,1))
#> Using nlminb to find conditional MLE
#> outer mgc: 31868.37
#> outer mgc: 29339.97
#> outer mgc: 18211.34
#> outer mgc: 9394.572
#> outer mgc: 3418.652
#> outer mgc: 1125.712
#> outer mgc: 305.367
#> outer mgc: 48.36082
#> outer mgc: 1.890408
#> outer mgc: 0.003210701
#> outer mgc: 9.310952e-09
#> Convergence codes from nlminb 0 relative convergence (4)
#> Finding conditional estimate of N
SPAS.print.model(mod..5)
#> Model Name: Physical complete pooling
#> Date of Fit: 2024-01-25 12:19
#> Version of OPEN SPAS used : SPAS-R 2023-03-31
#>
#> Raw data
#> V1 V2 V3 V4 V5
#> [1,] 160 127 72 82 3592
#> [2,] 24 66 13 10 532
#> [3,] 7960 9720 6264 7934 0
#>
#> Row pooling setup : 1 1
#> Col pooling setup : 1 1 1 1
#> Physical pooling : TRUE
#> Theta pooling : FALSE
#> CJS pooling : FALSE
#>
#>
#> Chapman estimator of population size 273430 (SE 10793 )
#>
#>
#> Raw data AFTER PHYSICAL (but not logical) POOLING
#> 1 V5
#> 1 554 4124
#> 31878 0
#>
#> Condition number of XX' where X= (physically) pooled matrix is 1
#> Condition number of XX' after logical pooling 1
#>
#> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
#>
#> Conditional Log-Likelihood: 331838.7 ; np: 3 ; AICc: -663671.3
#>
#> Code/Message from optimization is: 0 relative convergence (4)
#>
#> Estimates
#> 1 psi cap.prob exp factor Pop Est
#> 1 554 4124 0.017 57.5 273857
#> est unmarked 31878 0 NA NA 273857
#>
#> SE of above estimates
#> 1 psi cap.prob exp factor Pop Est
#> 1 23.5 64.2 0.001 2.5 10831
#> est unmarked NA 0.0 NA NA 10831
#>
#>
#> Chisquare gof cutoff : 0.1
#> Chisquare gof value : 1.562464e-19
#> Chisquare gof df : 0
#> Chisquare gof p : NA#> .id date model.id s.a.pool
#> 1 mod..3 2024-01-25 Physical pooling to single row 1
#> 2 mod..3a 2024-01-25 Logical pooling to single row 2
#> 3 mod..4 2024-01-25 Physical pooling all rows and last two columns 1
#> 4 mod..5 2024-01-25 Physical complete pooling 1
#> t.p.pool logL.cond np AICc gof.chisq gof.df gof.p Nhat Nhat.se
#> 1 4 287272.7 6 -574533.3 38.4 3 0 273857 10831
#> 2 4 285423.8 11 -570825.7 38.4 3 0 273857 10831
#> 3 2 309568.0 4 -619127.9 35.0 1 0 273857 10831
#> 4 1 331838.7 3 -663671.3 0.0 0 NA 273857 10831Notice that the estimates of the population size are identical under logical or physical row pooling.
Darroch, J. N. (1961). The two-sample capture-recapture census when tagging and sampling are stratified. Biometrika, 48, 241–260. https://www.jstor.org/stable/2332748
Plante, N., L.-P Rivest, and G. Tremblay. (1988). Stratified Capture-Recapture Estimation of the Size of a Closed Population. Biometrics 54, 47-60. https://www.jstor.org/stable/2533994
Schwarz, C. J., & Taylor, C. G. (1998). The use of the stratified-Petersen estimator in fisheries management with an illustration of estimating the number of pink salmon (Oncorhynchus gorbuscha) that return to spawn in the Fraser River. Canadian Journal of Fisheries and Aquatic Sciences, 55, 281–296. https://doi.org/10.1139/f97-238
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.