# Conne River 1991 Data

#### Carl James Schwarz

#### 2019-12-01

# 1 Introduction

This analyzes a dataset from the Conne River, Newfoundland collected in 1991. Smolts are tagged and released at one location in the river. A fence at a second station also captures fish. Data are stratified on a weekly level.

The Conne River 1992 vignette has larger sample sizes.

A stratified-Petersen estimator (Darroch, 1961; Plante et al. 1996) is found for the original data and some pooling of the rows/columns.

# 2 Load the SPAS package

    library(SPAS)
    #> Warning in checkMatrixPackageVersion(): Package version inconsistency detected.
    #> TMB was built with Matrix version 1.2.17
    #> Current Matrix version is 1.2.18
    #> Please re-install 'TMB' from source using install.packages('TMB', type = 'source') or ask CRAN for a binary version of 'TMB' matching CRAN's 'Matrix' package
    #> ***** SPAS: Stratified Petersen Analysis System - Version 2020.1.1 2020-01-01) *****
    #>
    #>       Help available with  help(package='SPAS')
    #>       Several vignettes are available. See browseVignettes(package="SPAS")
    #>
    #>       ***** CAUTION ***** Output object has significantly changed since last release ****
    #>       Use SPAS.print.legacy() to display previously created objects

This will load the SPAS fitting functions and any associated packages needed by SPAS.

# 3 Import the data.

The data should be stored as an \(s+1\) by \(t+1\) (before pooling) matrix where the upper \(s \times t\) matrix consists of the The \(s \times t\) upper left matrix is the number of animals released in row stratum \(i\) and recovered in column stratum \(j\). Row \(s+1\) contains the total number of UNMARKED animals recovered in column stratum \(j\). Column \(t+1\) contains the number of animals marked in each row stratum but not recovered in any column stratum. The \(rawdata[s+1, t+1]\) is not used and can be set to 0 or NA. The sum of the entries in each of the first \(s\) rows is then the number of animals marked in each row stratum. The sum of the entries in each of the first \(t\) columns is then the number of animals captured (marked and unmarked) in each column stratum. The row/column names of the matrix may be set to identify the entries in the output.

Here is the raw data for the Conne River. Notice that very small number of releases and recoveries in week 6.

    conne.data.csv <- textConnection("
    9  ,    21  ,     0  ,    0  ,    0  ,    0  ,   171
    0  ,   101  ,    22  ,    1  ,    0  ,    0  ,   763
    0  ,     0  ,   128  ,   49  ,    0  ,    0  ,   934
    0  ,     0  ,     0  ,   48  ,   12  ,    0  ,   434
    0  ,     0  ,     0  ,    0  ,    7  ,    0  ,    49
    0  ,     0  ,     0  ,    0  ,    0  ,    0  ,     4
    351,  2736  ,  3847  , 1818  ,  543  ,   191 ,     0")
    conne.data <- as.matrix(read.csv(conne.data.csv, header=FALSE))

A total of 201 fish were tagged and released in week 1. Of these fish, 9 were recovered down stream in column stratum 1; 21 were recovered in column stratum 2; and 171 were never seen again. A total of 351 UNMARKED fish were recovered in column stratum 1.

Notice the VERY small sample size in row 6 with only 4 fish released and non-recaptured. We will likely have to pool some rows in this problem because there is virtually no information provided by row 6 which makes the problem ill conditioned and may not converge properly.

You can add row and column names to the matrix which will used when the data are printed.

# 4 Fitting the Stratified-Petersen model

The Stratified-Petersen model can now be fit to the above data object.

    mod1 <- SPAS::SPAS.fit.model(conne.data,
                           model.id="No restrictions",
                           row.pool.in=1:6, col.pool.in=1:6)

The `row.pool.in` and `col.pool.in` inform the function on which rows or columns to physically pool before the analysis proceeds. Both parameters use a vector of codes (length \(s\) for row pooing and length \(t\) for column pooling) where rows/columns are pooled that share the same value.

For example. `row.pool.in=c(1,2,3,4,5,6)` would imply that no rows are pooled, while `row.pool.in=c('a','a','a','b','b','b')` would imply that the first three rows and last three rows are pooled. The entries in the vector can be numeric or character; however, using character entries implies that the final pooled matrix is displayed in the order of the character entries. I find that using entries such as `'123'` to represent pooling rows 1, 2, and 3 to be easiest to use.

The SPAS system only fits models were the number of rows after pooling is less than or equal to the number of columns after pooling.

## 4.1 Results from model 1 (no pooling).

The likelihood is rather flat, so don’t be too concerned about messages from the optimization on failing to converge. As long as the likelihood values seem to be have stabilized and estimates are sensible (i.e. capture rates not 0 or 1; estimates of movement sensible results), the estimates should be fine.

The results of the model fit is a LARGE list. But the function `SPAS.print.model` produces a nice report

    SPAS.print.model(mod1)
    #> Model Name: No restrictions
    #>    Date of Fit: 2019-12-01 21:22
    #>    Version of OPEN SPAS used : SPAS-R 2020-01-01
    #>
    #> Raw data
    #>       V1   V2   V3   V4  V5  V6  V7
    #> [1,]   9   21    0    0   0   0 171
    #> [2,]   0  101   22    1   0   0 763
    #> [3,]   0    0  128   49   0   0 934
    #> [4,]   0    0    0   48  12   0 434
    #> [5,]   0    0    0    0   7   0  49
    #> [6,]   0    0    0    0   0   0   4
    #> [7,] 351 2736 3847 1818 543 191   0
    #>
    #> Row pooling setup : 1 2 3 4 5 6
    #> Col pooling setup : 1 2 3 4 5 6
    #> Physical pooling  : TRUE
    #> Theta pooling     : FALSE
    #> CJS pooling       : FALSE
    #>
    #>
    #> Chapman estimator of population size  68228  (SE  3090  )
    #>
    #>
    #> Raw data AFTER PHYSICAL (but not logical) POOLING
    #>       pool1 pool2 pool3 pool4 pool5 pool6  V7
    #> pool1     9    21     0     0     0     0 171
    #> pool2     0   101    22     1     0     0 763
    #> pool3     0     0   128    49     0     0 934
    #> pool4     0     0     0    48    12     0 434
    #> pool5     0     0     0     0     7     0  49
    #> pool6     0     0     0     0     0     0   4
    #>         351  2736  3847  1818   543   191   0
    #>
    #> Condition number of XX' where X= (physically) pooled matrix is  Inf
    #> Condition number of XX' after logical pooling                   Inf
    #>
    #> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
    #>
    #>   Conditional   Log-Likelihood: 78115.99    ;  np: 48 ;  AICc: -156136
    #>
    #>   Code/Message from optimization is:  1 singular convergence (7)
    #>
    #> Estimates
    #>              pool1  pool2  pool3  pool4 pool5 pool6 psi cap.prob exp factor
    #> pool1          8.9   21.1    0.0    0.0   0.0   0.0 171    0.025       39.3
    #> pool2          0.0  101.2   21.8    1.0   0.0   0.0 763    0.050       18.9
    #> pool3          0.0    0.0  126.5   50.5   0.0   0.0 934    0.035       27.2
    #> pool4          0.0    0.0    0.0   48.5  11.5   0.0 434    0.103        8.7
    #> pool5          0.0    0.0    0.0    0.0   4.9   2.1  49    0.011       90.5
    #> pool6          0.0    0.0    0.0    0.0   0.0   0.0   4    0.269        2.7
    #> est unmarked 351.0 2736.0 3849.0 1816.0 546.0 189.0   0       NA         NA
    #>              Pop Est
    #> pool1           8103
    #> pool2          17611
    #> pool3          31308
    #> pool4           4808
    #> pool5           5126
    #> pool6             15
    #> est unmarked   66971
    #>
    #> SE of above estimates
    #>              pool1 pool2 pool3 pool4 pool5 pool6  psi cap.prob exp factor
    #> pool1            3   4.6   0.0   0.0   0.0   0.0 13.1    0.008       13.4
    #> pool2            0  10.0   4.6   1.0   0.0   0.0 27.6    0.010        3.8
    #> pool3            0   0.0  11.2   7.1   0.0   0.0 30.6    0.003        2.6
    #> pool4            0   0.0   0.0   6.9   3.3   0.0 20.8    0.053        5.0
    #> pool5            0   0.0   0.0   0.0   1.9   0.8  7.0    0.004       35.5
    #> pool6            0   0.0   0.0   0.0   0.0   0.0  2.0    0.000        0.0
    #> est unmarked    NA    NA    NA    NA    NA    NA  0.0       NA         NA
    #>              Pop Est
    #> pool1           2688
    #> pool2           3405
    #> pool3           2939
    #> pool4           2482
    #> pool5           1991
    #> pool6              0
    #> est unmarked    3496
    #>
    #>
    #> Chisquare gof cutoff  : 0.1
    #> Chisquare gof value   : 3.102442
    #> Chisquare gof df      : 0
    #> Chisquare gof p       : NA

The original data, the data after pooling, estimates and their standard errors are shown. Here the stratified-Petersen estimate of the total number of smolts passing the at the first sampling station is 66,971 with a standard error of 3,496.

Each entry in the output is available in fitted model object. You will need to explore the structure

    cat("Names of objects at highest level\n")
    #> Names of objects at highest level
    names(mod1)
    #>  [1] "version"        "date"           "input"          "kappa"
    #>  [5] "kappa.after.lp" "est"            "vcv"            "se"
    #>  [9] "fit.setup"      "conditional"    "model.info"     "gof"
    cat("\n\nNames of estimates (both beta and real)\n")
    #>
    #>
    #> Names of estimates (both beta and real)
    names(mod1$est)
    #> [1] "N.Chapman" "beta"      "real"
    cat("\n\nNames of real estimates\n")
    #>
    #>
    #> Names of real estimates
    names(mod1$est$real)
    #> [1] "psi"          "theta"        "cap"          "all.expanded" "N"
    #> [6] "N.stratum"    "exp.factor"

The `N` entries refer to the population size; the `N.stratum` entries refer to the individual stratum population sizes; the `cap` entries refer to the estimated probability of capture in each row stratum; the `exp.factor` entries refer to `(1-cap)/cap`, or the expansion factor for each row; the `psi` entries refer to the number of animals tagged but never seen again (the right most column in the input data); and the `theta` entries refer to the expected number of animals that were tagged in row stratum \(i\) and recovered in column stratum \(j\) (after pooling).

# 5 Physically pooling some rows and columns

As noted by Darroch (1961), the stratified-Petersen will fail if the matrix of movements is close to singular. This often happens if two rows are proportional to each other. In this case, there is no unique MLE for the probability of capture in the last two rows, and they should be pooled. A detailed discussion of pooling is found in Schwarz and Taylor (1998).

## 5.1 Pooling the first two rows, the last two rows, and the last two columns

Let us now pool the first two rows and the last two rows, but leave rows 3 and 4 alone. Similar, let us pool the last two columns.

The code is

    mod2 <- SPAS.fit.model(conne.data, model.id="Pooling some rows",
                           row.pool.in=c("12","12","3","4","56","56"),
                           col.pool.in=c(1,2,3,4,56,56))

Notice how we specify the pooling for rows and columns and the choice of entries for the two corresponding vectors. I used character entries for the row pooling to ensure that the pooled matrix is sorted properly (see below); the numeric entries for the columns is ok as is.

The results of the model fit is

    SPAS.print.model(mod2)
    #> Model Name: Pooling some rows
    #>    Date of Fit: 2019-12-01 21:22
    #>    Version of OPEN SPAS used : SPAS-R 2020-01-01
    #>
    #> Raw data
    #>       V1   V2   V3   V4  V5  V6  V7
    #> [1,]   9   21    0    0   0   0 171
    #> [2,]   0  101   22    1   0   0 763
    #> [3,]   0    0  128   49   0   0 934
    #> [4,]   0    0    0   48  12   0 434
    #> [5,]   0    0    0    0   7   0  49
    #> [6,]   0    0    0    0   0   0   4
    #> [7,] 351 2736 3847 1818 543 191   0
    #>
    #> Row pooling setup : 12 12 3 4 56 56
    #> Col pooling setup : 1 2 3 4 56 56
    #> Physical pooling  : TRUE
    #> Theta pooling     : FALSE
    #> CJS pooling       : FALSE
    #>
    #>
    #> Chapman estimator of population size  68228  (SE  3090  )
    #>
    #>
    #> Raw data AFTER PHYSICAL (but not logical) POOLING
    #>        pool1 pool2 pool3 pool4 pool56  V7
    #> pool12     9   122    22     1      0 934
    #> pool3      0     0   128    49      0 934
    #> pool4      0     0     0    48     12 434
    #> pool56     0     0     0     0      7  53
    #>          351  2736  3847  1818    734   0
    #>
    #> Condition number of XX' where X= (physically) pooled matrix is  541.5353
    #> Condition number of XX' after logical pooling                   541.5353
    #>
    #> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
    #>
    #>   Conditional   Log-Likelihood: 79052.91    ;  np: 28 ;  AICc: -158049.8
    #>
    #>   Code/Message from optimization is:  0 relative convergence (4)
    #>
    #> Estimates
    #>              pool1  pool2  pool3  pool4 pool56 psi cap.prob exp factor Pop Est
    #> pool12        11.7  119.3   21.9    1.0    0.0 934    0.042       23.0   26059
    #> pool3          0.0    0.0  127.6   49.4    0.0 934    0.037       26.2   30230
    #> pool4          0.0    0.0    0.0   48.2   11.8 434    0.088       10.4    5615
    #> pool56         0.8    0.0    0.0    0.0    6.2  53    0.010       98.6    5978
    #> est unmarked 347.0 2739.0 3847.0 1817.0  735.0   0       NA         NA   67882
    #>
    #> SE of above estimates
    #>              pool1 pool2 pool3 pool4 pool56  psi cap.prob exp factor Pop Est
    #> pool12         4.2  11.2   4.7   1.0    0.0 30.6    0.004        2.2    2402
    #> pool3          0.0   0.0  11.3   7.0    0.0 30.6    0.003        2.6    2837
    #> pool4          0.0   0.0   0.0   6.9    3.4 20.8    0.039        5.0    2458
    #> pool56         1.0   0.0   0.0   0.0    2.5  7.3    0.004       42.1    2524
    #> est unmarked    NA    NA    NA    NA     NA  0.0       NA         NA    3593
    #>
    #>
    #> Chisquare gof cutoff  : 0.1
    #> Chisquare gof value   : 1.637457
    #> Chisquare gof df      : 1
    #> Chisquare gof p       : 0.2006747

Here the stratified-Petersen estimate of the total number of smolts passing the at the first sampling station is 67,882 with a standard error of 3,593. which is a slight reduction from the unpooled estimates.

## 5.2 Physically pooling to a single row and complete pooling

You can pool to a single row (and multiple columns) or a single row and single column (which is equivalent to the pooled Petersen estimator). The code and output follow:

    mod3 <- SPAS.fit.model(conne.data, model.id="A single row",
                           row.pool.in=rep(1, nrow(conne.data)-1),
                           col.pool.in=c(1,2,3,4,56,56))

    SPAS.print.model(mod3)
    #> Model Name: A single row
    #>    Date of Fit: 2019-12-01 21:22
    #>    Version of OPEN SPAS used : SPAS-R 2020-01-01
    #>
    #> Raw data
    #>       V1   V2   V3   V4  V5  V6  V7
    #> [1,]   9   21    0    0   0   0 171
    #> [2,]   0  101   22    1   0   0 763
    #> [3,]   0    0  128   49   0   0 934
    #> [4,]   0    0    0   48  12   0 434
    #> [5,]   0    0    0    0   7   0  49
    #> [6,]   0    0    0    0   0   0   4
    #> [7,] 351 2736 3847 1818 543 191   0
    #>
    #> Row pooling setup : 1 1 1 1 1 1
    #> Col pooling setup : 1 2 3 4 56 56
    #> Physical pooling  : TRUE
    #> Theta pooling     : FALSE
    #> CJS pooling       : FALSE
    #>
    #>
    #> Chapman estimator of population size  68228  (SE  3090  )
    #>
    #>
    #> Raw data AFTER PHYSICAL (but not logical) POOLING
    #>   pool1 pool2 pool3 pool4 pool56   V7
    #> 1     9   122   150    98     19 2355
    #>     351  2736  3847  1818    734    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: 81857.54    ;  np: 7 ;  AICc: -163701.1
    #>
    #>   Code/Message from optimization is:  0 both X-convergence and relative convergence (5)
    #>
    #> Estimates
    #>              pool1  pool2  pool3  pool4 pool56  psi cap.prob exp factor Pop Est
    #> 1             14.5  115.1  160.9   77.2   30.3 2355     0.04       23.8   68368
    #> est unmarked 346.0 2743.0 3836.0 1839.0  723.0    0       NA         NA   68368
    #>
    #> SE of above estimates
    #>              pool1 pool2 pool3 pool4 pool56  psi cap.prob exp factor Pop Est
    #> 1                1     6   8.3   4.2    1.9 48.5    0.002        1.2    3105
    #> est unmarked    NA    NA    NA    NA     NA  0.0       NA         NA    3105
    #>
    #>
    #> Chisquare gof cutoff  : 0.1
    #> Chisquare gof value   : 13.65473
    #> Chisquare gof df      : 4
    #> Chisquare gof p       : 0.008482579

    mod4 <- SPAS.fit.model(conne.data, model.id="Pooled Peteren",
                           row.pool.in=rep(1, nrow(conne.data)-1),
                           col.pool.in=rep(1, ncol(conne.data)-1))

    SPAS.print.model(mod4)
    #> Model Name: Pooled Peteren
    #>    Date of Fit: 2019-12-01 21:22
    #>    Version of OPEN SPAS used : SPAS-R 2020-01-01
    #>
    #> Raw data
    #>       V1   V2   V3   V4  V5  V6  V7
    #> [1,]   9   21    0    0   0   0 171
    #> [2,]   0  101   22    1   0   0 763
    #> [3,]   0    0  128   49   0   0 934
    #> [4,]   0    0    0   48  12   0 434
    #> [5,]   0    0    0    0   7   0  49
    #> [6,]   0    0    0    0   0   0   4
    #> [7,] 351 2736 3847 1818 543 191   0
    #>
    #> Row pooling setup : 1 1 1 1 1 1
    #> Col pooling setup : 1 1 1 1 1 1
    #> Physical pooling  : TRUE
    #> Theta pooling     : FALSE
    #> CJS pooling       : FALSE
    #>
    #>
    #> Chapman estimator of population size  68228  (SE  3090  )
    #>
    #>
    #> Raw data AFTER PHYSICAL (but not logical) POOLING
    #>      1   V7
    #> 1  398 2355
    #>   9486    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: 95297.26    ;  np: 3 ;  AICc: -190588.5
    #>
    #>   Code/Message from optimization is:  0 relative convergence (4)
    #>
    #> Estimates
    #>                 1  psi cap.prob exp factor Pop Est
    #> 1             398 2355     0.04       23.8   68368
    #> est unmarked 9486    0       NA         NA   68368
    #>
    #> SE of above estimates
    #>                 1  psi cap.prob exp factor Pop Est
    #> 1            19.9 48.5    0.002        1.2    3105
    #> est unmarked   NA  0.0       NA         NA    3105
    #>
    #>
    #> Chisquare gof cutoff  : 0.1
    #> Chisquare gof value   : 8.319195e-16
    #> Chisquare gof df      : 0
    #> Chisquare gof p       : NA

Both models have an estimated abundance of 68,368 with a standard error of 3,105. which is a slight reduction from the unpooled estimates.

## 5.3 Comparing different physical pooling

Unfortunately, because pooling actually changes the data (internally), it is not possible to do likelihood ratio testing or to use AICc to compare different poolings. Methods to compare different poolings is under active investigation.

# 6 Logical pooling

In logical pooling, we do not physically pool rows or columns. As noted elsewhere, it is not possible to implement logical pooling for columns, so only row will be discussed here. In logical pooling, the capture probabilities in the stratum of release are forced equal. Logical pooling is specified using the same keywords for physical pooling, but the additional argument row.physical.pooling=FALSE should be set.

## 6.1 Logical pooling the first two rows, the last two rows, and physical pooling of the last two columns

Let us now logically pool the first two rows and the last two rows, but leave rows 3 and 4 alone. Similar, let us physically pool the last two columns.

The code is

    mod5 <- SPAS.fit.model(conne.data, model.id="Pooling some rows",
                           row.pool.in=c("12","12","3","4","56","56"),
                           row.physical.pool=FALSE,
                           col.pool.in=c(1,2,3,4,56,56))

This gives:

    SPAS.print.model(mod5)
    #> Model Name: Pooling some rows
    #>    Date of Fit: 2019-12-01 21:22
    #>    Version of OPEN SPAS used : SPAS-R 2020-01-01
    #>
    #> Raw data
    #>       V1   V2   V3   V4  V5  V6  V7
    #> [1,]   9   21    0    0   0   0 171
    #> [2,]   0  101   22    1   0   0 763
    #> [3,]   0    0  128   49   0   0 934
    #> [4,]   0    0    0   48  12   0 434
    #> [5,]   0    0    0    0   7   0  49
    #> [6,]   0    0    0    0   0   0   4
    #> [7,] 351 2736 3847 1818 543 191   0
    #>
    #> Row pooling setup : 12 12 3 4 56 56
    #> Col pooling setup : 1 2 3 4 56 56
    #> Physical pooling  : FALSE
    #> Theta pooling     : FALSE
    #> CJS pooling       : FALSE
    #>
    #>
    #> Chapman estimator of population size  68228  (SE  3090  )
    #>
    #>
    #> Raw data AFTER PHYSICAL (but not logical) POOLING
    #>         pool1 pool2 pool3 pool4 pool56  V7
    #> pool.12     9    21     0     0      0 171
    #> pool.12     0   101    22     1      0 763
    #> pool.3      0     0   128    49      0 934
    #> pool.4      0     0     0    48     12 434
    #> pool.56     0     0     0     0      7  49
    #> pool.56     0     0     0     0      0   4
    #>           351  2736  3847  1818    734   0
    #>
    #> Condition number of XX' where X= (physically) pooled matrix is  Inf
    #> Condition number of XX' after logical pooling                   541.5353
    #>
    #> Large value of kappa (>1000) indicate that rows are approximately proportional which is not good
    #>
    #>   Conditional   Log-Likelihood: 78538.08    ;  np: 40 ;  AICc: -156996.2
    #>
    #>   Code/Message from optimization is:  1 singular convergence (7)
    #>
    #> Estimates
    #>              pool1  pool2  pool3  pool4 pool56 psi cap.prob exp factor Pop Est
    #> pool.12       11.7   20.5    0.0    0.0    0.0 171    0.042       23.0    4814
    #> pool.12        0.0   98.8   21.9    1.0    0.0 763    0.042       23.0   21245
    #> pool.3         0.0    0.0  127.6   49.4    0.0 934    0.037       26.2   30230
    #> pool.4         0.0    0.0    0.0   48.2   11.8 434    0.088       10.4    5615
    #> pool.56        0.4    0.0    0.0    0.0    6.2  49    0.010       98.6    5580
    #> pool.56        0.4    0.0    0.0    0.0    0.0   4    0.010       98.6     399
    #> est unmarked 347.0 2739.0 3847.0 1817.0  735.0   0       NA         NA   67882
    #>
    #> SE of above estimates
    #>              pool1 pool2 pool3 pool4 pool56  psi cap.prob exp factor Pop Est
    #> pool.12        4.2   4.5   0.0   0.0    0.0 13.1    0.004        2.2     444
    #> pool.12        0.0  10.1   4.7   1.0    0.0 27.6    0.004        2.2    1958
    #> pool.3         0.0   0.0  11.3   7.0    0.0 30.6    0.003        2.6    2837
    #> pool.4         0.0   0.0   0.0   6.9    3.4 20.8    0.039        5.0    2458
    #> pool.56        0.5   0.0   0.0   0.0    2.5  7.0    0.004       42.1    2356
    #> pool.56        0.5   0.0   0.0   0.0    0.0  2.0    0.004       42.1     168
    #> est unmarked    NA    NA    NA    NA     NA  0.0       NA         NA    3593
    #>
    #>
    #> Chisquare gof cutoff  : 0.1
    #> Chisquare gof value   : 1.637457
    #> Chisquare gof df      : 1
    #> Chisquare gof p       : 0.2006747

# 7 References

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