# 1 Testing if a Pooled Petersen is appropriate

It is often of interest to know if a simple Pooled Petersen estimator, i.e., complete pooling over rows and columns, is appropriate.

As noted in Schwarz and Taylor (1998), the Pooled Petersen is unbiased under many conditions, but the most common are:

• Homogeneity of tagging probabilities, i.e. the probability of a fish being tagged in the release stratum is equal across all release strata.
• Homogeneity of recapture probabilities, i.e. the probability of a fish being recaptured is equal across all recovery strata.
• Complete Mixing, i.e. tagged fish mix completely with untagged fish.
• Correlation between tagging and recovery probabilities is zero, i.e. while probabilities are heterogeneous across fish, the two events are statistically independent.

We can examine the first of these conditions by examining the results of the stratified analysis and the results of a (logical) row pooling over all release strata.

# 3 Get the model objects fitted by tmb and create a report

model.list <- mget( ls()[grepl("^mod.$",ls())]) names(model.list) #> [1] "mod1" "mod2" "mod3" "mod4" "mod5" report <- plyr::ldply(model.list, function(x){ #browser() data.frame(#version=x$version,
date   = as.Date(x$date), model.id = x$model.info$model.id, s.a.pool =-1+nrow(x$fit.setup$pooldata), t.p.pool =-1+ncol(x$fit.setup$pooldata), logL.cond = x$model.info$logL.cond, np = x$model.info$np, AICc = x$model.info$AICc, gof.chisq = round(x$gof$chisq,1), gof.df = x$gof$chisq.df, gof.p = round(x$gof$chisq.p,3), Nhat = round(x$est$real$N),
Nhat.se          = round(x$se$real\$N))

})
report
#>    .id       date                                        model.id s.a.pool
#> 1 mod1 2019-12-02                                 No restrictions        3
#> 2 mod2 2019-12-02                   Logical pooling to single row        3
#> 3 mod3 2019-12-02                  Physical pooling to single row        1
#> 4 mod4 2019-12-02 Physical pooling all rows and last two colum ns        1
#> 5 mod5 2019-12-02                       Physical complete pooling        1
#>   t.p.pool logL.cond np      AICc gof.chisq gof.df gof.p  Nhat Nhat.se
#> 1        3  30329.53 15 -60629.07       2.6      0    NA 10369     467
#> 2        3  30304.40 13 -60582.80      53.9      2     0 10250     325
#> 3        3  31438.32  5 -62866.64      53.9      2     0 10250     325
#> 4        2  33180.75  4 -66353.49      42.1      1     0 10250     325
#> 5        1  36412.34  3 -72818.69       0.0      0    NA 10250     325

The AIC should be compared ONLY for the first two models because they are based on the same set of data. You cannot compare models that differ in the physical pooling

In this case, there is good evidence that the Pooled Petersen is too coarse because the goodness of fit statistic for the second model is very large (with a corresponding small goodness-of-fit p-value). Similarly, the AIC indicates that the model is 3x3 stratification (first model) is preferable to the model with complete row pooling (second model).

Notice that the estimates of the population size are identical under logical or physical row pooling (models 2 and 3). And how you pool columns (models 3, 4, 5) but assuming that the number of rows (after logical or physical pooling as long the number of rows is not larger than the number of columns) does not affect the population size estimate (or standard error).

# 4 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