Capture-recapture type experiments are used to estimate the total population size in situations when observing only a part of such population is feasible. In recent years these types of experiments have seen more interest.

Single source models are distinct from other capture-recapture models because we cannot estimate the population size based on how many units were observed in two or three sources which is the standard approach.

Instead in single source models we utilize count data regression models on positive distributions (i.e. on counts greater than 0) where the dependent variable is the number of times a particular unit was observed in source data.

This package aims to implement already existing and introduce new methods of estimating population size from single source to simplify the research process.

Currently we’ve implemented most of the frequentist approaches used in literature such as:

- Zero truncated Poisson, geometric and negative binomial regression.
- Zero truncated one inflated and one inflated zero truncated Poisson and geometric models. (Negative binomial is currently in development.)
- Zero one truncated Poisson geometric and negative binomial models.
- Generalized Chao and Zelterman’s models based on logistic regression.
- Three types of bootstrap parametric, semi-parametric and non parametric.
- And a wide range of additional functionalities associated with (vector) generalized linear models relevant to the topic.

You can install the current version of singleRcapture from main branch GitHub with:

```
# install.packages("devtools")
::install_github("ncn-foreigners/singleRcapture") remotes
```

or install the stable version from CRAN with:

`install.packages(singleRcapture)`

or development version from development branch on GitHub with:

```
# install.packages("devtools")
::install_github("github.com/ncn-foreigners/singleRcapture/tree/development") remotes
```

The main function of this package is `estimatePopsize`

which fitts regression on specified distribution and then uses fitted
regression to estimate the population size.

Lets look at a model from 2003 publication : Point and interval
estimation of the population size using the truncated Poisson regression
model Heijden, Peter GM van der et al. The call to
`estimatePopsize`

will look very similar to anyone who used
the `stats::glm`

function:

```
library(singleRcapture)
<- estimatePopsize(
model formula = capture ~ gender + age + nation, # specify formula
data = netherlandsimmigrant,
popVar = "analytic", # specify
model = "ztpoisson", # distribution used
method = "IRLS", # fitting method one of three currently supported
controlMethod = controlMethod(silent = TRUE) # ignore convergence at half step warning
)summary(model) # a summary method for singleR class with standard glm-like output and population size estimation resutls
#>
#> Call:
#> estimatePopsize.default(formula = capture ~ gender + age + nation,
#> data = netherlandsimmigrant, model = "ztpoisson", method = "IRLS",
#> popVar = "analytic", controlMethod = controlMethod(silent = TRUE))
#>
#> Pearson Residuals:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.486442 -0.486442 -0.298080 0.002093 -0.209444 13.910844
#>
#> Coefficients:
#> -----------------------
#> For linear predictors associated with: lambda
#> Estimate Std. Error z value P(>|z|)
#> (Intercept) -1.3411 0.2149 -6.241 4.35e-10 ***
#> gendermale 0.3972 0.1630 2.436 0.014832 *
#> age>40yrs -0.9746 0.4082 -2.387 0.016972 *
#> nationAsia -1.0926 0.3016 -3.622 0.000292 ***
#> nationNorth Africa 0.1900 0.1940 0.979 0.327398
#> nationRest of Africa -0.9106 0.3008 -3.027 0.002468 **
#> nationSurinam -2.3364 1.0136 -2.305 0.021159 *
#> nationTurkey -1.6754 0.6028 -2.779 0.005445 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> AIC: 1712.901
#> BIC: 1757.213
#> Residual deviance: 1128.553
#>
#> Log-likelihood: -848.4504 on 1872 Degrees of freedom
#> Number of iterations: 8
#> -----------------------
#> Population size estimation results:
#> Point estimate 12690.35
#> Observed proportion: 14.8% (N obs = 1880)
#> Std. Error 2808.169
#> 95% CI for the population size:
#> lowerBound upperBound
#> normal 7186.444 18194.26
#> logNormal 8431.275 19718.32
#> 95% CI for the share of observed population:
#> lowerBound upperBound
#> normal 10.332927 26.16037
#> logNormal 9.534281 22.29793
```

We implemented a method for `plot`

function to visualise
the model fit and other useful diagnostic information. One of which is
`rootogram`

, a type of plot that compares fitted and observed
marginal frequencies:

`plot(model, plotType = "rootogram")`

The possible values for `plotType`

argument are:

`qq`

- the normal quantile-quantile plot for pearson residuals.`marginal`

- a`matplot`

comparing fitted and observed marginal frequencies.`fitresid`

- plot of linear predictor values contrasted with pearson residuals.`bootHist`

- histogram of bootstrap sample.`rootogram`

- rootogram, example presented above.`dfpopContr`

- contrasting two deletion effects to identify presence of influential observations.`dfpopBox`

- boxplot of results from dfpopsize function see its documentation.`scaleLoc`

- scale-location plot.`cooks`

- plot of`cooks.values`

for distributions for which it is defined.`hatplot`

- plot of`hatvalues`

.`strata`

- plot of confidence intervals for selected such populations.

a user can also pass arguments to specify additional information such
as plot title, subtitle etc. similar to calling `plot`

on
some data. For more info check `plot.singleR`

method
documentation.

As we have seen there are some significant differences between fitted and observed marginal frequencies. To check our intuition let’s perform goodness of fit test between fitted and observed marginal frequencies.

To do it we call a `summary`

function of
`marginalFreq`

function which computes marginal frequencies
for the fitted `singleR`

class object:

```
summary(marginalFreq(model), df = 2, dropl5 = "group")
#> Test for Goodness of fit of a regression model:
#>
#> Test statistics df P(>X^2)
#> Chi-squared test 50.06 2 1.3e-11
#> G-test 34.31 2 3.6e-08
#>
#> --------------------------------------------------------------
#> Cells with fitted frequencies of < 5 have been grouped
#> Names of cells used in calculating test(s) statistic: 1 2 3
```

Finally let us check if we have any influential observations. We will do this by comparing the deletion effect of every observation on population size estimate by removing it entirely from the model (from population size estimate and regression) and by only omitting it in pop size estimation (this is what is called the contribution of an observation). If observation is not influential these two actions should have the approximately the same effect:

`plot(model, plotType = "dfpopContr")`

it is easy to deduce from the plot above that we have influential observations in our dataset (one in particular).

Lastly `singleRcapture`

offers some posthoc procedures for
example a function `stratifyPopsize`

that estimates sizes of
user specified sub populations and returns them in a
`data.frame`

:

```
stratifyPopsize(model, alpha = c(.01, .02, .03, .05), # different significance level for each sub population
stratas = list(
"Females from Surinam" = netherlandsimmigrant$gender == "female" & netherlandsimmigrant$nation == "Surinam",
"Males from Turkey" = netherlandsimmigrant$gender == "male" & netherlandsimmigrant$nation == "Turkey",
"Younger males" = netherlandsimmigrant$gender == "male" & netherlandsimmigrant$age == "<40yrs",
"Older males" = netherlandsimmigrant$gender == "male" & netherlandsimmigrant$age == ">40yrs"
))#> Observed Estimated ObservedPercentage StdError normalLowerBound
#> 1 20 931.469 2.147146 955.0676 -1528.62209
#> 2 78 1291.251 6.040652 741.0066 -432.58790
#> 3 1391 7337.072 18.958516 1282.1419 4554.70841
#> 4 91 1542.189 5.900703 781.4754 10.52561
#> normalUpperBound logNormalLowerBound logNormalUpperBound name
#> 1 3391.560 119.2661 8389.178 Females from Surinam
#> 2 3015.090 405.4127 4573.791 Males from Turkey
#> 3 10119.436 5134.8113 10834.792 Younger males
#> 4 3073.853 630.7551 3992.677 Older males
#> confLevel
#> 1 0.01
#> 2 0.02
#> 3 0.03
#> 4 0.05
```

`stratas`

argument may be specified in various ways for
example:

```
stratifyPopsize(model, stratas = ~ gender / age)
#> Observed Estimated ObservedPercentage StdError normalLowerBound
#> 1 398 3811.0924 10.443200 1153.9749 1549.34322
#> 2 1482 8879.2613 16.690578 1812.0813 5327.64720
#> 3 378 3169.8272 11.924940 880.9491 1443.19883
#> 4 1391 7337.0721 18.958516 1282.1419 4824.12025
#> 5 20 641.2651 3.118835 407.5268 -157.47265
#> 6 91 1542.1892 5.900703 781.4754 10.52561
#> normalUpperBound logNormalLowerBound logNormalUpperBound
#> 1 6072.842 2189.0439 6902.140
#> 2 12430.875 6090.7752 13354.889
#> 3 4896.456 1904.3121 5484.622
#> 4 9850.024 5306.3298 10421.088
#> 5 1440.003 212.3382 2026.728
#> 6 3073.853 630.7551 3992.677
#> name confLevel
#> 1 gender==female 0.05
#> 2 gender==male 0.05
#> 3 genderfemale:age<40yrs 0.05
#> 4 gendermale:age<40yrs 0.05
#> 5 genderfemale:age>40yrs 0.05
#> 6 gendermale:age>40yrs 0.05
```

The package was designed with convenience in mind, for example it is possible to specify that weights provided on call are to be interpreted as number of occurrences of units in each row:

```
<- netherlandsimmigrant[, c(1:3,5)]
df $ww <- 0
df### this is dplyr::count without dependencies
<- aggregate(ww ~ ., df, FUN = length)
df summary(estimatePopsize(
formula = capture ~ nation + age + gender,
data = df,
model = ztpoisson,
weights = df$ww,
controlModel = controlModel(weightsAsCounts = TRUE)
))#>
#> Call:
#> estimatePopsize.default(formula = capture ~ nation + age + gender,
#> data = df, model = ztpoisson, weights = df$ww, controlModel = controlModel(weightsAsCounts = TRUE))
#>
#> Pearson Residuals:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -317.6467 -2.4060 3.7702 0.0803 13.4920 183.2108
#>
#> Coefficients:
#> -----------------------
#> For linear predictors associated with: lambda
#> Estimate Std. Error z value P(>|z|)
#> (Intercept) -1.3411 0.2149 -6.241 4.35e-10 ***
#> nationAsia -1.0926 0.3016 -3.622 0.000292 ***
#> nationNorth Africa 0.1900 0.1940 0.979 0.327398
#> nationRest of Africa -0.9106 0.3008 -3.027 0.002468 **
#> nationSurinam -2.3364 1.0136 -2.305 0.021159 *
#> nationTurkey -1.6754 0.6028 -2.779 0.005445 **
#> age>40yrs -0.9746 0.4082 -2.387 0.016972 *
#> gendermale 0.3972 0.1630 2.436 0.014832 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> AIC: 1712.901
#> BIC: 1757.213
#> Residual deviance: 1128.553
#>
#> Log-likelihood: -848.4504 on 1872 Degrees of freedom
#> Number of iterations: 8
#> -----------------------
#> Population size estimation results:
#> Point estimate 12690.35
#> Observed proportion: 14.8% (N obs = 1880)
#> Std. Error 2808.169
#> 95% CI for the population size:
#> lowerBound upperBound
#> normal 7186.444 18194.26
#> logNormal 8431.275 19718.32
#> 95% CI for the share of observed population:
#> lowerBound upperBound
#> normal 10.332927 26.16037
#> logNormal 9.534281 22.29793
```

Methods such as regression diagnostics will be adjusted (values of weights will be reduced instead of rows being removed etc.)

We also included option to use common non standardargument such as significance levels different from usual 5%:

```
set.seed(123)
<- estimatePopsize(
modelInflated formula = capture ~ gender + age,
data = netherlandsimmigrant,
model = "oiztgeom",
method = "IRLS",
# control parameters for population size estimation check documentation of controlPopVar
controlPopVar = controlPopVar(
alpha = .01, # significance level
)
)summary(modelInflated)
#>
#> Call:
#> estimatePopsize.default(formula = capture ~ gender + age, data = netherlandsimmigrant,
#> model = "oiztgeom", method = "IRLS", controlPopVar = controlPopVar(alpha = 0.01,
#> ))
#>
#> Pearson Residuals:
#> Min. 1st Qu. Median Mean 3rd Qu. Max.
#> -0.357193 -0.357193 -0.357193 0.000343 -0.287637 10.233608
#>
#> Coefficients:
#> -----------------------
#> For linear predictors associated with: lambda
#> Estimate Std. Error z value P(>|z|)
#> (Intercept) -1.5346 0.1846 -8.312 < 2e-16 ***
#> gendermale 0.3863 0.1380 2.800 0.00512 **
#> age>40yrs -0.7788 0.2942 -2.648 0.00810 **
#> -----------------------
#> For linear predictors associated with: omega
#> Estimate Std. Error z value P(>|z|)
#> (Intercept) -1.7591 0.3765 -4.673 2.97e-06 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> AIC: 1736.854
#> BIC: 1759.01
#> Residual deviance: 1011.271
#>
#> Log-likelihood: -864.4272 on 3756 Degrees of freedom
#> Number of iterations: 6
#> -----------------------
#> Population size estimation results:
#> Point estimate 5661.522
#> Observed proportion: 33.2% (N obs = 1880)
#> Std. Error 963.9024
#> 99% CI for the population size:
#> lowerBound upperBound
#> normal 3178.674 8144.370
#> logNormal 3861.508 9096.681
#> 99% CI for the share of observed population:
#> lowerBound upperBound
#> normal 23.08343 59.14416
#> logNormal 20.66688 48.68564
```

and the option to estimate standard error of population size estimate by bootstrap, models with more than one distribution parameter being dependent on covariates and some non standard link functions for example:

```
<- estimatePopsize(
modelInflated2 formula = capture ~ age,
data = netherlandsimmigrant,
popVar = "bootstrap",
model = oiztgeom(omegaLink = "cloglog"),
method = "IRLS",
controlPopVar = controlPopVar(
B = 500,# number of boostrap samples
alpha = .01, # significance level
# type of bootstrap see documentation for estimatePopsize
bootType = "semiparametric",
# control regression fitting on bootstrap samples
bootstrapFitcontrol = controlMethod(
epsilon = .Machine$double.eps,
silent = TRUE,
stepsize = 2
)
),controlModel = controlModel(omegaFormula = ~ gender) # put covariates on omega i.e. the inflation parameter
)#> Warning in singleRcaptureinternalIRLSmultipar(dependent = y, covariates = X, :
#> Convergence at halfstepsize
#> Warning in estimatePopsize.default(formula = capture ~ age, data = netherlandsimmigrant, : The (analytically computed) hessian of the score function is not negative define.
#> NOTE: Second derivative test failing does not
#> necessarily mean that the maximum of score function that was found
#> numericaly is invalid since R^k is not a bounded space.
#> Additionally in one inflated and hurdle models second derivative test often fails even on valid arguments.
#> Warning in estimatePopsize.default(formula = capture ~ age, data =
#> netherlandsimmigrant, : Switching from observed information matrix to Fisher
#> information matrix because hessian of log-likelihood is not negative define.
popSizeEst(modelInflated2)
#> Point estimate: 5496.374
#> Variance: 1217985
#> 99% confidence intervals:
#> lowerBound upperBound
#> 4002.763 10231.717
```

the results are significantly different (the warning issued concerns the second derivative test for existence of local minimum, here it was inconclusive but we manually checked that fitting process found the optimal regression coefficients it’s here to provide more information to the user):

```
plot(modelInflated2,
plotType = "bootHist",
labels = TRUE,
ylim = c(0, 175),
breaks = 15)
```

and information criteria support the second model:

```
#> First model: AIC = 1736.854 BIC = 1759.01
#> Second model: AIC = 1734.803 BIC = 1756.959
```

Work on this package is supported by the the National Science Center, OPUS 22 grant no. 2020/39/B/HS4/00941.