1.1 Software for capture-recapture with a single source

The majority of SSCR methods assume zero-truncated distributions or their extensions (e.g., inclusion of one-inflation). The Distributions.jl (Besançon et al. 2021) (in Julia), StatsModels (Seabold and Perktold 2010) (in Python), countreg (Zeileis, Kleiber, and Jackman 2008), VGAM (T. Yee 2015) or distributions3 (Hayes et al. 2024) (in R) implement some of those truncated distributions (e.g. distributions3::ZTPoisson or countreg::zerotrunc) and the most general distributions, such as Generally Altered, Inflated, Truncated and Deflated, can be found in the VGAM package (e.g. VGAM::gaitdpoisson for the Poisson distribution, see Thomas W. Yee and Ma (2024) for a detailed description). However, the estimation of parameters of a given truncated (and possibly inflated) distribution is just the first step (as in the case of log-linear models in capture-recapture with two sources) and, to the best of our knowledge, there is no open-source software that can be used to estimate population size using on SSCR methods and includes variance estimators or diagnostics.

Therefore, the purpose of the singleRcapture, an R package, is to bridge this gap by providing scientists and other practitioners with a tool for estimating population size with SSCR methods. We have implemented state-of-the-art methods, as recently described by Böhning, Bunge, and Heijden (2018) or Böhning and Friedl (2024) and provided their extensions (e.g., inclusion of covariates, different treatment of one-inflation), which will be covered in detail in Section 2. The package implements variance estimation based on various methods, can be used to create custom models and diagnostic plots (e.g. rootograms) with parameters estimated using a modified iteratively reweighted least squares (IRLS) algorithm we have implemented for estimation stability. To the best of our knowledge, no existing open-source package allows the estimation of population size by selecting an appropriate SSCR model, conducting the estimation, and providing informative diagnostics of the results.

The remaining part of the paper is structured as follows. Section 2 contains a brief description of the theoretical background and information about fitting methods and available methods of variance estimation. Section 3 introduces the main functions of the package. Section 4 presents a case study and contains an assessment of its results, diagnostics and estimates of specific sub-populations. Section 5 describes classes and S3methods implemented in the package. The paper ends with conclusions and an appendix showing how to use the estimatePopsizeFit function which is aimed to mimic the glm.fit or similar functions. In replication materials we show how to implement a custom model as this option could be of interest to users wishing to apply any new bootstrap methods not implemented in the package.

2.1 How to estimate population size with a single source?

Let \(Y_{k}\) represent the number of times the \(k\)-th unit was observed in a single source (e.g. register). Clearly, we only observe \(k:Y_{k}>0\) and do not know how many units have been missed (i.e. \(Y_{k}=0\)), so the population size, denoted by \(N\), needs to be estimated. In general, we assume that the conditional distribution of \(Y_{k}\) given a vector of covariates \(\boldsymbol{x}_{k}\) follows a version of the zero-truncated count data distribution (and its extensions). When we know the parameters of the distribution we can estimate the population size using a Horvitz-Thompson type estimator given by:

\[ \begin{equation} \hat{N}= \sum_{k=1}^{N}\frac{I_{k}}{\mathbb{P}[Y_{k}>0|\boldsymbol{X}_{k}]}= \sum_{k=1}^{N_{obs}}\frac{1}{\mathbb{P}[Y_{k}>0|\boldsymbol{X}_{k}]}, \end{equation} \]

where \(I_{k}:=\mathcal{I}_{\mathbb{N}}(Y_{k})\), \(N_{obs}\) is the number of observed units and \(\mathcal{I}\) is the indicator function, while the maximum likelihood estimate of \(N\) is obtained after substituting regression parameters \(\boldsymbol{\beta}\) for \(\mathbb{P}[Y_{k}>0|\boldsymbol{x}_{k}]\) in the above equation.

The basic SSCR assumes independence between counts, which is a rather naive assumption, since the first capture may significantly influence the behavior of a given unit or limit the possibility of subsequent captures (e.g. due to incarceration).

To solve these issues, Ryan T. Godwin and Böhning (2017b) and Ryan T. Godwin and Böhning (2017a) introduced one-inflated distributions, which explicitly model the probability of singletons by giving additional mass \(\omega\) to singletons denoted as \(\mathcal{I}_{\{1\}}(y)\) (cf. Böhning and Friedl 2024). In other words they considered a new random varialbe \(Y^{\ast}\) that corresponds to the data collection process which exhibits one inflation:

\[ \begin{equation*} \mathbb{P}\left[Y^{\ast}=y|Y^{\ast}>0\right] = \omega\mathcal{I}_{\{1\}}(y)+(1-\omega)\mathbb{P}[Y=y|Y>0]. \end{equation*} \]

Analytic variance estimation is then performed by computing two parts of the decomposition according to the law of total variance given by:

\[ \begin{equation}\label{eq-law_of_total_variance_decomposition} \text{var}[\hat{N}] = \mathbb{E}\left[\text{var} \left[\hat{N}|I_{1},\dots,I_{n}\right]\right] + \text{var}\left[\mathbb{E}[\hat{N}|I_{1},\dots,I_{n}]\right], \end{equation} \]

where the first part can be estimated using the multivariate \(\delta\) method given by:

\[ \begin{equation*} \mathbb{E}\left[\text{var} \left[\hat{N}|I_{1},\dots,I_{n}\right]\right] = \left.\left(\frac{\partial(N|I_1,\dots,I_N)}{\partial\boldsymbol{\beta}}\right)^\top \text{cov}\left[\hat{\boldsymbol{\beta}}\right] \left(\frac{\partial(N|I_1,\dots,I_N)}{\partial\boldsymbol{\beta}}\right) \right|_{\boldsymbol{\beta}=\hat{\boldsymbol{\beta}}}, \end{equation*} \]

while the second part of the decomposition above, assuming independence of \(I_{k}\)’s and after some omitted simplifications, is optimally estimated by:

\[ \begin{equation*} \text{var}\left[\mathbb{E}(\hat{N}|I_{1},\dots,I_{n})\right] = \text{var}\left[\sum_{k=1}^{N}\frac{I_{k}}{\mathbb{P}(Y_{k}>0)}\right] \approx\sum_{k=1}^{N_{obs}}\frac{1-\mathbb{P}(Y_{k}>0)}{\mathbb{P}(Y_{k}>0)^{2}}, \end{equation*} \]

which serves as the basis for interval estimation. Confidence intervals are usually constructed under the assumption of (asymptotic) normality of \(\hat{N}\) or asymptotic normality of \(\ln(\hat{N}-N)\) (or log normality of \(\hat{N}\)). The latter is an attempt to address a common criticism of student type confidence intervals in SSCR, namely a possibly skewed distribution of \(\hat{N}\), and results in the \(1-\alpha\) confidence interval given by:

\[ \begin{equation*} \left(N_{obs}+\frac{\hat{N}-N_{obs}}{\xi},N_{obs} + \left(\hat{N}-N_{obs}\right)\xi\right), \end{equation*} \]

where:

\[ \begin{equation*} \xi = \exp\left(z\left(1-\frac{\alpha}{2}\right) \sqrt{\ln\left(1+\frac{\widehat{\text{Var}}(\hat{N})}{\left(\hat{N}-N_{obs}\right)^{2}}\right)}\right), \end{equation*} \]

and where \(z\) is the quantile function of the standard normal distribution. The estimator \(\hat{N}\) is best interpreted as being an estimator of the total number of units in the population, since we have no means of estimating the number of units in the population for which the probability of being included in the data is \(0\) (Heijden et al. 2003).

2.2 Available models

The full list of models implemented in singleRcapture along with corresponding expressions for probability density functions and point estimates can be found in the collective help file for all family functions:

?ztpoisson

For the sake of simplicity, we only list the family functions together with brief descriptions. For more detailed information, please consult the relevant literature.

The current list of these family functions includes:

• Generalized Chao’s (Chao 1987) and Zelterman’s (Zelterman 1988) estimators via logistic regression on variable \(Z\) defined as \(Z=1\) if \(Y=2\) and \(Z=0\) if \(Y=1\) with \(Z\sim b(p)\) where \(b(\cdot)\) is the Bernoulli distribution and \(p\) can be modeled for each unit \(k\) by \(\text{logit}(p_k)=\ln(\lambda_k/2)\) with Poisson parameter \(\lambda_k=\boldsymbol{x}_{k}\boldsymbol{\beta}\) (for a covariate extension see Böhning et al. (2013) and Böhning and Heijden (2009)):

\[ \hat{N}_{\text{Chao}} = N_{obs}+ \sum_{k=1}^{\boldsymbol{f}_{1}+\boldsymbol{f}_{2}} \left(2\exp\left(\boldsymbol{x}_{k}\hat{\boldsymbol{\beta}}\right)+ 2\exp\left(2\boldsymbol{x}_{k}\hat{\boldsymbol{\beta}}\right)\right)^{-1}, \tag{Chao's estimator} \]

\[ \hat{N}_{\text{Zelt}}=\sum_{k=1}^{N_{obs}} \left(1-\exp\left(-2\exp\left(\boldsymbol{x}_{k}\hat{\boldsymbol{\beta}}\right)\right)\right)^{-1}. \tag{Zelterman's estimator} \]

where \(\boldsymbol{f}_{1}\) and \(\boldsymbol{f}_{2}\) denotes number of units observed once and twice.

• Zero-truncated (zt\(^\ast\)) and zero-one-truncated (zot\(^\ast\)) Poisson (Böhning and Heijden 2019), geometric, NB type II (NB2) regression, where the non-truncated distribution is parameterized as:

\[ \mathbb{P}[Y=y|\lambda,\alpha] = \frac{\Gamma\left(y+\alpha^{-1}\right)}{\Gamma\left(\alpha^{-1}\right)y!} \left(\frac{\alpha^{-1}}{\alpha^{-1}+\lambda}\right)^{\alpha^{-1}} \left(\frac{\lambda}{\lambda + \alpha^{-1}}\right)^{y}. \]

• Zero-truncated one-inflated (ztoi\(^\ast\)) modifications, where the count data variable \(Y^{\ast}\) is defined such that its distribution statisfies:

\[ \mathbb{P}\left[Y^{\ast}=y\right]= \begin{cases} \mathbb{P}[Y=0] & y=0, \\ \omega\left(1-\mathbb{P}[Y=0]\right)+(1-\omega)\mathbb{P}[Y=1] & y=1, \\ (1-\omega)\mathbb{P}[Y=y] & y>1, \end{cases} \]

\[ \mathbb{P}\left[Y^{\ast}=y|Y^{\ast}>0\right]=\omega\mathcal{I}_{\{1\}}(y)+(1-\omega)\mathbb{P}[Y=y|Y>0]. \]

• One-inflated zero-truncated (oizt\(^\ast\)) modifications, where the count data variable \(Y^{\ast}\) is defined as:

\[ \mathbb{P}\left[Y^{\ast}=y\right] = \omega \mathcal{I}_{\{1\}}(y)+(1-\omega)\mathbb{P}[Y=y], \]

\[ \mathbb{P}\left[Y^{\ast}=y|Y^{\ast}>0\right] = \omega\frac{\mathcal{I}_{\{1\}}(y)}{1-(1-\omega)\mathbb{P}[Y=0]}+ (1-\omega)\frac{\mathbb{P}[Y=y]}{1-(1-\omega)\mathbb{P}[Y=0]}. \]

Note that ztoi\(^\ast\) and oizt\(^\ast\) distributions are equivalent, in the sense that the maximum value of the likelihood function is equal for both of those distributions given any data, as shown by (Böhning 2023) but population size estimators are different.

In addition, we propose two new approaches to model singletons in a similar way as in hurdle models. In particular, we have proposed the following:

• The zero-truncated hurdle model (ztHurdle*) for Poisson, geometric and NB2 is defined as:

\[ \mathbb{P}[Y^{\ast}=y]=\begin{cases} \frac{\mathbb{P}[Y=0]}{1-\mathbb{P}[Y=1]} & y=0, \\ \pi(1-\mathbb{P}[Y=1]) & y=1, \\ (1-\pi) \frac{\mathbb{P}[Y=y]}{1-\mathbb{P}[Y=1]} & y>1, \end{cases} \]

\[ \mathbb{P}[Y^{\ast}=y|Y^{\ast}>0]=\pi\mathbf{1}_{\{1\}}(y)+ (1-\pi)\mathbf{1}_{\mathbb{N}\setminus\{1\}}(y)\frac{\mathbb{P}[Y=y]}{1-\mathbb{P}[Y=0]-\mathbb{P}[Y=1]}. \]

where \(\pi\) denotes the conditional probability of observing singletons.

• The hurdle zero-truncated model (Hurdlezt*) for Poisson, geometric and NB2 is defined as:

\[ \mathbb{P}[Y^{\ast}=y]=\begin{cases} \pi & y=1, \\ (1-\pi) \frac{\mathbb{P}[Y=y]}{1-\mathbb{P}[Y=1]} & y\neq1, \end{cases} \]

\[ \mathbb{P}[Y^{\ast}=y|Y^{\ast}>0]=\begin{cases} \pi\frac{1-\mathbb{P}[Y=1]}{1-\mathbb{P}[Y=0]-\mathbb{P}[Y=1]} & y=1,\\ (1-\pi)\frac{\mathbb{P}[Y=y]}{1-\mathbb{P}[Y=0]-\mathbb{P}[Y=1]} & y>1, \end{cases} \]

where \(\pi\) denotes the unconditional probability of observing singletons.

The approaches presented above differ in their assumptions, computational complexity, or in the way they treat heterogeneity of captures and singletons. For instance, the dispersion parameter \(\alpha\) in the NB2 type models is often interpreted as measuring the severity of unobserved heterogeneity in the underlying Poisson process (Cruyff and Heijden 2008). When using any truncated NB model, the hope is that given the class of models considered, the consistency is not lost despite the lack of information.

While not discussed in the literature, the interpretation of heterogeneous \(\alpha\) across the population (specified in controlModel) would be that the unobserved heterogeneity affects the accuracy of the prediction for the dependent variable \(Y\) more severely than others. The geometric model (NB with \(\alpha=1\)) is singled out in the package and often considered in the literature because of inherent computational issues with NB models, which are exacerbated by the fact that data used for SSCR are usually of rather low quality. Data sparsity is a particularly common problem in SSCR and a big challenge for all numerical methods for fitting the (zero-truncated) NB model.

The extra mass \(\omega\) in one-inflated models is an important extension to the researcher’s toolbox for SSCR models, since the inflation at \(y=1\) is likely to occur in many types of applications. For example, when estimating the number of active people who committed criminal acts in a given time period, the fact of being captured for the first time following an arrest is associated with the risk of no longer being able to be captured a second time. One constraint present in modelling via inflated models is that attempts to include both the possibility of one inflation and one deflation lead to both numerical and inferential problems since the parameter space (of \((\omega, \lambda)\) or \((\omega, \lambda, \alpha)\)) is then given by \(\{(\omega, \lambda, \alpha) | \forall x\in \mathbb{N}: p(x|\omega, \lambda, \alpha)\geq0\}\) for the probability mass function \(p\). The boundary of this set is then a \(1\) or \(2-\)dimentional manifold, transforming this parameter space into \(\mathbb{R}^{3}\) would require using link functions that depend on more than one parameter.

Hurdle models represent another approach to modelling one-inflation. They can also model deflation as well as inflation and deflation simultaneously, so they are more flexible and, in the case of hurdle zero-truncated models, appear to be more numerically stable.

Although the question of how to interpret regression parameters tends to be somewhat overlooked in SSCR studies, we should point out that the interpretation of the \(\omega\) inflation parameter (in ztoi\(^\ast\) or oizt\(^\ast\)) is more convenient than the interpretation of the \(\pi\) probability parameter (in hurdle models). Additionally, the interpretation of the \(\lambda\) parameter in (one) inflated models conforms to the following intuition: given that unit \(k\) comes from the non-inflated part of the population, it follows a Poisson distribution (respectively geometric or negative binomial) with the \(\lambda\) parameter (or \(\lambda,\alpha\)); no such interpretation exists for hurdle models. Interestingly, estimates from hurdle zero-truncated and one-inflated zero-truncated models tend to be quite close to one another, although more rigorous studies are required to confirm this observation.

2.3 Fitting method

As previously noted, the singleRcapture package can be used to model the (linear) dependence of all parameters on covariates. A modified IRLS algorithm is employed for this purpose as presented in Algorithm 1; full details are available in T. Yee (2015). In order to apply the algorithm, a modified model matrix \(\boldsymbol{X}_{\text{vlm}}\) is created when the estimatePopsize function is called. In the context of the models implemented in singleRcapture, this matrix can be written as:

\[ \begin{equation}\label{X_vlm-definition} \boldsymbol{X}_{\text{vlm}}= \begin{pmatrix} \boldsymbol{X}_{1} & \boldsymbol{0} &\dotso &\boldsymbol{0}\\ \boldsymbol{0} & \boldsymbol{X}_{2} &\dotso &\boldsymbol{0}\\ \vdots & \vdots & \ddots & \vdots\\ \boldsymbol{0} & \boldsymbol{0} &\dotso &\boldsymbol{X}_{p} \end{pmatrix} \end{equation} \]

where each \(\boldsymbol{X}_{i}\) corresponds to a model matrix associated with a user specified formula.

2.4 Bootstrap variance estimators

We have implemented three types of bootstrap algorithms: parametric (adapted from theory in Zwane and Van der Heijden (2003), Norris and Pollock (1996) for multiple source setting with covariates), semi-parametric (see e.g. Böhning and Friedl (2021)) and nonparametric. The nonparametric version is the usual bootstrap algorithm; which will typically underestimate the variance of \(\hat{N}\). In this section, the focus is on the first two approaches.

The idea of semi-parametric bootstrap is to modify the usual bootstrap to include the additional uncertainty resulting from the fact that the sample size is a random variable. This type of bootstrap is performed in steps listed in Algorithm 2.

3.1 The estimatePopsize function

The singleRcapture package is built around the estimatePopsize function. The main design objective was to make using estimatePopsize as similar as possible to the standard stats::glm function or packages for fitting zero-truncated regression models, such as countreg (e.g. countreg::zerotrunc function). The estimatePopsize function is used to first fit an appropriate (vector) generalized linear model and to estimate the population size along with its variance. It is assumed that the response vector (i.e. the dependent variable) corresponds to the number of times a given unit was observed in the source. The most important arguments are given in Table below; the obligatory ones are formula, data, model.

An important step in using estimatePopsize is specifying the model parameter, which indicates the type of model that will be used for estimating the unobserved part of the population. For instance, to fit Chao’s or Zelterman’s model one should select chao or zelterman and, assuming that one-inflation is present, one can select one of the zero-truncated one-inflated (ztoi\(^\ast\)) or one-inflated zero-truncated (oizt\(^\ast\)) models, such as oiztpoisson for Poisson or ztoinegbin for NB2.

If it is assumed that heterogeneity is observed for NB2 models, one can specify the formula in the controlModel argument with the controlModel function and the alphaFormula argument. This enables the user to provide a formula for the dispersion parameter in the NB2 models. If heterogeneity is assumed for ztoi\(^\ast\) or oizt\(^\ast\), one can specify the omegaFormula argument, which corresponds to the \(\omega\) parameter in these models. Finally, if covariates are assumed to be available for the hurdle models (ztHurdle\(^\ast\) or Hurdlezt\(^\ast\)), then piFormula can be specified, as it provides a formula for the probability parameter in these models.

3.2 Controlling variance estimation with controlPopVar

The estimatePopsize function makes it possible to specify the variance estimation method via popVar (e.g. analytic or variance bootstrap) and control the estimation process by specifying controlPopVar. In the control function controlPopVar the user can specify the bootType argument, which has three possible values: "parametric", "semiparametric" and "nonparametric". Additional arguments accepted by the contorlPopVar function, which are relevant to bootstrap, include:

• alpha, B – the significance level and the number of bootstrap samples to be performed, respectively, with \(0.05\) and \(500\) being the default options.
• cores – the number of process cores to be used in bootstrap (1 by default); parallel computing is enabled by doParallel (Microsoft and Weston 2022a), foreach (Microsoft and Weston 2022b) and parallel packages (R Core Team 2023).
• keepbootStat – a logical value indicating whether to keep a vector of statistics produced by the bootstrap.
• traceBootstrapSize, bootstrapVisualTrace – logical values indicating whether sample and population size should be tracked (FALSE by default); these work only when cores = 1.
• fittingMethod, bootstrapFitcontrol – the fitting method (by default the same as the one used in the original call) and control parameters (controlMethod) for model fitting in the bootstrap.

In addition, the user can specify the type of confidence interval by means of confType and the type of covariance matrix by using covType for the analytical variance estimator (observed or the Fisher information matrix).

In the next sections we present a case study involving the use of a simple zero-truncated Poisson regression and a more advanced model: one-inflated zero-truncated geometric regression with the cloglog link function. First, we present the example dataset, then we describe how to estimate the population size and assess the quality and diagnostics measures. Finally, we show how to estimate the population size in user-specified sub-populations.

4.1 Dataset

We use a dataset from Heijden et al. (2003), which contains information about immigrants in four Dutch cities (Amsterdam, Rotterdam, The Hague and Utrecht), who were staying in the country without a legal permit in 1995 and appeared in police records for that year. This dataset is included in the package called netherlandsimmigrant:

data(netherlandsimmigrant)
head(netherlandsimmigrant)

The number of times each individual appeared in the records is included in the capture variable. The available covariates include gender, age, reason, nation; the last two represent the reason for being captured and the region of the world a given person comes from:

summary(netherlandsimmigrant)

##     capture         gender         age                reason    
##  Min.   :1.000   female: 398   <40yrs:1769   Illegal stay: 259  
##  1st Qu.:1.000   male  :1482   >40yrs: 111   Other reason:1621  
##  Median :1.000                                                  
##  Mean   :1.162                                                  
##  3rd Qu.:1.000                                                  
##  Max.   :6.000                                                  
##                     nation    
##  American and Australia: 173  
##  Asia                  : 284  
##  North Africa          :1023  
##  Rest of Africa        : 243  
##  Surinam               :  64  
##  Turkey                :  93

One notable characteristic of this dataset is that it contains a disproportionately large number of individuals who were observed only once (i.e. 1645).

table(netherlandsimmigrant$capture)

## 
##    1    2    3    4    5    6 
## 1645  183   37   13    1    1

The basic syntax of estimatePopsize is very similar to that of glm, the same can be said about the output of the summary method except for additional results of population size estimates (denoted as Population size estimation results).

basicModel <- estimatePopsize(
  formula = capture ~ gender + age + nation,
  model   = ztpoisson(),
  data    = netherlandsimmigrant,
  controlMethod = controlMethod(silent = TRUE)
)
summary(basicModel)

## 
## Call:
## estimatePopsize.default(formula = capture ~ gender + age + nation, 
##     data = netherlandsimmigrant, model = ztpoisson(), 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 2808.165
## 95% CI for the population size:
##           lowerBound upperBound
## normal      7186.449   18194.25
## logNormal   8431.277   19718.31
## 95% CI for the share of observed population:
##           lowerBound upperBound
## normal     10.332933   26.16035
## logNormal   9.534288   22.29793

The output regarding the population size contains the point estimate, the observed proportion (based on the input dataset), the standard error and two confidence intervals: one relating to the point estimate, the second – to the observed proportion.

According to this simple model, the population size is about 12,500, with about 15% of units observed in the register. The 95% CI under normality indicates that the true population size is likely to be between 7,000-18,000, with about 10-26% of the target population observed in the register.

Since there is a reasonable suspicion that the act of observing a unit in the dataset may lead to undesirable consequences for the person concerned (in this case, a possible deportation, detention or something similar). For these reasons, the user may consider one-inflated models, such as one-inflated zero-truncated geometric model (specified by oiztgeom family) and those presented below.

set.seed(123456)
modelInflated <- estimatePopsize(
    formula = capture ~ nation,
    model   = oiztgeom(omegaLink = "cloglog"),
    data    = netherlandsimmigrant,
    controlModel = controlModel(
        omegaFormula = ~ gender + age
    ),
    popVar = "bootstrap",
    controlPopVar = controlPopVar(bootType = "semiparametric",
                                  B = 50)
)
summary(modelInflated)

## 
## Call:
## estimatePopsize.default(formula = capture ~ nation, data = netherlandsimmigrant, 
##     model = oiztgeom(omegaLink = "cloglog"), popVar = "bootstrap", 
##     controlModel = controlModel(omegaFormula = ~gender + age), 
##     controlPopVar = controlPopVar(bootType = "semiparametric", 
##         B = 50))
## 
## Pearson Residuals:
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -0.41643 -0.41643 -0.30127  0.00314 -0.18323 13.88376 
## 
## Coefficients:
## -----------------------
## For linear predictors associated with: lambda 
##                      Estimate Std. Error z value  P(>|z|)    
## (Intercept)           -1.2552     0.2149  -5.840 5.22e-09 ***
## nationAsia            -0.8193     0.2544  -3.220  0.00128 ** 
## nationNorth Africa     0.2057     0.1838   1.119  0.26309    
## nationRest of Africa  -0.6692     0.2548  -2.627  0.00862 ** 
## nationSurinam         -1.5205     0.6271  -2.425  0.01532 *  
## nationTurkey          -1.1888     0.4343  -2.737  0.00619 ** 
## -----------------------
## For linear predictors associated with: omega 
##             Estimate Std. Error z value  P(>|z|)    
## (Intercept)  -1.4577     0.3884  -3.753 0.000175 ***
## gendermale   -0.8738     0.3602  -2.426 0.015267 *  
## age>40yrs     1.1745     0.5423   2.166 0.030326 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## AIC: 1677.125
## BIC: 1726.976
## Residual deviance: 941.5416
## 
## Log-likelihood: -829.5625 on 3751 Degrees of freedom 
## Number of iterations: 10
## -----------------------
## Population size estimation results: 
## Point estimate 6699.953
## Observed proportion: 28.1% (N obs = 1880)
## Boostrap sample skewness: 0.5209689
## 0 skewness is expected for normally distributed variable
## ---
## Bootstrap Std. Error 1579.235
## 95% CI for the population size:
## lowerBound upperBound 
##   5107.533  10738.242 
## 95% CI for the share of observed population:
## lowerBound upperBound 
##   17.50752   36.80838

According to this approach, the population size is about 7,000, which is about 5,000 less than in the case of the naive Poisson approach. A comparison of AIC and BIC suggests that the one-inflation model fits the data better with BIC for oiztgeom 1727 and 1757 for ztpoisson.

We can access population size estimates using the following code, which returns a list with numerical results.

popSizeEst(basicModel)    # alternative: basicModel$populationSize

## Point estimate: 12690.35
## Variance: 7885790
## 95% confidence intervals:
##           lowerBound upperBound
## normal      7186.449   18194.25
## logNormal   8431.277   19718.31

popSizeEst(modelInflated) # alternative: modelInflated$populationSize

## Point estimate: 6699.953
## Variance: 2493985
## 95% confidence intervals:
## lowerBound upperBound 
##   5107.533  10738.242

The decision whether to use a zero-truncated Poisson or one-inflated zero-truncated geometric model should be based on the assessment of the model and the assumptions regarding the data generation process. One possible method of selection is based on the likelihood ratio test, which can be computed quickly and conveniently with the lmtest (Zeileis and Hothorn (2002)) interface:

library(lmtest)

lrtest(basicModel, modelInflated, 
       name = function(x) {
    if (family(x)$family == "ztpoisson")
        "Basic model"
    else "Inflated model"
})

However, the above is not a standard method of model selection in SSCR. The next sections are dedicated to a detailed description of how to assess the results using standard statistical tests and diagnostics.

4.2 Testing marginal frequencies

A popular method of testing the model fit in single source capture-recapture studies consists in comparing the fitted marginal frequencies \(\displaystyle\sum_{j=1}^{N_{obs}}\hat{\mathbb{P}}\left[Y_{j}=k|\boldsymbol{x}_{j}, Y_{j} > 0\right]\) with the observed marginal frequencies \(\displaystyle\sum_{j=1}^{N}\mathcal{I}_{\{k\}}(Y_{j})=\sum_{j=1}^{N_{obs}}\mathcal{I}_{\{k\}}(Y_{j})\) for \(k\geq1\). If the fitted model bears sufficient resemblance to the real data collection process, these quantities should be quite close and both \(G\) and \(\chi^{2}\) tests can be used to test the statistical significance of the discrepancy with the following singleRcapture syntax for the Poisson model (rather poor fit):

margFreq <- marginalFreq(basicModel)
summary(margFreq, df = 1, dropl5 = "group")

## Test for Goodness of fit of a regression model:
## 
##                  Test statistics df P(>X^2)
## Chi-squared test           50.06  1 1.5e-12
## G-test                     34.31  1 4.7e-09
## 
## -------------------------------------------------------------- 
## Cells with fitted frequencies of < 5 have been grouped 
## Names of cells used in calculating test(s) statistic: 1 2 3

and for the one-inflated model (better fit):

margFreq_inf <- marginalFreq(modelInflated)
summary(margFreq_inf, df = 1, dropl5 = "group")

## Test for Goodness of fit of a regression model:
## 
##                  Test statistics df P(>X^2)
## Chi-squared test            1.88  1    0.17
## G-test                      2.32  1    0.13
## 
## -------------------------------------------------------------- 
## Cells with fitted frequencies of < 5 have been grouped 
## Names of cells used in calculating test(s) statistic: 1 2 3 4

where the dropl5 argument is used to indicate how to handle cells with less than \(5\) fitted observations. Note, however, that currently there is no continuity correction.

4.3 Diagnostics

The singleRStaticCountData class has a plot method implementing several types of quick demonstrative plots, such as the rootogram (Kleiber and Zeileis 2016), for comparing fitted and marginal frequencies, which can be generated with the following syntax:

plot(   basicModel, plotType = "rootogram", main = "ZT Poisson model")
plot(modelInflated, plotType = "rootogram", main = "OI ZT Geometric model")

The above plots suggest that the oiztgeom model fits the data better. Another important issue in population size estimation is to conduct model diagnostics in order to verify whether influential observations are present in the data. For this purpose the leave-one-out (LOO) diagnostic implemented in the dfbeta from the stats package has been adapted as shown below (multiplied by a factor of a hundred for better readability):

dfb <- dfbeta(basicModel)
round(t(apply(dfb, 2, quantile)*100), 4)

##                           0%     25%     50%    75%    100%
## (Intercept)          -0.9909 -0.1533  0.0191 0.0521  8.6619
## gendermale           -9.0535 -0.0777 -0.0283 0.1017  2.2135
## age>40yrs            -2.0010  0.0179  0.0379 0.0691 16.0061
## nationAsia           -9.5559 -0.0529  0.0066 0.0120 17.9914
## nationNorth Africa   -9.6605 -0.0842 -0.0177 0.0087  3.1260
## nationRest of Africa -9.4497 -0.0244  0.0030 0.0083 10.9787
## nationSurinam        -9.3138 -0.0065  0.0021 0.0037 99.3383
## nationTurkey         -9.6198 -0.0220  0.0079 0.0143 32.0980

dfi <- dfbeta(modelInflated)
round(t(apply(dfi, 2, quantile)*100), 4)

##                            0%     25%     50%     75%    100%
## (Intercept)           -1.4640  0.0050  0.0184  0.0557  9.0600
## nationAsia            -6.6331 -0.0346  0.0157  0.0347 12.2406
## nationNorth Africa    -7.2770 -0.0768 -0.0170  0.0085  1.9415
## nationRest of Africa  -6.6568 -0.0230  0.0081  0.0262  7.1710
## nationSurinam         -6.2308 -0.0124  0.0162  0.0421 62.2045
## nationTurkey          -6.4795 -0.0273  0.0204  0.0462 21.1338
## (Intercept):omega     -6.8668 -0.0193  0.0476  0.0476  9.3389
## gendermale:omega      -2.2733 -0.2227  0.1313  0.2482 11.1234
## age>40yrs:omega      -30.2130 -0.2247 -0.1312 -0.0663  2.0393

The result of the dfbeta can be further used in the dfpopsize function, which can be used to quantify LOO on the population size. Note the warning when the bootstap variance estimation is applied.

dfb_pop <- dfpopsize(basicModel, dfbeta = dfb)
dfi_pop <- dfpopsize(modelInflated, dfbeta = dfi)
summary(dfb_pop)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -4236.407     2.660     2.660     5.445    17.281   117.445

summary(dfi_pop)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -456.6443   -3.1121   -0.7243    3.4333    5.1535  103.5949

Figure 2 shows a comparison of the effect of deleting an observation on the population size estimate and inverse probability weights, which refer to the contribution of a given observation to the population size estimate:

plot(basicModel, plotType = "dfpopContr", 
     dfpop = dfb_pop, xlim = c(-4500, 150))
plot(modelInflated, plotType = "dfpopContr", 
     dfpop = dfi_pop, xlim = c(-4500, 150))

These plots show how the population size changes if a given observation is removed. For instance, if we remove observation 542, then the population size will increase by about 4236 for the ztpoisson model. In the case of oiztgeom, the largest change is equal to 457 for observation 900.

The full list of plot types along with the list of optional arguments that can be passed from the call to the plot method down to base R and graphics functions can be found in the help file of the plot method.

?plot.singleRStaticCountData

4.4 The stratifyPopsize function

Researchers may be interested not only in the total population size but also in the size of specific sub-populations (e.g. males, females, particular age groups). For this reason we have created the stratifyPopsize function, which estimates the size by strata defined by the coefficients in the model (the default option). The following output presents results based on the ztpoisson and oiztgeom models.

popSizestrata <- stratifyPopsize(basicModel)
cols <- c("name", "Observed", "Estimated", "logNormalLowerBound", 
          "logNormalUpperBound")
popSizestrata_report <- popSizestrata[, cols]
cols_custom <- c("Name", "Obs", "Estimated", "LowerBound", "UpperBound")
names(popSizestrata_report) <- cols_custom
popSizestrata_report

popSizestrata_inflated <- stratifyPopsize(modelInflated)
popSizestrata_inflated_report <- popSizestrata_inflated[, cols]
names(popSizestrata_inflated_report) <- cols_custom
popSizestrata_inflated_report

The stratifyPopsize function prepared to handle objects of the singleRStaticCountData class, accepts three optional parameters strata, alpha, cov, which are used for specifying sub-populations, significance levels and the covariance matrix to be used for computing standard errors. An example of the full call is presented below.

library(sandwich)
popSizestrataCustom <- stratifyPopsize(
  object  = basicModel,
  strata = ~ gender + age, 
  alpha   = rep(c(0.1, 0.05), each=2), 
  cov     = vcovHC(basicModel, type = "HC4")
)

popSizestrataCustom_report <- popSizestrataCustom[, c(cols, "confLevel")]
names(popSizestrataCustom_report) <- c(cols_custom, "alpha")
popSizestrataCustom_report

We have provided integration with the sandwich (Zeileis, Köll, and Graham 2020) package to correct the variance-covariance matrix in the \(\delta\) method. In the code we have used the vcovHC method for singleRStaticCountData class from the sandwich package, different significance levels for confidence intervals in each stratum and a formula to specify that we want estimates for both males and females to be grouped by nation and age. The strata parameter can be specified either as:

• a formula with the empty left hand side, as shown in the example above (e.g. ~ gender * age),
• a logical vector with the number of entries equal to the number of rows in the dataset, in which case only one stratum will be created (e.g. netherlandsimmigrant$gender == "male"),
• a vector of names of explanatory variables, which will result in every level of the explanatory variable having its own sub-population for each variable specified (e.g. c("gender", "age")),
• not supplied at all, in which case strata will correspond to levels of each factor in the data without any interactions (string vectors will be converted to factors for the convenience of the user),
• a (named) list where each element is a logical vector; names of the list will be used to specify variable names in the returned object, for example:

list(
  "Stratum 1" = netherlandsimmigrant$gender == "male"   & 
    netherlandsimmigrant$nation == "Suriname", 
  "Stratum 2" = netherlandsimmigrant$gender == "female" & 
    netherlandsimmigrant$nation == "North Africa"
)

One can also specify plotType = "strata" in the plot function, which results in a plot with point and CI estimates of the population size.

plot(basicModel, plotType = "strata")
plot(modelInflated, plotType = "strata")

Only the logNormal type of confidence interval is used for plotting since the studentized confidence intervals often result in negative lower bounds.

8.1 The estimatePopsizeFit function

In this section we provide a step-by-step description of how to prepare data in order to use the estimatePopsizeFit function, which may be useful to some users, e.g. those wishing to make modifications to the \(\hat{N}\) estimate or to the bootstrap. In order to show how to apply the function we will fit a zero truncated geometric model on the data from Böhning et al. (2013) with covariate dependency:

\[\begin{align*} \log(\lambda) &= \beta_{1, 1} + \beta_{1, 2} \text{log\_distance} + \beta_{1, 3} \text{C\_TYPE} + \beta_{1, 4} \text{log\_size}, \\ \text{logit}(\omega) &= \beta_{2, 1} + \beta_{2, 2} \text{log\_distance} + \beta_{2, 3} \text{C\_TYPE}. \end{align*}\]

This would be equivalent to the following esimatePopsize call:

estimatePopsize(
  TOTAL_SUB ~ .,
  data = farmsubmission,
  model = ztoigeom(),
  controlModel(
    omegaFormula = ~ 1 + log_size + C_TYPE
  )
)

1. Create a data matrix \(\boldsymbol{X}_{\text{vlm}}\)

X <- matrix(data = 0, nrow = 2 * NROW(farmsubmission), ncol = 7)

2. Fill the first \(n\) rows with model.matrix according to the specified formula and specify the attribute attr(X, "hwm") that informs the function which elements of the design matrix correspond to which linear predictor (covariates for counts and covariates for one-inflation)

X[1:NROW(farmsubmission), 1:4] <- model.matrix(
  ~ 1 + log_size + log_distance + C_TYPE, 
  farmsubmission
)
X[-(1:NROW(farmsubmission)), 5:7] <- model.matrix(
  ~ 1 + log_distance + C_TYPE, 
  farmsubmission
)
attr(X, "hwm") <- c(4, 3)

3. Obtain starting \(\boldsymbol{\beta}\) parameters using the glm.fit function.

start <- glm.fit(
  y = farmsubmission$TOTAL_SUB, 
  x = X[1:NROW(farmsubmission), 1:4], 
  family = poisson()
)$coefficients
start

## [1] -0.82583943  0.33254499 -0.03277732  0.32746933

4. Use the estimatePopsizeFit function to fit the model assuming a zero-truncated one-inflated geometric distribution as specified in the family argument.

res <- estimatePopsizeFit(
  y            = farmsubmission$TOTAL_SUB, 
  X            = X, 
  method       = "IRLS", 
  priorWeights = 1, 
  family       = ztoigeom(), 
  control      = controlMethod(silent = TRUE), 
  coefStart    = c(start, 0, 0, 0),
  etaStart     = matrix(X %*% c(start, 0, 0, 0), ncol = 2),
  offset       = cbind(rep(0, NROW(farmsubmission)), 
                       rep(0, NROW(farmsubmission)))
)

5. Compare our results with those obtained by applying the stats::optim function.

ll <- ztoigeom()$makeMinusLogLike(y = farmsubmission$TOTAL_SUB, X = X)

res2 <- estimatePopsizeFit(
  y = farmsubmission$TOTAL_SUB, 
  X = X, 
  method = "optim", 
  priorWeights = 1, 
  family = ztoigeom(), 
  coefStart = c(start, 0, 0, 0),
  control = controlMethod(silent = TRUE, maxiter = 10000),
  offset = cbind(rep(0, NROW(farmsubmission)), rep(0, NROW(farmsubmission)))
)

data.frame(IRLS  = round(c(res$beta, -ll(res$beta), res$iter), 4),
           optim = round(c(res2$beta, -ll(res2$beta), res2$iter[1]), 4))

The default maxiter parameter for "optim" fitting is \(1000\), but we needed to increase it since the optim does not converge in \(1000\) steps and “gets stuck” at a plateau, which results in a lower log-likelihood value compared to the standard "IRLS".

The above situation is rather typical. While we did not conduct any formal numerical analyses, it seems that when one attempts to model more than one parameter of the distribution as covariate dependent optim algorithms, both "Nelder-Mead" and "L-BFGS-B" seem to be ill-suited for the task despite being provided with the analytically computed gradient. This is one of the reasons why "IRLS" is the default fitting method.

8.2 Structure of a family function

In this section we provide details regarding the family object for the singleRcapture package. This object contains additional parameters in comparison to the standard family object from the stats package.