---
title: "singleRcapture: An R Package for Single-Source Capture-Recapture Models"
author: "Piotr Chlebicki, Maciej Beręsewicz"
date: "`r Sys.Date()`"
output: 
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        toc: true
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vignette: >
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  %\VignetteEngine{knitr::rmarkdown_notangle}
bibliography: bibliography.bib
link-citations: true
editor_options: 
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header-includes:
  - \usepackage{amsmath}
  - \usepackage{amsthm}
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  - \usepackage{calc}
  - \usepackage{ragged2e}
  - \usepackage[ruled]{algorithm2e}
  - \DeclareMathAlphabet{\mathmybb}{U}{bbold}{m}{n}
  - \newcommand{\1}{\mathcal{I}}
  - \newcommand{\bZero}{\boldsymbol{0}}
  - \newcommand{\bx}{\boldsymbol{x}}
  - \newcommand{\bX}{\boldsymbol{X}}
  - \newcommand{\bbeta}{\boldsymbol{\beta}}
  - \newcommand{\boeta}{\boldsymbol{\eta}}
  - \newcommand{\bW}{\boldsymbol{W}}
  - \newcommand{\bo}{\boldsymbol{o}}  
---

```{=html}
<style>
pre {
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}
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```

```{r plot-parameters, include=FALSE}
par(mar = c(2.5, 8.5, 4.1, 2.5), cex.main = .7, cex.lab = .6)
```

# Introduction {#sec-introduction}

Population size estimation is a methodological approach employed across
multiple scientific disciplines, which serves as the basis for research,
policy formulation, and decision-making processes [cf.
@bohning2018capture]. In the field of statistics, particularly official
statistics, precise population estimates are essential in order to
develop robust economic models, optimize resource allocation, and inform
evidence-based policy [cf. @baffour2013modern]. Social scientists
utilize advanced population estimation techniques to investigate
*hard-to-reach* populations, such as homeless individuals or illicit
drug users in an effort to overcome the inherent limitations of
conventional census methodologies. These techniques are crucial for
obtaining accurate data on populations that are typically
under-represented or difficult to access using traditional sampling
methods [cf. @vincent2022estimating]. In ecology and epidemiology,
researchers focus on estimating the size of individual species or
disease-affected populations within defined geographical regions as part
of conservation efforts, ecosystem management, and public health
interventions.

Population size estimation can be approached using various
methodologies, each with distinct advantages and limitations.
Traditional approaches include full enumeration (e.g. censuses) and
comprehensive sample surveys, which, while providing detailed data, are
often resource-intensive and may result in delayed estimates,
particularly for human populations. Alternative methods leverage
existing data sources, such as administrative registers or carefully
designed small-scale studies in wildlife research, or census coverage
surveys [cf. @wolter1986some; @zhang2019note]. Information from these
sources is often extracted by applying statistical methods, known as
*capture-recapture* or *multiple system estimation*, which rely on data
from multiple enumerations of the same population [cf.
@dunne2024system]. This approach can be implemented using either a
single source with repeated observations, two sources, or multiple
sources.

In this paper we focus on methods that involve a single data source with
multiple enumerations of the same units [cf. @ztpoisson]. In human
population studies, such data can be derived from police records, health
system databases, or border control logs; in the case of non-human
populations, data of this kind can come from veterinary records or
specialized field data. These methods are often applied to estimate
hard-to-reach or hidden populations, where standard sampling methods may
be inappropriate because of prohibitive costs or problems with
identifying population members.

While methods for two or more sources are implemented in various
open-source software packages, for instance **CARE-4** [@yang2006care4]
(in **GAUSS**), **Rcapture** [@baillargeon2007rcapture], **marked**
[@laake2013marked] or **VGAM** [@yee2015vgam] (all in **R**),
single-source capture-recapture (SSCR) methods are either not available
at all or are only partially implemented in existing **R** packages or
other software. Therefore, the paper attempts to bridge this gap by
introducing the **singleRcapture** package, which implement
*state-of-the-art* SSCR methods and offer a user friendly API resembling
existing **R** functions (e.g., `glm`). In the next subsection we
describe existing **R** packages and other software that could be used
for estimating population size using SSCR methods.

## Software for capture-recapture with a single source {#sec-software}

The majority of SSCR methods assume zero-truncated distributions or
their extensions (e.g., inclusion of one-inflation). The
**Distributions.jl** [@Distributionsjl] (in **Julia**), **StatsModels**
[@seabold2010statsmodels] (in **Python**), **countreg** [@countreg],
**VGAM** [@VGAM-main] or **distributions3** [@distributions3] (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 @gaitdcount 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
@bohning2018capture or @bohning2024one 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.

# Theoretical background {#sec-theory}

## 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 $\bx_{k}$ 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:

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

where $I_{k}:=\1_{\mathbb{N}}(Y_{k})$, $N_{obs}$ is the number of
observed units and $\1$ is the indicator function, while the maximum
likelihood estimate of $N$ is obtained after substituting regression
parameters $\bbeta$ for $\mathbb{P}[Y_{k}>0|\bx_{k}]$ 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, @godwin2017estimation and @ztoi-oizt-poisson
introduced one-inflated distributions, which explicitly model the
probability of singletons by giving additional mass $\omega$ to
singletons denoted as $\1_{\{1\}}(y)$ [cf. @bohning2024one]. In other
words they considered a new random variable $Y^{\ast}$ that corresponds
to the data collection process which exhibits one inflation:

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

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:

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

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 \underline{observable} 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 $0$ [@ztpoisson].

## 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:

```{r, eval=FALSE}
?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 [@chao1987estimating] and Zelterman's
    [@zelterman1988robust] 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
    $\lambda_k=\bx_{k}\bbeta$ (for a covariate extension see
    @chao-generalization and @zelterman):

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

$$
\hat{N}_{\text{Zelt}}=\sum_{k=1}^{N_{obs}}
\left(1-\exp\left(-2\exp\left(\bx_{k}\hat{\bbeta}\right)\right)\right)^{-1}.
\tag{Zelterman's estimator}
$$

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 [@zotmodels], 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 as:

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

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

-   One-inflated zero-truncated (`oizt`$^\ast$) modifications, where the
    count data variable $Y^{\ast}$ is defined such that its distribution
    statisfies:

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

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

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
[@bohning2023equivalence] 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.

-   A modified Chao estimator robust to one-inflation (`oichao`), which
    discards singletons and runs logistic regression on the indicator
    $Z=1$ if $Y=3$ and $Z=0$ if $Y=2$ with
    $\text{logit}(p_k)=\ln(\lambda_k/3)$, so that
    $p_k=\lambda_k/(3+\lambda_k)$ matches
    $\mathbb{P}[Y=3\mid Y\in\{2,3\}]$ for $Y\sim\mathrm{Poisson}(\lambda_k)$.
    The associated population-size estimator is

$$
\hat{N}_{\text{oiChao}} = N_{obs}+
\sum_{k=1}^{\boldsymbol{f}_{2}+\boldsymbol{f}_{3}}
\left(\tfrac{9}{2}\exp\left(2\bx_{k}\hat{\bbeta}\right)+
\tfrac{9}{2}\exp\left(3\bx_{k}\hat{\bbeta}\right)\right)^{-1},
\tag{oiChao estimator}
$$

where $\boldsymbol{f}_{2}$ and $\boldsymbol{f}_{3}$ denote the number of
units observed twice and three times. The inflation factor
$(\lambda_k^{2}/2+\lambda_k^{3}/6)^{-1}$ equals
$\mathbb{P}[Y=0]/\mathbb{P}[Y\in\{2,3\}]$ under $\mathrm{Poisson}(\lambda_k)$,
so the estimator is unaffected by extra mass at $Y=1$.

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
[@ztnegbin]. 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 $2-$dimensional 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
zero-truncated one-inflated models tend to be quite close to one
another, although more rigorous studies are required to confirm this
observation.

## 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 @VGAM-main. In order to apply the algorithm, a
modified model matrix $\bX_{\text{vlm}}$ is created when the
`estimatePopsize` function is called. In the context of the models
implemented in **singleRcapture**, this matrix can be written as:

$$
\begin{equation}\label{X_vlm-definition}
  \bX_{\text{vlm}}=
  \begin{pmatrix}
    \bX_{1} & \boldsymbol{0} &\dotso &\boldsymbol{0}\\
    \boldsymbol{0} & \bX_{2} &\dotso &\boldsymbol{0}\\
    \vdots & \vdots & \ddots & \vdots\\
    \boldsymbol{0} & \boldsymbol{0} &\dotso &\bX_{p}
  \end{pmatrix}
\end{equation}
$$

where each $\bX_{i}$ corresponds to a model matrix associated with a
user specified formula.

### Algorithm 1: The modified IRLS algorithm used in the **singleRcapture** package

1.  Initialize with `iter` ← 1, $\boeta$ ← `start`, $\bW$ ← I, $\ell$ ←
    $\ell(\bbeta)$

2.  Store values from the previous step:\
    $\ell_{-}$ ← $\ell$, $\bW_{-}$ ← $\bW$, $\bbeta_{-}$ ← $\bbeta$ (the
    last assignment is omitted during the first iteration), and assign
    values in the current iteration:

    $\displaystyle\boeta$ ← $\bX_{\text{vlm}}\bbeta+\bo$

    $\bW_{(k)}$ ←
    $\mathbb{E}\left[-\frac{\partial^{2}\ell}{\partial\boeta_{(k)}^\top\partial\boeta_{(k)}}\right]$

    $\boldsymbol{Z}_{(k)}$ ←
    $\boeta_{(k)}+\frac{\partial\ell}{\partial\boeta_{(k)}}\bW_{(k)}^{-1}-\bo_{(k)}$

    where $\bo$ denotes offset.

3.  Assign the current coefficient value:\
    $\bbeta$ ←
    $\left(\bX_{\text{vlm}}\bW\bX_{\text{vlm}}\right)^{-1}\bX_{\text{vlm}}\bW\boldsymbol{Z}$

4.  If $\ell(\bbeta)<\ell(\bbeta_{-})$ try selecting the smallest value
    $h$ such that for $\bbeta_{h}$ ←
    $2^{-h}\left(\bbeta+\bbeta_{-}\right)$ the inequality
    $\ell(\bbeta_{h})>\ell(\bbeta_{-})$ holds if this is successful
    $\bbeta$ ← $\bbeta_{h}$, else stop the algorithm.

5.  If convergence is achieved or `iter` is higher than `maxiter`, stop
    the algorithm, else `iter` ← 1 + `iter` and return to step 2.

In the case of multi-parameter families, we get a matrix of linear
predictors $\boeta$ instead of a vector, with the number of columns
matching the number of parameters in the distribution. "Weights" (matrix
$\bW$) are then modified to be information matrices
$\displaystyle\mathbb{E}\left[-\frac{\partial^{2}\ell}{\partial\boeta_{(k)}^\top\partial\boeta_{(k)}}\right]$,
where $\ell$ is the log-likelihood function and $\boeta_{(k)}$ is the
$k$-th row of $\boeta$, while in the typical IRLS they are scalars
$\displaystyle\mathbb{E}\left[-\frac{\partial^{2}\ell}{\partial\eta_{k}^{2}}\right]$,
which is often just
$\displaystyle-\frac{\partial^{2}\ell}{\partial\eta^{2}}$.

## Bootstrap variance estimators {#sec-boostrap}

We have implemented three types of bootstrap algorithms: parametric
(adapted from theory in @zwane, @norrpoll for multiple source setting
with covariates), semi-parametric (see e.g. @BoehningFriedl2021) 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.

### Algorithm 2: Semi-parametric bootstrap

1.  Draw a sample of size
    $N_{obs}'\sim\text{Binomial}\left(N', \frac{N_{obs}}{N'}\right)$,
    where
    $N'=\lfloor\hat{N}\rfloor+\text{Bernoulli}\left(\lfloor\hat{N}\rfloor-\hat{N}\right)$
2.  Draw $N_{obs}'$ units from the data uniformly without replacement
3.  Obtain a new population size estimate $N_b$ using bootstrap data
4.  Repeat 1-3 steps $B$ times

In other words, we first draw a sample size and then a sample
conditional on the sample size. Note that when using the semi-parametric
bootstrap one implicitly assumes that the population size estimate
$\hat{N}$ is accurate. The last implemented bootstrap type is the
parametric algorithm, which first draws a finite population of size
$\approx\hat{N}$ from the superpopulation model and then samples from
this population according to the selected model, as described in
Algorithm 3.

### Algorithm 3: Parametric bootstrap

1.  Draw the number of covariates equal to
    $\lfloor\hat{N}\rfloor+\text{Bernoulli}\left(\lfloor\hat{N}\rfloor-\hat{N}\right)$
    proportional to the estimated contribution
    $(\mathbb{P}\left[Y_{k}>0|\bx_{k}\right])^{-1}$ with replacement
2.  Using the fitted model and regression coefficients $\hat{\bbeta}$
    draw for each covariate the $Y$ value from the corresponding
    probability measure on $\mathbb{N}\cup\{0\}$
3.  Truncate units with the drawn $Y$ value equal to $0$
4.  Obtain a population size estimate $N_b$ based on the truncated data
5.  Repeat 1-4 steps $B$ times

Note that in order for this type of algorithm to result in consistent
standard error estimates, it is imperative that the estimated model for
the entire superpopulation probability space is consistent, which may be
much less realistic than in the case of the semi-parametric bootstrap.
The parametric bootstrap algorithm is the default option in
**singleRcapture**.

# The main function {#sec-main}

## 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`.

| Argument | Description |
|--------------------|----------------------------------------------------|
| `formula` | The main formula (i.e., for the Poisson $\lambda$ parameter); |
| `data` | A `data.frame` (or `data.frame` coercible) object; |
| `model` | Either a function, a string, or a family class object specifying which model should be used; possible values are listed in the documentation. The supplied argument should have the form `model = "ztpoisson"`, `model = ztpoisson`, or if a link function should be specified, then `model = ztpoisson(lambdaLink = "log")` can be used; |
| `method` | A numerical method used to fit regression `IRLS` or `optim`; |
| `popVar` | A method for estimating variance of $\hat{N}$ and creating confidence intervals (either bootstrap, analytic, or skipping the estimation entirely); |
| `controlMethod`, `controlModel`, or `controlPopVar` | Control parameters for numerical fitting, specifying additional formulas (inflation, dispersion) and population size estimation, respectively; |
| `offset` | A matrix of offset values with the number of columns matching the number of distribution parameters, providing offset values to each of the linear predictors; |
| `...` | Additional optional arguments passed to other methods, e.g., `estimatePopsizeFit`; |

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.

## 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**
    [@doParallel], **foreach** [@foreach] and **parallel** packages
    [@parallel].
-   `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:
zero-truncated one-inflated 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.

# Data analysis example {#sec-study}

The package can be installed in the standard manner using:

```{r, installation, eval=FALSE}
install.packages("singleRcapture")
```

Then, we need to load the package using the following code:

```{r, loading-package, warning=FALSE}
library(singleRcapture)
```

## Dataset

We use a dataset from @ztpoisson, 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`:

```{r, loading_singleRcapture}
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:

```{r, netherlandsimmigrant_summary}
summary(netherlandsimmigrant)
```

One notable characteristic of this dataset is that it contains a
disproportionately large number of individuals who were observed only
once (i.e. `r sum(netherlandsimmigrant$capture==1)`).

```{r, table_netherlands}
table(netherlandsimmigrant$capture)
```

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`).

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

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 zero-truncated one-inflated geometric model (specified by
`ztoigeom` family) and those presented below.

```{r, model_inflated_netherlands, warning=FALSE, cache=T}
set.seed(123456)
modelInflated <- estimatePopsize(
    formula = capture ~ nation,
    model   = ztoigeom(omegaLink = "cloglog"),
    data    = netherlandsimmigrant,
    controlModel = controlModel(
        omegaFormula = ~ gender + age
    ),
    popVar = "bootstrap",
    controlPopVar = controlPopVar(bootType = "semiparametric",
                                  B = 50)
)
summary(modelInflated)
```

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 `ztoigeom` `r round(BIC(modelInflated))` and
`r round(BIC(basicModel))` for `ztpoisson`.

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

```{r, extract_pop_size}
popSizeEst(basicModel)    # alternative: basicModel$populationSize
popSizeEst(modelInflated) # alternative: modelInflated$populationSize
```

The decision whether to use a zero-truncated Poisson or zero-truncated
one-inflated 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**
(@lmtest) interface:

```{r, message=FALSE}
library(lmtest)
```

```{r}
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.

## Testing marginal frequencies

A popular method of testing the model fit in single source
capture-recapture studies consists in comparing the fitted marginal
frequencies
$\displaystyle\sum_{j=1}^{N_{obs}}\hat{\mathbb{P}}\left[Y_{j}=k|\bx_{j}, Y_{j} > 0\right]$
with the observed marginal frequencies
$\displaystyle\sum_{j=1}^{N}\1_{\{k\}}(Y_{j})=\sum_{j=1}^{N_{obs}}\1_{\{k\}}(Y_{j})$
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):

```{r, marginal_freq_basic_model}
margFreq <- marginalFreq(basicModel)
summary(margFreq, df = 1, dropl5 = "group")
```

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

```{r, marginal_freq_inflated_model}
margFreq_inf <- marginalFreq(modelInflated)
summary(margFreq_inf, df = 1, dropl5 = "group")
```

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.

## Diagnostics

The `singleRStaticCountData` class has a `plot` method implementing
several types of quick demonstrative plots, such as the rootogram
[@rootogram], for comparing fitted and marginal frequencies, which can
be generated with the following syntax:

```{r, rootogram, fig.dim=c(5,5), fig.pos="ht", fig.show='hold', fig.cap="Rootograms for ztpoisson (left) and ztoigeom (right) models"}
plot(   basicModel, plotType = "rootogram", main = "ZT Poisson model")
plot(modelInflated, plotType = "rootogram", main = "ZTOI Geometric model")
```

The above plots suggest that the `ztoigeom` 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):

```{r, dfbeta_basic_model}
dfb <- dfbeta(basicModel)
round(t(apply(dfb, 2, quantile)*100), 4)
```

```{r, df_beta_inflated_model}
dfi <- dfbeta(modelInflated)
round(t(apply(dfi, 2, quantile)*100), 4)
```

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.

```{r, dfpopsize_basic_model}
dfb_pop <- dfpopsize(basicModel, dfbeta = dfb)
dfi_pop <- dfpopsize(modelInflated, dfbeta = dfi)
summary(dfb_pop)
summary(dfi_pop)
```

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:

```{r dfpopsize_plot, fig.dim=c(5,5), fig.pos="ht", fig.show='hold', fig.cap="Results for ztpoisson (left) and ztoigeom (right) model"}
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
`r which.min(dfb_pop)`, then the population size will increase by about
`r abs(round(min(dfb_pop)))` for the `ztpoisson` model. In the case of
`ztoigeom`, the largest change is equal to `r abs(round(min(dfi_pop)))`
for observation `r which.min(dfi_pop)`.

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.

```{r, eval=FALSE}
?plot.singleRStaticCountData
```

## 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 `ztoigeom` models.

```{r, strata}
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
```

```{r, strata-inflated}
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.

```{r, custom-strata, message=F, warning=F}
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** [@sandwich] 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:

```{r, eval=FALSE}
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.

```{r, strata_plot, fig.dim=c(6,6), fig.show='hold', fig.pos = "ht", fig.cap="Population size by covariates for ztpoisson (left) and ztoigeom (right) model"}
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.

# Classes and `S3Methods` {#sec-methods}

We have created a number of classes. The main ones are:
`singleRStaticCountData`, `singleRfamily`, and supplementary are:
`popSizeEstResults`, `summarysingleRmargin` and \newline
`summarysingleRStaticCountData`, which make it possible to extract
relevant information regarding the population size.

For instance, the `popSizeEst` function can be used to extract
information about the estimated size of the population as given below:

```{r, popsize_extract}
(popEst <- popSizeEst(basicModel))
```

and the resulting object `popEst` of the `popSizeEstResults` class
contains the following fields:

-   `pointEstimate`, `variance` -- numerics containing the point
    estimate and variance of this estimate.
-   `confidenceInterval` -- a `data.frame` with confidence intervals.
-   `boot` -- If the bootstrap was performed a numeric vector containing
    the $\hat{N}$ values from the bootstrap, a character vector with
    value `"No bootstrap performed"` otherwise.
-   `control` -- a `controlPopVar` object with controls used to obtain
    the object.

The only explicitly defined method for `popSizeEstResults`,
`summarysingleRmargin` and `summarysingleRStaticCountData` classes is
the `print` method, but the former one also accepts **R** primitives
like `coef`:

```{r, extract-coefs}
coef(summary(basicModel))
```

analogously to `glm` from **stats**. The `singleRfamily` inherits the
`family` class from **stats** and has explicitly defined `print` and
`simulate` methods. Example usage is presented below

| Function | Description |
|-------------------|-----------------------------------------------------|
| `fitted` | It works almost exactly like `glm` counterparts but returns more information, namely on fitted values for the truncated and non-truncated probability distribution; |
| `logLik` | Compared to `glm` method, it has the possibility of returning not just the value of the fitted log-likelihood but also the entire function (argument `type = "function"`) along with two first derivatives (argument `deriv = 0:2`); |
| `model.matrix` | It has the possibility of returning the $X_{\text{vlm}}$ matrix defined previously; |
| `simulate` | It calls the `simulate` method for the chosen model and fitted $\boeta$; |
| `predict` | It has the possibility of returning either fitted distribution parameters for each unit (`type = "response"`), or just linear predictors (`type = "link"`), or means of the fitted distributions of $Y$ and $Y|Y>0$ (`type = "mean"`) or the inverse probability weights (`type = "contr"`). It is possible to set the `se.fit` argument to `TRUE` in order to obtain standard errors for each of those by using the $\delta$ method. Also, it is possible to use a custom covariance matrix for standard error computation (argument `cov`); |
| `redoPopEstimation` | A function that applies all post-hoc procedures that were performed (such as heteroscedastic consistent covariance matrix estimation via **countreg**) to estimate the population size and standard errors; |
| `residuals` | Used for obtaining residuals of several types, we refer interested readers to the manual `?singleRcapture:::residuals.singleRStaticCountData`; |
| `stratifyPopsize, summary` | Compared to the `glm` class, summary has the possibility of adding confidence intervals to the coefficient matrix (argument `confint = TRUE`) and using a custom covariance matrix (argument `cov = someMatrix`); |
| `plot` | It has been discussed above; |
| `popSizeEst` | An extractor showcased above; |
| `cooks.distance` | It works only for single predictor models; |
| `dfbeta, dfpopsize` | Multi-threading in `dfbeta` is available and `dfpopsize` calls `dfbeta` if no `dfbeta` object was provided in the call; |
| `bread, estfun, vcovHC` | For (almost) full **sandwich** compatibility; |
| `AIC, BIC, extractAIC, family, confint, df.residual, model.frame, hatvalues, nobs, print, sigma, influence, rstudent, rstandard` | These work exactly like `glm` counterparts. |

```{r, simulate-method}
set.seed(1234567890)
N <- 10000
gender <- rbinom(N, 1, 0.2)
eta <- -1 + 0.5*gender
counts <- simulate(ztpoisson(), eta = cbind(eta), seed = 1)
summary(data.frame(gender, eta, counts))
```

The full list of explicitly defined methods for `singleRStaticCountData`
methods is presented in Table above.

# Concluding remarks

In this paper we have introduced the **singleRcapture** package for
single source capture-recapture models. The package implement
state-of-the-art methods for estimating population size based on a
single data set with multiple counts. The package implements different
methods to account for heterogeneity in capture probabilities, modelled
using covariates, as well as behavioural change, modelled using
one-inflation. We have built the package to facilitate the
implementation of new models using `family` objects; their application
is exemplified in the Section 7. An example of implementing a custom
family described in Section 8 is presented in replication materials.

Furthermore, since many **R** users are familiar with **countreg** or
**VGAM** packages, we have implemented a lightweight extension called
**singleRcaptureExtra**, available through Github
(\url{https://github.com/ncn-foreigners/singleRcaptureExtra}), which can
be used to integrate **singleRcapture** with these packages.

In future work we plan to implement Bayesian estimation using **Stan**
(e.g. via the **brms** package; @carpenter2017stan, @brms) and for
one-inflation models we can use the recent approach proposed by
@tuoto2022bayesian and implement our own families using the **brms**
package.

# Acknowledgements {#Acknowledgements}

The authors' work has been financed by the National Science Centre in
Poland, OPUS 20, grant no. 2020/39/B/HS4/00941.

The authors would like to thank Peter van der Heijden, Maarten Cruyff,
Dankmar Böhning, Łukasz Chrostowski and Layna Dennett for useful
comments that have helped to improve the functionality of the package.
In addition, we would like to thank Marcin Szymkowiak and Tymon
Świtalski for their valuable comments that have considerably improved
the paper.

# Detailed information {#sec-details}

## The `estimatePopsizeFit` function {#sec-estimatePopsizeFit short-title="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 one-inflated zero-truncated geometric model on the data
from @chao-generalization 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:

```{r, example-code-farm, eval=FALSE}
estimatePopsize(
  TOTAL_SUB ~ .,
  data = farmsubmission,
  model = oiztgeom(),
  controlModel(
    omegaFormula = ~ 1 + log_size + C_TYPE
  )
)
```

1.  Create a data matrix $\bX_{\text{vlm}}$

```{r, estimatePopsizeFit-1}
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)

```{r, estimatePopsizeFit-2}
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 $\bbeta$ parameters using the `glm.fit` function.

```{r, estimatePopsizeFit-3}
start <- glm.fit(
  y = farmsubmission$TOTAL_SUB, 
  x = X[1:NROW(farmsubmission), 1:4], 
  family = poisson()
)$coefficients
start
```

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

```{r, estimatePopsizeFit-4}
res <- estimatePopsizeFit(
  y            = farmsubmission$TOTAL_SUB, 
  X            = X, 
  method       = "IRLS", 
  priorWeights = 1, 
  family       = oiztgeom(), 
  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.

```{r, estimatePopsizeFit-5}
ll <- oiztgeom()$makeMinusLogLike(y = farmsubmission$TOTAL_SUB, X = X)
```

```{r, estimatePopsizeFit-6}
res2 <- estimatePopsizeFit(
  y = farmsubmission$TOTAL_SUB, 
  X = X, 
  method = "optim", 
  priorWeights = 1, 
  family = oiztgeom(), 
  coefStart = c(start, 0, 0, 0),
  control = controlMethod(silent = TRUE, maxiter = 10000),
  offset = cbind(rep(0, NROW(farmsubmission)), rep(0, NROW(farmsubmission)))
)
```

```{r, estimatePopsizeFit-7}
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.

## Structure of a family function {#sec-family}

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.

| Function | Description |
|------------------|------------------------------------------------------|
| `makeMinusLogLike` | A factory function for creating the following functions: <br> $\ell(\bbeta)$, $\frac{\partial\ell}{\partial\bbeta}$, $\frac{\partial^{2}\ell}{\partial\bbeta^\top\partial\bbeta}$ <br> from the $\boldsymbol{y}$ vector and the $\bX_{vlm}$ matrix. The `deriv` argument has possible values in `c(0, 1, 2)` that determine which derivative to return; the default value is `0`, which represents the minus log-likelihood. |
| `links` | A list with link functions; |
| `mu.eta, variance` | Functions of linear predictors that return the expected value and variance. The `type` argument with 2 possible values (`"trunc"` and `"nontrunc"`) specifies whether to return $\mathbb{E}[Y|Y>0], \text{var}[Y|Y>0]$ or $\mathbb{E}[Y], \text{var}[Y]$ respectively. The `deriv` argument with values in `c(0, 1, 2)` is used for indicating the derivative with respect to the linear predictors, which is used for providing standard errors in the `predict` method. |
| `family` | A string that specifies the model name; |
| `valideta, validmu` | For now, it only returns `TRUE`. In the near future, it will be used to check whether applied linear predictors are valid (i.e., are transformed into some elements of the parameter space subjected to the inverse link function). |
| `funcZ, Wfun` | Functions that create pseudo residuals and working weights used in the IRLS algorithm; |
| `devResids` | A function that returns deviance residuals given a vector of prior weights of linear predictors and the response vector; |
| `pointEst, popVar` | Functions that return the point estimate for the population size and analytic estimation of its variance given prior weights of linear predictors and, in the latter case, also estimates of $\text{cov}(\hat{\bbeta})$ and $\bX_{\text{vlm}}$ matrix. There is an additional boolean parameter `contr` in the former function, which, if set to `TRUE`, returns the contribution of each unit. |
| `etaNames` | Names of linear predictors; |
| `densityFunction` | A function that returns the value of PMF at values of `x` given linear predictors. The `type` argument specifies whether to return $\mathbb{P}[Y|Y>0]$ or $\mathbb{P}[Y]$; |
| `simulate` | A function that generates values of a dependent vector given linear predictors; |
| `getStart` | An expression for generating starting points; |

# Implementing a custom **singleRcapture** family function {#sec-custom short-title="Implementing custom singleRcapture family function"}

Suppose we want to implement a very specific zero truncated family
function in the **singleRcapture**, which corresponds to the following
"untruncated" distribution:

$$
  \mathbb{P}[Y=y|\lambda, \pi] = \begin{cases}
    1 - \frac{1}{2}\lambda - \frac{1}{2}\pi & \text{when: } y=0\\
    \frac{1}{2}\pi & \text{when: } y=1\\
    \frac{1}{2}\lambda & \text{when: } y=2,
  \end{cases}
$$

with $\lambda, \pi\in\left(0, 1\right)$ being dependent on covariates.

Below we provide a possible way of implementing the above model, with
`lambda, pi` meaning $\frac{1}{2}\lambda,\frac{1}{2}\pi$. We provide a
simple example that shows that the proposed approach works as expected.

```{r custom-family-example}
myFamilyFunction <- function(lambdaLink = c("logit", "cloglog", "probit"),
                             piLink     = c("logit", "cloglog", "probit"),
                             ...) {
  if (missing(lambdaLink)) lambdaLink <- "logit"
  if (missing(piLink))         piLink <- "logit"

  links <- list()
  attr(links, "linkNames") <- c(lambdaLink, piLink)

  lambdaLink <- switch(lambdaLink,
                       "logit"   = singleRcapture:::singleRinternallogitLink,
                       "cloglog" = singleRcapture:::singleRinternalcloglogLink,
                       "probit"  = singleRcapture:::singleRinternalprobitLink
  )

  piLink <- switch(piLink,
                   "logit"   = singleRcapture:::singleRinternallogitLink,
                   "cloglog" = singleRcapture:::singleRinternalcloglogLink,
                   "probit"  = singleRcapture:::singleRinternalprobitLink
  )

  links[1:2] <- c(lambdaLink, piLink)

  mu.eta <- function(eta, type = "trunc", deriv = FALSE, ...) {
    pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
    lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2

    if (!deriv) {
      switch (type,
              "nontrunc" = pi + 2 * lambda,
              "trunc" = 1 + lambda / (pi + lambda)
      )
    } else {
      # Only necessary if one wishes to use standard errors in predict method
      switch (type,
              "nontrunc" = {
                matrix(c(2, 1) * c(
                  lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) / 2,
                  piLink(eta[, 2], inverse = TRUE, deriv = 1) / 2
                ), ncol = 2)
              },
              "trunc" = {
                matrix(c(
                  pi / (pi + lambda) ^ 2,
                  -lambda / (pi + lambda) ^ 2
                ) * c(
                  lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) / 2,
                  piLink(eta[, 2], inverse = TRUE, deriv = 1) / 2
                ), ncol = 2)
              }
      )
    }
  }

  variance <- function(eta, type = "nontrunc", ...) {
    pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
    lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2

    switch (type,
            "nontrunc" = pi * (1 - pi) + 4 * lambda * (1 - lambda - pi),
            "trunc" = lambda * (1 - lambda) / (pi + lambda)
    )
  }

  Wfun <- function(prior, y, eta, ...) {
    pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
    lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2

    G01 <- ((lambda + pi) ^ (-2)) * piLink(eta[, 2], inverse = TRUE, deriv = 1) *
      lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) * prior / 4

    G00 <- ((lambda + pi) ^ (-2)) - (pi ^ (-2)) - lambda / ((lambda + pi) * (pi ^ 2))
    G00 <- G00 * prior * (piLink(eta[, 2], inverse = TRUE, deriv = 1) ^ 2) / 4

    G11 <- ((lambda + pi) ^ (-2)) - (((lambda + pi) * lambda) ^ -1)
    G11 <- G11 * prior * (lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) ^ 2) / 4

    matrix(
      -c(G11, # lambda
         G01, # mixed
         G01, # mixed
         G00  # pi
      ),
      dimnames = list(rownames(eta), c("lambda", "mixed", "mixed", "pi")),
      ncol = 4
    )
  }

  funcZ <- function(eta, weight, y, prior, ...) {
    pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
    lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2

    weight <- weight / prior

    G0 <- (2 - y) / pi     - ((lambda + pi) ^ -1)
    G1 <- (y - 1) / lambda - ((lambda + pi) ^ -1)

    G1 <- G1 * lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) / 2
    G0 <- G0 *     piLink(eta[, 2], inverse = TRUE, deriv = 1) / 2

    uMatrix <- matrix(c(G1, G0), ncol = 2)

    weight <- lapply(X = 1:nrow(weight), FUN = function (x) {
      matrix(as.numeric(weight[x, ]), ncol = 2)
    })

    pseudoResid <- sapply(X = 1:length(weight), FUN = function (x) {
      #xx <- chol2inv(chol(weight[[x]])) # less computationally demanding
      xx <- solve(weight[[x]]) # more stable
      xx %*% uMatrix[x, ]
    })
    pseudoResid <- t(pseudoResid)
    dimnames(pseudoResid) <- dimnames(eta)
    pseudoResid
  }

  minusLogLike <- function(y, X, offset,
                           weight    = 1,
                           NbyK      = FALSE,
                           vectorDer = FALSE,
                           deriv     = 0,
                           ...) {
    y <- as.numeric(y)
    if (is.null(weight)) {
      weight <- 1
    }
    if (missing(offset)) {
      offset <- cbind(rep(0, NROW(X) / 2), rep(0, NROW(X) / 2))
    }

    if (!(deriv %in% c(0, 1, 2)))
      stop("Only score function and derivatives up to 2 are supported.")
    deriv <- deriv + 1

    switch (deriv,
            function(beta) {
              eta <- matrix(as.matrix(X) %*% beta, ncol = 2) + offset
              pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
              lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2
              -sum(weight * ((2 - y) * log(pi) + (y - 1) * log(lambda) - log(pi + lambda)))
            },
            function(beta) {
              eta <- matrix(as.matrix(X) %*% beta, ncol = 2) + offset
              pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
              lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2

              G0 <- (2 - y) / pi     - ((lambda + pi) ^ -1)
              G1 <- (y - 1) / lambda - ((lambda + pi) ^ -1)

              G1 <- G1 * weight * lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) / 2
              G0 <- G0 * weight *     piLink(eta[, 2], inverse = TRUE, deriv = 1) / 2

              if (NbyK) {
                XX <- 1:(attr(X, "hwm")[1])
                return(cbind(as.data.frame(X[1:nrow(eta), XX]) * G1,
                             as.data.frame(X[-(1:nrow(eta)), -XX]) * G0))
              }
              if (vectorDer) {
                return(cbind(G1, G0))
              }

              as.numeric(c(G1, G0) %*% X)
            },
            function (beta) {
              lambdaPredNumber <- attr(X, "hwm")[1]
              eta <- matrix(as.matrix(X) %*% beta, ncol = 2) + offset
              pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
              lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2

              res <- matrix(nrow = length(beta), ncol = length(beta),
                            dimnames = list(names(beta), names(beta)))

              # pi^2 derivative
              dpi <- (2 - y) / pi - (lambda + pi) ^ -1
              G00 <- ((lambda + pi) ^ (-2)) - (2 - y) / (pi ^ 2)

              G00 <- t(as.data.frame(X[-(1:(nrow(X) / 2)), -(1:lambdaPredNumber)] *
                                       (G00 * ((piLink(eta[, 2], inverse = TRUE, deriv = 1) / 2) ^ 2) +
                                          dpi * piLink(eta[, 2], inverse = TRUE, deriv = 2) / 2) * weight)) %*%
                as.matrix(X[-(1:(nrow(X) / 2)), -(1:lambdaPredNumber)])
              # mixed derivative
              G01 <- (lambda + pi) ^ (-2)

              G01 <- t(as.data.frame(X[1:(nrow(X) / 2), 1:lambdaPredNumber]) *
                         G01 * (lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) / 2) *
                         (piLink(eta[, 2], inverse = TRUE, deriv = 1) / 2) * weight) %*%
                as.matrix(X[-(1:(nrow(X) / 2)), -(1:lambdaPredNumber)])
              # lambda^2 derivative
              G11 <- ((lambda + pi) ^ (-2)) - (y - 1) / (lambda ^ 2)
              dlambda <- (y - 1) / lambda - ((lambda + pi) ^ -1)

              G11 <- t(as.data.frame(X[1:(nrow(X) / 2), 1:lambdaPredNumber] *
                                       (G11 * ((lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) / 2) ^ 2) +
                                          dlambda * lambdaLink(eta[, 1], inverse = TRUE, deriv = 2) / 2) * weight)) %*%
                X[1:(nrow(X) / 2), 1:lambdaPredNumber]

              res[-(1:lambdaPredNumber), -(1:lambdaPredNumber)] <- G00
              res[1:lambdaPredNumber, 1:lambdaPredNumber] <- G11
              res[1:lambdaPredNumber, -(1:lambdaPredNumber)] <- t(G01)
              res[-(1:lambdaPredNumber), 1:lambdaPredNumber] <- G01

              res
            }
    )
  }

  validmu <- function(mu) {
    (sum(!is.finite(mu)) == 0) && all(0 < mu) && all(2 > mu)
  }

  # this is optional
  devResids <- function(y, eta, wt, ...) {
    0
  }

  pointEst <- function (pw, eta, contr = FALSE, ...) {
    pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
    lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2
    N <- pw / (lambda + pi)
    if(!contr) {
      N <- sum(N)
    }
    N
  }

  popVar <- function (pw, eta, cov, Xvlm, ...) {
    pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
    lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2

    bigTheta1 <- -pw / (pi + lambda) ^ 2 # w.r to pi
    bigTheta1 <- bigTheta1 * piLink(eta[, 2], inverse = TRUE, deriv = 1) / 2
    bigTheta2 <- -pw / (pi + lambda) ^ 2 # w.r to lambda
    bigTheta2 <- bigTheta2 * lambdaLink(eta[, 1], inverse = TRUE, deriv = 1) / 2 # w.r to lambda

    bigTheta <- t(c(bigTheta2, bigTheta1) %*% Xvlm)

    f1 <- t(bigTheta) %*% as.matrix(cov) %*% bigTheta

    f2 <- sum(pw * (1 - pi - lambda) / ((pi + lambda) ^ 2))

    f1 + f2
  }

  dFun <- function (x, eta, type = c("trunc", "nontrunc")) {
    if (missing(type)) type <- "trunc"
    pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
    lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2

    switch (type,
            "trunc" = {
              (pi * as.numeric(x == 1) + lambda * as.numeric(x == 2)) / (pi + lambda)
            },
            "nontrunc" = {
              (1 - pi - lambda) * as.numeric(x == 0) +
                pi * as.numeric(x == 1) + lambda * as.numeric(x == 2)
            }
    )
  }

  simulate <- function(n, eta, lower = 0, upper = Inf) {
    pi     <-     piLink(eta[, 2], inverse = TRUE) / 2
    lambda <- lambdaLink(eta[, 1], inverse = TRUE) / 2
    CDF <- function(x) {
      ifelse(x == Inf, 1,
             ifelse(x < 0, 0,
                    ifelse(x < 1, 1 - pi - lambda,
                           ifelse(x < 2, 1 - lambda, 1))))
    }
    lb <- CDF(lower)
    ub <- CDF(upper)
    p_u <- stats::runif(n, lb, ub)
    sims <- rep(0, n)
    cond <- CDF(sims) <= p_u
    while (any(cond)) {
      sims[cond] <- sims[cond] + 1
      cond <- CDF(sims) <= p_u
    }
    sims
  }

  getStart <- expression(
    if (method == "IRLS") {
      etaStart <- cbind(
        family$links[[1]](mean(observed == 2) * (1 + 0 * (observed == 2))), # lambda
        family$links[[2]](mean(observed == 1) * (1 + 0 * (observed == 1)))  # pi
      ) + offset
    } else if (method == "optim") {
      init <- c(
        family$links[[1]](weighted.mean(observed == 2, priorWeights) * 1 + .0001),
        family$links[[2]](weighted.mean(observed == 1, priorWeights) * 1 + .0001)
      )
      if (attr(terms, "intercept")) {
        coefStart <- c(init[1], rep(0, attr(Xvlm, "hwm")[1] - 1))
      } else {
        coefStart <- rep(init[1] / attr(Xvlm, "hwm")[1], attr(Xvlm, "hwm")[1])
      }
      if ("(Intercept):pi" %in% colnames(Xvlm)) {
        coefStart <- c(coefStart, init[2], rep(0, attr(Xvlm, "hwm")[2] - 1))
      } else {
        coefStart <- c(coefStart, rep(init[2] / attr(Xvlm, "hwm")[2], attr(Xvlm, "hwm")[2]))
      }
    }
  )

  structure(
    list(
      makeMinusLogLike = minusLogLike,
      densityFunction  = dFun,
      links     = links,
      mu.eta    = mu.eta,
      valideta  = function (eta) {TRUE},
      variance  = variance,
      Wfun      = Wfun,
      funcZ     = funcZ,
      devResids = devResids,
      validmu   = validmu,
      pointEst  = pointEst,
      popVar    = popVar,
      family    = "myFamilyFunction",
      etaNames  = c("lambda", "pi"),
      simulate  = simulate,
      getStart  = getStart,
      extraInfo = c(
        mean       = "pi / 2 + lambda",
        variance   = paste0("(pi / 2) * (1 - pi / 2) + 2 * lambda * (1 - lambda / 2 - pi / 2)"),
        popSizeEst = "(1 - (pi + lambda) / 2) ^ -1",
        meanTr     = "1 + lambda / (pi + lambda)",
        varianceTr = paste0("lambda * (1 - lambda / 2) / (pi + lambda)")
      )
    ),
    class = c("singleRfamily", "family")
  )
}
```

A quick tests shows us that this implementation in fact works:

```{r}
set.seed(123)
Y <- simulate(
  myFamilyFunction(lambdaLink = "logit", piLink = "logit"),
  nsim = 1000, eta = matrix(0, nrow = 1000, ncol = 2),
  truncated = FALSE
)
mm <- estimatePopsize(
  formula = Y ~ 1,
  data = data.frame(Y = Y[Y > 0]),
  model = myFamilyFunction(lambdaLink = "logit",
                           piLink = "logit"),
  # the usual observed information matrix
  # is ill-suited for this distribution
  controlPopVar = controlPopVar(covType = "Fisher")
)
summary(mm)
```

where the link functions, such as
`singleRcapture:::singleRinternalcloglogLink`, are just internal
functions in `singleRcapture` that compute link functions, their
inverses and derivatives of both links and inverse links up to the third
order:

```{r single-link}
singleRcapture:::singleRinternalcloglogLink
```

One could, of course, include the code for computing them manually.

# References
