R: Single source capture-recapture models

estimatePopsize {singleRcapture}

R Documentation

Single source capture-recapture models

Description

estimatePopsize first fits appropriate (v)glm model and then estimates full (observed and unobserved) population size. In this types of models 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. Population size is then usually estimated by Horvitz-Thompson type estimator:

\[\hat{N} = \sum_{k=1}^{N}\frac{I_{k}}{\mathbb{P}(Y_{k}>0)} = \sum_{k=1}^{N_{obs}}\frac{1}{1-\mathbb{P}(Y_{k}=0)}\]

where \(I_{k}=I_{Y_{k} > 0}\) are indicator variables, with value 1 if kth unit was observed at least once and 0 otherwise.

Usage

estimatePopsize(formula, ...)

## Default S3 method:
estimatePopsize(
  formula,
  data,
  model = c("ztpoisson", "ztnegbin", "ztgeom", "zotpoisson", "ztoipoisson",
    "oiztpoisson", "ztHurdlepoisson", "Hurdleztpoisson", "zotnegbin", "ztoinegbin",
    "oiztnegbin", "ztHurdlenegbin", "Hurdleztnegbin", "zotgeom", "ztoigeom", "oiztgeom",
    "ztHurdlegeom", "ztHurdlegeom", "zelterman", "chao"),
  ratioReg = FALSE,
  weights = NULL,
  subset = NULL,
  naAction = NULL,
  method = c("optim", "IRLS"),
  popVar = c("analytic", "bootstrap", "noEst"),
  controlMethod = NULL,
  controlModel = NULL,
  controlPopVar = NULL,
  modelFrame = TRUE,
  x = FALSE,
  y = TRUE,
  contrasts = NULL,
  offset,
  ...
)

Arguments

`formula`	formula for the model to be fitted, only applied to the "main" linear predictor. Only single response models are available.
`...`	additional optional arguments passed to other methods eg. `estimatePopsizeFit`.
`data`	data frame or object coercible to data.frame class containing data for the regression and population size estimation.
`model`	model for regression and population estimate full description in `singleRmodels()`.
`ratioReg`	Not yet implemented
`weights`	optional object of prior weights used in fitting the model. Can be used to specify number of occurrences of rows in data see `controlModel()`
`subset`	a logical vector indicating which observations should be used in regression and population size estimation. It will be evaluated on `data` argument provided on call.
`naAction`	Not yet implemented.
`method`	method for fitting values currently supported: iteratively reweighted least squares (`IRLS`) and maximum likelihood (`optim`).
`popVar`	a method of constructing confidence interval either analytic or bootstrap. Bootstrap confidence interval type may be specified in `controlPopVar.` There is also the third possible value of `noEst` which skips the population size estimate all together.
`controlMethod`	a list indicating parameters to use in fitting the model may be constructed with `singleRcapture::controlMethod` function. More information included in `controlMethod()`.
`controlModel`	a list indicating additional formulas for regression (like formula for inflation parameter/dispersion parameter) may be constructed with `singleRcapture::controlModel` function. More information will eventually be included in `controlModel()`.
`controlPopVar`	a list indicating parameters to use in estimating variance of population size estimation may be constructed with `singleRcapture::controlPopVar` function. More information included in `controlPopVar()`.
`modelFrame`, `x`, `y`	logical value indicating whether to return model matrix, dependent vector and model matrix as a part of output.
`contrasts`	not yet implemented.
`offset`	a matrix of offset values with number of columns matching the number of distribution parameters providing offset values to each of linear predictors.

Details

The generalized linear model is characterized by equation \[\boldsymbol{\eta}=\boldsymbol{X}\boldsymbol{\beta}\] where \(\boldsymbol{X}\) is the (lm) model matrix. The vector generalized linear model is similarly characterized by equations \[\boldsymbol{\eta}_{k}=\boldsymbol{X}_{k}\boldsymbol{\beta}_{k}\] where \(\boldsymbol{X}_{k}\) is a (lm) model matrix constructed from appropriate formula (specified in controlModel parameter).

The \(\boldsymbol{\eta}\) is then a vector constructed as:

\[\boldsymbol{\eta}=\begin{pmatrix} \boldsymbol{\eta}_{1} \cr \boldsymbol{\eta}_{2} \cr \dotso \cr \boldsymbol{\eta}_{p} \end{pmatrix}^{T}\]

and in cases of models in our package the (vlm) model matrix is constructed as a block matrix:

\[\boldsymbol{X}_{vlm}= \begin{pmatrix} \boldsymbol{X}_{1} & \boldsymbol{0} &\dotso &\boldsymbol{0}\cr \boldsymbol{0} & \boldsymbol{X}_{2} &\dotso &\boldsymbol{0}\cr \vdots & \vdots & \ddots & \vdots\cr \boldsymbol{0} & \boldsymbol{0} &\dotso &\boldsymbol{X}_{p} \end{pmatrix}\]

this differs from convention in VGAM package (if we only consider our special cases of vglm models) but this is just a convention and does not affect the model, this convention is taken because it makes fitting with IRLS (explanation of algorithm in estimatePopsizeFit()) algorithm easier. (If constraints matrixes in vglm match the ones we implicitly use the vglm model matrix differs with respect to order of kronecker multiplication of X and constraints.) In this package we use observed likelihood to fit regression models.

As mentioned above usually the population size estimation is done via: \[\hat{N} = \sum_{k=1}^{N}\frac{I_{k}}{\mathbb{P}(Y_{k}>0)} = \sum_{k=1}^{N_{obs}}\frac{1}{1-\mathbb{P}(Y_{k}=0)}\]

where \(I_{k}=I_{Y_{k} > 0}\) are indicator variables, with value 1 if kth unit was observed at least once and 0 otherwise. The \(\mathbb{P}(Y_{k}>0)\) are estimated by maximum likelihood.

The following assumptions are usually present when using the method of estimation described above:

The specified regression model is correct. This entails linear relationship between independent variables and dependent ones and dependent variable being generated by appropriate distribution.
No unobserved heterogeneity. If this assumption is broken there are some possible (admittedly imperfect) workarounds see details in singleRmodels().
The population size is constant in relevant time frame.
Depending on confidence interval construction (asymptotic) normality of \(\hat{N}\) statistic is assumed.

There are two ways of estimating variance of estimate \(\hat{N}\), the first being "analytic" usually done by application of law of total variance to \(\hat{N}\):

\[\text{var}(\hat{N})=\mathbb{E}\left(\text{var} \left(\hat{N}|I_{1},\dots,I_{n}\right)\right)+ \text{var}\left(\mathbb{E}(\hat{N}|I_{1},\dots,I_{n})\right)\]

and then by \(\delta\) method to \(\hat{N}|I_{1},\dots I_{N}\):

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

and the \(\text{var}\left(\mathbb{E}(\hat{N}|I_{1},\dots,I_{n})\right)\) term may be derived analytically (if we assume independence of observations) since \(\hat{N}|I_{1},\dots,I_{n}\) is just a constant.

In general this gives us: \[ \begin{aligned} \text{var}\left(\mathbb{E}(\hat{N}|I_{1},\dots,I_{n})\right)&= \text{var}\left(\sum_{k=1}^{N}\frac{I_{k}}{\mathbb{P}(Y_{k}>0)}\right)\cr &=\sum_{k=1}^{N}\text{var}\left(\frac{I_{k}}{\mathbb{P}(Y_{k}>0)}\right)\cr &=\sum_{k=1}^{N}\frac{1}{\mathbb{P}(Y_{k}>0)^{2}}\text{var}(I_{k})\cr &=\sum_{k=1}^{N}\frac{1}{\mathbb{P}(Y_{k}>0)^{2}}\mathbb{P}(Y_{k}>0)(1-\mathbb{P}(Y_{k}>0))\cr &=\sum_{k=1}^{N}\frac{1}{\mathbb{P}(Y_{k}>0)}(1-\mathbb{P}(Y_{k}>0))\cr &\approx\sum_{k=1}^{N}\frac{I_{k}}{\mathbb{P}(Y_{k}>0)^{2}}(1-\mathbb{P}(Y_{k}>0))\cr &=\sum_{k=1}^{N_{obs}}\frac{1-\mathbb{P}(Y_{k}>0)}{\mathbb{P}(Y_{k}>0)^{2}} \end{aligned} \]

Where the approximation on 6th line appears because in 5th line we sum over all units, that includes unobserved units, since \(I_{k}\) are independent and \(I_{k}\sim b(\mathbb{P}(Y_{k}>0))\) the 6th line is an unbiased estimator of the 5th line.

The other method for estimating variance is "bootstrap", but since \(N_{obs}=\sum_{k=1}^{N}I_{k}\) is also a random variable bootstrap will not be as simple as just drawing \(N_{obs}\) units from data with replacement and just computing \(\hat{N}\).

Method described above is referred to in literature as "nonparametric" bootstrap (see controlPopVar()), due to ignoring variability in observed sample size it is likely to underestimate variance.

A more sophisticated bootstrap procedure may be described as follows:

Compute the probability distribution as: \[ \frac{\hat{\boldsymbol{f}}_{0}}{\hat{N}}, \frac{\boldsymbol{f}_{1}}{\hat{N}}, \dots, \frac{\boldsymbol{f}_{\max{y}}}{\hat{N}}\] where \(\boldsymbol{f}_{n}\) denotes observed marginal frequency of units being observed exactly n times.
Draw \(\hat{N}\) units from the distribution above (if \(\hat{N}\) is not an integer than draw \(\lfloor\hat{N}\rfloor + b(\hat{N}-\lfloor\hat{N}\rfloor)\)), where \(\lfloor\cdot\rfloor\) is the floor function.
Truncated units with \(y=0\).
If there are covariates draw them from original data with replacement from uniform distribution. For example if unit drawn to new data has \(y=2\) choose one of covariate vectors from original data that was associated with unit for which was observed 2 times.
Regress \(\boldsymbol{y}_{new}\) on \(\boldsymbol{X}_{vlm new}\) and obtain \(\hat{\boldsymbol{\beta}}_{new}\), with starting point \(\hat{\boldsymbol{\beta}}\) to make it slightly faster, use them to compute \(\hat{N}_{new}\).
Repeat 2-5 unit there are at least B statistics are obtained.
Compute confidence intervals based on alpha and confType specified in controlPopVar().

To do step 1 in procedure above it is convenient to first draw binary vector of length \(\lfloor\hat{N}\rfloor + b(\hat{N}-\lfloor\hat{N}\rfloor)\) with probability \(1-\frac{\hat{\boldsymbol{f}}_{0}}{\hat{N}}\), sum elements in that vector to determine the sample size and then draw sample of this size uniformly from the data.

This procedure is known in literature as "semiparametric" bootstrap it is necessary to assume that the have a correct estimate \(\hat{N}\) in order to use this type of bootstrap.

Lastly there is "paramteric" bootstrap where we assume that the probabilistic model used to obtain \(\hat{N}\) is correct the bootstrap procedure may then be described as:

Draw \(\lfloor\hat{N}\rfloor + b(\hat{N}-\lfloor\hat{N}\rfloor)\) covariate information vectors with replacement from data according to probability distribution that is proportional to: \(N_{k}\), where \(N_{k}\) is the contribution of kth unit i.e. \(\dfrac{1}{\mathbb{P}(Y_{k}>0)}\).
Determine \(\boldsymbol{\eta}\) matrix using estimate \(\hat{\boldsymbol{\beta}}\).
Generate \(\boldsymbol{y}\) (dependent variable) vector using \(\boldsymbol{\eta}\) and probability mass function associated with chosen model.
Truncated units with \(y=0\) and construct \(\boldsymbol{y}_{new}\) and \(\boldsymbol{X}_{vlm new}\).
Regress \(\boldsymbol{y}_{new}\) on \(\boldsymbol{X}_{vlm new}\) and obtain \(\hat{\boldsymbol{\beta}}_{new}\) use them to compute \(\hat{N}_{new}\).
Repeat 1-5 unit there are at least B statistics are obtained.
Compute confidence intervals based on alpha and confType specified in controlPopVar()

It is also worth noting that in the "analytic" method estimatePopsize only uses "standard" covariance matrix estimation. It is possible that improper covariance matrix estimate is the only part of estimation that has its assumptions violated. In such cases post-hoc procedures are implemented in this package to address this issue.

Lastly confidence intervals for \(\hat{N}\) are computed (in analytic case) either by assuming that it follows a normal distribution or that variable \(\ln(N-\hat{N})\) follows a normal distribution.

These estimates may be found using either summary.singleRStaticCountData method or popSizeEst.singleRStaticCountData function. They're labelled as normal and logNormal respectively.

Value

Returns an object of class c("singleRStaticCountData", "singleR", "glm", "lm") with type list containing:

y – Vector of dependent variable if specified at function call.
X – Model matrix if specified at function call.
formula – A list with formula provided on call and additional formulas specified in controlModel.
call – Call matching original input.
coefficients – A vector of fitted coefficients of regression.
control – A list of control parameters for controlMethod and controlModel, controlPopVar is included in populationSize.
model – Model which estimation of population size and regression was built, object of class family.
deviance – Deviance for the model.
priorWeights – Prior weight provided on call.
weights – If IRLS method of estimation was chosen weights returned by IRLS, otherwise same as priorWeights.
residuals – Vector of raw residuals.
logL – Logarithm likelihood obtained at final iteration.
iter – Numbers of iterations performed in fitting or if stats::optim was used number of call to loglikelihood function.
dfResiduals – Residual degrees of freedom.
dfNull – Null degrees of freedom.
fittValues – Data frame of fitted values for both mu (the expected value) and lambda (Poisson parameter).
populationSize – A list containing information of population size estimate.
modelFrame – Model frame if specified at call.
linearPredictors – Vector of fitted linear predictors.
sizeObserved – Number of observations in original model frame.
terms – terms attribute of model frame used.
contrasts – contrasts specified in function call.
naAction – naAction used.
which – list indicating which observations were used in regression/population size estimation.
fittingLog – log of fitting information for "IRLS" fitting if specified in controlMethod.

Author(s)

Piotr Chlebicki, Maciej Beręsewicz

References

General single source capture recapture literature:

Zelterman, Daniel (1988). ‘Robust estimation in truncated discrete distributions with application to capture-recapture experiments’. In: Journal of statistical planning and inference 18.2, pp. 225–237.

Heijden, Peter GM van der et al. (2003). ‘Point and interval estimation of the population size using the truncated Poisson regression model’. In: Statistical Modelling 3.4, pp. 305–322. doi: 10.1191/1471082X03st057oa.

Cruyff, Maarten J. L. F. and Peter G. M. van der Heijden (2008). ‘Point and Interval Estimation of the Population Size Using a Zero-Truncated Negative Binomial Regression Model’. In: Biometrical Journal 50.6, pp. 1035–1050. doi: 10.1002/bimj.200810455

Böhning, Dankmar and Peter G. M. van der Heijden (2009). ‘A covariate adjustment for zero-truncated approaches to estimating the size of hidden and elusive populations’. In: The Annals of Applied Statistics 3.2, pp. 595–610. doi: 10.1214/08-AOAS214.

Böhning, Dankmar, Alberto Vidal-Diez et al. (2013). ‘A Generalization of Chao’s Estimator for Covariate Information’. In: Biometrics 69.4, pp. 1033– 1042. doi: 10.1111/biom.12082

Böhning, Dankmar and Peter G. M. van der Heijden (2019). ‘The identity of the zero-truncated, one-inflated likelihood and the zero-one-truncated likelihood for general count densities with an application to drink-driving in Britain’. In: The Annals of Applied Statistics 13.2, pp. 1198–1211. doi: 10.1214/18-AOAS1232.

Navaratna WC, Del Rio Vilas VJ, Böhning D. Extending Zelterman's approach for robust estimation of population size to zero-truncated clustered Data. Biom J. 2008 Aug;50(4):584-96. doi: 10.1002/bimj.200710441.

Böhning D. On the equivalence of one-inflated zero-truncated and zero-truncated one-inflated count data likelihoods. Biom J. 2022 Aug 15. doi: 10.1002/bimj.202100343.

Böhning, D., Friedl, H. Population size estimation based upon zero-truncated, one-inflated and sparse count data. Stat Methods Appl 30, 1197–1217 (2021). doi: 10.1007/s10260-021-00556-8

Bootstrap:

Zwane, PGM EN and Van der Heijden, Implementing the parametric bootstrap in capture-recapture models with continuous covariates 2003 Statistics & probability letters 65.2 pp 121-125

Norris, James L and Pollock, Kenneth H Including model uncertainty in estimating variances in multiple capture studies 1996 in Environmental and Ecological Statistics 3.3 pp 235-244

Vector generalized linear models:

Yee, T. W. (2015). Vector Generalized Linear and Additive Models: With an Implementation in R. New York, USA: Springer. ISBN 978-1-4939-2817-0.

Examples


# Model from 2003 publication 
# Point and interval estimation of the
# population size using the truncated Poisson regression mode
# Heijden, Peter GM van der et al. (2003)
model <- estimatePopsize(
   formula = capture ~ gender + age + nation, 
   data = netherlandsimmigrant, 
   model = ztpoisson
)
summary(model)
# Graphical presentation of model fit
plot(model, "rootogram")
# Statistical test
# see documentation for summary.singleRmargin
summary(marginalFreq(model), df = 1, "group")

# We currently support 2 methods of numerical fitting
# (generalized) IRLS algorithm and via stats::optim
# the latter one is faster when fitting negative binomial models 
# (and only then) due to IRLS having to numerically compute
# (expected) information matrixes, optim is also less reliable when
# using alphaFormula argument in controlModel
modelNegBin <- estimatePopsize(
    formula = TOTAL_SUB ~ ., 
    data = farmsubmission, 
    model = ztnegbin, 
    method = "optim"
)
summary(modelNegBin)
summary(marginalFreq(modelNegBin))

# More advanced call that specifies additional formula and shows
# in depth information about fitting procedure
pseudoHurdleModel <- estimatePopsize(
    formula = capture ~ nation + age,
    data = netherlandsimmigrant,
    model = Hurdleztgeom,
    method = "IRLS",
    controlMethod = controlMethod(verbose = 5),
    controlModel = controlModel(piFormula = ~ gender)
)
summary(pseudoHurdleModel)
# Assessing model fit
plot(pseudoHurdleModel, "rootogram")
summary(marginalFreq(pseudoHurdleModel), "group", df = 1)


# A advanced input with additional information for fitting procedure and
# additional formula specification and different link for inflation parameter.
Model <- estimatePopsize(
 formula = TOTAL_SUB ~ ., 
 data = farmsubmission, 
 model = oiztgeom(omegaLink = "cloglog"), 
 method = "IRLS", 
 controlMethod = controlMethod(
   stepsize = .85, 
   momentumFactor = 1.2,
   epsilon = 1e-10, 
   silent = TRUE
 ),
 controlModel = controlModel(omegaFormula = ~ C_TYPE + log_size)
)
summary(marginalFreq(Model), df = 18 - length(Model$coefficients))
summary(Model)

[Package singleRcapture version 0.2.1.2 Index]