Combining link-tracing sampling and cluster sampling to estimate the size of a hidden population in presence of heterogeneous link-probabilities 4. Confidence intervalsCombining link-tracing sampling and cluster sampling to estimate the size of a hidden population in presence of heterogeneous link-probabilities 4. Confidence intervals

We will consider two types of confidence intervals (CIs) for the population sizes: profile likelihood and bootstrap CIs.

4.1 Profile likelihood confidence intervals

Several authors such as Cormack (1992), Evans, Kim and O’Brien (1996), Coull and Agresti (1999) and Gimenes, Choquet, Lamor, Scofield, Fletcher, Lebreton and Pradel (2005) have indicated that, in the context of capture-recapture sampling, profile likelihood confidence intervals (PLCIs) perform better than traditional Wald CIs when the sample size is not large. Some factors that affect the performance of Wald CIs are biases in the estimators of the population size, biases in the estimators of the variances and asymmetries in the distributions of the estimators of the population size. Besides, a Wald CI for the population size might present the drawback that its lower bound might be less than the number of captured elements. Notice that, with the exception of the first listed factor, none of the others affect the performance of PLCIs. Furthermore, Evans et al. (1996), based on Ratkowsky (1988), indicate that the nonlinear nature of the capture-recapture estimators is approximated by likelihood-based CIs better than by Wald CIs.

Since the proposed estimators resemble those used in capture-recapture sampling and based on the previous comments, we should expect that also in our case PLCIs performance better than Wald CIs. It is worth noting that if we wanted to use Wald CIs, we would need to compute estimators of the variances of the proposed estimators. One alternative is to construct estimators of their asymptotic variances by using Sanathanan’s (1972) results; however, for $n$ large, say 20 or greater, obtaining these type of estimator is computationally very expensive because for each estimator is required the construction of a $(n + 1) \times (n + 1)$ symmetric matrix whose elements are sums of $2^{n}$ terms.

To get PLCIs for $τ_{1}, τ_{2}$ and $τ$ we will follow Coull and Agresti’s (1999) approach. Thus, for fixed values $τ_{1}, τ_{2}$ and $τ$ of the population sizes, let $r_{10}, r_{20}$ and $r_{00}$ be non-negative real numbers such that $τ_{1} = m + r_{1} + r_{10}, τ_{2} = r_{2} + r_{20}$ and $τ = m + r_{1} + r_{2} + r_{00},$ where $m, r_{1}$ and $r_{2}$ are the observed values of the random variables $M, R_{1}$ and $R_{2} .$ Then $100 (1 - α) %$ PLCIs for $τ_{1}, τ_{2}$ and $τ$ are defined as the following sets: ${τ_{1} = m + r_{1} + r_{10} : - 2 \ln [Λ_{1} (r_{10})] \leq χ_{1,1 - α}^{2}},$ ${τ_{2} = r_{2} + r_{20} : - 2 \ln [Λ_{2} (r_{20})] \leq χ_{1,1 - α}^{2}}$ and ${τ = m + r_{1} + r_{2} + r_{00} : - 2 \ln [Λ (r_{00})] \leq χ_{1,1 - α}^{2}},$ respectively, where

$\begin{array}{l} Λ_{1} (r_{10}) & = & \max_{α_{1}, σ_{1}} L_{(1)} (m + r_{1} + r_{10}, α_{1}, σ_{1}) / L_{(1)} ({\hat{τ}}_{1}, {\hat{α}}_{1}, {\hat{σ}}_{1}), \\ Λ_{2} (r_{20}) & = & \max_{α_{2}, σ_{2}} L_{(2)} (r_{2} + r_{20}, α_{2}, σ_{2}) / L_{(2)} ({\hat{τ}}_{2}, {\hat{α}}_{2}, {\hat{σ}}_{2}) \end{array}$

and

$\begin{array}{l} Λ (r_{00}) & = & \max_{r_{10}, α_{1}, α_{2}, σ_{1}, σ_{2}} L (m + r_{1} + r_{10}, r_{2} + r_{00} - r_{10}, α_{1}, α_{2}, σ_{1}, σ_{2}) / L ({\hat{τ}}_{1}, {\hat{τ}}_{2}, {\hat{α}}_{1}, {\hat{α}}_{2}, {\hat{σ}}_{1}, {\hat{σ}}_{2}), \end{array}$

${\hat{τ}}_{k}, {\hat{α}}_{k}$ and ${\hat{σ}}_{k}$ are either the UMLEs or CMLEs of $τ_{k}, α_{k}$ and $σ_{k}, k = 1, 2,$ and $χ_{1,1 - α}^{2}$ is the $100 {(1 - α)}^{th}$ quantile of the chi-square distribution with 1 degree of freedom.

Although PLCIs for $τ_{2}$ are not affected by a possible extra-Poisson variation of the $M_{i} ’ s$ [or strictly speaking extra-multinomial dispersion of $(M_{1}, \dots, M_{n})]$ because they are obtained from the likelihood function $L_{(2)} (τ_{2}, α_{2}, σ_{2})$ which does not depend on these variables, we do not expect that the PLCIs for $τ_{1}$ and $τ$ be robust to extra-Poisson variation of the $M_{i} ’ s;$ therefore we will consider adjusted PLCIs for $τ_{1}$ and $τ$ that take into account this extra variation. Following the suggestion of Gimenes et al. (2005), the adjusted PLCIs are constructed as the previous ones but replacing the value $χ_{1,1 - α}^{2}$ by the value $(s_{M}^{2} / \bar{m}) F_{1, n - 1,1 - α},$ where $\bar{m} = m / n$ and $s_{M}^{2} = \sum_{1}^{n} {(m_{i} - \bar{m})}^{2} / (n - 1)$ are the sample mean and variance of the $m_{i} ’ s,$ and $F_{1, n - 1,1 - α}$ is the $100 {(1 - α)}^{th}$ quantile of the $F$ distribution with 1 and $n - 1$ degrees of freedom. Observe that $s_{M}^{2} / \bar{m}$ is obtained by dividing by $n - 1$ the value of the Pearson chi-square test statistic to test the hypothesis that the conditional distribution of the observed $M_{i} ’ s,$ given that $\sum_{1}^{n} M_{i} = m,$ is multinomial with parameter of size $m$ and vector of probabilities $(1 / n, \dots, 1 / n) .$ The adjusted PLCIs should be used if the null hypothesis of the conditional multinomial distribution of the $M_{i} ’ s$ were rejected at the $100 α %$ level of significance, that is, if $s_{M}^{2} / \bar{m} > χ_{n - 1,1 - α}^{2} / (n - 1),$ otherwise the unadjusted PLCIs should be used.

It is worth noting that the calculation of PLCIs is a computationally expensive task; therefore, efficient numerical algorithms need to be used, such as the one proposed by Venzon and Moolgavkar (1988).

4.2 Bootstrap confidence intervals

We will present a variant of bootstrap to construct CIs for the population sizes $τ_{1}, τ_{2}$ and $τ$ based on either the UMLEs or the CMLEs. The proposed variant is obtained by combining the bootstrap version for finite populations proposed by Booth, Butler and Hall (1994) and the parametric bootstrap variant (see Davison and Hinkley 1997, Chapter 2). This version of bootstrap is an extension of the one used by Félix-Medina and Monjardin (2006) in the case of homogeneous link-probabilities.

Since our proposed version of bootstrap is a parametric variant, we need to have estimates of all the parameters associated with the assumed models. Until now, the only parameters that have not yet been estimated are the random effects $β_{j}^{(k)} ’ s .$ We will now derive a predictor of $β_{j}^{(k)} .$ Thus, given the subset $U_{k} - S_{0}$ or $A_{i} \in S_{A}$ that contains the element $j,$ the conditional joint pdf of $X_{j}^{(k)}$ and $β_{j}^{(k)}$ is

$\begin{array}{l} f (x_{j}^{(k)}, β_{j}^{(k)} | j \in U_{k} - S_{0}, S_{A}) & = & \Pr (X_{j}^{(k)} = x_{j}^{(k)} | β_{j}^{(k)}, j \in U_{k} - S_{0}, S_{A}) f (β_{j}^{(k)}) \\ \propto & \prod_{i = 1}^{n} {[p_{i j}^{(k)}]}^{x_{i j}^{(k)}} {[1 - p_{i j}^{(k)}]}^{1 - x_{i j}^{(k)}} \exp [- {(β_{j}^{(k)})}^{2} / 2 σ_{k}^{2}] \\ if j \in U_{k} - S_{0}, k = 1, 2, \end{array}$

$\begin{array}{l} f (x_{j}^{(1)}, β_{j}^{(1)} | j \in A_{i^{'}} \in S_{A}, S_{A}) & \propto & \prod_{i \neq i^{'}}^{n} {[p_{i j}^{(1)}]}^{x_{i j}^{(1)}} {[1 - p_{i j}^{(1)}]}^{1 - x_{i j}^{(1)}} \exp [- {(β_{j}^{(1)})}^{2} / 2 σ_{1}^{2}] \\ if j \in A_{i^{'}} \in S_{A}, i^{'} = 1, \dots, n . \end{array}$

We will use as a prediction or estimate of $β_{j}^{(k)}$ the value ${\hat{β}}_{j}^{(k)}$ that maximizes the conditional joint pdf of $X_{j}^{(k)}$ and $β_{j}^{(k)}$ with the parameters $α_{k}$ and $σ_{k}$ set at either their UMLEs or their CMLEs. This procedure yields that ${\hat{β}}_{j}^{(k)}$ is given as the solution to the following equation:

$\begin{array}{l} \sum_{i = 1}^{n} x_{i j}^{(k)} - \sum_{i = 1}^{n} \frac{\exp [{\hat{α}}_{i}^{(k)} + β_{j}^{(k)}]}{1 + \exp [{\hat{α}}_{i}^{(k)} + β_{j}^{(k)}]} - \frac{1}{{\hat{σ}}_{k}^{2}} β_{j}^{(k)} = 0 & if j \in U_{k} - S_{0}, k = 1, 2, \end{array}$

$\begin{array}{l} \sum_{i \neq i^{'}}^{n} x_{i j}^{(1)} - \sum_{i \neq i^{'}}^{n} \frac{\exp [{\hat{α}}_{i}^{(1)} + β_{j}^{(1)}]}{1 + \exp [{\hat{α}}_{i}^{(1)} + β_{j}^{(1)}]} - \frac{1}{{\hat{σ}}_{1}^{2}} β_{j}^{(1)} = 0 & if j \in A_{i^{'}} \in S_{A}, i^{'} = 1, \dots, n, \end{array}$

where ${\hat{α}}_{i}^{(k)}$ and ${\hat{σ}}_{k}$ denote either the UMLEs or the CMLEs of $α_{i}^{(k)}$ and $σ_{k}, i = 1, \dots, n; k = 1, 2.$ Note that this equation implies that the predictor ${\hat{β}}_{j}^{(k)}$ of $β_{j}^{(k)}$ depends on the number of clusters that are linked to the element $j,$ but not on the particular clusters to which that element is linked. Thus, if two persons $j$ and $j^{'}$ in $U_{k} - S_{0}$ are linked to the same number of clusters in $S_{A},$ the predictors ${\hat{β}}_{j}^{(k)}$ and ${\hat{β}}_{j^{'}}^{(k)}$ are equal one another. The same happens for two persons in $A_{i} \in S_{A} .$

Hereinafter, we will denote by $[{\hat{τ}}_{k}],$ the nearest integer to ${\hat{τ}}_{k},$ where ${\hat{τ}}_{k}$ denotes either the UMLE or the CMLE of $τ_{k}, k = 1, 2.$ The steps of the proposed bootstrap procedure are the following. (i) Construct a population vector $m_{Boot}$ of $N$ values of $m_{i} ’ s$ by repeating $N / n$ times, assuming that $N / n$ is an integer, the observed sample of $n$ cluster sizes $m_{s} = {m_{1}, \dots, m_{n}} .$ If $N / n$ is not an integer, that is, if $N = a n + b,$ where $a$ and $b, b < n,$ are positive integers, then repeat $a$ times $m_{s}$ and add to this set a SRSWOR of $b$ values of $m_{i} ’ s$ selected from $m_{s} .$ (ii) For each $k = 1, 2,$ construct a population vector ${\hat{α}}_{Boot}^{(k)}$ of dimension $N$ whose elements are the estimates ${\hat{α}}_{i}^{(k)} ’ s$ of the $α_{i}^{(k)} ’ s$ associated with the clusters whose sizes $m_{i} ’ s$ are in $m_{Boot} .$ (iii) Construct a population vector ${\hat{β}}_{Boot}^{(0)}$ whose elements are the estimates ${\hat{β}}_{j}^{(1)} ’ s$ of the $β_{j}^{(1)} ’ s$ associated with the people who belong to the clusters whose sizes $m_{i} ’ s$ are in $m_{Boot} .$ Observe that the dimension of this vector is not necessarily $[{\hat{τ}}_{1}],$ but it equals the sum of the $m_{i} ’ s$ in $m_{Boot} .$ $(iv)$ Construct a population vector ${\hat{β}}_{Boot}^{(1)}$ of dimension $[{\hat{τ}}_{1}]$ whose first $m$ elements are the estimates ${\hat{β}}_{j}^{(1)} ’ s$ of the $β_{j}^{(1)} ’ s$ associated with the people in $S_{0};$ the remaining $[{\hat{τ}}_{1}] - m$ elements are the $r_{1}$ estimates ${\hat{β}}_{j}^{(1)} ’ s$ of the $β_{j}^{(1)} ’ s$ associated with the people in $S_{1}$ and the $[{\hat{τ}}_{1}] - m - r_{1}$ estimates ${\hat{β}}_{j}^{(1)} ’ s$ of the $β_{j}^{(1)} ’ s$ associated with the non sampled people in $U_{1} .$ These $[{\hat{τ}}_{1}] - m$ elements ${\hat{β}}_{j}^{(1)} ’ s$ are randomly placed after the first $m$ elements ${\hat{β}}_{j}^{(1)}$ of ${\hat{β}}_{Boot}^{(1)} .$ $(v)$ Construct a population vector ${\hat{β}}_{Boot}^{(2)}$ of dimension $[{\hat{τ}}_{2}]$ whose first $r_{2}$ elements are the estimates ${\hat{β}}_{j}^{(2)} ’ s$ of the $β_{j}^{(2)} ’ s$ associated with the people in $S_{2}$ and the remaining $[{\hat{τ}}_{2}] - r_{2}$ elements are the estimates ${\hat{β}}_{j}^{(2)} ’ s$ of the $β_{j}^{(2)} ’ s$ associated with the non sampled people in $U_{2} .$ (vi) Select a SRSWOR of $n$ values $m_{i}$ from $m_{Boot} .$ Let $S_{A}^{Boot} = {i_{1}, \dots, i_{n}}$ be the set of indices of the $m_{i} ’ s$ in the sample. In addition, let $A_{i}^{Boot} = (\sum_{t = 1}^{i - 1} m_{t}, \sum_{t = 1}^{i} m_{t}) \cap ℤ$ be the set of indices $j$ associated with the elements in the cluster whose index is $i \in S_{A}^{Boot},$ where $m_{t}$ is the $t^{th}$ element of $m_{Boot}$ and $ℤ$ is the set of the integer numbers. Finally, let $S_{0}^{Boot} = \cup_{i \in S_{A}^{Boot}} A_{i}^{Boot} .$ (vii) For each $i \in S_{A}^{Boot}$ and $j \in {1, \dots, [{\hat{τ}}_{2}]}$ generate a value $x_{i j}^{(2)}$ by sampling from the Bernoulli distribution with mean ${\hat{p}}_{i j}^{(2)}$ given by (3.2), but replacing $α_{i}^{(2)}$ and $β_{j}^{(2)}$ by their estimates ${\hat{α}}_{i}^{(2)}$ and ${\hat{β}}_{j}^{(2)} .$ Similarly, for each $i \in S_{A}^{Boot}$ and $j \in {1, \dots, [{\hat{τ}}_{1}]} - A_{i}^{Boot}$ generate a value $x_{i j}^{(1)}$ by sampling from the Bernoulli distribution with mean ${\hat{p}}_{i j}^{(1)},$ where the value of ${\hat{β}}_{j}^{(1)}$ that is used to compute ${\hat{p}}_{i j}^{(1)}$ is obtained from ${\hat{β}}_{Boot}^{(0)}$ if $j \in S_{0}^{Boot},$ and from ${\hat{β}}_{Boot}^{(1)}$ otherwise. (viii) Compute the estimates of $τ_{1}, τ_{2}$ and $τ$ using the same procedure as that used to compute the original estimates ${\hat{τ}}_{1}, {\hat{τ}}_{2}$ and $\hat{τ} .$ $(ix)$ Repeat the steps $(vi) - (viii)$ a large enough number $B$ of times. Let ${\hat{τ}}_{1, b}^{Boot}, {\hat{τ}}_{2, b}^{Boot}$ and ${\hat{τ}}_{b}^{Boot}$ be the estimates obtained in the $b^{th}$ bootstrap sample, $b = 1, \dots, B .$

The final step of our proposed bootstrap variant consists in constructing the CIs for the population sizes. There exist several alternatives to do this. One is to construct them without assuming any distributions for the estimators ${\hat{τ}}_{1}, {\hat{τ}}_{2}$ and $\hat{τ} .$ As examples of this alternative are the basic and the percentile method. (See Davison and Hinkley 1997, Chapter 5, for descriptions of these methods.) In the basic method a $100 (1 - α) %$ CI for $τ$ is $[2 \hat{τ} - {\hat{τ}}_{1 - α / 2}^{Boot},2 \hat{τ} - {\hat{τ}}_{α / 2}^{Boot}],$ and in the percentile method the CI is $[{\hat{τ}}_{α / 2}^{Boot}, {\hat{τ}}_{1 - α / 2}^{Boot}],$ where ${\hat{τ}}_{α / 2}^{Boot}$ and ${\hat{τ}}_{1 - α / 2}^{Boot}$ are the lower and upper $α / 2$ points of the empirical distribution obtained from ${\hat{τ}}_{b}^{Boot}, b = 1, \dots, B .$ Although this type of alternative has good properties of robustness, it requires a large number $B$ of bootstrap samples, say $B = 1,000,$ and this might be a serious problem if $\hat{τ}$ is costly to compute.

Another alternative to construct CIs is to assume a distribution for $\hat{τ}$ and use the bootstrap sample to estimate the parameters of that distribution. In this case the number $B$ of required bootstrap samples is not so large, say $50 \leq B \leq 200$ is generally enough. Examples of this alternative are the assumption that $\hat{τ}$ is normally distributed and the one that $\hat{τ} - ν$ is lognormally distributed, where $ν$ is the number of sampled elements. In the first case a $100 (1 - α) %$ CI for $τ$ is the well known Wald CI given by $\hat{τ} \pm z_{α / 2} \sqrt{\hat{V} (\hat{τ})},$ whereas in the second case the CI is $[ν + (\hat{τ} - ν) / c, ν + (\hat{τ} - ν) \times c],$ where $c = \exp {z_{α / 2} \sqrt{\ln [1 + \hat{V} (\hat{τ}) / {(\hat{τ} - ν)}^{2}]}},$ $z_{α / 2}$ is the upper $α / 2$ point of the standard normal distribution and $\hat{V} (\hat{τ})$ is an estimate of the variance of $\hat{τ} .$ (See Williams, Nichols and Conroy 2002, Section 14.2, for a description of this type of CI.) It is worth noting that in the lognormal based CIs for $τ_{1}, τ_{2}$ and $τ$ the values of $ν$ are $m + r_{1}, r_{2}$ and $m + r_{1} + r_{2},$ respectively.

An estimator $\hat{V} (\hat{τ})$ of the variance of $\hat{τ}$ could be computed using the sample variance of the bootstrap sample ${\hat{τ}}_{b}^{Boot}, b = 1, \dots, B .$ However, this estimator is not robust to extreme values of ${\hat{τ}}_{b}^{Boot},$ which are likely to occur with the proposed estimators when the sampling rates are not large enough. Therefore, to use a robust estimator of $V (\hat{τ})$ is a better strategy. One possibility is to use Huber’s proposal 2 to jointly estimate the parameters of location and scale from the bootstrap sample. (See Staudte and Sheather 1990, Section 4.5, for a description of this method.) In particular, the estimate of the parameter of scale is an estimate of the standard deviation $\sqrt{\hat{V} (\hat{τ})}$ of $\hat{τ} .$

Editorial policy

Survey Methodology publishes articles dealing with various aspects of statistical development relevant to a statistical agency, such as design issues in the context of practical constraints, use of different data sources and collection techniques, total survey error, survey evaluation, research in survey methodology, time series analysis, seasonal adjustment, demographic studies, data integration, estimation and data analysis methods, and general survey systems development. The emphasis is placed on the development and evaluation of specific methodologies as applied to data collection or the data themselves. All papers will be refereed. However, the authors retain full responsibility for the contents of their papers and opinions expressed are not necessarily those of the Editorial Board or of Statistics Canada.

Submission of Manuscripts

Survey Methodology is published twice a year in electronic format. Authors are invited to submit their articles in English or French in electronic form, preferably in Word to the Editor, (statcan.smj-rte.statcan@canada.ca, Statistics Canada, 150 Tunney’s Pasture Driveway, Ottawa, Ontario, Canada, K1A 0T6). For formatting instructions, please see the guidelines provided in the journal and on the web site (www.statcan.gc.ca/SurveyMethodology).

Note of appreciation

Canada owes the success of its statistical system to a long-standing partnership between Statistics Canada, the citizens of Canada, its businesses, governments and other institutions. Accurate and timely statistical information could not be produced without their continued co-operation and goodwill.

Standards of service to the public

Statistics Canada is committed to serving its clients in a prompt, reliable and courteous manner. To this end, the Agency has developed standards of service which its employees observe in serving its clients.

Copyright

Published by authority of the Minister responsible for Statistics Canada.

Catalogue no. 12-001-X

Frequency: semi-annual

Ottawa

Date modified:: 2017-09-20

Language selection

Search and menus

Search

4.1 Profile likelihood confidence intervals

4.2 Bootstrap confidence intervals