An Efficient Poisson-Distributed Adaptive Cluster Sampling Model Using Randomized Response Strategy

The new model is a hybrid of randomised response model and adaptive two-stage cluster sampling, to estimate rare sensitive attributes. We provide estimation method for the mean of the population with a rare sensitive attribute (MNSA) using modified Horvitz-Thompson type estimator. We investigate the known and unknown conditions of a unique non-sensitive trait that is unrelated. In the second stage of sampling, we collect responses from the elementary units using the randomization method suggested by Singh and Tarray [24], as well as the recent work [25, 26].

2.1 When the rare attribute that is unrelated is known

Participants are requested to engage with and encounter the randomization device without prior knowledge of whether the distinctive characteristic will be revealed or concealed, contingent upon the identification of the proportion of individuals possessing the unrelated rare attribute.

The respondents selected from the i^th cluster would be given a deck of k_i cards as the randomization mechanism and card picked has one of the questions to determine if the respondent has (i) rare sensitive attribute be “F” with probability P_1i, (ii) rare unrelated attribute "U" with probability P_2i, and (iii) Blank Card. 

The respondent must repeat the preceding steps without taking the card out if the statement (iii) is drawn. When the statement (iii) is drawn, he or she is compelled to report "No" at the second level. The suggested deck's total number of cards and the likelihood that "Yes" will be answered are both provided.

$\phi_{i 0}=\left[P_{1 i} \beta_{i f}+P_{2 i} \beta_{i u}\right]+P_{2 i}\left(1-\beta_{i f}\right)$            (6)

Let's assume that, $m_i \rightarrow \infty, m_i \phi_{i 0}=v_{i 0}>0$, as $\phi_{i 0} \rightarrow$ $0, m_i \beta_{i f}=v_{i f}>0, a s \beta_{i f} \rightarrow 0$, and $m_i \beta_{i u}=v_{i u}>0 a s \rightarrow$ $\infty, \beta_{i u} \rightarrow 0$,

where,

$v_{i 0}=\left[P_{1 i} v_{i f}+P_{2 i} v_{i u}\right]+P_{2 i}\left(1-v_{i f}\right)$           (7)

$\begin{gathered} f\left(v_{i 0}\right)=\prod_{j=1}^{m_i} \frac{e^{-v_{i 0}} v_{i 0}{ }^{y_{i j}}}{y_{i j}!}, \\\text{Let}\, Y \sim Pois \left(v_{i 0}\right) \,\text{a random sample of}\,\, m_i \,\,\text{from the}\,\, i^{{th }} \text{cluster}. \\L\left(v_{i 0}\right)=e^{-m_i v_{i o} }v_{i 0}{ }^{\sum_{j=1}^{m_i} y i j} \prod_{j=1}^{m_i} \frac{1}{y i j !},\end{gathered}$               (8)

where, Eq. (8) is the likelihood function.

From Eq. (7) and Eq. (8), the MLE $v_{i f}$

$\hat{v}_{i f}=\frac{1}{P_{1 i}}\left[\frac{\sum_{j=1}^{m_i} y_{i j}}{P_{2 i}\left(1-v_{i u}\right)}-P_{2 i} v_{i u}\right]$           (9)

To show that $\hat{v}_{i f}$ is unbiased parameter $v_{i f}$, following Eq. (2) check [7, 25].

Theorem 1: The estimator $\hat{v}_{i f}$ is an unbiased estimator of the parameter $v_{i f}$.

Proof: following Eq. (2) [24]. 

If for any cluster, w_k≠1, and indicator variable (IV)=0, it implies that the k^thcluster was not chosen in the initial because it falls short of the requirement. So, j_k=1. A the two-stage adaptive sampling strategy, a improved type of Horvitz-Thompson estimator of MNSA in the population is represented as:

$\widehat{v}_{H T}=\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}$             (10)

The estimator $\widehat{v}_{H T}$ is computed as a weighted sum of the observed values, where the weights are inversely proportional to the inclusion probabilities, ensuring unbiased estimation of the population total. 

Theorem 2: The estimator $\widehat{v}_{H T}$ of the MNSA $v_f$ is unbiased.

Proof: consider

$E_1 E_2\left(\hat{v}_{H T}\right)=E_1 E_2\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}\right)$

where, E₁ is the anticipated total number of first-round picks and E₂ is the anticipated total number of second-round picks.

$E_1 E_2\left(\hat{v}_{H T}\right)=E_1\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}\right)$     (11)

See Mansour (2021).

$E_1 E_2\left(\hat{v}_{H T}\right)=\left(\frac{1}{N} \sum_{k=1}^\tau v_{i f}\right)=v_f$     (12)

Theorem 3: The variance of the estimator $\widehat{v}_{f H T}$ is:

$\begin{gathered}V\left(\hat{v}_{H T}\right)=\frac{1}{N^2}\left[\sum_{k=1}^K \sum_{b=1}^K v_{k f^*} v_{b f^*}\left(v_{k b}-v_k v_b\right) /\left(v_k v_b v_{k b}\right)+\right. \left.\sum_{k=1}^N m_k \frac{\psi_k}{\alpha_k}\right]\end{gathered}$, 

where,

$\psi_{r k}=\left[\frac{P_{1 k} v_{k f}+P_{2 k} v_{k u}+P_{2 k}\left(1-v_{k u}\right)}{P_{1 k}^2 v_{k f}-P_{2 k}^2 v_{k u}}\right]$    (13)

Proof. The detailed proof can be found in the study by Singh and Tarray [24]. The abridged proof is as follows: 

${Var}\left(\hat{v}_{H T}\right)={Var}_1 E_2\left(\hat{v}_{H T}\right)+E_1 {Var}_2\left(\hat{v}_{H T}\right)$     (14)

where, var₁- variance of the adaptive, and var₂- variance over the second stage

The first term of (14) is:

$\begin{gathered}{Var}_1 E_2\left(\widehat{v}_{H T}\right)={Var}_1 E_2\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}\right)= {Var}_1\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}\right)\end{gathered}$     (15)

Thus,

$\begin{gathered}{Var}_1\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}\right)= \frac{1}{N^2} \sum_{j-1}^{\xi} \sum_{h=1}^{\xi} v_{j f *} v_{h f *} {cov}\left(v_j, v_h\right) /\left(v_j v_h\right) \\ {Var}_1\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}\right)= \frac{1}{N^2} \sum_{j-1}^{\xi} \sum_{h=1}^{\xi} v_{j f *} v_{h f *}\left(v_j, v_h\right) /\left(v_j v_h\right)\end{gathered}$     (16)

Variance of an unbiased estimator:

$\begin{gathered}{Var}_1\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}\right)=\frac{1}{N^2} \sum_{j-1}^{\xi} \sum_{h=1}^{\xi} v_{j f *} v_{h f *}\left(v_{k b}-\right. \left.v_k v_b\right) /\left(v_k v_k v_{k b}\right)\end{gathered}$     (17)

The second term of (14) is:

$\begin{gathered}E_1 {Var}_2\left(\hat{v}_{f H T}\right)=E_1 {Var}_2\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f} j_k}{\alpha_k}\right) =E_1\left(\frac{1}{N^2} \sum_{k=1}^\tau \frac{j_k {Var}_2\left(\widehat{v}_{k f}\right)}{\alpha_k^2}\right) \\ \hat{v}_{i f}=\frac{1}{P_{1 i}}\left[\frac{\sum_{j=1}^{m_i} y_{i j}}{P_{2 i}\left(1-v_{i u}\right)}-P_{2 i} v_{i u}\right] \\ =E_1\left[\frac{1}{N^2} \sum_{k=1}^\tau \frac{j_k}{\alpha_k^2} {Var}_2\left(\frac{1}{P_{1 k}}\left(\frac{\sum_{j=1}^{m_i} y_{i j}}{m_k P_{2 i}\left(1-v_{i u}\right)}-P_{2 i} v_{i u}\right)\right)\right] \\ =E_1\left[\frac{1}{N^2} \sum_{k=1}^\tau \frac{j_k}{\alpha_k^2}\left(\frac{1}{P_{1 k}^2}\left(\frac{\sum_{j=1}^{m_k} V a r_2\left(y_{k j}\right)}{m_k^2 P_{2 i}\left(1-v_{i u}\right)}\right)\right)\right] \\ =E_1\left[\frac{1}{N^2} \sum_{k=1}^\tau \frac{j_k}{\alpha_k^2}\left(\frac{1}{P_{1 k}^2}\left(\frac{\sum_{j=1}^{m_k} v_{k 0}}{m_k^2 P_{2 i}\left(1-v_{i u}\right)^2}\right)\right)\right] \\ =E_1\left[\frac{1}{N^2} \sum_{k=1}^\tau \frac{j_k}{\alpha_k^2}\left(\frac{1}{P_{1 k}^2}\left(\frac{v_{k 0}}{m_k P_{2 i}\left(1-v_{i u}\right)^2}\right)\right)\right] \\ =E_1\left[\frac{1}{N^2} \sum_{k=1}^\tau \frac{j_k}{\alpha_k^2}\left(\frac{1}{P_{1 k}^2}\left(\frac{\left(P_{1 k} v_{k f}+P_{2 k} v_{k u}\right) P_{2 i}\left(1-v_{i u}\right)}{m_k P_{2 i}\left(1-v_{i u}\right)^2}\right)\right)\right] \\ =E_1\left[\frac{1}{N^2} \sum_{k=1}^\tau m_k \frac{1}{\alpha_k^2}\left(\frac{\left(P_{1 k} v_{k f}+P_{2 k} v_{k u}\right)}{P_{1 k}^2 P_{2 i}\left(1-v_{i u}\right)}\right)\right]\end{gathered}$     (18)

$E_1 {Var}_2\left(\hat{v}_{f H T}\right)=\frac{1}{N^2} \sum_{k=1}^N m_k \frac{\psi_k}{\alpha_k}$     (19)

where, $\psi_{r k}=\left[\frac{P_{1 k} v_{k f}+P_{2 k} v_{k u}}{P_{1 k}^2 P_{2 i}\left(1-v_{i u}\right)}\right]$.

The $\hat{v}_{f H T}$ in Eq. (13) earlier given was obtained by substituting the equation in (11) and (19) into (14).

2.2 The case when unrelated rare non-sensitive attribute is unknown

In the case of randomized response (RR) strategies combined with adaptive cluster sampling (ACS), the use of an unrelated rare non-sensitive attribute can play a crucial role in maintaining respondent confidentiality and enhancing the accuracy of data collection. Typically, RR techniques rely on respondents answering questions about sensitive attributes indirectly, often by referencing unrelated non-sensitive attributes to introduce randomness. 

Unknown rare non-sensitive attributes can lead to increased response bias, difficulty in designing the RR mechanism, complexity in data analysis, and increased operational challenges. Respondents may feel less secure about the confidentiality of their responses, potentially undermining the effectiveness of the RR strategy. Designing the RR mechanism requires precise probability distributions, which can be challenging when the attribute is unknown. Data analysis becomes more complex, potentially leading to biased or inaccurate conclusions. To address these challenges, innovative approaches include pre-survey piloting, statistical adjustment techniques, and adaptive RR mechanisms. 

Each respondent is asked twice for a response in order to estimate the MNSA. While the unrelated, unusual, harmless trait is unidentified. These randomization tools are made up of decks of cards and cards that are comparable, as explained in Section 2.1. The initial randomization device based on the i^th cluster's respondents asks them to respond "yes" or "no" at start. We follow the procedure outlined in section 2.1. 

Using a second randomization system consisting of the outlined statements with probabilities T_1i, T_2i, and T_3i instead of P_1i, P_2i, and P_3i, the respondents chosen from the i^th cluster are required to answer the same questions. Responders in the i^th cluster have "yes" responses with the following probabilities:

$\phi_{i 1}=\left[P_{1 i} \beta_{i f}+P_{2 i} \beta_{i u}\right]+P_{2 i}\left(1-\beta_{i u}\right)$           (20)

and

$\phi_{i 2}=\left[T_{1 i} \beta_{i f}+T_{2 i} \beta_{i u}\right]+T_{2 i}\left(1-\beta_{i u}\right)$          (21)

If $m_i \rightarrow \infty, \phi_{i 1} \rightarrow \infty$, and $\phi_{i 2} \rightarrow 0, m_i \phi_{i 1}=v_{i 1}>0$, and $m_i \phi_{i 2}=v_{i 2}>0$, as $\beta_{i u} \rightarrow 0$ where $m_i \beta_{i u}=v_{i u}>0$, and $m_i \beta_{i f}=v_{i f}>0$. Subsequently, (20) and (21) $m_i \beta_{i u}=v_{i u}>$ 0.

where,

$v_{i 1}=\left[P_{1 i} v_{i f}+P_{2 i} v_{i u}\right]+P_{2 i}\left(1-\beta_{i u}\right)$           (22)

and

$v_{i 2}=\left[T_{1 i} v_{i f}+T_{2 i} v_{i u}\right]+T_{2 i}\left(1-\beta_{i u}\right)$            (23)

Simplifying (16) and (17) results in

$\frac{1}{m_i} \sum_{j=1}^{m_i} y_{i 1 j}=\left[P_{1 i} \hat{v}_{i f}+P_{2 i} \hat{v}_{i u}\right]+P_{2 i}\left(1-\beta_{i u}\right)$     (24)

$\frac{1}{m_i} \sum_{j=1}^{m_i} y_{i 2 j}=\left[T_{1 i} \hat{v}_{i f}+T_{2 i} \hat{v}_{i u}\right]+T_{2 i}\left(1-\beta_{i u}\right)$     (25)

Solving Eq (24) and Eq (25), we then have the estimators of $v_{i f}$ and $v_{i u}$ to be:

$\hat{v}_{i f 2}=\frac{1}{m_i\left(T_{2 i} P_{1 i}+T_{1 i} P_{2 i}\right)}\left(\frac{T_{2 i} \sum_{j=1}^{m_i} y_{i 1 j}}{P_{2 i}\left(1-\beta_{i u}\right)}-\frac{P_{2 i} \sum_{j=1}^{m_i} y_{i 2 j}}{T_{2 i}\left(1-\beta_{i u}\right)}\right)$     (26)

where, T_2i P_1i≠T_1i P_2i.

$\hat{v}_{i u 2}=\frac{1}{m_i\left(T_{1 i} P_{2 i}+T_{2 i} P_{1 i}\right)}\left(\frac{T_{1 i} \sum_{j=1}^{m_i} y_{i 1 j}}{P_{2 i}\left(1-\beta_{i u}\right)}-\frac{P_{1 i} \sum_{j=1}^{m_i} y_{i 2 j}}{T_{2 i}\left(1-\beta_{i u}\right)}\right)$     (27)

where, $T_{2 i} P_{1 i} \neq T_{1 i} P_{2 i}$.

The computation of the estimator for the population mean of the overall count of individuals with a rare and sensitive trait is subsequently carried out using the designated method.

$\widehat{v}_{H T 2}=\left(\frac{1}{N} \sum_{k=1}^\tau \frac{\widehat{v}_{k f 2} j_k}{\alpha_k}\right)$     (28)

Theorem 4: The estimator or $\hat{v}_{f H T 2}$ of the MNSA attribute is unbiased.

Proof.

Since $\mathrm{E}\left(E_1\right)=\mathrm{E}\left(E_2\right)$ we conclude that $E\left(\hat{v}_{f H T 2}\right)=v_f$. where, $E_1$, and $E_2$ are the expected total number of first and second -stage selections receptively and $E(\cdot)$ is the expectation.

The variance of the estimator $\hat{v}_{f H T 2}$ is:

$V\left(\hat{v}_{H T 2}\right)=\frac{1}{N^2}\left[\sum_{k=1}^K \sum_{b=1}^K v_{k f^*} v_{b f^*}\left(v_{k b}-v_k v_b\right) /\left(v_k v_b v_{k b}\right)+\sum_{k=1}^N m_k \frac{\phi_k}{\alpha_k}\right]$,     (29)

This equation represents the variance $V\left(\hat{v}_{H T 2}\right)$ of the Horvitz-Thompson estimator (HT2) for a Poisson-distributed adaptive cluster sampling model. The first term inside the brackets involves double summation over clusters, accounting for interactions between pairs of clusters k and b. The second term sums over all clusters, adjusting for specific sampling parameters mk, ϕ_k and α_k.

$\begin{gathered} H_k & =\frac{T_{2 k}}{P_{2 i}\left(1-\beta_{i u}\right)}, \\ Q_k & =\frac{P_{2 k}}{T_{2 i}\left(1-\beta_{i u}\right)}\end{gathered}$

These equations define two parameters, H_k and Q_k, used in the adaptive cluster sampling model. Here, T_2k and P_2k rrepresents specific values related to cluster k, while P_2i and T_2i are reference values for another index i. The term (1-β_iu) adjusts for a factor β_iu influencing the relationship between clusters and the overall sample.

This research endeavors to bridge the gap between adaptive sampling techniques and respondent confidentiality, presenting a novel Poisson-distributed adaptive cluster sampling model enhanced by a randomized response strategy. Through theoretical development and empirical validation, we aim to demonstrate the efficacy of this integrated approach in yielding high-quality, reliable data from clustered and sensitive populations.

In conclusion, the utilization of cluster sampling and the randomized response technique, coupled with the introduction of the adaptive cluster sampling randomized response model featuring a Poisson distribution, has brought forth valuable insights into estimating the prevalence of sensitive groups within the population. The demonstrated lower variance in the proposed model not only signifies increased efficiency but also highlights its potential for more accurate estimations. this study employed cluster sampling and the randomized response technique to gauge the proportion of the population associated with a sensitive group. Introducing the adaptive cluster sampling randomized response model with a Poisson distribution proved to be a significant advancement. The results showcased that this proposed model exhibited lower variance, indicating greater efficiency compared to existing models. Notably, the study demonstrated the superiority of the proposed model over its counterparts, marking a substantial contribution to the field. This innovative methodology holds promise for advancing the field of statistical sampling and improving the outcomes of various research endeavors reliant on accurate and unbiased data.