Enhancing Estimation Precision in Sensitive Public Health Data: Implications for Evidence-Based Decision-Making

Abstract

Introduction

Reliable estimation of sensitive public health outcomes, such as underreported COVID-19 cases and vaccine hesitancy, is frequently hindered by social stigma and nonresponse bias, particularly in culturally sensitive contexts like Pakistan. Traditional surveys often underestimate prevalence, limiting the effectiveness of public health interventions.

Methods

Multi-stage stratified paired response frameworks (MSPRFs) incorporating two and three-stage randomization with dual scrambling mechanisms under stratified simple random sampling were applied. A cross-sectional survey of 1,200 participants (600 urban, 600 rural) from Faisalabad, Lahore, Multan, and Rawalpindi (June-August 2021) was conducted using anonymous questionnaires to collect data on outbreak cases and vaccine hesitancy.

Results

The MSPRFs demonstrated substantially enhanced estimation efficiency (PRE 290–399%) relative to conventional methods. MSPRF-I estimated COVID-19 prevalence at 16.8% (urban) and 20.8% (rural), compared with reported rates of 12.1% and 13.6%; MSPRF-II estimated 16.8% and 19.2% versus reported 12.6% and 14.6%. Vaccine hesitancy was estimated at 33.8% (MSPRF-I) and 29.5% (MSPRF-II), relative to 24.3% reported. Urban respondents exhibited higher reporting sensitivity () than rural respondents (), indicating geographic disparities in disclosure behavior.

Conclusion

MSPRFs provide a robust, privacy-preserving methodology for accurately estimating underreported COVID-19 cases and vaccine hesitancy, supporting evidence-based public health decision-making. MSPRF-II offers marginally higher adjustment efficiency due to inclusion of a truthful-response component. These frameworks are appropriate for sensitive epidemiological surveys in contexts where conventional self-reporting underestimates prevalence.

Introduction

Reliable measurement of sensitive public health information remains a persistent methodological challenge. Specifically, variables such as vaccine hesitancy, disease occurrence, and health-related behaviors are frequently affected by concealment, inaccurate reporting, or refusal to participate, particularly when respondents fear social judgment or disclosure. Consequently, these challenges undermine conventional survey designs and produce biased public health indicators. To address this, the present study develops MSPRFs that enhance both respondent privacy and estimator precision. Building on this foundation, recent research has refined randomized response technique (RRT) to reduce non-response bias and improve estimator performance. Shahid and Hussain [41], along with Hussain and Shahid [10], further refined unbiased estimation techniques, enhancing both additive and optional scrambling-based models. While Kumar et al. [21] introduced randomized scrambling procedures. In addition, Gupta et al. [8] developed variance coefficients for population estimation, and Saleem et al. [34] proposed efficient estimators under randomized assessment designs. Furthermore, Azeem et al. [3] recently introduced a weighted measure that incorporates both efficiency and privacy, yielding a more effective assessment of randomized response methods, and Siddiqui et al. [39] studied correlated variable frameworks. Collectively, these contributions [2, 5, 7, 14, 47] highlight the growing methodological sophistication in privacy-preserving survey estimation. Therefore, these advances provide a strong foundation for developing MSPRF-I and MSPRF-II, ensuring that sensitive health data can be collected ethically while maintaining statistical reliability.

While prior studies have emphasized randomized response under equal probability sampling [21, 41], the proposed MSPRF frameworks are inherently adaptable to unequal probability designs through the use of sampling weights. This flexibility ensures that MSPRF-I and MSPRF-II maintain unbiased, high-efficiency estimation of sensitive population parameters, even when selection probabilities vary, thereby reinforcing therobustness and practical applicability of the methodology across diverse survey designs.

Moreover, the COVID-19 pandemic has further emphasized the need for accurate, privacy-preserving public health data. Beyond initial respiratory symptoms, severe infections caused organ damage, mortality, and long-term complications such as sleep disturbances, fatigue, cognitive decline, and loss of taste [13, 15, 23, 25–28, 32, 36, 37, 40]. Consequently, persistent complications among survivors exert significant pressure on healthcare systems. As a result, randomized response techniques have become crucial tools to protect respondent privacy while improving data reliability.

In this context, RRT models have been demonstrated to balance privacy and accuracy in health surveys. For example, Liu et al. [24] compared multiple RRT models, while Rodriguez and Ramirez [33] explored privacy-enhancing technologies. Similarly, Perez and Wang [29] examined vaccination estimates, and Johnson and Thompson [16] highlighted ethical considerations. In addition, Gomez and Sanchez [9] and Batista and Lima [4] emphasized real-world applicability, and Williams and Lee [46] demonstrated how privacy concerns shape survey responses. Moreover, Patel and Kumar [30], Singh and Yadav [43], Zhou and Guo [48], and Liu and Zhang [22] contributed further privacy-focused, bias-reducing approaches. Collectively, these studies underscore the importance of RRT for producing high-quality sensitive health data. Shah et al. [38] further demonstrated that multi-stage RRT frameworks achieve high efficiency, supporting the adequacy of stratified split-sample designs across urban and rural strata.

Furthermore, the literature can be categorized into three interconnected streams. First, methodological advances in randomized response models for sensitive quantitative variables emphasize privacy, robustness, and estimator efficiency [1, 12, 35]. Second, health survey methodology highlights how privacy concerns, nonresponse, and survey design impact the accuracy of health data [6, 11, 42, 45]. Third, applied public health research in Pakistan documents variation in vaccine hesitancy, acceptance, and related behaviors [17–20, 31, 44]. Accordingly, these interconnected strands highlight the necessity of privacy-preserving, statistically efficient, and context-sensitive survey frameworks capable of producing reliable public health estimates where direct questioning leads to biased responses.

Conventional single-stage randomized response methods often underperform for highly sensitive variables and heterogeneous populations. The proposed MSPRF-I and MSPRF-II frameworks use two- and three-stage designs with dual scrambling and stratified paired responses, ensuring unbiased, efficient estimation even with unequal probability sampling, and enhancing privacy, robustness, and applicability across urban and rural strata.

Finally, to address these challenges, we developed two MSPRFs using dual scrambling factors ( and ) within a stratified response system, enabling simultaneous estimation of population means and sensitivity levels via a cost-effective dual-response protocol. Consequently, this approach improves estimate reliability without increasing data collection costs. We empirically applied MSPRF-I and MSPRF-II through a cross-sectional survey in Faisalabad, Lahore, Multan, and Rawalpindi, Pakistan (June–August 2021), targeting underreported outbreak cases and vaccination reluctance. Ensuring respondent anonymity facilitated ethical data collection, supported evidence-based policy, and strengthened public health monitoring. The article is organized as follows: Section 2 presents MSPRF formulations, Section 3 details the epidemiological application, and Section 4 concludes with implications.

Methods

Framework of the Study

The study implements two MSPRFs to estimate population means and sensitivity levels for stigmatized public health variables. MSPRF-I uses a two-stage dual-scrambling procedure () within each stratum, where respondents provide paired responses to maintain privacy while capturing true behavioral measures. The scrambling variables and are generated from Exponential distributions with parameters corresponding to the stratum-level sensitivity () and fraction assigned to scramble (). Its mechanism is illustrated in Fig. 1, showing the dual-response randomization and stratified sampling flow. MSPRF-II extends this to a three-stage procedure, combining truthful reporting with controlled scrambling for improved precision. Figure 2 depicts MSPRF-II’s structural mechanism, highlighting enhanced privacy and estimator efficiency. Stratified SRS selects participants across urban and rural strata, accounting for socio-economic heterogeneity. Together, the two models provide a robust conceptual and procedural framework for privacy-preserving estimation of sensitive public health outcomes.

Fig. 1.

Mechanism of MSPRF-I illustrating the dual-stage randomization and stratified sampling procedure

Fig. 2.

Mechanism of MSPRF-II illustrating the three-stage randomization, combining truthful and scrambled responses

Study Population and Sampling

The survey targeted 1,200 adults (≥ 18 years) from urban and rural areas of Faisalabad, Lahore, Multan, and Rawalpindi, Punjab, Pakistan, during June-August 2021. Stratified SRS selected participants across urban/rural and socio-economic subgroups, with 600 participants per stratum. Non-response was addressed by replacing unavailable participants with eligible individuals from the same stratum. Demographic data included age, sex, education, household size, and income. Structured questionnaires (see supplementary material) collected information on health service utilization, COVID-19 illness, underreporting, and vaccine hesitancy. Sensitive items were administered using MSPRF-I and MSPRF-II to ensure confidentiality, with trained enumerators providing guidance in the local language.

In the proposed MSPRF study, equal allocation across urban and rural strata ensures balanced precision, preserves estimator efficiency, and aligns with the MSPRF randomization design, making it the most suitable choice. Equal allocation avoids complexity, maintains respondent clarity, and supports unbiased estimation of sensitive public health outcomes.

The MSPRF frameworks use multi-stage randomized response to protect privacy and ensure unbiased estimation. MSPRF-I employs dual-stage scrambling ( and ), while MSPRF-II adds a third stage with a truthful-response component. This design enhances efficiency, accommodates stratum heterogeneity, and uses exponential random numbers to mask sensitive responses without bias.

While the double-response procedure in MSPRF-I and MSPRF-II requires respondents to answer twice, it is designed to be practical for real-world surveys. This procedure enhances privacy, reduces social desirability bias, and encourages honest reporting. Clear instructions, assurances of anonymity, and trained enumerators minimize respondent burden and refusals. Consequently, the double-response design remains feasible, reliable, and effective for collecting sensitive public health data without adding extra costs or complexity.

MSPRF-I Design

The suggested technique, which depends on SRS, requires selecting individual respondents from the -sized sample in the h-th stratum. This framework estimates the mean of the stigmatizing factor, , and the sensitivity, . The participants are requested to provide a couple of responses to the following question: In the first answer, at stage 1, a certain fraction of respondents, denoted as , are instructed to scramble their reply, whilst the remaining respondents, with a probability of , advance to stage 2. Regarding stage 2, two distinct randomization devices generate the scrambling variables and , which are drawn from certain exponential distributions and then divided and added to the genuine answer, . Respondents who consider the issue non-sensitive provide the accurate answer, . The second response follows the same procedure, except that the ratio is subtracted from . The i-th respondent selected from the sample of the h-th stratum is required to provide two responses, and , as specified below:

1

2

MSPRF-II Design

The MSPRF-II is implemented by stratifying the population to reduce within-stratum variance and increase among-stratum variation. From each stratum, an SRSWR sample of size is drawn to achieve precise estimates at minimal cost. This framework extends the 3-stage MSPRF-II by estimating the mean of the stigmatizing factor, , and the sensitivity level, . Each respondent provides two responses. In the first response, a fraction reports the true value, while the remainder follows MSPRF-I. The second response is similar, except that the division of the scrambling variables is subtracted from . Each respondent reports both replies, denoted and , in sequence as:

3

4

Here, follow Exponential distributions parameterized by , and , ensuring that scrambling reflects stratum-level sensitivity and assigned proportions. Figures 1 and 2 illustrate the operational mechanisms of MSPRF-I and MSPRF-II.

Results and Discussion

Empirical Application of the Proposed MSPRFs

The proposed MSPRF-I and MSPRF-II models were applied to a cross-sectional survey conducted in Faisalabad, Lahore, Multan, and Rawalpindi, Punjab, Pakistan, from June to August 2021. The empirical analysis focused on two sensitive public health outcomes: underreported COVID-19 outbreak cases and vaccine hesitancy. These outcomes are prone to concealment because respondents may fear stigma, social judgment, or disclosure of private health information. The MSPRF approach was employed to obtain more reliable estimates while maintaining respondent anonymity. The total sample consisted of n = 1,200 respondents, equally distributed across urban and rural strata. The stratified design allowed the models to account for differences in reporting behavior, sensitivity levels, and socio-demographic conditions between population groups. Table 1 presents the stratum-specific distributional settings and true response parameters used in the empirical implementation.

Table 1.

Mean, variation, and actual response estimates across both rural and urban strata are summarized

Strata	Mean values	Variance values	True responses
Urban (stratum-1)	, (5, 6); , (1, 1)	, (3, 2); , (3, 2 )	(6), (3)
Rural (stratum-2)	, (6, 6); , (1, 1)	, (3, 3); , (2, 1)	(7), (3)

Underreporting of COVID-19 Outbreak Cases

Table 2 compares reported and estimated outbreak cases using MSPRF-I and MSPRF-II across urban and rural strata. The results indicate underreporting in both population groups. Under MSPRF-I, the estimated infection rate was 16.8% (± 1.2) in urban areas and 20.8% (± 1.4) in rural areas, compared with reported rates of 12.1% and 13.6%, respectively, highlighting a substantial underestimation in direct reporting.

Table 2.

Comparison of outbreak underreporting analysis using MSPRF-I and MSPRF-II in the districts of Faisalabad, Lahore, Multan, and Rawalpindi

Factors	MSPRF-I		MSPRF-II
	Urban (Stratum-I)	Rural (Stratum-II)	Urban (Stratum-I)	Rural (Stratum-II)
Sample size (n)	600 ()	600 ()	600 ()	600 ()
Sensitivity level (W)	0.60 ()	0.40 ()	0.62 ()	0.48 ()
Truthful response (T)	N/A	N/A	0.25 ()	0.30 ()
Scrambled response (F)	0.20 ()	0.25 ()	0.20 ()	0.25 ()
Reported cases (%)	12.1	13.6	12.6	14.6
Estimated mean	16.8 (±1.2)	20.8 (±1.4)	16.8 (±0.8)	19.2 (±1.3)
Estimated sensitivity	0.60	0.40	0.62	0.48
PRE of mean	375.2	290.4	387.6	325.9
PRE of sensitivity	395.8	252.1	255.2	295.6

A similar pattern was observed under MSPRF-II. The estimated infection rate was 16.8% (± 0.8) in urban areas and 19.2% (± 1.3) in rural areas, exceeding reported rates of 12.6% and 14.6%. The sensitivity level was higher in urban strata than in rural strata , suggesting that disclosure behavior varied by location. Higher PRE values for MSPRF-II, particularly for mean estimates, indicate improved estimation efficiency and stronger adjustment for response bias.

Overall Estimates of Underreporting and Vaccine Hesitancy

Table 3 presents overall estimates for underreported cases and vaccine hesitancy in the selected districts. Both models produced higher estimates than direct reporting, confirming the presence of hidden public health burden. For underreported cases, reported prevalence was 13.4%, whereas MSPRF-I and MSPRF-II estimated 16.6% (± 1.5) and 16.5% (± 1.3), respectively.

Table 3.

MSPRF-I and MSPRF-II provide extensive estimates of vaccination hesitancy as well as outbreak underreporting across Pakistan

Variables	Reported (Overall)	MSPRF-I	MSPRF-II	W (Overall)	T (Overall)	F (Overall)	PREs (Overall)
Underreported cases	13.4	16.6 (± 1.5)	16.5 (± 1.3)	0.60	0.25	0.20	399.5%
Vaccine hesitancy	24.3	33.8 (± 1.6)	29.5 (± 1.4)	0.40	0.30	0.25	366.4%

The gap was more pronounced for vaccine hesitancy. The reported hesitancy rate was 24.3%, while MSPRF-I estimated 33.8% (± 1.6) and MSPRF-II estimated 29.5% (± 1.4). These findings indicate that direct questioning fails to capture a substantial proportion of hesitant individuals. High PRE values further demonstrate the statistical advantage of the proposed frameworks in estimating sensitive outcomes with improved reliability.

Temporal and Regional Variation

Figure 3 presents the temporal pattern of reported COVID-19 cases and the distribution of average response sensitivity level across the selected districts. Figure 3(a) shows considerable variation in daily reported cases over epidemiological weeks, reflecting heterogeneous outbreak dynamics across districts. Figure 3(b) shows that average response sensitivity level varied across districts, supporting the use of stratified privacy-preserving models.

Fig. 3.

Temporal pattern of reported COVID-19 cases and regional response sensitivity. Sub-figure (a) shows the variation in daily reported cases across epidemiological weeks for the selected districts, while sub-figure (b) presents the proportional distribution of average sensitivity levels across districts

Comparative Performance of MSPRF Models

Figure 4 illustrates the distribution of reported cases and the comparative performance of MSPRF-I and MSPRF-II. Figure 4(a) shows a wide dispersion of daily reported cases, confirming regional heterogeneity. Figure 4(b) compares MSPRF-I and MSPRF-II estimates, with MSPRF-II producing slightly higher values, indicating stronger adjustment for underreporting and response sensitivity.

Fig. 4.

Distribution of reported cases and comparative performance of the proposed MSPRF models. Sub-figure (a) shows the frequency distribution of daily reported cases across districts, while sub-figure (b) compares the average case estimates obtained from MSPRF-I and MSPRF-II across the selected districts

Public Health Interpretation and Implications

Collectively, these findings demonstrate that conventional survey responses underestimate both COVID-19 outbreak cases and vaccine hesitancy. The difference between reported and MSPRF-estimated values highlights the influence of social desirability bias and privacy concerns on survey responses, which can weaken surveillance systems and reduce the effectiveness of targeted interventions. Reporting behavior differed across urban and rural strata, as reflected by variation in estimated sensitivity levels. These differences emphasize the need for context-sensitive survey strategies. By incorporating stratification, paired responses, and controlled scrambling, the proposed MSPRFs improve data accuracy while protecting respondent privacy. Overall, MSPRF-II showed slightly stronger adjustment capacity than MSPRF-I due to the inclusion of a truthful-response component. Both models outperformed direct reporting, providing robust evidence for identifying hidden public health risks.

Study Limitations

Although the study is limited to selected districts of Punjab, Pakistan, the use of stratified sampling and MSPRF-I and MSPRF-II models enhances the generalizability of results. The cross-sectional design does not allow causal inference; however, the randomized response frameworks minimize bias in sensitive reporting. Sample size and demographic diversity provide robust estimates despite geographic constraints. Self-reported data may introduce reporting errors, yet the dual-scrambling and truthful-response components substantially mitigate misreporting. While RRT designs expose participants to randomization, optional pathways allowing direct or scrambled responses could further reduce nonresponse. The current MSPRF frameworks remain robust due to stratification, dual-scrambling, and anonymity, but future implementations may integrate respondent choice to enhance participation without compromising estimation efficiency.

Conclusion

This study demonstrates that MSPRFs provide a robust, privacy-preserving approach for accurately estimating sensitive public health outcomes. By integrating dual exponential scrambling with additive and subtractive randomization in two and three-stage protocols, MSPRFs deliver unbiased estimators for population means and sensitivity levels. Empirical validation using a cross-sectional survey of 1,200 respondents across urban and rural strata in Faisalabad, Lahore, Multan, and Rawalpindi revealed substantial underreporting: COVID-19 prevalence was 16.8-20.8% versus reported 12.1-14.6%, and vaccine hesitancy was 29.5–33.8% versus reported 24.3%, with urban respondents exhibiting higher reporting sensitivity () than rural (). MSPRFs achieved 300-450% relative efficiency gains over conventional randomized response methods, demonstrating superior statistical performance. These findings highlight the potential of MSPRFs to uncover hidden disease burdens and health behaviors, informing evidence-based public health interventions, targeted vaccination strategies.

Author Contributions

The data analysis and interpretation, provided critical manuscript review, and supervised the study via LH. The study’s conception and design, manuscript drafting, and data analysis and/or interpretation were conducted by FS. MIS contributed to data acquisition, analysis, and manuscript revision. Drafting the manuscript and interpretation of data by KA.

Funding

This research is supported by a project of the National Social Science Fund: Collection, Sorting, and Research of the Centennial Medical and Health Historical Data of the CPC (21&ZD036).

Data Availability

The findings of this study are supported by data obtained from a primary cross-sectional survey conducted from the districts of Faisalabad, Lahore, Multan, and Rawalpindi in Punjab, Pakistan, between June and August 2021. In accordance with ethical guidelines and to protect participant confidentiality, the datasets generated during this study are not accessible to the public. Upon reasonable request, the identified data can be provided by the corresponding author.

Declarations

Competing Interests

The authors declare that they do not have any competing interests to disclose.

Ethics Approval and Consent to Participate

This study involved anonymous questionnaire-based data collection only and did not include biological samples, human tissues, clinical procedures, or any intervention involving human participants. The study was reviewed and approved by Henan Normal University. In line with ethical standards for social science research in Punjab, Pakistan, formal approval from any Institutional Board was not required for this study as it involved non-invasive procedures and anonymized data collection through questionnaires, with no collection of identifiable personal information. Participations were entirely voluntary, and verbal informed consent was obtained from all participants prior to their inclusion in the study. All procedures were carried out in accordance with the ethical principles of the Declaration of Helsinki.

Consent for Publication

Not applicable.