Statistical Methods And Applications Of Nine RRT Models From A (Stratified) Three-stage Sampling Design Dealing With Sensitive Questions

Objective：Sampling survey is usually referred to as an essential method for scientific researchin medicine and health. Almost inevitably, sampling survey related to sensitive subjectsis encountered. Getting truthful answers to survey questions about sensitive matters is achallenge. Sensitive topics are perceived as threatening to preserving privacy or makingany public statement. The direct inquiry method often leads to refusals or untruthfulreplies. Untruthful reporting is social desirable or undesirable responding not accurateresponse. Answers to sensitive questions are distorted by nonresponse bias or lying bias.The randomized response technique (RRT) was first conceived by Warner in1965andintroduced as a method for guaranteeing respondents to maintain privacy and improvingthe accuracy of estimates about sensitive dichotomous characteristic.Since Warner published his first paper on randomized response, many researchershave improved and further developed this technique. Efforts have been made to proposevarious forms of RRT during the last forty years. In research of RRT, much attention hasbeen paid to both dichotomous and quantitative sensitive questions. Correspondingly,there is less study on the polychotomous sensitive questions. Simple random samplingis the most widely used for surveys on sensitive topics. Sampling strategy is usuallyrestricted on simple random sampling when surveys are dealing with sensitive topics.Respondents, invited to participate in research on sensitive topics, are always confinedto small area and are usually drawn by simple random sampling. What is more, onlysimple random sampling might be taken into account in the analysis of data fromcomplex sampling survey. The design of sampling is a particular important aspect of sampling survey; determination method of sample size is the key link in samplingdesign. However, sample size determination for complex sampling survey in sensitivetopics using randomized response model is not yet available.The selection of eighteen survey methods, which constituted varied combinationsof nine randomized response model and two sampling methods, was in the presentresearch. Based on the premise that the estimators of the population parameters forrandomized response model in (stratified) three-stage sampling survey were given,sample sizes formulae for (stratified) three-stage sampling survey were deduced in thisstudy, so as to minimize the cost of survey implementation for a specified level ofprecision and to provide reasonably precise estimates under the constraint of a fixedbudget. These formulae were suitable for complex sample survey on a large scale.Preliminary investigation into sensitive behaviors among female sex workers (FSW) inXichang showed a meaningful trends in sexually transmitted diseases (STD)/acquiredimmunodeficiency syndrome (AIDS) and provided related statistic value needed forsample size formulae. According to the deduced formulae in this research, requiredsample size at each stage was calculated to estimate sensitive features of FSW inXichang in a field investigation which will be carried out in2015. As to six surveymethods, which were comprised of varied combinations of three randomized responsemodel and two sampling methods, we built sampling simulation with SAS programsbased on Monte Carlo method. According to sample size calculated based onpreliminary simulation experiments, simulated survey was conducted to estimate thevalue of simulated population parameters using point estimate and interval estimate. Wecompared this value with predetermined simulated population proportion/mean so as toevaluate the validity and reliability of survey methods and statistical formulae andsample size formulae.Method：1. Statistical formulae used in eighteen survey methods were showed whichconsisted of varied combinations of nine randomized response model (e.g. Warner RRTmodel, Simmons RRT model, Greenberg RRT model, improved RRT model,multiple-choice sensitive question with single response RRT model, multiple-choicesensitive question with indirect response RRT model, unrelated question RRT model,additive constant model and multiplicative RRT model) and two sampling methods (e.g. three-stage sampling and stratified three-stage sampling). These statistical formulaewhich were some ways of producing estimators of population proportion/mean andpopulation variance had been deduced from Cochran’s sampling theory as well asstatistics and probability theory (e.g. total probability theorem, etc).2. By the Cauchy-Schwarz inequality and the Lagrange function, to theminimization of cost for specified sampling errors and the minimization of samplingerrors under the constraint of a fixed cost, the formulae for the optimum sample size forthree-stage and stratified three-stage sampling survey were deduced.3. Randomizing devices in RRT models were designed and questionnaire itemsasking sensitive topics needed to be crafted. Behavioral characteristics wereinvestigated in three-stage sampling study of FSW in Xichang from May to July in2011.Following the statistical formulae, we conducted a preliminary analysis to estimaterelated statistics value needed in the sample size formulae.4. For all of ten sensitive questions in three-stage sampling survey research usingthree RRT model (e.g. Simmons RRT model, multiple-choice sensitive question withsingle response RRT model and additive constant model), optimum sample size weredetermined when we limited sampling errors for the minimum of cost and limited costfor the minimum of sampling errors respectively. All these were based on the bothrelated statistics value and sample size formulae deduced in this study.5. Based on the survey results on the behavioral characteristics of FSW, simulatedpopulation was built with SAS program. Sample mean/proportion or sample proportionin the different categorize from survey data collected in Xichang was considered assimulated population parameter. We simulated a stratified three-stage sampling processand then select three RRT model (e.g. Simmons RRT model, multiple-choice sensitivequestion with single response RRT model and additive constant model) to investigatevirtual FSW. This process was called simulated sampling preliminary survey. Followingthe sample size formulae, SAS gave sample sizes within each stratum at each stagewhich were needed for simulated sampling formal survey. We simulated a stratifiedthree-stage sampling once more according to the calculated sample size, that is,simulated sampling formal survey. On the basis of the statistical formulae, we calculatedsimulated sample statistics and then computed95%confidence interval (CI) ofsimulated population proportion/mean or simulated population proportion in differentsensitive categorizes. This process was repeated100times. If almost all100CIs include the predetermined population proportion/mean, survey methods, statistical formulae andsample size formulae proved to be strong validity. When100simulated sample statisticsalmost got close to a fixed value (predetermined population proportion/mean was truevalue), survey methods, statistical formulae and sample size formulae indicated a highdegree of reliability.Results：1. This study proposed eighteen types of survey methods which were combinationof nine RRT models and two sampling methods. For each survey method, formulae forestimators of population proportion/mean and population variance were given.2. The formulae for the optimum sample sizes with eighteen types of surveymethods were deduced when the cost was minimized for specified sampling errors andthe sampling errors was minimized under the constraint of a fixed budget.3. Using three types of RRT models from a three-stage sampling survey toinvestigate behavioral features of FSW in Xichang, the preliminary survey resultsshowed as follows: FSW provided their first paid sex services in the age21.45years,with the standard error of0.8162; FSW provided sex services41.66times a month, withthe standard error of1.4550; the average price per sex service was213.67RMB, withthe standard error of8.2475; the proportion of having spouse or steady sex partner was55.94%, with the standard error of3.87%; the proportion of ceasing trading as a sexservice provider when FSW were diagnosed as STD was75.85%, with the standarderror of3.00%; the proportion of supporting legal prostitution in China was56.77%,with standard error of4.12%; the proportion of results on STD test, including failure toget STD test in government hospitals, negative cases and positive cases, were62.12%,21.36%,5.57%in the most recent year, with the standard error of4.00%,3.87%,2.24%respectively; the proportion of results on HIV test, including not being tested for HIV,negative and positive, were57.11%,23.54%,2.35%within one year, with the standarderror of4.00%,4.00%,1.00%respectively; the proportion of FSW reported condombreaking during the act was8.27%, and the standard error was2.65%; the proportion ofcondom usage, which was classified into never used, sometimes used and always used,were11.40%,14.21%,74.40%last month, with the standard error of2.65%,3.16%,4.69%respectively.4. National Natural Science Fund Project will be launched in2015to conduct three-stage sampling survey concerning sensitive characteristics of FSW in Xichang.Taking preliminary survey data into consideration, required sample size at each stagewere shown. In the first stage, five districts should be selected (n15). Then in thesecond stage, an average of six venues should be drawn from each chosen district(n26). Finally, in the third stage, twenty nine FSW, on average, should be sampledfrom each chosen venue (n329).5. The simulation on6types of methods for surveying sensitive topics displayedthe following results:5.1For Simmons RRT model applied to dichotomous sensitive questions in astratified three-stage sampling survey,96CIs for simulated population proportion from100CIs contained predetermined simulated population proportion (from preliminarysurvey and accepted true value). Therefore, sampling survey method, statisticalformulae and sample size formulae for Simmons RRT model indicated strong validityand reliability.5.2For multiple-choice sensitive question with single response RRT model from astratified three-stage sampling survey, there were97,97, and96CIs for simulatedpopulation proportion out of100CIs contained the real population proportion of eachsensitive categorize respectively. Thus, sampling survey method, statistical formulaeand sample size formulae for multiple-choice sensitive question with single responseRRT model demonstrated a good degree of validity and reliability.5.3For quantitative additive RRT model under a stratified three-stage samplingdesign,99CIs for simulated population mean from100CIs contained the true value ofpopulation mean of sensitive quantitative characteristics. As a consequence, samplingsurvey method, statistical formulae and sample size formulae for additive RRT modelshowed high validity and reliability.Conclusion：1. Statistical formulae about eighteen survey methods for the estimators ofpopulation proportions/mean and corresponding population variance were given in thisstudy. And three RRT models in a three-stage sampling design were successfullyemployed to preliminarily investigate into FSW those who were at high risk of STD andAIDS in Xichang. Survey methods and statistical formulae proved to be effective and reliable, and got broad perspectives on application.2. We presented the preliminary results. FSW provided sexual services to theirclients multiple times and charged low fees for each sexual service. A great number ofFSW argued that prostitution should be legalized. Many FSW reported that they did notreceive HIV testing and condom had broken during sexual service. Government andhealth authorities should pay more attention to the less optimistic situation and look forsuitable way of settlement.3. Formulae for optimum sample size were deduced to provide sampling surveydesign on sensitive topic with scientific methods. Optimum sample size at each stagewas calculated in order to carry out three-stage sampling formal survey on sensitivefeatures among FSW using three RRT models, which enjoyed a broad prospect ofpopularization.4. For six survey methods, combination of three RRT models and two samplingmethods, sample size was determined on the basis of preliminary simulation data, andthen formal simulated sampling survey was conducted. The simulation results showedthat almost all95%CIs contained true value of population proportion/mean, indicatingsurvey methods, statistical formulae and sample size formulae were accurate andreliable.