Unbiased estimator modeling in unrelated dichotomous randomized response

Adediran, Adetola Adedamola; Adebola, Femi Barnabas; Ewemooje, Olusegun Sunday Article Unbiased estimator modeling in unrelated dichotomous randomized response Statistics in Transition New Series Provided in Cooperation with: Polish Statistical Association Suggested Citation: Adediran, Adetola Adedamola; Adebola, Femi Barnabas; Ewemooje, Olusegun Sunday (2020) : Unbiased estimator modeling in unrelated dichotomous randomized response, Statistics in Transition New Series, ISSN 2450-0291, Exeley, New York, Vol. 21, Iss. 5, pp. 119-132, https://doi.org/10.21307/stattrans-2020-058 This Version is available at: https://hdl.handle.net/10419/236808 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ STATISTICS IN TRANSITION new series, December 2020 Vol. 21, No. 5, pp. 119–132, DOI 10.21307/stattrans-2020-058 Received – 24.03.2019; accepted – 17.06.2020 Unbiased estimator modeling in unrelated dichotomous randomized response Adetola Adedamola Adediran1, Femi Barnabas Adebola2, Olusegun Sunday Ewemooje3 ABSTRACT The unrelated design has been shown to improve the efficiency of a randomized response method and reduces respondents’ suspicion. In the light of this, the paper proposes a new Unrelated Randomized Response Model constructed by incorporating an unrelated question into the alternative unbiased estimator in the dichotomous randomized response model proposed by Ewemooje in 2019. An unbiased estimate and variance of the model are thus obtained. The variance of the proposed model decreases as the proportion of the sensitive attribute π_A and the unrelated attribute π_U increases, in contrast to the earlier Ewemooje model, whose variance increases as the proportion of the sensitive attribute increases. The relative efficiency of the proposed model over the earlier Ewemooje model decreases as π_U increases when 0.1≤π_A≤ 0.3 and increases as π_U increases when 0.35≤π_A≤ 0.45. Application of the proposed model also revealed its efficiency over the direct method in estimating the prevalence of examination malpractices among university students;the direct method gave an estimate of 19.0%, compared to the proposed method’s estimate of 23.0%. Hence, the proposed model is more efficient than the direct method and the earlier Ewemooje model as the proportion of people belonging to the sensitive attribute increases. Key words: dichotomous, relative efficiency, sensitive attribute. 1. Introduction One of the problems in a survey is non-response; this is referred to as failure of getting the required information from a respondent. Non-response reduces the sample size as some respondents do not give the needed information and thereby making the 1 Department of Statistics, Federal University of Technology Akure, Nigeria. E-mail: [email protected]. ORCID: https://orcid.org/0000-0003-3176-7872. 2 Department of Statistics, Federal University of Technology Akure, Nigeria. E-mail: fbadeb[email protected]. ORCID: https://orcid.org/0000-0001-7790-1331. 3 Department of Statistics, Federal University of Technology Akure, Nigeria. E-mail: [email protected]. ORCID: https://orcid.org/0000-0003-3236-6018. 120 A. A. Adediran et al.: Unbiased estimator modeling … accuracy of the estimate to be compromised. Obtaining information about sensitive attributes lead to non-response or false response as participants in the sample may give false response or decide not to give an answer for diverse reasons. In order to reduce error due to this non-response bias, Warner in 1965 developed the Randomized Response Model (RRM) for estimating the proportion of people that belong to a sensitive attribute. Quite a number of authors have reviewed and expanded the work of Warner, including Horvitz et al. (1967) Unrelated Question Design, Greenberg et al. (1969) Unrelated Question Design with known distribution, Mangat and Singh (1990) Randomized Response Model (RRM), Hussain-Shabbir (2007) Dichotomous Randomized Response Model (DRRM), Adebola and Adepetun (2011), Tripartite Randomized Response Model (TRRM), Ewemooje (2017) Equal Probabilities of Protection, Adebola et al. (2017) Hybrid Tripartite Randomized Response Technique, Ewemooje et al. (2019a) Dichotomous Randomized Response Technique, Ewemooje et al. (2018) Stratified Hybrid Tripartite Randomized Response Technique. Also, Yu et al. (2008) worked on the Crosswise Model (CM) and Triangular Model (TM) while Fox et al. (2019) proposed Generalized Linear Mixed Models for Randomized Responses (GLMRR), among others. To test the applicability of the RRM; Jann et al. (2012) applied a modified RRM (Crosswise Model by Yu et al., 2008) to elicit information on plagiarism among German and Swiss students. They found out that RRM elicited more socially undesirable answers than direct questioning. Ewemooje et al., (2017) also used Improved Randomized Response Technique for two sensitive attributes (IRRT2) to show that RRM performs better that Direct Method of questioning (DM) by estimating prevalence of induced abortion and multiple sexual partners. Cobo et al., (2016) used RRM to investigate cannabis use by Spanish University students and then compared the result with DM. Their results revealed that RRM increases the response rate for cannabis use and that it is an efficient method. Furthermore, Ewemooje et al., (2019b) measured substance use disorder prevalence using RRM and DM; their findings showed that RRM estimated the disorder better with lower error than DM. Conversely, Hoglinger and Jann (2018) evaluated the variability of several variants of RRM and the crosswise model by comparing the respondents’ self-reports on cheating in dice games to actual cheating behaviour; their result showed that the RRM fails to reduce the level of misinterpreting compared to DM and none of the RRMs evaluated outperformed the conventional DM. Therefore, in this work we consider dichotomous randomized response design in the presence of unrelated questions; the estimator and variance are obtained and compared with the Dichotomous Randomized Response Model by Ewemooje et al., (2019a) using relative efficiency. Also, to verify more-is-better assumption, the proposed method and Direct Method (DM) were applied to the same subpopulation in a survey. STATISTICS IN TRANSITION new series, December 2020 121 2. Dichotomous Randomized Response Model by Ewemooje et al., (2019a) In their model, respondents were asked sensitive question directly, if he/she responds “yes” then he/she is not allowed to use the randomized device while if “no”, he/she is required to use the randomized device. Two randomized devices were used each consisting of two questions with different selection probabilities. A simple random sample with replacement sampling was adopted in their selection of the sample of size, n with α and β as any two positive real numbers such that q =   is the probability of using the first randomized device and 1q =   is the probability of using the second randomized device. If all respond truthfully, their population proportion of “yes” answers is given by: 𝑃󰇛𝑦𝑒𝑠󰇜𝛳 π  (1P󰇜󰇛1π 󰇜  󰇛1  P󰇜󰇛1  π󰇜 (1) where P is the probability of the sensitive attribute in randomized devices R and P is the probability of the sensitive attribute in randomized devices R. This yielded an unbiased estimate of the population proportion as: 𝜋 𝛉 󰇛󰇜   (2) The variance of their estimate was given as 𝑉󰇛𝜋󰇜󰇛󰇜 󰇛󰇜󰇛󰇜 󰇛󰇜 (3) 3. Proposed Model In sampling a finite population, the simple random sample with replacement was used to obtain the sample size of respondents who respond to sensitive questions using Randomized Response Model. Sensitive question was asked directly from the respondents. If “yes” answer is obtained, he/she does not need to use the randomized device but if he/she answers “no”, then he/she uses the randomized device. The two randomized devices R and R consists of two unrelated questions (the sensitive question A in which the interviewer is interested in with probability P, and nonsensitive attribute question B that is unrelated to the sensitive question A with probability, 1-P) each. Say: Sensitive question: “do you belong to a sensitive attribute A?” Non-sensitive question: “do you love soccer?” Two responses were considered for each of the two unrelated questions: “yes” and “no”, where α and β are positive real numbers such that q   ,αβ is the 122 A. A. Adediran et al.: Unbiased estimator modeling … probability of using R and 1q   ,αβ is the probability of using R with preset probabilities P and P respectively for each of the devices. Let 𝜋 be the true proportion of people that belongs to the sensitive attribute and 𝜋, the proportion of people that belongs to the unrelated non-sensitive attribute. If all respond truthfully as the devices provide protection for respondents, the population proportion of “yes” answers is given by: 𝑃󰇛𝑦𝑒𝑠󰇜𝛳 𝜋    󰇟P𝜋󰇛1P 󰇜𝜋󰇠   󰇟P𝜋󰇛1P 󰇜𝜋󰇠 (4) where P is the probability of the sensitive attribute in randomized devices R while P is the probability of the sensitive attribute in randomized devices R. Solving equation (4) further yield the estimate of the population proportion of the sensitive attribute 𝜋 󰇛 󰇜  󰇛󰇛󰇜 󰇜 󰇛  󰇜 (5) where 𝜃 𝑛𝑛 , 𝑛 is the number of respondents that answered "yes" to sensitive question while 𝑛 is the sample size. The proposed estimator, 𝜋, is an unbiased estimator of the population parameter 𝜋. 3.1. Variance Estimation The variance of the model is obtained as follows: v󰇛𝜋󰇜v󰇛 θ 󰇛𝛼  𝛽 󰇜  𝜋󰇛󰇛𝛼𝛽󰇜 𝛼𝑃 β𝑃 󰇜 󰇛𝛼  𝛽  𝛼𝑃β𝑃 󰇜 󰇜 v󰇛𝜋󰇜󰇛 󰇜 󰇛 󰇜 󰇛  󰇜 (6) where vθ   󰇛 󰇜  recall that 𝛳󰇛       󰇛󰇜 󰇛󰇜   󰇜, substituting this in equation (6), the variance of the proposed unbiased estimator is given as: v󰇛𝜋󰇜󰇝󰇛 󰇜󰇛  󰇜󰇞 󰇛  󰇜  󰇛 󰇜󰇛󰇛  󰇜󰇜 󰇛  󰇜 (7) Therefore, the variance of the proposed unbiased estimator can be estimated using: v󰇛𝜋󰇜 󰇝󰇛 󰇜 󰇛  󰇜󰇞 󰇛󰇜󰇛  󰇜  󰇛 󰇜󰇛 󰇛  󰇜󰇜 󰇛󰇜󰇛  󰇜 (8) STATISTICS IN TRANSITION new series, December 2020 123 4. Efficiency Comparison The proposed model will be more efficient than the conventional one if the condition for the relative efficiency holds: RE =         1 The relative efficiency of the proposed model over the conventional model were gotten for varying sample sizes (n), varying probabilities P and P of using the randomized devices at different values of 𝜋 and 𝜋. The comparison between the proposed estimator and Ewemooje et al. (2019a) estimator at different sample sizes in Table 1 shows that the proposed estimator is approximately ten (10) times more efficient than that due to Ewemooje et al. (2019a). As the sample size increases from 50 to 500, the variances due to Ewemooje et al. (2019a) estimator reduces from 0.0053 to 0.0005 while the proposed estimator reduces from 0.0005 to 0.0001. Therefore, as the sample sizes increases the variability reduces, this implies consistency of the two models. Considering a constant sample size at varying probabilities of selecting the randomized device, the variances due to Ewemooje et al. (2019a) estimator increases from 0.00131 to 0.00138, the proposed estimator increases from 0.00018 to 0.00022 while the relative efficiency reduces from 7.089 to 6.227 as shown in Table 2. Table 1. Relative efficiency comparison between the proposed model and Ewemooje et al. (2019a) model when π = 0.5; π= 0.5; P= 0.5; P= 0.5; α= 25; β= 35 for varying sample sizes (n). n 𝝅𝑨 𝝅𝑼 𝐏𝟏 𝐏𝟐 α β 𝐯󰇛𝝅  󰇜 𝐯󰇛𝝅  𝑨󰇜 RE 50 0.5 0.5 0.5 0.5 25 35 0.005333 0.000537 9.931034 100 0.5 0.5 0.5 0.5 25 35 0.002667 0.000269 9.931034 150 0.5 0.5 0.5 0.5 25 35 0.001778 0.000179 9.931034 200 0.5 0.5 0.5 0.5 25 35 0.001333 0.000134 9.931034 250 0.5 0.5 0.5 0.5 25 35 0.001067 0.000107 9.931034 300 0.5 0.5 0.5 0.5 25 35 0.000889 0.0000895 9.931034 350 0.5 0.5 0.5 0.5 25 35 0.000762 0.0000767 9.931034 400 0.5 0.5 0.5 0.5 25 35 0.000667 0.0000671 9.931034 450 0.5 0.5 0.5 0.5 25 35 0.000593 0.0000597 9.931034 500 0.5 0.5 0.5 0.5 25 35 0.000533 0.0000537 9.931034 124 A. A. Adediran et al.: Unbiased estimator modeling … Table 2. Relative efficiency comparison between the proposed model and Ewemooje et al. (2019a) model when π = 0.5; π= 0.5; α= 25; β= 35; n=200 for varying P and P n 𝝅𝑨 𝝅𝑼 𝐏𝟏 𝐏𝟐 α β 𝐯󰇛𝝅  󰇜 𝐯󰇛𝝅  𝑨󰇜 RE 200 0.5 0.5 0.1 0.9 25 35 0.001306 0.000184 7.089034 200 0.5 0.5 0.2 0.8 25 35 0.001312 0.000188 6.966797 200 0.5 0.5 0.3 0.7 25 35 0.001318 0.000193 6.848074 200 0.5 0.5 0.4 0.6 25 35 0.001325 0.000197 6.733115 200 0.5 0.5 0.5 0.5 25 35 0.001333 0.000201 6.622212 200 0.5 0.5 0.6 0.4 25 35 0.001342 0.000206 6.515705 200 0.5 0.5 0.7 0.3 25 35 0.001352 0.000211 6.413994 200 0.5 0.5 0.8 0.2 25 35 0.001363 0.000216 6.317551 200 0.5 0.5 0.9 0.1 25 35 0.001376 0.000221 6.226937 Table 3 shows that for varying π and π, P= 0.3; P= 0.7, the variance of the Ewemooje et al. (2019a) model increases at all values of π while the variance of the proposed model increases as π increases when 0.1  𝜋  0.3 and decreases as πU increases when 0.35  𝜋  0.45. The relative efficiency of the proposed model over Ewemooje et al. (2019a) reduces as π increases when 0.1  𝜋 0.3 and increases as π increases when 0.35  𝜋 0.45. However, as the sensitive character, 𝜋 increases, the relative efficiency increases with the values ranging from 1.0135 to 21.4409. The relative efficiency (RE) is greater than 1 for 𝜋 0.1 when 0.1  𝜋 0.4, RE greater than 1 for 𝜋 0.15 when 0.1  𝜋 0.7 and RE greater than 1 when 0.2  𝜋  0.45 at all values of π. This shows that the proposed model is more efficient than the Ewemooje et al. (2019a) model as the proportion of people belonging to the sensitive attribute increases. In Table 4, the probability of selecting the sensitive attribute was increased to 0.4 i.e. P= 0.4 while P= 0.6. The relative efficiency of the proposed model over Ewemooje et al. (2019a) also reduces as π increases when 0.1  𝜋 0.3 and increases as π increases when 0.35  𝜋 0.45. The relative efficiencies range between 1. 0284 and 18.8538. This shows that there is increase in efficiency as P increases. STATISTICS IN TRANSITION new series, December 2020 125 Table 3. Relative efficiency comparison between the proposed model and Ewemooje et al. (2019a) model when P= 0.3; P= 0.7; α= 25; β= 35; n=200 for varying π and π. 𝝅𝑨 𝝅𝑼 𝐯󰇛𝝅  󰇜 𝐯󰇛𝝅  𝑨󰇜 RE 𝝅𝑨𝝅𝑼𝐯󰇛𝝅  󰇜𝐯󰇛𝝅  𝑨󰇜 RE 0.1 0.000573 0.000345 1.662303 0.1 0.001146 0.000536 2.137366 0.2 0.000573 0.000413 1.387377 0.2 0.001146 0.000543 2.108094 0.3 0.000573 0.000481 1.191302 0.3 0.001146 0.000551 2.080862 0.4 0.000573 0.000549 1.044416 0.4 0.001146 0.000557 2.055543 0.1 0.5 0.000573 0.000616 0.930275 0.3 0.5 0.001146 0.000564 2.032025 0.6 0.000573 0.000683 0.839031 0.6 0.001146 0.00057 2.010205 0.7 0.000573 0.00075 0.764424 0.7 0.001146 0.000576 1.989992 0.8 0.000573 0.000816 0.702286 0.8 0.001146 0.000581 1.971302 0.9 0.000573 0.000882 0.649735 0.9 0.001146 0.000586 1.954063 1 0.000573 0.000948 0.604711 1 0.001146 0.000591 1.938206 0.1 0.000754 0.00043 1.752585 0.1 0.001226 0.000521 2.352242 0.2 0.000754 0.000483 1.559987 0.2 0.001226 0.000514 2.387847 0.3 0.000754 0.000536 1.406395 0.3 0.001226 0.000505 2.426134 0.4 0.000754 0.000588 1.281057 0.4 0.001226 0.000497 2.46731 0.15 0.5 0.000754 0.00064 1.176839 0.35 0.5 0.001226 0.000488 2.511608 0.6 0.000754 0.000692 1.088822 0.6 0.001226 0.000479 2.559291 0.7 0.000754 0.000744 1.013506 0.7 0.001226 0.00047 2.610657 0.8 0.000754 0.000795 0.948329 0.8 0.001226 0.00046 2.666043 0.9 0.000754 0.000846 0.891377 0.9 0.001226 0.00045 2.725833 1 0.000754 0.000896 0.841189 1 0.001226 0.000439 2.790465 0.1 0.000909 0.00049 1.854419 0.1 0.001282 0.000482 2.661543 0.2 0.000909 0.000528 1.721451 0.2 0.001282 0.000459 2.79495 0.3 0.000909 0.000566 1.607214 0.3 0.001282 0.000435 2.944671 0.4 0.000909 0.000603 1.508023 0.4 0.001282 0.000412 3.113841 0.2 0.5 0.000909 0.00064 1.421098 0.4 0.5 0.001282 0.000388 3.306451 0.6 0.000909 0.000676 1.344305 0.6 0.001282 0.000363 3.52767 0.7 0.000909 0.000713 1.275977 0.7 0.001282 0.000339 3.784304 0.8 0.000909 0.000749 1.214796 0.8 0.001282 0.000314 4.085509 0.9 0.000909 0.000784 1.159703 0.9 0.001282 0.000288 4.443899 1 0.000909 0.000819 1.109838 1 0.001282 0.000263 4.877343 0.1 0.00104 0.000526 1.978355 0.1 0.001313 0.000417 3.147851 0.2 0.00104 0.000548 1.896602 0.2 0.001313 0.000379 3.465365 0.3 0.00104 0.000571 1.822394 0.3 0.001313 0.00034 3.857866 0.4 0.00104 0.000593 1.754752 0.4 0.001313 0.000301 4.355411 0.25 0.5 0.00104 0.000614 1.692864 0.45 0.5 0.001313 0.000262 5.006603 0.6 0.00104 0.000636 1.636043 0.6 0.001313 0.000223 5.895497 0.7 0.00104 0.000657 1.583709 0.7 0.001313 0.000183 7.181136 0.8 0.00104 0.000677 1.53537 0.8 0.001313 0.000143 9.205181 0.9 0.00104 0.000698 1.490601 0.9 0.001313 0.000102 12.85955 1 0.00104 0.000718 1.449035 1 0.001313 0.0000612 21.44086 126 A. A. Adediran et al.: Unbiased estimator modeling … Table 4. Relative efficiency comparison between the proposed model and Ewemooje et al. (2019a) model when P= 0.4; P= 0.6; α= 25; β= 35; n=200 for varying π and π 𝝅𝑨 𝝅𝑼 𝐯󰇛𝝅  󰇜 𝐯󰇛𝝅  𝑨󰇜 RE 𝝅𝑨𝝅𝑼𝐯󰇛𝝅  󰇜𝐯󰇛𝝅  𝑨󰇜 RE 0.1 0.000586 0.000353 1.660952 0.1 0.001156 0.000548 2.107674 0.2 0.000586 0.000425 1.377199 0.2 0.001156 0.000557 2.073896 0.3 0.000586 0.000498 1.177079 0.3 0.001156 0.000566 2.042447 0.4 0.000586 0.00057 1.02837 0.4 0.001156 0.000574 2.013165 0.1 0.5 0.000586 0.000641 0.913521 0.3 0.5 0.001156 0.000582 1.985906 0.6 0.000586 0.000713 0.822151 0.6 0.001156 0.000589 1.960538 0.7 0.000586 0.000783 0.747731 0.7 0.001156 0.000597 1.936947 0.8 0.000586 0.000854 0.685946 0.8 0.001156 0.000603 1.915028 0.9 0.000586 0.000924 0.633833 0.9 0.001156 0.00061 1.894686 1 0.000586 0.000994 0.589287 1 0.001156 0.000616 1.875838 0.1 0.000766 0.000439 1.74396 0.1 0.001236 0.000535 2.310815 0.2 0.000766 0.000496 1.544414 0.2 0.001236 0.000528 2.341486 0.3 0.000766 0.000552 1.386723 0.3 0.001236 0.00052 2.37458 0.4 0.000766 0.000608 1.258975 0.4 0.001236 0.000513 2.410267 0.15 0.5 0.000766 0.000664 1.153387 0.35 0.5 0.001236 0.000505 2.448743 0.6 0.000766 0.000719 1.064657 0.6 0.001236 0.000496 2.490223 0.7 0.000766 0.000774 0.989051 0.7 0.001236 0.000487 2.534952 0.8 0.000766 0.000829 0.923862 0.8 0.001236 0.000478 2.583208 0.9 0.000766 0.000883 0.867078 0.9 0.001236 0.000469 2.635303 1 0.000766 0.000937 0.817176 1 0.001236 0.000459 2.691595 0.1 0.000921 0.0005 1.839616 0.1 0.001291 0.000496 2.601386 0.2 0.000921 0.000541 1.700959 0.2 0.001291 0.000473 2.727499 0.3 0.000921 0.000582 1.582692 0.3 0.001291 0.00045 2.868693 0.4 0.000921 0.000622 1.480635 0.4 0.001291 0.000426 3.027789 0.2 0.5 0.000921 0.000662 1.391678 0.4 0.5 0.001291 0.000402 3.208359 0.6 0.000921 0.000701 1.313461 0.6 0.001291 0.000378 3.414995 0.7 0.000921 0.00074 1.244157 0.7 0.001291 0.000353 3.653698 0.8 0.000921 0.000779 1.182331 0.8 0.001291 0.000328 3.932472 0.9 0.000921 0.000817 1.126842 0.9 0.001291 0.000303 4.262223 1 0.000921 0.000855 1.076768 1 0.001291 0.000277 4.658218 0.1 0.001051 0.000537 1.956943 0.1 0.00132 0.000432 3.053186 0.2 0.001051 0.000562 1.870325 0.2 0.00132 0.000394 3.354703 0.3 0.001051 0.000586 1.79212 0.3 0.00132 0.000354 3.725976 0.4 0.001051 0.00061 1.72118 0.4 0.00132 0.000315 4.194316 0.25 0.5 0.001051 0.000634 1.656556 0.45 0.5 0.00132 0.000275 4.80343 0.6 0.001051 0.000658 1.59746 0.6 0.00132 0.000235 5.627905 0.7 0.001051 0.000681 1.543228 0.7 0.00132 0.000194 6.80632 0.8 0.001051 0.000704 1.4933 0.8 0.00132 0.000153 8.628625 0.9 0.001051 0.000726 1.447199 0.9 0.00132 0.000112 11.82045 1 0.001051 0.000748 1.404517 1 0.00132 0.00007 18.85377 Table 5 shows that as PP  0.5, the variance of the Ewemooje et al. (2019a) model increases at all values of π from 0.00040 to 0.00133 while the variance of the proposed model decreases as π increases with values ranging from 0.00108 to 0.00008.