Appendix

```
!***************************************************************************
! This code was written to calculate the relative efficiencies in Table 2
!***************************************************************************
INCLUDE 'link_fnl_static.h'
REAL(8) THETA,C, PIA, K,E, VAR_RRT, VAR_NRRT
REAL(8) R(21,21)
INTEGER I,J,ITENO,N
OPEN (3,FILE="C:\Users\Oral\Documents\Fortran results\Table2.txt")
WRITE(3,*) ""
WRITE(3,*) ""
WRITE(3,50) "Calculated Relative Efficiencies for various p=theta and Popula-
tion Proportion (Pi_A) combinations"
WRITE(3,*)""
WRITE(3,*)"*****RE MATRIX*****"
WRITE(3,*)""
WRITE(3,*)"p=Theta value "
DO K=0.0,1.05,0.05
WRITE(3, 100,advance='no') K
END DO
WRITE(3, *) "
DO J=1,148
WRITE(3, 200,advance='no') '_'
END DO
WRITE(3, *) "
THETA=0.0
PIA=0.0
DO I=1,11
THETA=0.0
DO J=1,21
WRITE (*,*) 'theta=p=', THETA, 'Pi_A=', PIA
VAR_RRT=(PIA*(1-PIA))+((THETA*(1-THETA))/(((2*THETA)-1)**2.0))
VAR_NRRT=2*((((PIA*(1-PIA)))*(1-(2*THETA)+(2*(THETA**2.0))))+(2*(1-
PIA)*THETA*(1-THETA)))
WRITE (*,*) 'VAR_RRT=',VAR_RRT, 'VAR_NRRT=', VAR_NRRT
R(I,J)=VAR_RRT/VAR_NRRT
THETA=THETA+0.05
WRITE(3,100,advance='no') R(I,J);
END DO
WRITE (*,*)"
WRITE(3, *) "
PIA=PIA+0.1
END DO
100 FORMAT (F15.5,1X)
200 FORMAT (A1)
CLOSE(3)
END
```
Author details

LSUHSC School of Public Health, Biostatistics program, New Orleans, LA, USA

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: eoral@lsuhsc.edu

Surveying Sensitive Topics with Indirect Questioning DOI: http://dx.doi.org/10.5772/intechopen.84524

provided the original work is properly cited.

Evrim Oral

23

Surveying Sensitive Topics with Indirect Questioning DOI: http://dx.doi.org/10.5772/intechopen.84524

Appendix

Statistical Methodologies

INCLUDE 'link\_fnl\_static.h'

tion Proportion (Pi\_A) combinations"

WRITE(3,\*)"\*\*\*\*\*RE MATRIX\*\*\*\*\*"

WRITE(3,\*)"p=Theta value "

WRITE(3, 100,advance='no') K

WRITE(3, 200,advance='no') '\_'

PIA)\*THETA\*(1-THETA)))

THETA=THETA+0.05

100 FORMAT (F15.5,1X) 200 FORMAT (A1)

END DO WRITE (\*,\*)" WRITE(3, \*) " PIA=PIA+0.1 END DO

CLOSE(3) END

22

R(I,J)=VAR\_RRT/VAR\_NRRT

WRITE(3,100,advance='no') R(I,J);

WRITE (\*,\*) 'theta=p=', THETA, 'Pi\_A=', PIA

REAL(8) R(21,21) INTEGER I,J,ITENO,N

WRITE(3,\*) "" WRITE(3,\*) ""

WRITE(3,\*)""

WRITE(3,\*)""

END DO WRITE(3, \*) " DO J=1,148

END DO WRITE(3, \*) " THETA=0.0 PIA=0.0 DO I=1,11 THETA=0.0 DO J=1,21

DO K=0.0,1.05,0.05

!\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\* ! This code was written to calculate the relative efficiencies in Table 2 !\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

OPEN (3,FILE="C:\Users\Oral\Documents\Fortran results\Table2.txt")

WRITE(3,50) "Calculated Relative Efficiencies for various p=theta and Popula-

VAR\_RRT=(PIA\*(1-PIA))+((THETA\*(1-THETA))/(((2\*THETA)-1)\*\*2.0)) VAR\_NRRT=2\*((((PIA\*(1-PIA)))\*(1-(2\*THETA)+(2\*(THETA\*\*2.0))))+(2\*(1-

WRITE (\*,\*) 'VAR\_RRT=',VAR\_RRT, 'VAR\_NRRT=', VAR\_NRRT

REAL(8) THETA,C, PIA, K,E, VAR\_RRT, VAR\_NRRT

## Author details

Evrim Oral LSUHSC School of Public Health, Biostatistics program, New Orleans, LA, USA

\*Address all correspondence to: eoral@lsuhsc.edu

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## References

[1] Lensvelt-Mulders G. Surveying sensitive topics. In: de Leeuw ED, Hox JJ, Dillman DA, editors. International Handbook of Survey Methodology. New York: LEA, Taylor & Francis; 2008. pp. 1-17

[2] DeLeeuw E, Callegaro M, Hox J, Korendijk E, Lensvelt-Mulders G. The influence of advanced letters on response in telephone surveys: A metaanalysis. Public Opinion Quarterly. 2007;71(3):413-443

[3] Groves RM, Fowler FJ, Couper MP, Lepkowski JM, Singer E, Tourangeau R. Survey Methodology. 2nd ed. Hoboken, NJ: Wiley; 2009

[4] Link M, Mokdad A, Town M, Weiner J, Roe D. Improving Response Rates for the BRFSS: Use of Lead Letters and Answering Machine Messages. Paper presented at the annual conference of the American Association for Public Opinion Research, Nashville, TN; 2003

[5] Singer E, Groves RM, Dillman DA, Eltinger JL, Little RJA, editors. The Use of Incentives to Reduce Nonresponse in Household Surveys. Wiley-Interscience; 2002. pp. 163-177

[6] Spiers S, Oral E, Fontham E, Peters ES, Mohler JL, Bensen JT, et al. Modelling attrition and nonparticipation in a longitudinal study of prostate cancer. BMC Medical Research Methodology. 2018;18:60. DOI: 10.1186/ s 12874-018-0518-6

[7] Oral E, Simonsen N, Brennan C, Berken J, Su LJ, Mohler JL, et al. Unit nonresponse in a population-based study of prostate cancer. PLoS One. 2016;11(12):e0168364. DOI: 10.1371/ journal.pone.0168364

[8] Lessler JT, O'Reilly JM. Mode of interview and reporting of sensitive issues: Design and implementation of audio computer assisted self interviewing. NIDA Research Monograph. 1997;167:366-382

[9] van Griensven F, Naorat S, Kilmarx PH, et al. Palmtop-assisted selfinterviewing for the collection of sensitive behavioral data: Randomized trial with drug use urine testing. American Journal of Epidemiology. 2006;163(3):271-278

[16] Miller JD. A new survey technique for studying deviant behavior [PhD thesis]. The George Washington

Surveying Sensitive Topics with Indirect Questioning DOI: http://dx.doi.org/10.5772/intechopen.84524

> survey. International Journal of Statistics and Applications. 2016;6:

[26] Zimmerman RS, Langer LM. Improving estimates of prevalence rates of sensitive behaviors: The randomized lists technique and consideration of selfreported honesty. The Journal of Sex

Research. 1995;32(2):107-117

[27] Bernard HR, Johnsen EC, Killworth PD, Robinson S. Estimating the size of an average personal network and of an event subpopulation. In: Kochen M, editor. The Small World. Norwood, NJ: Albex Pub. Corp.; 1989. pp. 159-175

[28] Bernard HR, Johnsen EC, Killworth PD. Estimating the size of an average personal network and of an event subpopulation: Some empirical results. Social Science Research. 1991;20:

[29] Killworth PD, Johnsen EC, McCarty C, Shelley GA, Bernard HR. A social network approach to estimating seroprevalence in the United States. Social Networks. 1998;20:23-50

[30] Russell HB, Hallett T, Iovita A, Johnsen EC, Lyerla R, McCarty C, et al. Counting hard-to-count populations: The network scale-up method for public health sexually transmitted infections.

[31] Swensson B. Combined questions: A new survey technique for eliminating evasive answer bias (I)—Basic theory. Report No. 70 of the Errors in Surveys Research Project. Institute of Statistics,

2010;86(Supp. 2):ii11-ii15

University of Stockholm; 1974

[32] Takahasi K, Sakasegawa H. A randomized response technique without making use of any randomizing device. Annals of the Institute of Statistical Mathematics. 1977;29(1):1-8

[33] Tian GL, Tang M-L. Incomplete

Categorical Data Design:

177-187

109-121

[17] Miller J, Cisin I, Harrell A. A new technique for surveying deviant behavior: Item-count estimates of marijuana, cocaine, and heroin. Paper presented at the Annual Meeting of the American Association for Public Opinion Research. St. Petersburg, FL; 1986

[18] LaBrie JW, Earleywine M. Sexual risk behaviors and alcohol: Higher base rates revealed using the unmatchedcount technique. The Journal of Sex Research. 2000;37(4):321-326

[19] Biemer P, Brown G. Model-based estimation of drug use prevalence using item count data. Journal of Official Statistics. 2005;21(2):287-308

[20] Wolter F, Laier B. The effectiveness of the item count technique in eliciting valid answers. To sensitive questions: An evaluation in the context of selfreported delinquency. Survey Research

[21] Gervais WM, Najle MB. How many atheists are there? Social Psychological and Personality Science. 2018;9(1):3-10.

[22] Tsuchiya T. Domain estimators for the item count technique. Survey Methodology. 2005;31(1):41-51

[23] Chaudhuri A, Christofides TC. Item count technique in estimating the proportion of people with a sensitive feature. Journal of Statistical Planning and Inference. 2007;137:589-593

[24] Hussain Z, Ali Shah E, Shabir J. An alternative item count technique in sensitive surveys. Revista Colombiana

[25] Ibrahim F. An alternative modified item count technique in sampling

Estadística. 2012;35:39-54

25

Methods. 2014;8(3):153-168

DOI: 10.1177/1948550617707015

University; 1984

[10] Lind LH, Schober MF, Conrad FG, Reichert H. Why do survey respondents disclose more when computers ask the questions? Public Opinion Quarterly. 2013;77:888-935

[11] Schober MF, Conrad FG, Antoun C, Ehlen P, Fail S, Hupp AL, et al. Precision and disclosure in text and voice interviews on smartphones. PLoS One. 2015;10(6):e0128337

[12] Beach SR, Schulz R, Degenholtz HB, Castle NG, Rosen J, Fox AR, et al. Using audio computer-assisted selfinterviewing and interactive voice response to measure elder mistreatment in older adults: Feasibility and effects on prevalence estimates. Journal of Official Statistics. 2010;26(3):507-533

[13] Couper MP, Rowe B. Evaluation of a computer-assisted self-interview component in a computer-assisted personal interview survey. Public Opinion Quarterly. 1996;60:89-105

[14] Smith TW, Kim J. A review of survey data collection modes: With a focus on computerizations. Sociological Theory and Methods. 2015;20(2):185-200

[15] Raghavarao D, Federer WT. Block total response as an alternative to the randomized response method in surveys. Journal of the Royal Statistical Society: Series B: Methodological. 1979; 41:40-45

Surveying Sensitive Topics with Indirect Questioning DOI: http://dx.doi.org/10.5772/intechopen.84524

[16] Miller JD. A new survey technique for studying deviant behavior [PhD thesis]. The George Washington University; 1984

References

Statistical Methodologies

pp. 1-17

2007;71(3):413-443

NJ: Wiley; 2009

2002. pp. 163-177

s 12874-018-0518-6

journal.pone.0168364

24

[1] Lensvelt-Mulders G. Surveying sensitive topics. In: de Leeuw ED, Hox JJ, Dillman DA, editors. International Handbook of Survey Methodology. New York: LEA, Taylor & Francis; 2008.

audio computer assisted self interviewing. NIDA Research Monograph. 1997;167:366-382

2006;163(3):271-278

2013;77:888-935

2015;10(6):e0128337

[9] van Griensven F, Naorat S, Kilmarx PH, et al. Palmtop-assisted selfinterviewing for the collection of sensitive behavioral data: Randomized trial with drug use urine testing. American Journal of Epidemiology.

[10] Lind LH, Schober MF, Conrad FG, Reichert H. Why do survey respondents disclose more when computers ask the questions? Public Opinion Quarterly.

[11] Schober MF, Conrad FG, Antoun C, Ehlen P, Fail S, Hupp AL, et al. Precision

[12] Beach SR, Schulz R, Degenholtz HB, Castle NG, Rosen J, Fox AR, et al. Using audio computer-assisted selfinterviewing and interactive voice response to measure elder mistreatment in older adults: Feasibility and effects on prevalence estimates. Journal of Official

[13] Couper MP, Rowe B. Evaluation of a

[14] Smith TW, Kim J. A review of survey data collection modes: With a focus on computerizations. Sociological Theory and Methods. 2015;20(2):185-200

[15] Raghavarao D, Federer WT. Block total response as an alternative to the randomized response method in surveys. Journal of the Royal Statistical Society: Series B: Methodological. 1979;

41:40-45

and disclosure in text and voice interviews on smartphones. PLoS One.

Statistics. 2010;26(3):507-533

computer-assisted self-interview component in a computer-assisted personal interview survey. Public Opinion Quarterly. 1996;60:89-105

[2] DeLeeuw E, Callegaro M, Hox J, Korendijk E, Lensvelt-Mulders G. The influence of advanced letters on response in telephone surveys: A metaanalysis. Public Opinion Quarterly.

[3] Groves RM, Fowler FJ, Couper MP, Lepkowski JM, Singer E, Tourangeau R. Survey Methodology. 2nd ed. Hoboken,

[4] Link M, Mokdad A, Town M, Weiner J, Roe D. Improving Response Rates for the BRFSS: Use of Lead Letters and Answering Machine Messages. Paper presented at the annual conference of the American Association for Public Opinion Research, Nashville, TN; 2003

[5] Singer E, Groves RM, Dillman DA, Eltinger JL, Little RJA, editors. The Use of Incentives to Reduce Nonresponse in Household Surveys. Wiley-Interscience;

[6] Spiers S, Oral E, Fontham E, Peters ES, Mohler JL, Bensen JT, et al.

Modelling attrition and nonparticipation in a longitudinal study of prostate cancer. BMC Medical Research

Methodology. 2018;18:60. DOI: 10.1186/

[7] Oral E, Simonsen N, Brennan C, Berken J, Su LJ, Mohler JL, et al. Unit nonresponse in a population-based study of prostate cancer. PLoS One. 2016;11(12):e0168364. DOI: 10.1371/

[8] Lessler JT, O'Reilly JM. Mode of interview and reporting of sensitive issues: Design and implementation of [17] Miller J, Cisin I, Harrell A. A new technique for surveying deviant behavior: Item-count estimates of marijuana, cocaine, and heroin. Paper presented at the Annual Meeting of the American Association for Public Opinion Research. St. Petersburg, FL; 1986

[18] LaBrie JW, Earleywine M. Sexual risk behaviors and alcohol: Higher base rates revealed using the unmatchedcount technique. The Journal of Sex Research. 2000;37(4):321-326

[19] Biemer P, Brown G. Model-based estimation of drug use prevalence using item count data. Journal of Official Statistics. 2005;21(2):287-308

[20] Wolter F, Laier B. The effectiveness of the item count technique in eliciting valid answers. To sensitive questions: An evaluation in the context of selfreported delinquency. Survey Research Methods. 2014;8(3):153-168

[21] Gervais WM, Najle MB. How many atheists are there? Social Psychological and Personality Science. 2018;9(1):3-10. DOI: 10.1177/1948550617707015

[22] Tsuchiya T. Domain estimators for the item count technique. Survey Methodology. 2005;31(1):41-51

[23] Chaudhuri A, Christofides TC. Item count technique in estimating the proportion of people with a sensitive feature. Journal of Statistical Planning and Inference. 2007;137:589-593

[24] Hussain Z, Ali Shah E, Shabir J. An alternative item count technique in sensitive surveys. Revista Colombiana Estadística. 2012;35:39-54

[25] Ibrahim F. An alternative modified item count technique in sampling

survey. International Journal of Statistics and Applications. 2016;6: 177-187

[26] Zimmerman RS, Langer LM. Improving estimates of prevalence rates of sensitive behaviors: The randomized lists technique and consideration of selfreported honesty. The Journal of Sex Research. 1995;32(2):107-117

[27] Bernard HR, Johnsen EC, Killworth PD, Robinson S. Estimating the size of an average personal network and of an event subpopulation. In: Kochen M, editor. The Small World. Norwood, NJ: Albex Pub. Corp.; 1989. pp. 159-175

[28] Bernard HR, Johnsen EC, Killworth PD. Estimating the size of an average personal network and of an event subpopulation: Some empirical results. Social Science Research. 1991;20: 109-121

[29] Killworth PD, Johnsen EC, McCarty C, Shelley GA, Bernard HR. A social network approach to estimating seroprevalence in the United States. Social Networks. 1998;20:23-50

[30] Russell HB, Hallett T, Iovita A, Johnsen EC, Lyerla R, McCarty C, et al. Counting hard-to-count populations: The network scale-up method for public health sexually transmitted infections. 2010;86(Supp. 2):ii11-ii15

[31] Swensson B. Combined questions: A new survey technique for eliminating evasive answer bias (I)—Basic theory. Report No. 70 of the Errors in Surveys Research Project. Institute of Statistics, University of Stockholm; 1974

[32] Takahasi K, Sakasegawa H. A randomized response technique without making use of any randomizing device. Annals of the Institute of Statistical Mathematics. 1977;29(1):1-8

[33] Tian GL, Tang M-L. Incomplete Categorical Data Design:

Non-Randomized Response Techniques for Sensitive Questions in Surveys. Boca Raton, FL: Chapman & Hall/CRC; 2014

[34] Warner SL. Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statistical Association. 1965; 60:63-69

[35] Chaudhuri A, Mukerjee R. Randomized Response: Theory and Techniques. Statistics: Textbooks and Monographs. Vol. 85. New York: Marcel Dekker, Inc.; 1988

[36] Greenberg RG, Abul-Ela ALA, Simmons WR, Horvitz DG. The unrelated question randomized response model: Theoretical framework. Journal of the American Statistical Association. 1969;64(326):520-539

[37] Greenberg RG, Keubler RT, Abernathy JR, Horvitz DG. Application of randomized response technique in obtaining quantitative data. Journal of the American Statistical Association. 1971;66:243-250

[38] Gupta SN. Qualifying the sensitivity level of binary response personal interview survey questions. Journal of Combinatorics, Information & System Sciences. 2001;26:101-109

[39] Gupta SN, Gupta RC, Singh S. Estimation of sensitivity level of personal interview survey questions. Journal of Statistical Planning and Inference. 2002;100:239-247

[40] Gupta SN, Thornton B, Shabbir J, Singhal S. A comparison of multiplicative and additive optional RRT models. Journal of Statistical Theory and Applications. 2006;5: 226-239

[41] Gupta SN, Shabbir J, Sehra S. On the estimation of population mean and sensitivity in a two-stage optional randomized response model. Journal of

the Indian Society of Agricultural Statistics. 2010;61:164-168

[42] Yu B, Jin Z, Tian J, Gao G. Estimation of sensitive proportion by randomized response data in successive sampling. Computational and Mathematical Methods in Medicine. 2015;18(2015):172918

population use of the randomized response technique to estimate prevalence. The Journal of Adolescent

Surveying Sensitive Topics with Indirect Questioning DOI: http://dx.doi.org/10.5772/intechopen.84524

[51] Srivastava R, Nigam AK, Singh N. Application of randomized response techniques in estimation of prevalence of child sexual abuse. Statistics and

Health. 1992;13:281-285

Applications. 2015;13:37-45

and Statistics. 2016;2:1-9

[52] Chhabra A, Dass BK, Gupta S. Estimating prevalence of sexual abuse by an acquaintance with an optional unrelated question RRT model. The North Carolina Journal of Mathematics

[53] van den Hout A, Bockenholt U, Van der Heijden PGM. Estimating the prevalence of sensitive behavior and cheating with a dual design for direct questioning and randomized response. Applied Statistics. 2010;59:723-736

[54] Jing L, Lu Q, Cui Y, Yu H, Wang T. Combining the randomized response technique and the network scale-up method to estimate the female sex worker population size: An exploratory study. Public Health. 2018;160:81-86

[55] Erdmann A. Non-randomized response models: An experimental application of the triangular model as an

indirect questioning method for sensitive topics. Methods, Data Analyses. 2018;13(1):139-167. DOI:

10.12758/mda.2018.07

27

[43] Sihm JS, Chhabra A, Gupta S. An optional unrelated question RRT model. Involve: A Journal of Mathematics. 2016;2(9):195-209

[44] Gupta SN, Shabbir J. Sensitivity estimation for personal interview survey questions. Statistica. 2004;64:643-653

[45] Ardah IH, Oral E. Model selection in randomized response techniques for binary responses. Communication in Statistics-Theory and Methods. 2018; 47(14):3305-3323

[46] Chaudhuri A. Randomized Response and Indirect Questioning Techniques in Surveys. Boca Raton, FL: Chapman and Hall/CRC Taylor and Francis Group. 2011

[47] Chaudhuri A, Christofides TC. Indirect Questioning in Sample Surveys. Berlin, Heidelberg: Springer-Verlag; 2013. DOI: https://doi.org/10.1007/987- 3-642-36276-7

[48] Lensvelt-Mulders G, Hox JJ, van der Heijden P, Maas C. Meta-analysis of randomized response: 35 years of validation studies. Sociological Methods & Research. 2005;33:319-348

[49] Kirtadze I, Otiashvili D, Tabatadze M, Vardanashvili I, Stutua L, Zabransky T, et al. Republic of Georgia estimates for prevalence of drug use: Randomized response techniques suggest underestimation. Drug and Alcohol Dependence. 2018;187:300-304

[50] Fisher M, Kupferman LB, Lesser M. Substance use in a school-based clinic

Surveying Sensitive Topics with Indirect Questioning DOI: http://dx.doi.org/10.5772/intechopen.84524

population use of the randomized response technique to estimate prevalence. The Journal of Adolescent Health. 1992;13:281-285

Non-Randomized Response Techniques for Sensitive Questions in Surveys. Boca Raton, FL: Chapman & Hall/CRC; 2014

the Indian Society of Agricultural Statistics. 2010;61:164-168

[42] Yu B, Jin Z, Tian J, Gao G. Estimation of sensitive proportion by randomized response data in successive

sampling. Computational and Mathematical Methods in Medicine.

[43] Sihm JS, Chhabra A, Gupta S. An optional unrelated question RRT model. Involve: A Journal of Mathematics.

[44] Gupta SN, Shabbir J. Sensitivity estimation for personal interview survey questions. Statistica. 2004;64:643-653

[45] Ardah IH, Oral E. Model selection in randomized response techniques for binary responses. Communication in Statistics-Theory and Methods. 2018;

2015;18(2015):172918

2016;2(9):195-209

47(14):3305-3323

Francis Group. 2011

3-642-36276-7

[46] Chaudhuri A. Randomized Response and Indirect Questioning Techniques in Surveys. Boca Raton, FL: Chapman and Hall/CRC Taylor and

[47] Chaudhuri A, Christofides TC. Indirect Questioning in Sample Surveys. Berlin, Heidelberg: Springer-Verlag; 2013. DOI: https://doi.org/10.1007/987-

[48] Lensvelt-Mulders G, Hox JJ, van der Heijden P, Maas C. Meta-analysis of randomized response: 35 years of validation studies. Sociological Methods

[49] Kirtadze I, Otiashvili D, Tabatadze M, Vardanashvili I, Stutua L, Zabransky T, et al. Republic of Georgia estimates for prevalence of drug use: Randomized response techniques suggest underestimation. Drug and Alcohol Dependence. 2018;187:300-304

[50] Fisher M, Kupferman LB, Lesser M. Substance use in a school-based clinic

& Research. 2005;33:319-348

[34] Warner SL. Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statistical Association. 1965;

[35] Chaudhuri A, Mukerjee R. Randomized Response: Theory and Techniques. Statistics: Textbooks and Monographs. Vol. 85. New York: Marcel

[36] Greenberg RG, Abul-Ela ALA, Simmons WR, Horvitz DG. The unrelated question randomized

[37] Greenberg RG, Keubler RT,

response model: Theoretical framework. Journal of the American Statistical Association. 1969;64(326):520-539

Abernathy JR, Horvitz DG. Application of randomized response technique in obtaining quantitative data. Journal of the American Statistical Association.

[38] Gupta SN. Qualifying the sensitivity level of binary response personal interview survey questions. Journal of Combinatorics, Information & System

60:63-69

Dekker, Inc.; 1988

Statistical Methodologies

1971;66:243-250

Sciences. 2001;26:101-109

Singhal S. A comparison of

226-239

26

[39] Gupta SN, Gupta RC, Singh S. Estimation of sensitivity level of personal interview survey questions. Journal of Statistical Planning and Inference. 2002;100:239-247

[40] Gupta SN, Thornton B, Shabbir J,

[41] Gupta SN, Shabbir J, Sehra S. On the estimation of population mean and sensitivity in a two-stage optional randomized response model. Journal of

multiplicative and additive optional RRT models. Journal of Statistical Theory and Applications. 2006;5:

[51] Srivastava R, Nigam AK, Singh N. Application of randomized response techniques in estimation of prevalence of child sexual abuse. Statistics and Applications. 2015;13:37-45

[52] Chhabra A, Dass BK, Gupta S. Estimating prevalence of sexual abuse by an acquaintance with an optional unrelated question RRT model. The North Carolina Journal of Mathematics and Statistics. 2016;2:1-9

[53] van den Hout A, Bockenholt U, Van der Heijden PGM. Estimating the prevalence of sensitive behavior and cheating with a dual design for direct questioning and randomized response. Applied Statistics. 2010;59:723-736

[54] Jing L, Lu Q, Cui Y, Yu H, Wang T. Combining the randomized response technique and the network scale-up method to estimate the female sex worker population size: An exploratory study. Public Health. 2018;160:81-86

[55] Erdmann A. Non-randomized response models: An experimental application of the triangular model as an indirect questioning method for sensitive topics. Methods, Data Analyses. 2018;13(1):139-167. DOI: 10.12758/mda.2018.07

Chapter 3

Abstract

be suggested to use it.

1. Introduction

for determining acreage.

29

Subject Classification: 62D05

Development of Estimation

Procedure of Population Mean in

Two-Phase Stratified Sampling

Partha Parichha, Kajla Basu and Arnab Bandyopadhyay

This article describes the problem of estimation of finite population mean in two-phase stratified random sampling. Using information on two auxiliary variables, a class of product to regression chain type estimators has been proposed and its characteristic is discussed. The unbiased version of the proposed class of estimators has been constructed and the optimality condition for the proposed class of estimators is derived. The efficacy of the proposed methodology has been justified through empirical investigations carried over the data set of natural population as well as the data set of artificially generated population. The survey statistician may

Keywords: stratified random sampling, double sampling, auxiliary variables, chain type estimators, bias, mean square error, efficiency, AMS 2000 Mathematics

In this present paper we have made use of Auxiliary information extracted from the variables having correlation with study variable. Auxiliary information may be utilized at planning, design and estimation stages to develop improved estimation procedures in sample surveys. Sometimes, information on auxiliary variable may be readily available for all the units of population; for example, tonnage (or seat capacity) of each vehicle or ship is known in survey sampling of transportation and number of beds available in different hospitals may be known well in advance in health care surveys. If such information lacks, it is sometimes, relatively cheap to take a large preliminary sample where auxiliary variable alone is measured, such practice is applicable in two-phase (or double) sampling. Two-phase stratified sampling happens to be a powerful and cost effective (economical) technique for obtaining the reliable estimate in first-phase (preliminary) sample for the unknown parameters of the auxiliary variables. For example, Sukhatme [1] mentioned that in a survey to estimate the production of lime crop based on orchards as sampling units, a comparatively larger sample is drawn to determine the acreage under the crop while the yield rate is determined from a sub sample of the orchards selected

In order to construct an efficient estimator of the population mean of the auxiliary variable in first-phase (preliminary) sample, Chand [2] introduced a technique

#### Chapter 3
