1. About a survey method on sensitive matters in human life Jong-Min Kim Discipline of Statistics Division of Science & Mathematics University of Minnesota, Morris

2. Introduction  Why do we need Randomized Response (RR) technique ? Direct question queries often fail to get reliable data on such sensitive matters of human life. Warner (1965) invented RR technique which is designed to encourage cooperation and truthful replies to socially undesirable questions.

3.  Qualitative RR technique (Dichotomous nature). Warner’s RR technique (1965, JASA). Do you belong to a sensitive group? Don’t you belong to a sensitive group? An unrelated question RR technique (1969, JASA). Do you belong to a sensitive group? Do you belong to a non-sensitive group? Introduction (cont’d)

4.  How does RR technique work ? Campbell and Joiner (1973) - Apply an unrelated question RR technique to estimate the proportion of regular “pot” smokers. - Use the table of random numbers(00-99). Introduction (cont’d) 41 % of students use pot at least once a week (from RR Technique.) 38 % of students were regular pot smokers. (the students privately verify the technique.) P(“yes” answer) = P(Sensitive question is chosen)*P(Yes | Sensitive question) + P(Nonsensitive question is chosen)*P(Yes | Nonsensitive question).

5. Introduction (cont’d)  Quantitative RR technique Unrelated question RR technique (1971, JASA). - Modify a qualitative unrelated question RR technique to get a quantitative data. 1. How many abortions have you had during your lifetime? 2. If a woman has to work full-time to make a living, how many children do you think she should have? - The survey can be designed more efficiently when the mean and variance of the innocuous question are known in advance.

6. Introduction (cont’d)  Quantitative RR technique A new model for RR (Eriksson, 1973). - Modify a quantitative unrelated question RR technique. - Use a deck of two kinds of cards. 1. Give a true answer ! 2. Say the figure that is on your card. A new discrete RR model (Liu and Chow, 1976). - Use the Hopkins’ Randomizing Device which consists of red balls, and white balls with a discrete number. - Use the predetermined combination of balls instead of asking innocuous question.

7. Introduction (cont’d) 1 1 2 4 1 3 3 2 2 1 25 1 1 3 2 1 1 2 1 1 2 4 1 3 3 2 2 1 2 5 1 1 3 2 1 1 2 Hopkins’ Randomizing Device.

8. Introduction (cont’d) Tracy and Fox (1981) - Conduct a field-validation of a quantitative RR technique. - Compare self-report of past criminal behavior (Arrests) Direct question vs. RR technique. - Use the Hopkins’ Device presented by Liu and Chow(1976).  Validation of RR technique Mean Reported Arrests Mean Official Arrests Two or more Arrests Sample size (n) RR technique 84 2.7286 3.2024 Direct question 40 1.5500 3.3500

9. Research Topics  The First Research Topic: A Stratified Warner’s Randomized Response Technique.  The Second Research Topic: A Mixed Randomized Response Technique.  The Third Research Topic: A Multinomial Distribution Approach to A Quantitative Randomized Response Technique.

10. The Third Research  The Third Research Topic: A Multinomial Distribution Approach to A Quantitative Randomized Response Technique. Why must the problem be addressed? By using the Hopkins’ randomizing device, we can derive a multinomial distribution for sensitive categories. It means that we can easily estimate the proportion of each category. Thus the attempt is useful to treat ordinal data in quantitative manner by assigning values to the categories. For example, if we apply the model to a criminology then we can assign a value like this; one arrest (1), two arrests (2), more than two arrests (3).

11. Focus on the Third Research  Estimate the proportions of sensitive categories using a multinomial distribution.  Derive the true estimates of the sensitive categories through the random transformation of a lying model.  Apply quantitative RR to a multiple contrast method by Goodman (1964).  Present a Pearson product-moment Correlation between two sensitive variables.

12. Contents of the Third Research, and as follow: ~ MULT.. Our RR technique utilizes the Hopkins’ device to derive a multinomial distribution for sensitive categories (A 1,A 2,…A k+1 ), non-sensitive categories (B 1,B 2,…B k+1 ) and total categories (T 1,T 2,…T k+1 ). Then we can define a multinomial distribution of

13. Contents of the Third Research The estimate of a proportion of sensitive category h is where and. The estimate of variance is. The estimate of covariance is where.

14. Contents of the Third Research What is the lying model of Mukhopadhay (1980) ? Assume that there is a sensitive category for such that has no social stigma and is more social stigma as i increases. Let R ij denote the probability that a person of category A i announces himself or herself as one of category A j. Intuitively, we can stipulate the following:.

15. Contents of the Third Research Then Rewriting the above one like this:.. If R is nonsingular,. The estimator ofis given by.

16. Contents of the Third Research Observed outcomes (Estimated Expected Outcomes) Total Female Group155(57)10(10.8)10(7.2)75 Female Group275(76)21(14.4)4(9.6)100 Female Group360(57)5(10.8)10(7.2)75 Total1903624250 Table 1. Observed and Estimated expected outcomes for a three-sample.

17. Contents of the Third Research We want to test the null hypothesis. The test statistics is The set of 95% confidence intervals about two different proportions is given by

18. Contents of the Third Research Not Significant Significant Not Significant Significant

19. Contents of the Third Research How to estimate the correlation between two sensitive variables ? For two different sensitive questions, we are going to use two Hopkins’ devices which have different ratios of red balls and green balls with a designated number. Bourke (1981) presented a multivariate RR design for category proportions. By using the multivariate RR design, we can derive the true proportion that a respondent belongs in the category i for the first question and belongs in the category j for the second one.

20. Contents of the Third Research The product moment correlation between two sensitive variables and is where and.,, and

21. Contents of the Third Research Observed Outcomes A(c 1 )=1A(c 2 )=2A(c 3 )=3Total A(r 1 )=1459660 A(r 2 )=2185225 A(r 3 )=3103215 731710100 Table 2. The number of respondents who belong to two different sensitive categories.

22. Contents of the Third Research The proportion that a respondent of category i and category j announces a respondent as one of category k and one of category l is The proportioncan be expressed as the Matrix form:..

23. Contents of the Third Research From Table 2, each can be obtained as follows: By rewriting in terms of and, Supposeand...

24. Contents of the Third Research Sinceis nonsingular and is known, each true proportion is..

25. Contents of the Third Research Suppose that we get and from the randomized randomized response technique. Then the correlation between two sensitive variables is. The relationship between two sensitive variables strongly correlated each other..

26. Discussion on three researches New qualitative randomized response technique - A stratified Warner’s RR technique. - A mixed randomized response technique. - A stratified mixed RR technique. New quantitative randomized response technique. - A discrete RR technique using a multinomial distribution. The purpose of this research is to cover various randomized response techniques and investigate the properties of RR technique.

27. Thank you.