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Communications of the Korean Statistical Society 2009, Vol. 16, No. 1, 169–183

Unrelated Question Model in Sensitive Multi-Character Surveys Sukhjinder Singh Sidhua , Mohan Lal Bansal a , Jong-Min Kim 1,b , Sarjinder Singhc a

b

Dept.of Mathematics, Statistics and Physics, Punjab Agricultural Univ., Div. of Science and Mathematics, Univ. of Minnesota, c Dept. of Mathematics, Texas A&M Univ.

Abstract The simplicity and wide application of Greenberg et al. (1971) prompts to propose a set of alternative estimators of population total for multi-character surveys that elicit simultaneous information on many sensitive study variables. The proposed estimators take into account the already known rough value of the correlation coefficient between Y(the characteristic under study) and p(the measure of size). These estimators are biased, but it is expected that the extent of bias will be smaller, since the proposed estimators are suitable for situations in between those optimum for the usual estimators and the estimators based on multi-characters for no correlation. The relative efficiency of the proposed estimators has been studied under a super population model through empirical study. It has been found through simulation study that a choice of an unrelated variable in the Greenberg et al. (1971) model could be made based on its correlation with the auxiliary variable used at estimation stage in multi-character surveys.

Keywords: Total estimation, RRT, sensitive multi-characteristics, mean square error, super population model, cost aspects and empirical study.

1. Introduction The well-known Hansen and Hurwitz (1943) estimator of population total for probability proportional to size and with replacement sampling (PPSWR) is given by 1 X yi b YHH = , n i=1 pi n

 N −1 X  xi  . where pi = xi 

(1.1)

i=1

In sample surveys of many variables, some of the study variables may be poorly correlated with the selection probabilities. In this the use of usual estimators available in literature results in larger variance. Rao (1966) has provided alternative estimators when the study variable and size measure are unrelated and demonstrated that these alternative estimators are more efficient though biased. But Rao’s (1966) model is not commonly encountered in practice since the correlation is not always zero. Bansal and Singh (1985) developed a transformed estimator of population total suitable for the characteristics covering entire range of positive correlation. Amahia et al. (1989) suggested simple alternatives to the transformations in Bansal and Singh (1985). The transformations of selection probabilities used are as follows: 1

Corresponding author: Associate Professor, Statistics Discipline, Division of Science and Mathematics, University of Minnesota, Morris, MN 56267-2134, USA. E-mail: [email protected]

Sukhjinder Singh Sidhu, Mohan Lal Bansal, Jong-Min Kim, Sarjinder Singh

170

p∗i0 =

1 , N

p∗i1 = 1 +

1 N

[Rao (1966)]

(1.2)

[Bansal and Singh (1985)]

(1.3)

[Amahia et al. (1989)]

(1.4)

[Amahia et al. (1989)]

(1.5)

[Amahia et al. (1989)]

(1.6)

[Grewal et al. (1997)].

(1.7)

!1−ρ (1 + pi )ρ − 1,

1 (1 − ρ) + ρpi , N !1−ρ 1 ∗ pρi , pi3 = N ( )−1 ρ p∗i4 = N (1 − ρ) + , pi 1 1 1  p∗i5 = 1 − ρ 3 + ρ 3 pi , N p∗i2 =

On the basis of these transformations, following types of estimators of population total Y under PPSWR sampling are available in the literature: n   1 X yi b Ypps = , h n i=1 p∗ih

h = 1, 2, 3, 4, 5.

(1.8)

The transformations p∗ih (h = 1, 2, 3, 4, 5) at (1.3) to (1.7) of the selection probabilities pi are useful for positive correlation between yi and pi variables, whereas transformation (1.2) is useful under no correlation situation. Interestingly for ρ = 0, p∗ih (h = 1, 2, 3, 4, 5) reduce to p∗i0 at (1.2) and for ρ = 1 these transformations reduce to original selection probabilities pi . For detail one can refer to Arnab (2001) and Singh (2003). The surveys on human population had established the fact that the direct question about sensitive characters often result in either refusal to respond or falsification of the answer. This can bias the estimates. Warner (1965) developed an interviewing procedure designed to reduce or eliminate this bias and called it as Randomized Response Technique(RRT). It is beneficial to combine multi-characteristics and RRT. Bansal et al. (1994) and Grewal et al.(1997) had discussed the multicharacteristics in RRT to estimate population total. It was felt that the confidence of the respondents in anonymity provided by RRT and hence reliability of their responses, might be further enhanced if one of the two question belong to non sensitive, innocuous attribute unrelated to the sensitive characteristics. Greenberg et al. (1971) developed the work for quantitative responses and found that his unrelated question technique was more efficient than the Warner (1965) model.

2. UQ Model In the quantitative unrelated question (UQ) random response model, using two questions, the overall distribution of responses is comprised of numerical answers to both questions, the answers being indistinguishable as to question. This distribution is a mixture of two pure distributions, which must be statistically separated to provide meaningful estimates of the parameters of interest. The population means of both the sensitive(Y) and unrelated non-sensitive(U) variables are µy and µU with their respective variances σ2y and σ2U .

Unrelated Question Model in Sensitive Multi-Character Surveys

171

When the value of U(total of unrelated character) is known in advance we select one sample of size n. The respondent in the sample is provided with a randomization device, with probability T and (1 − T ), respectively, consisting of sensitive and non sensitive statements: (i) About how much money in dollars did the head of household, earn last year? (ii) About how much average money in dollars do you think the head of a household of your size earns in a year? The respondent selects randomly one of the two statements, unobserved by the interviewer, and reports the answer. Let response from ith individual in the sample for the characteristic under study be denoted by ri . ri = T yi + (1 − T ) µi (2.1) with

V(ri ) = T (1 − T ) (yi − µi )2 .

(2.2)

Keeping in view the importance of this model, we extend the method to multi-character surveys to propose estimators of population total. The behavior of the proposed estimators has been examined under the super population model given below.

3. Super Population Model A general super population model for sensitive characteristic under study is: Yi = βpi + ei ,

i = 1, 2, . . . , N,

(3.1)

where ei ’s are the error terms such that:

Em and



Em (ei | pi ) = 0,  ei e j | pi p j = 0

  Em e2i | pi = apgi ,

(3.2) (3.3) a > 0, g ≥ 0.

(3.4)


Here Em e2i | pi is the residual variances of Y for given pi . The expected value of this residual variance in the super population model is given by:   Em apgi = aEm (pgi ) (3.5) and when the infinite super population is simulated by a finite large population of N units having the same characteristics it will be reduced to: N   a X E apgi = pg . N i=1 i

  Also the expected value of residual variance is known to be given by σ2y 1 − ρ2 . Thus we have: N   a X g pi = σ2y 1 − ρ2 N i=1

(3.6)

(3.7)

Sukhjinder Singh Sidhu, Mohan Lal Bansal, Jong-Min Kim, Sarjinder Singh

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or σ2y =

N a X g p N i=1 i

1 − ρ2

.

(3.8)

The value of the regression coefficient is given by: σ2y

ρ2 β =ρ 2 = σ p 1 − ρ2 2

where

2

  N  a X pgi    , N i=1 σ2p 

 !2  N   P X pi  N   1  i=1  . σ2p =  p2i − N  i=1 N   

(3.9)

(3.10)

The super population model for unrelated non-sensitive question is: Ui = β∗ pi + e∗i ,

i = 1, 2, . . . , N,

(3.11)

where e∗i ’s are the error terms satisfying all the conditions at (3.2), (3.3) and (3.4). It is assumed for simplicity that means of Yi and Ui are different but the residual variances of U  for p = pi , i.e. E e∗2 | p is same as of Y. i i Similarly  g N σ2 ρ∗2  a X pi  ∗2 ∗2 U (3.12) β =ρ =  . σ2p 1 − ρ∗2  N i=1 σ2p  We first obtain the estimator of population total for PPSWR.


4. Estimator b Y1 When the value of U(total of unrelated character) is known in advance we select one sample of size n. The respondents in the sample are provided with a randomization device consisting of sensitive and non sensitive statements with probability T and (1 − T ) respectively. The respondent selects randomly one of the two statements, unobserved by the interviewer and reports the answer. Let response from ith individual in the sample for the characteristic under study be denoted by ri . ri = T yi + (1 − T ) µi . The estimator of population total (b Y1 ) for PPSWR is obtained as  n    X ri   1 1   b . − (1 − T ) U  Y1 =     T n pi

(4.1)

(4.2)

i=1

The variance of the randomized response of ith individual is V (ri ) = T (1 − T ) (yi − µi )2 .

(4.3)

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Using this result it can be shown easily that the estimator (b Y1 ) is unbiased. The variance of the estimator (b Y1 ) is given in the following theorem. Theorem 1. The variance of the estimator (b Y1 ) given in (4.2) is given by ( ! !)2 N N   1−T X (Yi − Ui )2 Yi 1 X Ui b (1 ) V Y1 = + pi T −Y + −T −U . nT i=1 pi pi pi nT 2 i=1

(4.4)

Proof: Please see the Appendix A. We now extend the theory for the estimator obtained above to propose the estimators of population total in case of multi-character surveys. ¤


5. Proposed Estimator b Y2 The proposed estimators of population total (b Y2 )h for multi-characteristics are given by  n    X ri     1 1   b Y2 =  − (1 − T ) U  , ∗   h  T n pih

(5.1)

i=1

where p∗ih are defined in (1.2) to (1.7). The proposed estimators are biased and the bias in the the estimators (b Y2 )h is given by !( ) N X   (1 − T ) pi b B Y2 = − 1 Yi + Ui . (5.2) h p∗ih T i=1 One can easily see that the variance of the estimator (b Y2 )h is given by N   1 − T X (Yi − Ui )2 pi V b Y2 = h nT i=1 p∗2 ih   N 2  N    1 X (T Yi + (1 − T ) Ui )2 pi  X (T Yi + (1 − T ) Ui ) pi    + − .    ∗  2 ∗2    pih nT i=1 pih i=1

(5.3)

Please see the full derivation of the equation (5.3) in the Appendix A. To obtain the expected Mean Square Error (MSE) of proposed estimators (b Y2 )h under super population model we have the following theorem. Theorem 2. The expected value of MSE of (b Y2 )h under the superpopulation model is h   i 1 Em MSE b Y2 = A1 + A2 , h n where   ! N N X X   1 − T  pg+1 p3i i 2 ∗ (β − β )  + 1 + 2a A1 =  ∗2 ∗2 T T2 p pih i=1 ih i=1    N 2 2  N X X pi    p3i  ∗ 2    + (βT + (1 − T ) β )  −   ∗   ∗2   p p  i=1 ih i=1 ih

(5.4)

  N g+1  N  X  X   pi pg+2   2 i  2   a T + (1 − T )  −   ∗2 ∗2   pih pih i=1 i=1 (5.5)

Sukhjinder Singh Sidhu, Mohan Lal Bansal, Jong-Min Kim, Sarjinder Singh

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and  N   ! 2 ! N  g+2 2      pg+1 (1 − T )2 X  pi 1−T ∗  X pi  g i  A2 =  +a 1+   ∗2 + pi − 2 ∗  .  β+ T β  ∗ − 1   2   p p T pih ih i=1 ih i=1

(5.6)

Proof: Please see the Appendix A. When the value of non-sensitive question is known in advance. We choose the strategy, which for a fixed cost can estimate Y with maximum accuracy. For this we find the minimum expected mean square error for fixed cost under super population model. This we do in the following theorem. ¤ Theorem 3. Under superpopulation model, for the fixed cost C0 the minimum expected mean square error of estimator (b Y2 )h is given by h   i C1 Em MSE b Y2 = A1 + A2 , h C0

(5.7)

where A1 and A2 are defined in (5.5) and (5.6) and C1 cost of processing per unit in the sample. Proof: Please see the Appendix A.

¤

6. Empirical Study To investigate into the performance of the proposed estimators we resort to an empirical study under super population model given in Section 3. For this the relative efficiency under unrelated question model(RE h) of the proposed estimators (b Y2 )h for h = 1, 2, 3, 4, 5 with respect to (b Y2 )0 is given by h   i Em MSE b Y2 h  0 i × 100, (RE h) = (6.1) Em MSE b Y2 h

where symbols have their usual meanings. The probability associated with the statements in the device is 0.7 and 0.3, respectively. The choice of T = 0.7 in the Greenberg et al. (1971) seems to be a reasonable choice, because a very high value of T may effect the respondents’ coopeartion while asking a question through the randomization device. The density functions for the auxiliary variable, which is assumed to have correlation of values ρ in the range 0 to 1 with the study variables, are presented in Table 1. For the sensitive character value of correlation coefficient between X and Y is ρ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, but for unrelated question, correlation coefficient ρ∗ = 0.15, 0.65, 0.95 is used. Note that ρ∗ is the value of correlation coefficient between the unrelated character variable and the auxiliary variable or say the selection probabilities pi . Thus, it could always be feasible to select an unrelated character variable which may have either low, moderate or high value of correlation coefficient with the selection probabilities p, thus we considered only three such values of ρ∗ . As the nature of the sensitive variables yi s remains unpredictable, thus we decided to consider entire range of values of the correlation coefficient ρ between 0 and 1. A PPSWR sample of size 20 is considered as drawn from a population consisting of 100 respondents. The computations are given in Appendix B. The results obtained from these computations are given below in Table 2. From this table it is clear that the proposed estimators fare better than the usual estimator for all the p∗ih (h = 1, 2, 3, 4, 5) in majority of the cases. It is to pointed out that if the value of ρ∗ is low

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Table 1: Density functions for various probability distributions Sr. No.

Distribution

1

Right Triangular

2

Exponential

Density function

Range

f (x) = 2 (1 − x)

0≤x≤1

f (x) = e−x

3

Chi-Square at ν = 6

f (x) =

4

Gamma, α = 2, β = 1

f (x) =

5

Normal

f (x) =

6

Log Normal

f (x) =

7

Beta, α = 3, β = 2

f (x) =

1

x≥0 − 2x

ν−2 2

e x ν 22 Γν 2 1 α−1 − βx x e βα Γα  x−µ 2 1 −1 √ e 2 σ σ 2π 2 1 1 √ e− 2 (log(x)) x 2π 1 xα−1 (1 − x)β−1 B (α, β)

x≥0 x≥0 −∞ < x < +∞ x>0 0≤x≤1

Table 2: Correlation range for which the proposed estimator (b Y2 )h of population total of a sensitive character is

more efficient than usual estimator for p∗ih given in (1.2) to (1.7) . Distribution ρ∗ Right Triangular Exponential Normal Chi-sq, ν = 6 Gamma (2, 1) Log Normal Beta (3, 2)

0.15 0.2–0.9 0.2–0.9 0.2–0.9 0.6–0.9 0.2–0.9 0.2–0.9 0.5–0.9

g=0 0.65 0.1–0.9 0.1–0.9 0.2–0.9 0.5–0.9 0.1–0.9 0.1–0.9 0.3–0.9

0.95 0.1–0.9 0.1–0.9 0.2–0.9 0.1–0.9 0.1–0.9 0.1–0.9 0.1–0.9

0.15 0.2–0.9 0.2–0.9 0.2–0.9 0.5–0.9 0.2–0.9 0.1–0.9 0.4–0.9

g=1 0.65 0.1–0.9 0.1–0.9 0.2–0.9 0.5–0.9 0.1–0.9 0.1–0.9 0.3–0.9

0.95 0.1–0.9 0.1–0.9 0.2–0.9 0.1–0.9 0.1–0.9 0.1–0.9 0.1–0.9

0.15 0.2–0.9 0.1–0.9 0.2–0.9 0.5–0.9 0.1–0.9 0.1–0.9 0.4–0.9

g=2 0.65 0.1–0.9 0.1–0.9 0.2–0.9 0.4–0.9 0.1–0.9 0.1–0.9 0.3–0.9

0.95 0.1–0.9 0.1–0.9 0.2–0.9 0.1–0.9 0.1–0.9 0.1–0.9 0.1–0.9

or mderate, then a high value of correlation coefficient ρ is required for the proposed estimators to efiicient in case of Chi-Square (ν = 6) and Beta (3, 2) distribution. If the value of ρ∗ is high then the proposed estimator remains always more efficient. Interestingly, the choice of unrelated variable in the randomization device could be decided based on its correlation with the auxiliary variable used in the selection stage. Table 2 indicates that the value ρ could be any value in the range [0.1, 0.9], the the proposed estimators performs better for ρ∗ = 0.95 in case of all the seven distributions considered in the simulation study.

Acknowledgements The authors are thankful to the Editor, Associate Editor and two referees for their comments on the original version of the manuscript.

Appendix A: Proof: (Proof of Theorem 1.) Let E1 and E2 denote the expected values with respect to sampling design and over randomization device respectively and let V1 and V2 be the corresponding variances, then       V b Y1 = E 1 V 2 b Y1 + V1 E2 b Y1     n   n   X ri X ri  1    1      1 1      + V1 E2   − (1 − T ) U  − (1 − T ) U  = E1 V2   .       T n pi T n pi i=1

i=1

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On using (4.3) and substituting E (ri ) = yi T + (1 − T )ui , we have    n    1     1 X T (1 − T ) (yi − ui )2   b  V Y1 = E1  2      2 2   T n i=1 pi    n  X yi T + (1 − T )ui  1    1   − (1 − T ) U  + V1       T  n i=1 pi ( ! !)2 N N 1 − T X (Yi − Ui )2 1 X Yi Ui . = + pi T − Y + (1 − T ) −U nT i=1 pi pi pi nT 2 i=1 ¤ Full derivation of the equation 5.3: Let E1 and E2 denote the expected values with respect to sampling design and over randomization device respectively and let V1 and V2 be the corresponding variances, then       V b Y2 = E 1 V 2 b Y2 + V1 E2 b Y2 h     n   n   X ri X ri  1    1      1     = E1 V2   − (1 − T ) U  − (1 − T ) U   + V1 E2        ∗ ∗   T  i=1 pi T  n i=1 pi       n n   1   1       1 X V2 (ri )    1 X E2 (ri )    (1 ) = E1  2  + V − − T U   .     1   ∗      T  n i=1 pih T  n2 i=1 p∗2 hi On using (4.3) and substituting E (ri ) = yi T + (1 − T )ui , we have    n    1     1 X T (1 − T ) (yi − ui )2   b  V Y2 = E1  2      h  T  n2 i=1 p∗2 ih    n  X yi T + (1 − T )ui  1    1   (1 ) + V1   − − T U   ∗     T n i=1 pih   n    n 2  X yi T + (1 − T )ui  X  1       (1 ) (y ) 1 1 T − T − u     i i = E1  2   + 2 2 V1      ∗   2 ∗2    p T  n i=1 n T pih ih i=1  2 N N N   1 − T X (Yi − Ui )2 pi 1 X  T Yi + (1 − T )Ui X T Yi + (1 − T )Ui   = pi  + 2 − pi ∗ ∗   ∗2  nT i=1 pih p nT i=1  pih ih i=1 =

N 1 − T X (Yi − Ui )2 pi nT i=1 p∗2 ih   N 2  N   X   (T Yi + (1 − T )Ui )2 pi  1  X (T Yi + (1 − T ) Ui ) pi    + − ,         p∗ih nT 2  i=1 p∗2 ih i=1

which proves the equation. Proof: (Proof of Theorem 2.) We know that n n  o n   o2   o Em MSE b Y2 = E m V b Y2 + E m B b Y2 . h

h

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Thus we have n   o (1 − T ) Em (B1 ) Em (B2 ) − Em (B3 ) n   o2 Em MSE b Y2 = + + E B b Y2 , m h h nT nT 2

(A.1)

where B1 =

N X (Yi − Ui )2 i=1

p∗2 ih

pi ,

B2 =

N X {T Yi + (1 − T ) Ui }2

p∗2 ih

i=1

pi

and  N 2    X T Yi + (1 − T ) Ui   B3 =  pi  . ∗     pih i=1

Under the superpopulation model, we have  2   2  N N (β − β∗ )2 p2 + e − e∗ + 2 (β − β∗ ) p e − e∗ X X βpi + ei − β∗ pi − e∗i i i i i i i B1 = pi = pi . p∗2 p∗2 ih ih i=1 i=1 Thus Em (B1 ) = (β − β∗ )2

N X p3i i=1

p∗2 ih

+ 2a

N X pg+1 i i=1

p∗2 ih

.

Now  N  2 2 2 2    X T Yi + (1 − T ) Ui + 2T (1 − T )Yi Ui   B2 =  pi    ∗2   p ih i=1  2   N T 2 (βp + e )2 + (1 − T )2 β∗ p + e∗ + 2T (1 − T ) (βp + e ) β∗ p + e∗ X i i i i i i i i = pi p∗2 ih i=1     N T 2 β2 p2 + e2 + 2βp e + (1 − T )2 β∗2 p2 + e∗2 + 2β∗ p e∗ X i i i i i i i i = pi p∗2 ih i=1   N T (1 − T ) ββ∗ p2 + βp e∗ + β∗ p e + e e∗ X i i i i i i i +2 pi . p∗2 ih i=1 Thus, we have N N X n o X pg+1 p3i i ∗ 2 (1 ) + + − T β . Em (B2 ) = T 2 + (1 − T )2 a {βT } p∗2 p∗2 ih i=1 ih i=1

Now  N 2  N 2 N   X     Ui pi   X Yi pi X T Yi + (1 − T ) Ui    T = . B3 =  pi  ∗ ∗ + (1 − T ) ∗          p p p ih ih ih i=1 i=1 i=1

Sukhjinder Singh Sidhu, Mohan Lal Bansal, Jong-Min Kim, Sarjinder Singh

178

Using the superpopulation models, we have Using the superpopulation models, we have    2  N 2  N    X (βpi + ei ) pi  X β∗ pi + e∗i pi          B3 = T 2  + (1 − T )2    ∗ ∗         p p   i=1 ih ih i=1      ∗ ∗   N N       X (βpi + ei ) pi   X β pi + ei pi  + 2T (1 − T )     ∗ ∗         pih pih  i=1  i=1  N 2  N 2 2 2 N N X X  ∗ X pi  X pi e∗i pi  ei pi  2   (1 ) = T 2 β + − T β + +   p∗ p∗ih  p∗ p∗ih  i=1 ih i=1 i=1 ih i=1  N 2  N  N N X X  X pi e∗i pi  ei pi   ∗ X p2i   + 2T (1 − T ) β  β ∗ + ∗  ∗ + ∗  p p p p ih ih i=1 ih i=1 i=1 ih i=1   2  N 2 N   N N N 2 2 2   X X X X X       p e p p p p e e p e    i j i j   2  i i i  i i i    β + + = T2  + 2β      ∗  ∗ ∗ ∗ ∗    ∗2   p p p p p  i=1 pih ih jh ih  ih i=1 j,i=1 i=1 ih i=1   2  N 2 N   N N N p p e∗ e∗   X X pi X pi e∗i    X p2i  X e∗2 p2i i j i j i 2  ∗2  ∗  +   β  + (1 − T )  + + 2β  ∗ ∗ ∗ ∗ ∗    p pih p jh p pih  p∗2   ih i=1 ih i=1 i=1 ih i=1 j,i=1   N   N 2 2  N p2 e∗ p   X p3 e∗ X  i j j  ∗ X pi   i i  ββ  + 2T (1 − T )  +  + β  ∗  ∗ ∗    ∗2  p p p p  ih i=1 ih i=1 j,i=1 ih jh  N   N  N N e e∗ p p  2  X p3 ei X  X ei e∗ p2 X p e p i j i j j j   i i i i ∗  +  . +β  + +  ∗ ∗  ∗ ∗    ∗2 ∗2  p p p p p p ih jh  ih ih i=1 i=1 j,i=1 ih jh j,i=1 Taking expected value, we have       2 2     N N N N 2     X X   X   pg+2 pg+2 p2i   2 X pi    i i 2  ∗2    (B ) (1 ) β Em 3 = T  β  + − T + a   + a      ∗ ∗     ∗2  ∗2     p p p p   ih  ih  i=1 ih i=1 ih i=1 i=1    N 2 2       ∗ X pi    ββ  + 2T (1 − T )    ∗      p  i=1 ih  2

 N g+2   N 2 2  N 2 2 n o X X pi  X pi  pi   + T 2 β2   + (1 − T )2 β∗2   = a T 2 + (1 − T )2  ∗ p p∗  p∗2 ih i=1 i=1 ih i=1 ih  N 2 2 X pi  ∗  (1 ) + 2T − T ββ  p∗  i=1 ih  N g+2   2 N X n o X pi   p2i   ∗ 2 2      . = a T + (1 − T )   +  βT + (1 − T ) β p∗  p∗2 ih i=1 ih i=1

Unrelated Question Model in Sensitive Multi-Character Surveys

179

Now we have  N  n   o2  X b E m B Y2 = Em   h  i=1  N   X = Em    i=1  N   X = Em   

2 ! !   1−T pi  − 1 Yi + Ui    p∗ih T

2 ! !  N   pi 1 − T X pi  − 1 Yi + − 1 Ui    p∗ih T i=1 p∗ih  N 2 2 !  !       (1 − T )2  pi  X pi  − 1 Yi  + − 1 Ui        p∗ih p∗ih T2  i=1 i=1    !2 !   N N  X  p j    2 (1 − T )  pi X pi   Yi U j   + − 1 Y U + − 1 − 1   i i ∗ ∗      T p p  i=1 p∗ih  ih jh j,i=1

2   N 2  N  g+2 X X pi   pg+1 pi  g i   + a  + pi − 2 ∗  = β  ∗ − 1 ∗2 p pih pih i=1 i=1 ih    2   N N  g+2 2  X     pi pg+1 (1 − T )2    ∗2 X pi g i     β + a + − 1 + p − 2   ∗2      i ∗ ∗  2   p pih  T pih   i=1 ih i=1 2

 N 2  1 − T X p2i  ββ∗ +2  ∗ − 1 T p ih i=1   N   N 2 2 2  X  1 − T ∗ X    pi pi      + =  β − 1 β − 1       ∗ ∗      p T p i=1 ih i=1 ih  ( ) N  g+2  pg+1 (1 − T )2 X  pi g i  . +a 1+ + p − 2   i ∗ pih  T2 p∗2 ih i=1 On substituting E(B1 ), E(B2 ), E(B3 ) and Em (B(b Y2 )h )2 in (A.1) and on re-arranging, we can get the following one: h   i 1 Em MSE b Y2 = A 1 + A 2 , h n where   ! g+1  N N 3 X X  1 − T    p p   i i 2 (β − β∗ ) A1 =  + 2a    ∗2 ∗2    T p p ih i=1 ih i=1   N g+1  N X  X pi   pg+2 1    2 i  2   a T + (1 − T )  + 2 −  ∗2 ∗2  T pih pih  i=1 i=1    N 2 2  N 3  X  X pi    p  i ∗ 2    + (βT + (1 − T ) β )  −  p∗  p∗2  i=1 ih i=1 ih

Sukhjinder Singh Sidhu, Mohan Lal Bansal, Jong-Min Kim, Sarjinder Singh

180

and  N   ! 2 ( ) N  g+2 2      pg+1 (1 − T ) ∗  (1 − T )2 X  pi X pi  g i   . A2 =  − 1 β + β + a 1 + + p − 2        i ∗ ∗    2 ∗2   p T p T p ih ih ih i=1 i=1 ¤ Proof: (Proof of Theorem 3.) Let C1 be the cost per unit of collecting information of each individual. The cost C0 of observing the sample of size n is given by C0 = nC1 .

(A.2)

To minimize Em [MSE(b Y2 )h ] subject to condition (A.2), consider the function L=

1 A1 + A2 + λ (nC1 − C0 ) . n

(A.3)

Differentiating (A.3) partially with respect to n and we get n = C0 /C1 and hence (5.4) becomes (5.7), which proves the theorem. ¤

Appendix B: Relative efficiencies(RE h), h = 1, 2, 3, 4, 5 of the proposed estimators for PPSWR sampling scheme using Unrelated Question Model under various distributions.

g

ρ

RE 1

RE 2

0

.1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9

75.76 100.0 140.1 200.2 285.0 398.3 540.4 706.6 897.9 76.51 100.6 140.1 200.9 290.8 418.3 589.1 799.4 1029 77.93 103.2 144.1 207.5 303.0 440.8 627.9 855.9 1088

75.78 100.1 140.2 200.4 285.4 398.8 541.0 707.2 898.3 76.54 100.7 140.2 201.1 291.2 418.8 589.7 799.9 1030 77.97 103.3 144.2 207.8 303.4 441.4 628.6 856.5 1088

1

2

ρ∗ = 0.15 RE 3 RE 4 72.30 90.32 119.4 161.9 220.9 300.8 407.6 553.0 775.1 73.08 91.85 122.4 169.1 238.9 342.0 491.3 698.9 963.8 74.07 93.63 125.2 174.2 248.8 361.6 528.0 758.6 1031

68.25 79.20 96.95 122.6 159.3 213.1 296.1 434.7 691.5 70.46 84.63 107.1 140.6 190.6 266.6 386.5 580.5 883.0 71.38 86.43 110.1 145.9 200.0 283.7 417.0 631.3 946.0

RE 5

RE 1

93.72 122.2 165.0 221.9 292.6 378.2 482.5 617.0 818.0 113.4 150.9 205.7 281.2 379.9 503.8 654.3 832.3 1035 128.3 173.7 238.2 326.9 442.8 586.4 755.0 939.5 1119

106.4 142.9 201.0 293.5 439.2 660.6 964.6 1297 1515 102.8 138.2 194.8 286.4 436.8 683.0 1065 1552 1867 101.9 137.9 195.5 290.0 448.3 716.3 1151 1712 2038

ρ∗ = 0.65 RE 2 RE 3 RE 4 Right Triangular 106.5 99.23 93.15 143.0 124.2 107.9 201.3 163.1 130.1 294.0 222.8 163.2 440.1 313.8 213.2 661.9 449.4 291.4 966.2 643.6 420.0 1298 904.3 645.1 1516 1231 1051 102.8 96.15 91.33 138.3 121.4 108.6 195.0 161.3 135.2 286.9 225.2 176.7 437.7 329.7 243.1 684.4 504.3 355.1 1067 796.6 555.3 1553 1245 924.6 1868 1698 1485 102.0 95.21 90.50 138.0 120.9 108.4 195.8 161.7 136.2 290.5 227.8 179.8 449.2 338.3 251.0 717.8 529.2 373.8 1153 864.2 600.1 1723 1399 1029 2039 1885 1645

RE 5

RE 1

RE 2

180.2 237.2 308.8 401.0 517.8 662.5 838.8 1053 1312 187.8 257.4 350.6 481.1 664.7 919.3 1255 1634 1868 197.1 276.9 385.9 543.1 772.5 1101 1538 1988 2117

124.8 165.7 224.2 310.9 446.2 672.1 1089 1977 3900 123.3 163.8 221.7 308.3 445.2 680.9 1145 2295 5883 122.8 163.3 221.6 309.3 449.2 693.8 1190 2497 7194

124.9 165.9 224.5 311.5 447.0 673.3 1091 1979 3904 123.4 164.0 222.1 308.8 446.1 682.2 1147 2298 5887 122.9 163.5 222.0 309.8 450.0 695.1 1192 2501 7199

ρ∗ = 0.95 RE 3 RE 4 115.5 143.6 183.9 243.6 336.4 489.8 766.4 1338 2726 114.2 142.3 182.9 244.1 342.0 513.2 857.0 1732 4752 113.7 141.9 182.7 244.8 345.0 523.2 892.2 1895 5890

108.6 126.8 152.1 189.0 245.6 339.2 513.4 905.4 2083 107.8 126.5 153.4 193.4 257.2 368.9 595.6 1189 3511 107.3 126.3 153.5 194.4 260.0 376.6 618.9 1284 4182

RE 5 258.7 334.9 416.0 514.2 643.9 831.4 1135 1717 3032 259.4 340.6 430.6 545.1 706.5 961.5 1433 2546 5856 261.9 346.8 442.4 566.1 743.8 1033 1590 3018 7886

Unrelated Question Model in Sensitive Multi-Character Surveys

g

ρ

RE 1

RE 2

0

.1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9

86.55 124.8 187.9 282.6 416.4 597.5 837.0 1163 1661 94.12 136.7 205.1 311.4 470.0 696.4 1005 1409 1939 102.6 152.9 231.1 354.0 543.4 824.5 1214 1702 1661

86.70 125.2 188.6 283.7 418.1 599.5 839.1 1165 1663 94.36 137.3 206.1 312.9 472.2 699.2 1008 1412 1940 103.0 153.7 232.5 356.1 546.6 828.7 1219 1706 1663

g

ρ

RE 1

RE 2

0

.1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9

80.39 109.7 158.1 230.7 332.9 469.2 643.1 862.4 1162 84.08 115.3 165.7 243.5 358.9 522.5 741.1 1014 1334 88.60 123.4 178.1 263.0 391.8 579.2 833.8 1144 1462

80.47 109.9 158.5 231.4 333.9 470.5 644.5 863.5 1162 84.19 115.5 166.1 244.3 360.1 524.0 742.7 1016 1335 88.76 123.7 178.7 264.0 393.3 581.1 835.8 1145 1462

1

2

1

2

ρ∗ = 0.15 RE 3 RE 4

RE 5

RE 1

73.71 89.55 114.7 151.8 207.6 295.7 445.3 725.4 1316 77.36 94.78 121.7 162.3 224.8 325.9 501.3 833.2 1503 80.25 98.95 127.4 170.9 239.1 351.9 552.3 936.4 1316

107.1 145.1 204.0 286.0 396.4 546.0 756.5 1073 1605 165.8 223.4 305.2 417.9 569.1 769.9 1039 1408 1928 239.7 333.7 456.0 620.0 834.6 1107 1443 1840 1605

127.3 191.9 292.4 446.6 675.2 997.2 1426 1980 2699 124.5 190.2 294.5 462.1 729.5 1145 1754 2552 3358 125.3 195.2 307.3 493.3 807.5 1337 2186 3324 2699

ρ∗ = 0.65 RE 2 RE 3 RE 4 Exponential 127.7 108.3 97.86 192.9 144.5 117.1 294.3 201.1 146.2 449.6 289.4 190.4 679.4 428.3 260.2 1002 648.8 376.9 1431 1001 589.0 1983 1566 1019 2701 2447 1971 125.0 105.6 95.6 191.2 142.9 116.3 296.5 201.9 147.4 465.3 296.9 195.1 734.5 453.8 271.9 1152 720.1 404.0 1761 1180 653.5 2558 1954 1182 3361 3020 2340 125.8 104.8 94.38 196.4 143.6 116.0 309.6 205.4 148.4 497.3 307.0 198.7 814.1 481.4 280.8 1347 794.2 425.8 2199 1376 709.4 3335 2415 1337 2701 2447 1971

ρ∗ = 0.15 RE 3 RE 4

RE 5

RE 1

RE 2

98.84 129.6 176.6 240.8 324.7 434.0 580.7 789.5 1115 134.3 179.2 243.6 331.9 448.2 597.4 787.3 1029 1333 170.6 232.9 317.3 431.7 580.9 767.1 989.3 1239 1493

114.7 161.9 236.0 351.6 527.8 783.6 1124 1526 1934 111.7 159.0 233.8 354.5 550.0 861.7 1328 1915 2374 111.6 160.8 239.1 367.7 583.5 947.1 1529 2284 2733

115.0 162.4 237.0 353.2 530.2 786.8 1127 1529 1935 112.0 159.5 234.8 356.1 552.6 865.4 1332 1919 2376 111.8 161.3 240.2 369.5 586.6 951.7 1534 2289 2735

78.87 104.1 145.6 208.7 302.7 442.7 653.6 982.6 1538 83.41 111.0 155.5 225.4 334.4 504.1 768.4 1176 1791 87.78 118.4 167.0 244.5 369.1 570.7 894.6 1391 1538

75.32 95.95 129.7 180.2 253.2 357.9 508.2 728.8 1073 77.92 100.2 136.2 192.0 276.7 404.4 594.5 870 1245 80.63 104.8 143.1 203.3 296.9 442.3 663.5 980.3 1368

71.17 84.60 106.3 138.4 185.7 257.9 374.4 576.9 957.3 73.87 89.33 113.5 149.9 204.9 291.1 432.9 678.4 1104 75.9 92.51 118.2 157.1 217.0 312.5 472.3 749.2 1204

ρ∗ = 0.65 RE 3 Normal 103.0 132.5 178.4 249.6 360.2 532.1 797.4 1196 1741 100.4 130.6 178.4 254.9 380.1 590.1 944.6 1508 2158 99.47 130.8 180.4 261.0 397.0 635.5 1062 1771 2487

181

RE 5

RE 1

RE 2

249.4 328.3 424.3 547.8 711.5 936.6 1263 1765 2560 290.5 406.7 550.3 739.3 994.3 1346 1838 2521 3312 345.4 511.5 724.8 1018 1430 2009 2795 3711 2560

150.5 230.8 349.2 526.6 797.0 1219 1910 3195 6132 148.8 228.9 348.1 530.6 819.1 1297 2158 3989 8786 148.0 229.1 351.1 541.2 850.5 1389 2439 4996 6132

151.1 232.3 351.7 530.6 802.8 1226 1919 3203 6138 149.4 230.3 350.7 534.7 825.3 1306 2169 4002 8796 148.6 230.6 353.8 545.6 857.3 1399 2453 5016 6138

RE 4

RE 5

RE 1

RE 2

95.27 112.3 138.3 177.9 239.7 340.8 517.7 850.6 1482 93.26 111.9 140.4 184.4 255.0 375.0 595.1 1027 1802 92.26 111.8 141.6 187.9 263.6 395.5 645.3 1154 2036

206.0 269.2 346.8 446.3 576.2 749.6 989.6 1334 1813 227.1 313.9 425.3 576.0 782.4 1066 1450 1932 2355 252.1 361.9 507.2 711.8 1004 1417 1972 2570 2812

135.2 191.3 271.9 391.3 574.7 870.3 1384 2401 4736 133.7 189.5 270.2 391.4 582.5 905.3 1515 2917 6885 133.0 189.3 271.2 395.5 594.8 1630 1630 3378 9293

135.5 192.0 273.1 393.1 577.4 874.1 1389 2407 4741 134.0 190.2 271.4 393.3 585.3 909.4 1521 2924 6893 133.3 190.0 272.5 397.5 597.8 1637 1637 3388 9305

ρ∗ = 0.95 RE 3 RE 4 124.4 167.2 231.1 329.3 487.1 755.5 1254 2325 5217 123.0 165.9 230.4 331.1 496.8 790.4 1373 2775 7192 122.0 165.3 230.7 334.0 507.3 824.3 1489 3267 5217

RE 5

110.9 133.3 165.3 213.0 288.8 420.1 679.4 1315 3562 109.8 132.5 165.1 214.1 293.0 432.3 716.9 1458 4415 108.9 131.8 164.9 214.9 296.2 441.9 747.1 1583 3562

430.8 576.5 721.1 888.2 1103 1413 1923 2951 5640 443.6 611.7 788.0 1001 1288 1719 2468 4086 8549 462.0 653.5 862.8 1126 1495 2076 3158 5760 5640

ρ∗ = 0.95 RE 3 RE 4

RE 5

119.2 153.6 203.6 278.7 397.0 595.7 963.6 1766 3948 118.0 152.5 202.9 280.0 404.3 622.0 1055 2126 5655 117.2 117.2 203.0 281.7 410.4 1120 1120 2407 7402

109.5 129.7 158.2 200.5 266.8 380.3 601.3 1133 2929 108.6 129.1 158.4 202.3 272.5 395.8 646.4 1299 3865 107.8 107.8 158.4 203.3 275.6 673.1 673.1 1408 4646

321.4 421.2 523.3 643.9 801.5 1031 1413 2196 4240 326.6 437.6 557.1 705.4 909.4 1224 1788 3068 6747 334.1 334.1 587.4 757.0 997.5 2115 2115 3940 9918

Sukhjinder Singh Sidhu, Mohan Lal Bansal, Jong-Min Kim, Sarjinder Singh

182

g

ρ

RE 1

RE 2

0

.1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9

63.36 66.06 75.05 87.37 99.55 109.7 117.5 123.0 126.5 63.68 66.84 76.08 88.54 100.8 110.9 118.5 123.7 126.9 64.00 67.52 76.98 89.57 101.9 112.0 119.4 124.3 127.2

63.37 66.08 75.08 87.41 99.60 109.8 117.5 123.0 126.5 63.69 66.86 76.11 88.59 100.9 111.0 118.6 123.8 126.9 64.00 67.54 77.01 89.61 101.9 112.0 119.4 124.4 127.2

g

ρ

RE 1

RE 2

0

.1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9 .1 .2 .3 .4 .5 .6 .7 .8 .9

82.76 110.4 156.6 223.5 319.3 447.7 611.2 812.3 1071 88.82 119.0 167.2 240.4 347.7 498.3 698.5 946.2 1229 94.97 131.0 185.3 267.2 388.3 560.7 790.0 1063 1337

82.93 110.8 156.4 224.8 321.2 450.1 613.7 814.5 1073 88.47 119.5 168.1 241.9 349.9 501.1 701.4 948.6 1230 95.34 131.8 186.6 269.2 391.3 564.4 793.9 1066 1338

1

2

1

2

ρ∗ = 0.15 RE 3 RE 4

RE 5

RE 1

62.66 64.40 72.18 83.40 95.04 105.4 113.9 120.5 125.2 63.05 65.33 73.42 84.82 96.51 106.8 115.0 121.3 125.6 63.40 66.08 74.42 85.94 97.68 107.9 116.0 121.9 126.0

65.58 68.46 77.84 90.35 102.3 111.9 118.9 123.7 126.7 68.05 71.78 81.48 93.85 105.3 114.3 120.6 124.7 127.2 70.13 74.59 84.54 96.77 107.8 116.2 121.9 125.5 127.5

91.89 92.64 94.28 98.58 109.9 134.0 158.9 159.1 145.9 91.50 92.66 94.65 99.56 111.9 137.8 163.4 161.8 146.8 91.56 92.70 95.00 100.4 113.7 141.0 167.1 164.0 147.5

ρ∗ = 0.65 RE 3 RE 4 Chi Square 91.91 91.38 90.83 92.67 91.51 90.27 94.33 92.36 90.18 98.65 95.58 92.08 110.0 105.3 99.85 134.1 127.8 119.9 159.0 153.5 146.0 159.2 1567 153.2 145.9 145.1 144.2 91.71 91.20 90.67 92.69 91.55 90.35 94.70 92.79 90.70 99.64 96.68 93.3 112.1 107.6 102.2 137.9 131.8 124.0 163.5 158.3 150.8 161.9 159.5 156.1 146.8 146.0 145.1 91.58 91.07 90.56 92.73 91.61 90.45 95.04 93.16 91.12 100.5 97.58 94.26 113.8 109.4 104.1 141.1 135.1 127.2 167.2 162.1 154.6 164.1 161.8 158.4 147.5 146.8 145.8

ρ∗ = 0.15 RE 3 RE 4

RE 5

RE 1

RE 2

99.20 134.2 179.5 241.1 320.7 422.1 554.3 737.2 1018 99.70 186.6 247.7 331.0 440.2 579.2 753.0 968.4 1230 99.80 252.2 332.7 439.0 575.2 742.5 938.8 1155 1368

113.9 158.4 227.6 335.2 500.0 742.5 1070 1449 1790 112.4 158.1 229.2 342.4 524.3 813.6 1248 1795 2194 113.8 162.6 238.8 361.8 565.4 904.2 1440 2122 2496

114.4 159.4 229.4 338.2 504.5 748.4 1076 1455 1792 112.9 159.1 231.1 345.5 529.2 820.6 1256 1802 2197 114.3 163.8 241.0 365.5 571.4 913.1 1451 2132 2500

63.03 65.28 73.71 85.52 97.48 107.8 115.8 121.8 125.9 63.38 66.11 74.82 86.81 98.84 109.1 117.0 122.6 126.3 63.70 66.82 75.76 87.88 99.96 110.1 117.8 123.2 126.6

77.84 97.49 129.6 177.7 246.9 344.7 482.2 678.8 979.2 81.60 103.6 138.5 192.1 273.0 393.7 570.7 821.3 1151 85.91 110.6 148.7 207.4 297.6 435.1 639.8 924.2 1259

73.73 86.24 106.6 136.7 181.0 248.1 355.2 538.0 875.4 77.30 92.41 115.9 151.2 204.4 287.3 422.0 649.9 1030 80.53 97.17 122.4 160.4 218.7 310.9 462.7 718.7 1120

RE 2

ρ∗ = 0.65 RE 3 Gamma 102.9 131.3 175.4 243.6 349.2 511.7 757.9 1116 1583 101.2 130.8 177.2 251.1 371.8 573.2 910.3 1435 2003 101.4 132.5 181.3 260.1 392.0 621.3 1026 1680 2291

RE 5

RE 1

RE 2

99.67 100.6 101.9 105.8 116.7 140.1 162.6 160.5 146.2 100.3 101.9 104.0 109.3 122.4 148.7 170.7 164.3 147.3 100.9 102.9 105.8 112.2 127.2 155.8 177.3 167.4 148.1

100.8 103.8 107.1 110.8 115.4 122.2 135.6 182.5 642.3 100.8 103.8 107.1 110.9 115.6 122.5 136.3 184.6 676.0 100.8 103.8 107.2 111.0 115.8 122.8 136.9 186.4 703.7

100.8 103.9 107.2 110.9 115.5 122.3 135.7 182.6 642.4 100.8 103.9 107.2 111.0 115.7 122.6 136.4 184.7 676.2 100.8 103.9 107.2 111.0 115.8 122.9 136.9 186.4 703.8

RE 4

RE 5

RE 1

RE 2

95.46 111.9 136.9 174.8 233.6 329.0 493.2 795.8 1352 94.2 112.6 140.5 183.6 252.8 370.3 584.0 995.6 1693 93.96 113.4 142.8 188.5 263.0 392.4 635.5 1119 1906

202.4 263.4 338.8 435.9 561.8 726.4 946.4 1249 1650 226.2 308.4 414.0 557.7 755.9 1029 1394 1837 2183 258.3 364.6 502.8 696.0 971.0 1358 1869 2402 2575

133.6 186.6 261.5 371.1 538.2 808.5 1286 2257 4466 132.5 185.6 261.1 372.7 546.7 839.3 1396 2704 6530 132.2 186.2 263.5 378.9 561.5 876.2 1501 3102 8617

134.2 188.0 263.8 374.6 543.2 815.3 1295 2268 4477 133.1 186.9 263.4 376.3 551.9 846.5 1406 2718 6545 132.8 187.6 265.9 382.6 567.0 884.3 1513 3120 8640

ρ∗ = 0.95 RE 3 RE 4 100.3 102.9 105.7 109.1 113.5 120.1 133.3 179.4 629.7 100.3 102.9 105.8 109.2 113.7 120.4 134.1 181.7 663.9 100.2 102.9 105.8 109.3 113.8 120.7 134.6 183.5 691.7

RE 5

99.79 101.8 104.2 107.2 111.2 117.4 130.2 174.8 604.7 99.76 101.8 104.3 107.3 111.4 117.8 131.0 177.1 637.5 99.73 101.8 104.3 107.4 111.6 118.1 131.6 178.9 663.9

110.0 112.8 115.1 117.5 120.8 126.3 138.7 185.3 651.2 110.1 113.0 115.3 117.8 121.3 127.0 139.9 188.5 639.7 110.2 113.1 115.5 118.1 121.6 127.6 140.9 190.9 728.7

ρ∗ = 0.95 RE 3 RE 4

RE 5

118.5 151.6 199.3 270.5 381.9 567.7 909.4 1647 3608 117.5 150.8 199.2 272.5 390.1 595.0 1002 2015 5387 117.0 150.7 200.0 275.2 397.4 615.1 1065 2273 6980

109.1 128.7 156.3 197.0 260.4 368.2 576.2 1071 2700 108.3 128.4 156.9 199.6 267.6 386.8 629.0 1261 3742 107.8 128.1 157.2 200.9 271.2 396.2 656.0 1367 4492

307.0 398.2 492.4 604.5 751.8 966.1 1322 2043 3876 313.1 413.9 522.5 658.0 846.0 1138 1669 2888 6436 322.7 432.6 553.4 707.3 926.1 1278 1950 3642 9258

Unrelated Question Model in Sensitive Multi-Character Surveys

183

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