An Endowed Randomized Response Model for Estimating a Rare Sensitive Attribute Using Poisson Distribution

INTRODUCTION

The randomized response (RR) data-gathering device to procure trustworthy data on sensitive issues by protecting privacy of the respondent was first developed by Warner¹. Feeling that the co-operation of the respondent might be further enhanced if one of the two questions referred to a non-sensitive, innocuous attribute, say Y, unrelated to the sensitive attribute A. Horvitz et al.² proposed an unrelated question randomized response model. Greenberg et al.³ provided theoretical framework for a rectification to the Warner’s¹ model envisaged by Horvitz et al.². Numerous randomized response techniques have been developed for reducing non sampling errors in sample surveys, protecting a respondent’s privacy and increasing response rates. The key references of the randomized response model are Singh and Mathur^4,5, Singh et al.⁶, Kim and Warde⁷, Kim and Elam^8,9 and Singh and Tarray^10,11.

Land et al.¹² have considered a different and unique problem where the number of persons possessing a rare sensitive attribute is very small and a huge sample size is required to estimate this number. They proposed a method to estimate the mean of the number of persons possessing a rare sensitive attribute by utilizing the Poisson distribution in survey sampling. Land et al.¹² have discussed two different situations that when the proportion of persons possessing a rare unrelated attribute is known and that when it is unknown. Lee et al.¹³ has extended the studies of Land et al.¹² to the stratified sampling.

Singh and Tarray¹⁴ have further considered the problem of estimating the mean of the number of persons possessing a rare sensitive attribute using the Poisson distribution in the situation where the proportion of persons possessing a rare unrelated attribute is known. Singh and Tarray¹⁴ have suggested an alternative randomized response model based on Singh et al.¹⁵ model and studied its properties in presence of known population proportion of rare unrelated attributes. The main problem with the use of the method due to Singh and Tarray¹⁴ is that sometimes the mean value of the rare unrelated attribute remains unknown.

ESTIMATION OF PROPORTION OF A RARE SENSITIVE ATTRIBUTE WHEN PROPORTION OF A RARE UNRELATED ATTRIBUTE IS UNKNOWN

Let π₁ be the true proportion of the rare sensitive attribute A₁ in the population U. For example, the proportion of AIDS/HIV patients who continue having affairs with strangers, the proportion of persons who have witnessed a murder, the proportion of persons who are told by the doctors that they will not survive long due to a ghastly disease, for more examples, the reader is referred to Land et al.¹². Consider selecting a large sample of n persons from the population such that as n → ∞ and π₁→0 then, lim (n π₁) = δ₁ (finite). Let π₂ be the true proportion of the population having the rare unrelated attribute A₂ such that as n → ∞ and π₂→0 then, lim (n π₂) = δ₂ (finite and known). For instance, π₂ might be the proportion of persons who are born exactly at 12:00 o’clock, the proportion of babies born blind, see Land et al.¹².

In the proposed procedure, each respondent in the sample of n persons, selecting using simple random sampling with replacement (SRSWR) from the given population, is requested to use the deck of cards marked as Deck-I and Deck-II. Each respondent in the sample is requested to use Deck-I consists of three types of cards bearing statements:

with probabilities P₁, P₂ and P₃, respectively such that . The respondent is required to draw one card randomly from Deck-I and give answer in term of “Yes” or “No” according to his/her actual status and the statement, (i) or (ii), drawn. However if the statement (iii) is drawn, he/she is required to repeat the above process without replacing that card. If the statement (iii) is drawn in the second phase, he/she is directed to report “No”. If m is the total number of cards in the Deck-I, the probability of a “Yes” answer is given by:

(1)

Note that both attributes A₁ and A₂ are very rare in population. Assuming that, as → ∞ and θ₁ → 0 such that (finite), thus it is clear that:

(2)

Next, the respondent is requested to use Deck-II consists of three types of cards bearing statements:

with probabilities T₁, T₂ and T₃, respectively such that . The respondent is required to draw one card randomly from Deck-II and give answer in term of “Yes” or “No” according to his/her actual status and the statement, (i) or (ii) drawn. However, if the statement (iii) is drawn, he/she is required to repeat the above process without replacing that card. If the statement (iii) is drawn in the second phase, he/she is instructed to report “No”. If m is the total number of cards in the Deck - II, the probability of a “Yes” answer is given by:

(3)

As before assuming that as → ∞ and θ₂ → 0 such that (finite). Thus it is obvious that:

(4)

By following the procedure as adopted by Singh and Tarray¹⁴, we have:

(5)

and:

(6)

where, y_1i and y_2i denotes the observed values in the first and the second response from the ith respondent, respectively. Solving Eq. 5 and 6 for it has established the following theorems.

Theorem 1: An unbiased estimator of the parameter δ₁ for the rare sensitive attribute A₁ is given by:

(7)

Where:

and:

Proof: Since y_1i ~ iid Poisson (δ^*₁) and y_2i ~ iid Poisson (δ^*₂), thus by taking expected value on both sides of the Eq. 7, we have:

which proves the theorem.

Theorem 2: The variance of the unbiased estimator of the parameter δ^*₁ is given by:

(8)

Proof: Since y_1i~ iid Poisson (δ^*₁) and y_2i~ iid Poisson (δ^*₂), thus V(y_1i) = δ^*₁ and V(y_2i) = δ^*₂. It is to be mentioned that both responses are not independent, thus we have Eq. 9 where:

(9)

(10)

(11)

and

(12)

Putting Eq. 10-12 in 9 it established the theorem.

Corollary 1: An unbiased estimator to estimate the parameter δ₂ for rare unrelated attribute A₂ is given by:

(13)

with the variance:

(14)

Proof: Analogous to the proof of the theorems 1 and 2.

Corollary 2: An unbiased estimator of the variance of the estimator is given by:

(15)

and an unbiased estimator of the variance of the estimator is given by:

(16)

where, and are, respectively defined in Eq. 7 and 13, respectively.

RELATIVE EFFICIENCY

The percentage of relative efficiency of the proposed estimator with respect to the Land et al.¹² estimator δ₁^* is given by:

(17)

Where:

See Land et al.¹², Eq. 14, p. 7.

It is observed from Eq. 17 that the percentage of relative efficiency of the proposed estimator with respect to Land et al.¹² estimator is free from the sample size n. To see the performance of the proposed estimator relative to Land et al.¹² estimator , it has computed the values of PRE (, ) using the formula given in Eq. 17 for fixed (m =100) and different parametric values as given in Table 1. The resulting values of PRE (,) are shown in Table 1.

Table 1 exhibited that the values of PREs are greater than 100 for all the parametric values considered here. Thus the proposed procedure is better than the Land et al.¹² procedure.

Table 1:	Percentage of relative efficiency of the proposed estimator 1 with respect to Land et al.12 estimator

For the choice of δ₁, δ₂ as 0.5, 1.50, the percentage of relative efficiency remains considerably larger than the other two cases, which reveals that it is appropriate to use the rare unrelated attribute Y, one with a mean value greater than that of the rare sensitive attribute A without affecting the cooperation of the respondents in using the suggested randomization device. The choice of (P_i, T_i ), I = 1, 2 should be made in such a way that the respondents should not feel that their privacy is threatened, while the difference (P₁T₂-P₂T₁) should be kept large as compared to P₁-T₁. Finally, our recommendation is to use the suggested estimator ₁ in practice.