An Efficient Alternative Mixed Randomized Response Procedure

Abstract

In this article, we have suggested a new modified mixed randomized response (RR) model and studied its properties. It is shown that the proposed mixed RR model is always more efficient than the Kim and Warde’s mixed RR model. The proposed mixed RR model has also been extended to stratified sampling. Numerical illustrations and graphical representations are also given in support of this study.

Keywords

randomized response technique dichotomous population estimation of proportion privacy of respondents innocuous variable sensitive characteristics

Introduction

In survey of human population, questions requiring personal or controversial assertions often run into trouble in terms of resistance. It is difficult to collect reliable data from interviewees and hard to raise the quality of responses when the survey topic is sensitive. The purpose is to formulate an effective random device so as to induce each respondent to give truthful answers to sensitive questions without exposing his or her true identity to the interviewer. A large number of rectifications on Warner’s (1965) pioneering technique have been cited in the literature, for instance, see Fox and Tracy (1986), Chaudhuri and Mukherjee (1987, 1988), Hedayat and Singh (1991), Ryu, Hong, and Lee (1993), Tracy and Mangat (1996), and Singh (2003), for the reviews. Some other developments on randomized response (RR) sampling in recent years include Mahmood, Singh, and Horn (1998), Ryu et al. (2005–2006), Hong (2005–2006), Grewal, Bansal, and Sidhu (2005–2006), Perri (2008), Mahajan, Sharma, and Gupta (2007), Chua and Tsui (2000), Singh, Singh, and Mangat (2000), Chang and Huang (2001), Chang, Wang, and Huang (2004), Huang (2004), Land, Singh, and Sedory (2011), Kim and Warde (2004), Kim and Elam (2005), and so on.

To implement the privacy problem with the Moors (1971) model, Mangat, Singh, and Singh (1997) and Singh et al. (2000) have given several strategies as an alternative to Moors (1971) model, but their models may lose a large portion of data information and require a high cost to obtain confidentiality of the respondents. These drawbacks with the previous alternative models for the Moors model motivated Kim and Warde (2005) to envisage a mixed RR model using simple random sampling with replacement that modifies the privacy problem.

In this article, we have suggested a modification of Kim and Warde’s (2005) model to estimate the proportion of a qualitative sensitive variable. It has been demonstrated that the suggested model performs better than the mixed RR model of Kim and Warde (2005). In the fourth section, we introduce the stratified version of the proposed mixed RR model along with its properties. Comparison of the proposed stratified mixed RR model with that of Kim and Warde’s (2005) model has been given in the fifth section. The empirical studies performed show that, the proposed mixed RR model is more efficient than Kim and Warde’s (2005) model in stratified sampling too.

The Suggested Model

A single sample with size n is selected by simple random sampling with replacement (SRSWR) from the population. Each respondent from the sample is instructed to answer the direct question, “I am a member of the innocuous trait group.” If a respondent answers “Yes” to direct question, then he or she is instructed to go to randomization device R₁ consisting of the statements (i) “I am a member of the sensitive trait group” and (ii) “I am a member of the innocuous trait group” with probabilities of selection P₁ and (1 − P₁), respectively. If a respondent answers “No” to the direct question, then the respondent is instructed to use a randomization procedure due to Mangat (1994). In the Mangat’s (1994) RR procedure, each respondent is instructed to say “Yes” if he or she is a member of the sensitive trait group. If he or she is not a member of the sensitive trait group, then the respondent is required to use the Warner’s (1965) randomization device R₂ consisting of the statements: (i) “I belong to the sensitive trait group” and (b) “I do not belong to the sensitive trait group” with preassigned probabilities P and (1 − P), respectively. Then he or she is to report “Yes” or “No” according to the outcome of the randomization device R₂ and the actual status that he or she has with respect to the sensitive trait group. The survey procedures are performed under the assumption that both the sensitive and the innocuous questions are unrelated and independent in a randomization device R₁. To protect the respondent’s privacy, the respondents should not disclose to the interviewer the question they answered from either R₁ or R₂.

Let n be the sample size confronted with a direct question, and n₁ and n₂ (= n − n₁) denote the number of “Yes” and “No” answers from the sample. Since all the respondents using a randomization device R₁ already responded “Yes” from the initial direct innocuous question, the proportion “Y” of getting “Yes” answers from the respondents using randomization device R₁ should be

Y = P_{1} π_{S} + (1 - P_{1}) π_{1} = P_{1} π_{S} + (1 - P_{1}),

where

π_{S}

is the proportion of “Yes” answers from the sensitive trait and

π_{1}

is the proportion of “Yes” answers from the innocuous question.

An unbiased estimator of $π_{S}$ , in terms of the sample proportion of “Yes” responses $\hat{Y}$ , becomes

{\hat{π}}_{h 1} = \frac{\hat{Y} - (1 - P_{1})}{P_{1}},

whose variance is given by:

V ({\hat{π}}_{h 1}) = \frac{Y (1 - Y)}{n_{1} P_{1}^{2}} = [\frac{[(1 - π_{S}) π_{S}}{n_{1}} + \frac{(1 - π_{S}) (1 - P_{1})]}{n_{1} P_{1}}] .

The proportion of “Yes” answers from the respondents using Mangat’s (1994) randomization procedure is given by:

X = π_{S} + (1 - π_{S}) (1 - P),

where X is the proportion of “Yes” responses.

An unbiased estimator of $π_{S}$ , in terms of the sample proportion of “Yes” responses $\hat{X}$ , becomes

{\hat{π}}_{h 2} = \frac{\hat{X} - (1 - P)}{P} .

Its variance is given by:

\begin{aligned} V ({\hat{π}}_{h 2}) & = \frac{X (1 - X)}{n_{2} P^{2}} \\ = \frac{1}{n_{2}} [π_{S} (1 - π_{S}) + \frac{(1 - P) (1 - π_{S})}{P}] . \end{aligned}

Now we shall pool the two unbiased estimators using weights, to formulate an estimator for $π_{S}$ as

\begin{aligned} {\hat{π}}_{h} & = \frac{n_{1}}{n} {\hat{π}}_{h 1} + \frac{n_{2}}{n} {\hat{π}}_{h 2} \\ = \frac{n_{1}}{n} {\hat{π}}_{h 1} + \frac{(n - n_{1})}{n} {\hat{π}}_{h 2}, f o r 0 < \frac{n_{1}}{n} < 1. \end{aligned}

Since the randomization device R₁ and the used randomization procedure due to Mangat (1994) are independent, therefore,

\begin{aligned} V ({\hat{π}}_{h}) & = \frac{n_{1}}{n^{2}} [π_{S} (1 - π_{S}) + \frac{(1 - π_{S}) (1 - P_{1})}{P_{1}}] \\ + \frac{(n - n_{1})}{n^{2}} [π_{S} (1 - π_{S}) + \frac{(1 - π_{S}) (1 - P)}{P}] . \end{aligned}

Horvitz, Shah, and Simmons (1967) presented Simmon’s method, when

π_{1}

is known and unknown. Under the situation that the Warner (1965) model and Simmon’s method (known

π_{1}

) are equally confidential to respondents, Lanke (1976) obtained a unique value of P as

P = \frac{1}{2} + \frac{P_{1}}{2 P_{1} + 4 (1 - P_{1}) π_{1}} f o r e v e r y P_{1} a n d e v e r y π_{1} .

Since the proposed mixed RR model also use Simmon’s method $π_{1} = 1$ , we can apply Lanke’s (1976) arguments to the suggested model. Putting $π_{1} = 1$ in (9), we get

P = {(2 - P_{1})}^{- 1} .

Setting $P = {(2 - P_{1})}^{- 1}$ in equation (8), we get the variance of ${\hat{π}}_{h}$ as

V ({\hat{π}}_{h}) = \frac{1}{n} [π_{S} (1 - π_{S}) + (1 - π_{S}) (1 - P_{1}) \{\frac{λ}{P_{1}} + (1 - λ)\}],

where

n = n_{1} + n_{2} a n d λ = \frac{n_{1}}{n} .

Thus, we established the following theorem.

Theorem 2.1: The variance of ${\hat{π}}_{h}$ is given by:

V ({\hat{π}}_{h}) = \frac{1}{n} [π_{S} (1 - π_{S}) + (1 - π_{S}) (1 - P_{1}) \{\frac{λ}{P_{1}} + (1 - λ)\}] .

Efficiency Comparisons

An efficiency comparison of the proposed mixed RR model, under completely truthful reporting case, has been done with Kim and Warde’s (2005) model.

From Kim and Warde’s (2005, eq. 2.10:213), the variance of the Kim and Warde (2005) estimator ${\hat{π}}_{k w}$ based on mixed RR model is given by:

V ({\hat{π}}_{k w}) = \frac{1}{n} [π_{S} (1 - π_{S}) + (1 - P_{1}) \{\frac{λ (1 - π_{S})}{P_{1}} + \frac{(1 - λ)}{P_{1}^{2}}\}] .

From equations (12) and (13), we have

n [V ({\hat{π}}_{k w}) - V ({\hat{π}}_{h})] = (1 - λ) (1 - P_{1}) [\frac{1}{P_{1}^{2}} - 1 + π_{S}],

which is always positive.

Thus, the proposed estimator ${\hat{π}}_{h}$ is more efficient than the Kim and Warde (2005) estimator ${\hat{π}}_{k w}$ .

To have the insight over the performance of the proposed mixed RR model, we have computed the percentage relative efficiency (PRE) of the proposed estimator ${\hat{π}}_{h}$ with respect to the Kim and Warde’s (2005) estimator ${\hat{π}}_{k w}$ using the formula:

\begin{aligned} P R E ({\hat{π}}_{h}, {\hat{π}}_{k w}) & = \frac{V ({\hat{π}}_{k w})}{V ({\hat{π}}_{h})} \times 100 \\ = \frac{[π_{S} (1 - π_{S}) + (1 - P_{1}) \{\frac{λ (1 - π_{S})}{P_{1}} + \frac{(1 - λ)}{P_{1}^{2}}\}]}{[π_{S} (1 - π_{S}) + (1 - P_{1}) (1 - π_{S}) \{\frac{λ}{P_{1}} + (1 - λ)\}]} \times 100, \end{aligned}

for different cases of

π_{S}, n, n_{1}, P_{1}

, and

λ = (0. 2, 0. 4, 0. 6, 0. 8)

. Findings are shown in Table 1 and diagrammatic representation is given in Figure 1.

Table 1.

Percentage Relative Efficiency of the Proposed Estimator ${\hat{π}}_{h}$ with Respect to Kim and Warde’s (2005) Estimator ${\hat{π}}_{k w}$ .

$π_{S}$	n = 1,000		$λ$ = n₁/n	P ₁
$π_{S}$	n ₁	n ₂	$λ$ = n₁/n	0.3	0.4	0.5	0.6	0.7	0.8	0.9
0.1	200	800	0.2	663.92	424.24	296.82	220.66	171.46	137.99	114.70
	400	600	0.4	427.87	301.88	229.16	182.53	150.54	127.60	110.90
	600	400	0.6	278.47	215.05	176.54	150.57	131.87	117.84	107.19
	800	200	0.8	175.39	150.23	134.44	123.39	115.12	108.66	103.55
0.2	200	800	0.2	688.40	433.67	300.00	221.08	170.80	137.19	114.37
	400	600	0.4	448.49	311.42	233.33	183.96	150.62	127.23	110.70
	600	400	0.6	291.96	222.01	180.00	152.04	132.24	117.73	107.08
	800	200	0.8	181.77	153.78	136.36	124.31	115.43	108.66	103.51
0.3	200	800	0.2	727.80	452.38	309.52	226.08	173.46	138.65	115.18
	400	600	0.4	477.82	326.53	241.42	188.31	152.92	128.43	111.32
	600	400	0.6	310.32	232.14	185.71	155.22	133.94	118.59	107.51
	800	200	0.8	190.26	158.73	139.28	125.99	116.35	109.12	103.73
0.4	200	800	0.2	787.64	483.05	326.66	236.11	179.41	142.07	116.84
	400	600	0.4	519.64	349.26	254.54	196.07	157.52	131.04	112.57
	600	400	0.6	335.82	246.75	194.44	160.49	137.07	120.37	108.34
	800	200	0.8	201.90	165.69	143.58	128.65	117.94	110.02	104.15

Figure 1.

Percentage relative efficiency of the proposed estimator ${\hat{π}}_{h}$ with respect to Kim and Warde’s (2005) estimator ${\hat{π}}_{k w}$ .

It is observed from Table 1 and Figure 1 that the values of $P R E ({\hat{π}}_{h}, {\hat{π}}_{k w})$ is more than 100. We can say that the envisaged estimator ${\hat{π}}_{h}$ is more efficient than the Kim and Warde’s (2005) estimator ${\hat{π}}_{k w}$ . Figure 1 show results for $π_{S} = (0. 1 a n d 0. 4)$ , $λ = (0. 2, 0. 4, 0. 6, 0. 8)$ , and different values of $P_{1}, n, a n d n_{1}$ , which also justify the previous fact.

We note from Table 1 that the values of the $P R E ({\hat{π}}_{h}, {\hat{π}}_{k w})$ decrease as the value of P₁ increases. Also the values of $P R E ({\hat{π}}_{h}, {\hat{π}}_{k w})$ increase as the value of λ decreases for fixed values of P₁. We further note from the results of Figure 1 that there is large gain in efficiency by using the suggested estimator ${\hat{π}}_{h}$ over the Kim and Warde’s (2005) estimator ${\hat{π}}_{k w}$ when the proportion of stigmatizing attribute is moderately large.

A Mixed RR Model Using Stratification

Stratified random sampling is generally obtained by dividing the population into nonoverlapping groups called strata and selecting a simple random sample from each stratum. The main advantage of the stratified approach is that the technique overcomes the limitation of the loss of individual characteristics of the respondents. An RR technique using a stratified random sampling gives the group characteristics related to each stratum estimator. Also, stratified sampling protects a researcher from the possibility of obtaining a poor sample. Hong, Yum, and Lee (1994) suggested a stratified RR technique using a proportional allocation. Kim and Warde (2004) presented a stratified RR model based on Warner (1965) model that has an optimal allocation and large gain in precision. Kim and Elam (2005) suggested a two-stage stratified Warner’s RR model using optimal allocation. Further Kim and Warde (2005) suggested a mixed stratified RR model.

In the proposed model, the assumptions for a stratified mixed RR model are similar to Kim and Warde (2004) and Kim and Elam (2005) model. We assume that the population is partitioned into “r” nonoverlapping strata, and a sample is selected by simple random sampling with replacement from each stratum. To get the full benefit from stratification, we assumed that the number of units in each stratum is known. In this proposed model, an individual respondent in a sample from each stratum is instructed to answer a direct question “I am a member of the innocuous trait group.” Respondents answer the direct question by “Yes” or “No.” If a respondent answers “Yes,” then he or she is instructed to go to the randomization device R_k₁ consisting of statements: (i) “I am the member of the sensitive trait group” and (ii) “I am a member of the innocuous trait group” with preassigned probabilities Q_k and (1 − Q_k), respectively. If a respondent answers “No,” then the respondent is instructed to use a randomization procedure due to Mangat (1994). In the Mangat’s (1994) RR procedure, each respondent is instructed to say “Yes” if he or she is a member of the sensitive trait group. If he or she is not a member of the sensitive trait group, then the respondent is required to use the Warner’s (1965) randomization device R_k₂ consisting of the statement: (i) “I belong to the sensitive trait group” and (b) “I do not belong to the sensitive trait group” with preassigned probabilities P_k and (1 − P_k), respectively. Then he or she is to report “Yes” or “No” according to the outcome of the randomization device R_k₂ and the actual status that he or she has with respect to the sensitive trait group. The survey procedures are performed under the assumption that both the sensitive and the innocuous questions are unrelated and independent in a randomization device R_k₁. To protect the respondent’s privacy, the respondents should not disclose to the interviewer the question they answered from either R_k₁ or R_k₂. Suppose we denote m_k as the number of units in the sample from stratum k and n as the total number of units in samples from all strata. Let m_k₁ be the number of people responding “Yes” when respondents in a sample m_k were asked the direct question and m_k₂ be the number of people responding “No” when respondents in a sample m_k were asked the direct question so that $n = \sum_{k = 1}^{r} m_{k} = \sum_{k = 1}^{r} (m_{k 1} + m_{k 2})$ . Under the assumption that these “Yes” or “No” reports are made truthfully, and Q_k and P_k (≠ 0.5) are set by the researcher, then the proportion of “Yes” answer from the respondents using the randomization device R_k₁ will be

Y_{k} = Q_{k} π_{S_{k}} + (1 - Q_{k}) π_{1_{k}} f o r k = 1, 2, \dots, r,

where Y_k is the proportion of “Yes” answers in stratum k,

π_{S_{k}}

is the proportion of respondents with the sensitive trait in stratum k,

π_{1_{k}}

is the proportion of respondents with the innocuous trait in stratum k, and Q_k is the probability that a respondent in the sample stratum k is asked a sensitive question.

Since the respondent performing a randomization device R_k₁ respond “Yes” to the direct question of the innocuous trait, if he or she chooses the same innocuous question from R_k₁, then $π_{1_{k}}$ is equal to one $(i . e ., π_{1_{k}} = 1)$ . Therefore, equation (16) becomes $Y_{k} = Q_{k} π_{S_{k}} + (1 - Q_{k})$ . The estimator of $π_{S_{k}}$ is given by:

{\hat{π}}_{h 1_{k}} = \frac{{\hat{Y}}_{k} - (1 - Q_{k})}{Q_{k}} f o r k = 1, 2, \dots, r,

where

{\hat{Y}}_{k}

is the proportion of “Yes” answers in a sample in stratum k and

{\hat{π}}_{h 1_{k}}

_is the proportion of respondents with the sensitive trait in a sample from stratum k. Since each

{\hat{Y}}_{k}

is a binomial distribution B (m_k₁, Y_k), the estimator

{\hat{π}}_{h 1_{k}}

is an unbiased estimator for

π_{S_{k}}

with the variance,

\begin{aligned} V ({\hat{π}}_{h 1_{k}}) & = \frac{Q_{k} (1 - π_{S_{k}}) [Q_{k} π_{S_{k}} + (1 - Q_{k})]}{m_{k 1} Q_{k}^{2}} \\ = \frac{(1 - π_{S_{k}}) [Q_{k} π_{S_{k}} + (1 - Q_{k})]}{m_{k 1} Q_{k}} . \end{aligned}

The proportion of “Yes” answers from the respondents using Mangat (1994) RR technique will be

X_{k} = π_{S_{k}} + (1 - π_{S_{k}}) (1 - P_{k}),

where X_k is the proportion of “Yes” responses in stratum k,

π_{S_{k}}

is the proportion of respondents with the sensitive trait in stratum k, and P_k is the probability that a respondent in the sample stratum k has a sensitive question card. The unbiased estimator in this case is given by:

{\hat{π}}_{h 2_{k}} = \frac{{\hat{X}}_{k} - (1 - P_{k})}{P_{k}},

where

{\hat{X}}_{k}

is the proportion of “Yes” responses in a sample from a stratum k and

{\hat{π}}_{h 2_{k}}

is the proportion of respondents with the sensitive trait in a sample from stratum k. By using m_k = m_k₁ + m_k₂ and P_k = (2 − Q_k)⁻¹, the variance of

{\hat{π}}_{h 2_{k}}

is given by:

V ({\hat{π}}_{h 2_{k}}) = \frac{1}{(m_{k} - m_{k 1})} [π_{S_{k}} (1 - π_{S_{k}}) + \frac{(1 - Q_{k}) (1 - π_{S_{k}})}{Q_{k}}] .

The unbiased estimator of $π_{S_{k}}$ , in terms of sample proportion of “Yes” responses ${\hat{Y}}_{k}$ and ${\hat{X}}_{k}$ , is

{\hat{π}}_{m S_{k}} = \frac{m_{k 1}}{m_{k}} {\hat{π}}_{h 1_{k}} + \frac{m_{k} - m_{k 1}}{m_{k}} {\hat{π}}_{h 2_{k}}, f o r 0 < \frac{m_{k 1}}{m_{k}} < 1.

Its variance is

V ({\hat{π}}_{m S_{k}}) = \frac{π_{S_{k}} (1 - π_{S_{k}})}{m_{k}} + \frac{1}{m_{k}} [(1 - π_{S_{k}}) (1 - Q_{k}) \{\frac{λ_{k}}{Q_{k}} + (1 - λ_{k})\}],

where m_k = m_k₁ + m_k₂ and

λ_{k} = m_{k 1} / m_{k} .

Thus, the unbiased estimator of $π_{S} = \sum_{k = 1}^{r} w_{k} π_{S_{k}}$ is given by:

{\hat{π}}_{m S} = \sum_{k = 1}^{r} w_{k} {\hat{π}}_{m S_{k}} = \sum_{k = 1}^{r} w_{k} [\frac{m_{k 1}}{m_{k}} {\hat{π}}_{h 1_{k}} + \frac{m_{k} - m_{k 1}}{m_{k}} {\hat{π}}_{h 2_{k}}],

where N is the number of units in the whole population, N_k is the total number of units in stratum k, and w_k = N_k/N for k = 1, 2, … , r, so that

\sum_{k = 1}^{r} w_{k} = 1.

It can be shown that the proposed estimator

{\hat{π}}_{m S}

is unbiased for the population proportion

π_{S}

. The variance of the estimator

{\hat{π}}_{m S}

is given by:

\begin{aligned} V ({\hat{π}}_{m S}) \end{aligned} = \sum_{k = 1}^{r} \frac{w_{k}^{2}}{m_{k}} \times [π_{S_{k}} (1 - π_{S_{k}}) + ((1 - π_{S_{k}}) (1 - Q_{k}) \{\frac{λ_{k}}{Q_{k}} + (1 - λ_{k})\})],

In order to do the optimal (Neyman) allocation of a sample size n, we need to know $λ_{k} = m_{k 1} / m_{k}$ and $π_{S_{k}}$ . Information on $λ_{k} = m_{k 1} / m_{k}$ and $π_{S_{k}}$ is usually unavailable. But if prior information about them is available from past experience, it will help to derive the following optimal (Neyman) allocation formula.

Theorem 4.1: The optimal (Neyman) allocation of m to m₁, m₂…m_{r − 1} and m_r to derive the minimum variance of the ${\hat{π}}_{m S}$ subject to $n = \sum_{k = 1}^{r} m_{k}$ is approximately given by:

\frac{m_{k}}{n} = \frac{w_{k} {[π_{S_{k}} (1 - π_{S_{k}}) + ((1 - π_{S_{k}}) (1 - Q_{k}) \{\frac{λ_{k}}{Q_{k}} + (1 - λ_{k})\})]}^{1 / 2}}{\sum_{k = 1}^{r} w_{k} {[π_{S_{k}} (1 - π_{S_{k}}) + ((1 - π_{S_{k}}) (1 - Q_{k}) \{\frac{λ_{k}}{Q_{k}} + (1 - λ_{k})\})]}^{1 / 2}},

where m_k = m_k₁ + m_k₂ and

λ_{k} = m_{k 1} / m_{k} .

The minimal variance of the estimator ${\hat{π}}_{m S}$ is given by:

\begin{array}{l} V ({\hat{π}}_{m S}) \\ = \frac{1}{n} {[\sum_{k = 1}^{r} {w_{k} \{π_{S_{k}} (1 - π_{S_{k}}) + ((1 - π_{S_{k}}) (1 - Q_{k}) \{\frac{λ_{k}}{Q_{k}} + (1 - λ_{k})\})\}}^{1 / 2}]}^{2}, \end{array}

where

n = \sum_{k = 1}^{r} m_{k}

, m_k = m_k₁ + m_k₂ and

λ_{k} = m_{k 1} / m_{k}

Efficiency Comparison

In this section, we do an efficiency comparison of Kim and Warde’s (2005) stratified mixed RR model with our proposed stratified mixed RR model by way of variance comparison.

The minimal variance of the Kim and Warde’s (2005) estimator ${\hat{π}}_{k w}$ based on stratified mixed RR model is given by:

\begin{aligned} V ({\hat{π}}_{k w}) = \frac{1}{n} {[\sum_{k = 1}^{r}, w_{k} {\{π_{S_{k}} (1 - π_{S_{k}}) + (\frac{(1 - Q_{k}) {λ_{k} Q_{k} (1 - π_{S_{k}}) + (1 - λ_{k})}}{Q_{k}^{2}})\}}^{1 / 2}]}^{2} = \frac{1}{n} {[\sum_{k = 1}^{2}, w_{k} {a_{k} + b_{k} + c_{k}}^{1 / 2}]}^{2}, \end{aligned}

(see Kim and Warde’s 2005, eq. (4.12):218), where

\begin{aligned} a_{k} = π_{S_{k}} (1 - π_{S_{k}}), b_{k} & = \frac{{λ_{k} (1 - π_{S_{k}}) (1 - Q_{k})}}{Q_{k}} \\ a n d c_{k} & = \frac{{(1 - Q_{k}) (1 - λ_{k})}}{Q_{k}^{2}} . \end{aligned}

The minimal variance of the proposed estimator ${\hat{π}}_{m S}$ in equation (27) is rewritten as

V ({\hat{π}}_{m S}) = \frac{1}{n} {[\sum_{k = 1}^{r}, w_{k} {a_{k} + b_{k} + d_{k}}^{1 / 2}]}^{2},

where a_k and b_k are same as defined in equation (28) and

d_{k} = \frac{{(1 - Q_{k}) (1 - λ_{k}) (1 - π_{S_{k}}}}{Q_{k}}

From equations (28) and (29), we have

\begin{array}{l} V ({\hat{π}}_{k w}) - V ({\hat{π}}_{m S}) \\ = \frac{1}{n} \{{[\sum_{k = 1}^{r} w_{k} {a_{k} + b_{k} + c_{k}}^{1 / 2}]}^{2} - {[\sum_{k = 1}^{r} w_{k} {a_{k} + b_{k} + d_{k}}^{1 / 2}]}^{2}\} \\ = \frac{1}{n} \{\sum_{k = 1}^{r} w_{k} {a_{k} + b_{k} + c_{k}}^{1 / 2} - \sum_{k = 1}^{r} w_{k} {a_{k} + b_{k} + d_{k}}^{1 / 2}\} \\ \times \{\sum_{k = 1}^{r} w_{k} {a_{k} + b_{k} + c_{k}}^{1 / 2} + \sum_{k = 1}^{r} w_{k} {a_{k} + b_{k} + d_{k}}^{1 / 2}\} \\ = \frac{1}{n} [\sum_{k = 1}^{r} \{w_{k} {a_{k} + b_{k} + c_{k}}^{1 / 2} - {a_{k} + b_{k} + d_{k}}^{1 / 2}\}] \\ \times [\sum_{k = 1}^{r} \{w_{k} {a_{k} + b_{k} + c_{k}}^{1 / 2} + {a_{k} + b_{k} + d_{k}}^{1 / 2}\}], \end{array}

which is positive if

\begin{array}{l} [{a_{k} + b_{k} + c_{k}}^{1 / 2} - {a_{k} + b_{k} + d_{k}}^{1 / 2}] > 0 \forall k = 1, 2, \dots, r \\ t h a t i s, i f {a_{k} + b_{k} + c_{k}} > {a_{k} + b_{k} + d_{k}} \forall k = 1, 2, \dots, r \\ t h a t i s, i f {a_{k} + b_{k} + c_{k}} > {a_{k} + b_{k} + d_{k}} \forall k = 1, 2, \dots, r \\ t h a t i s, i f c_{k} > d_{k} \forall k = 1, 2, \dots, r \\ t h a t i s, i f \frac{(1 - λ_{k}) (1 - Q_{k})}{Q_{k}^{2}} > \frac{(1 - Q_{k}) (1 - π_{S_{k}}) (1 - λ_{k})}{Q_{k}}, \\ \forall k = 1, 2, . . ., r \\ t h a t i s, i f \frac{1}{Q_{k}} > (1 - π_{S_{k}}), \forall k = 1, 2, \dots, r \\ t h a t i s, i f (\frac{1}{Q_{k}} - 1 + π_{S_{k}}) > 0, \forall k = 1, 2, \dots, r, \end{array}

which is always true.

It follows from equation (30) that our proposed stratified mixed RR model is superior to the one earlier considered by Kim and Warde (2005).

To have tangible idea about the performance of the proposed stratified estimator ${\hat{π}}_{m S}$ over Kim and Warde (2005) stratified estimator ${\hat{π}}_{k w}$ in case of two strata (i.e., r = 2), we have computed PRE by using the formula:

\begin{array}{l} P R E ({\hat{π}}_{m S}, {\hat{π}}_{k w}) = \frac{V ({\hat{π}}_{k w})}{V ({\hat{π}}_{m S})} \times 100 \\ = \frac{{[\sum_{k = 1}^{2}, w_{k} {\{π_{S_{k}} (1 - π_{S_{k}}) + (\frac{(1 - Q_{k}) {λ_{k} Q_{k} (1 - π_{S_{k}}) + (1 - λ_{k})}}{Q_{k}^{2}})\}}^{1 / 2}]}^{2}}{{[\sum_{k = 1}^{2}, w_{k} {\{π_{S_{k}} (1 - π_{S_{k}}) + ((1 - π_{S_{k}}) (1 - Q_{k}) \{\frac{λ_{k}}{Q_{k}} + (1 - λ_{k})\})\}}^{1 / 2}]}^{2}} \times 100, \end{array}

where

V ({\hat{π}}_{k w})

is given by Kim and Warde (2005, eq. (4.12):218),

π_{S} = w_{1} π_{S 1} + w_{2} π_{S 2} .

We have obtained the values of the PREs for different cases of $π_{S}, λ, n, n_{1}, a n d P_{1}$ . Findings are shown in Table 2, and diagrammatic representation is given in Figure 2, respectively.

Table 2.

Percentage Relative Efficiency of the Proposed Stratified Estimator ${\hat{π}}_{m S}$ With Respect to Kim and Warde’s (2005) Stratified Estimator ${\hat{π}}_{k w}$ .

$π_{S 1}$	$π_{S 2}$	$π_{S}$	w ₁	w ₂	λ	P₁ = Q₁ = Q₂
$π_{S 1}$	$π_{S 2}$	$π_{S}$	w ₁	w ₂	λ	0.9	0.91	0.92	0.93	0.94	0.95	0.96
0.88	0.92	0.896	0.6	0.4	0.1	349.31	339.00	328.76	318.58	308.44	298.33	288.23
			0.6	0.4	0.2	337.52	328.36	319.27	310.24	301.25	292.30	283.36
			0.6	0.4	0.3	325.55	317.57	309.66	301.80	293.98	286.21	278.45
			0.6	0.4	0.4	313.41	306.63	299.91	293.25	286.63	280.05	273.50
0.89	0.93	0.906	0.6	0.4	0.1	374.52	362.90	351.34	339.84	328.36	316.90	305.41
			0.6	0.4	0.2	361.31	350.97	340.69	330.46	320.27	310.09	299.91
			0.6	0.4	0.3	347.90	338.86	329.89	320.97	312.08	303.22	294.36
			0.6	0.4	0.4	334.27	326.57	318.93	311.34	303.79	296.26	288.76
0.90	0.94	0.916	0.6	0.4	0.1	405.96	392.69	379.47	366.29	353.12	339.94	326.71
			0.6	0.4	0.2	390.96	379.13	367.35	355.61	343.89	332.16	320.41
			0.6	0.4	0.3	375.72	365.36	355.05	344.78	334.53	324.30	314.05
			0.6	0.4	0.4	360.22	351.36	342.55	333.78	325.05	316.33	307.62
0.91	0.95	0.926	0.6	0.4	0.1	446.34	430.93	415.56	400.21	384.84	369.41	353.89
			0.6	0.4	0.2	429.03	415.26	401.54	387.83	374.11	360.36	346.55
			0.6	0.4	0.3	411.42	399.33	387.28	375.26	363.24	351.20	339.12
			0.6	0.4	0.4	393.48	383.11	372.78	362.48	352.19	341.91	331.61

Figure 2.

Percentage relative efficiency of the proposed stratified estimator ${\hat{π}}_{m S}$ with respect to Kim and Warde (2005) stratified estimator ${\hat{π}}_{k w}$ .

Table 2 exhibits that the PRE of the proposed stratified estimator ${\hat{π}}_{m S}$ with respect to the Kim and Wardes (2005) stratified estimator ${\hat{π}}_{k w}$ increases as sample size and value of P increase. Larger gain in efficiency is observed for small as well as moderately large sample sizes. However, the PRE is more than 100 for all parametric values considered here; therefore, the proposed estimator ${\hat{π}}_{m S}$ is better than the Kim and Warde’s (2005) stratified estimator ${\hat{π}}_{k w}$ . Also the values of the $P R E ({\hat{π}}_{m S}, {\hat{π}}_{k w})$ increase as the value of λ increase for fixed values of P₁. Figure 2 demonstrates that there is large gain in efficiency by using the suggested estimator ${\hat{π}}_{m S}$ based on the proposed stratified mixed RR model over the Kim and Warde’s (2005) stratified estimator ${\hat{π}}_{k w}$ when the proportion of stigmatizing attribute is moderately large.

Discussion

In this article, we have envisaged a mixed RR model as well as its stratified version to estimate the proportion of qualitative sensitive character. It has been shown both theoretically and empirically that the proposed mixed RR model and its stratified version are always better than the Kim and Warde’s (2005) mixed randomize response models with larger gain in efficiency. Thus, our recommendation is to prefer the proposed mixed RR model and its stratified version in practice.

Footnotes

Acknowledgements

The authors are thankful to the editor in chief—Professor Christopher Winship, and to the three anonymous learned referees for their valuable suggestions regarding improvement of the article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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