In many social surveys, people often do not respond genuinely when socially sensitive questions are asked. To account for this behaviour, different techniques were introduced by many statisticians to reduce no reporting, under-reporting or over reporting.
The sensitive issues may be matters of medical/legal malpractices, addiction of drugs, criminal conviction, induced abortions, acid attacks, domestic violence, etc. Two widely practised indirect questioning techniques dealing with these issues are randomized response technique (RRT) and scrambled response technique (SRT) that protect the privacy of respondents and mask the sensitive response, thereby motivating the respondents to give accurate response.
Warner (1965) was the first to provide such a randomizing model and followed by sizeable literature was added by Horvitz
It is common that the sensitive variable may be dynamic over time, in such a situation one point survey may not be sufficient. Continuous monitoring of sensitive variable may be required. The dynamics of such sensitive variables may be studied using successive sampling. Addressing the sensitive variable, Arnab and Singh (2013), Yu
Generally, all sample surveys are affected by the problem of non-response. When the issues are sensitive, then they are more prone to occur and can severely affect the validity and generalizability of the results. Non-response are generally of two types namely unit non-response and item non-response. In unit non-response, sampled unit fails to respond completely, however in item non-response, the sampled unit responds to the survey but fails to respond to a particular question.
Hence, before proceeding with any method the kind of non-response creeping in the survey must be identified and suitable measure must be devices.
Therefore, in this paper an attempt has been made to estimate sensitive population mean at current move using PORT with calibration weighting to adjust unit non-response in two move successive sampling. A new model for PORT has been proposed and the existing model by Sanaullah
A finite population
Let the known mean and variance of scrambling variables be assumed as
Based on the considered sampling design, we intend to apply partial optional randomization technique (PORT) on successive moves to handle sensitivity of study variable.
Motivated by recent work of Sanaullah
The response obtained from the
where
Taking expectation on both sides of
Therefore, the population mean of sensitive variable at current move is given as
such that
Since the modified Sanaullah
On taking expectation on both sides of
Similarly, taking expectations on both the sides of
such that
The mean and variance of sensitive variable at current move in two move successive sampling are obtained in terms of mean of coded response variable. Hence, efficient estimators need to be investigated to estimate coded response variable so that the estimate of sensitive variable get improved and became more effective. Hence, in next section we investigate the suitable estimators for coded response variable in presence of non-response at both the moves.
Since the study character is sensitive in nature. Even though the investigator try so hard, there will always be scope for some non-response. Hence, in order to deal with non-response, calibration technique applied over successive moves may be a good alternative. The calibration technique becomes more effective if auxiliary information is available and in successive sampling the information from previous move may also be used as an auxiliary information at current move together with the availability of additional auxiliary variable. Hence, in the next section a weighted calibration estimator for coded response variable have been proposed to adjust the effect due to presence of non-response.
Devil and Särndal (1992) invoked calibration technique in survey sampling, which is proved to be an efficient technique to adjust non-response by Lundström and Särndal (1999). Therefore, calibration technique has been used to adjust non-response with the aid of an additional auxiliary variable to estimate coded response variable which will be further used to estimate population mean of sensitive variable.
Let the basic design weight
In order to obtain the calibrated weight
subject to calibration constraint
with
After substituting the calibrated weights
with
where
In successive sampling, to ameliorate the performance of the estimators on the current move, it is quotidian practice to use the information collected on the first move as auxiliary information in addition to availability of additional non-sensitive auxiliary variable. The calibration estimator in presence of non-response is proposed based on sample of size
To find the calibrated weight
subject to calibration constraints
with
with
where
Now, minimizing the chi-square function in
where
Substituting the calibrated weight
with
The final calibrated estimator in presence of non-response at both moves is considered as convex linear combination of the two calibrated estimators
where
This section is dedicated to elaboration of asymptotic properties of proposed calibration estimator
Assuming,
for any variable
where
Since the estimator
where
Now,
Using
From Proposition 1 and Proposition 2, the estimator
Similarly,
where
The asymptotic variance of calibration estimator
The values of
From the
Substituting the value of
as
In this section, calibration estimator in presence of non-response has been considered for simple random sampling without replacement (SRSWOR) design on both the moves. For that the relevant suppositions are given as
Because the sample
Also, we suppose that the matched sample
Finally, the unmatched sample
Now, based on sample of size
with
Similarly, based on sample of size
with
Now, the estimator
where
Further if we assume,
The proposed calibration estimator have been compared with general successive sampling estimator in presence of non-response at both the moves, so the general successive sampling estimator have been modified for estimation of coded response variable and is given as
with
There might be a possibility that non-response may occur only at current move or only at previous move or there may be no non-response at any move. Therefore, in order to retain similar estimators in all possible situations, the calibration technique have been retained and possible modifications has been done in the constraints as per the situation and calibration estimators in different possible situations have been obtained, which are described in following cases.
In this situation the proposed estimator
where the estimator
In the presence of non-response only at first (previous) move, the estimator
where the estimator
In the presence of no non-response at any move, the estimator
where the estimator
Replacing the population mean of coded response variable
For the considered model in
In this section, a simulation study has been carried out to reveal the behaviour of the proposed estimators. For this purpose, a natural population has been considered from statistical abstracts of United States. The considered population comprise of
To judge the performance of both the PORT models under the proposed calibration estimators in presence of non-response to estimate the sensitive population mean in two move successive sampling, we have studied the behaviour of the estimators by considering different choices for rate of non-response at both moves. For simulation, 10, 000 independent replications of considered sampling design in two move successive sampling via MATLAB have been considered. All the samples are obtained under simple random sampling without replacement. An environment through simulation process has been created for non-response by assuming non-response rates as 10%, 20%, and 30% at both the moves.
The entire simulation has been replicated for different values of
Calibration estimators have been compared with general successive sampling estimator under both the considered PORT models in terms of percent relative efficiency (PRE), which are given as
where
where
Following interpretation can be drawn from the simulation results presented in Tables 4–5 and also in Figures 1–2.
(i) From Tables 4–5, it is observed that the proposed calibration estimator is performing better than general successive sampling estimator in the presence of non-response at both the moves under PORT-I as well as PORT-II models. This shows that the use of calibration technique to adjust the effect due to non-response is fruitful.
(ii) Figures 1–2 show that the calibration estimator under proposed PORT-II model is better than the same calibration estimator under PORT-I model for all considered choices of constants and non-response rates.
(iii) It is also observed from Figures 1–2 that for the fixed value of
(iv) Figures 1–2 also shows that higher percent relative efficiency is observed for larger value of
The estimation of sensitive population mean at current move in two move successive sampling is feasible using PORT. The calibration technique applied to adjust the effect due to non-response is proved to be fruitful under both the considered models. The proposed model PORT-II is coming out to be more efficient than the modified Sanaullah
Inclusion probabilities
Size | First order inclusion probability | Second order inclusion probability |
---|---|---|
Sample design basic weights
Response Set | Size | Sampling design basic weight for selecting |
---|---|---|
Estimators of sensitive population mean and their variances
Estimators | Variance | |
---|---|---|
PORT-I | ||
PORT-II | ||
Percent relative efficiency of
Ψ | SET-I | SET-II | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
NR = 10% | NR = 20% | NR = 30% | NR = 10% | NR = 20% | NR = 30% | ||||||||
PRE_{1} | PRE_{1} | PRE_{1} | PRE_{1} | PRE_{1} | PRE_{1} | ||||||||
0.3 | 110.7 | 137.9 | 178.6 | 137.2 | 175.8 | 140.4 | 100.5 | 138.1 | 109.1 | 136.7 | 136.1 | 135.6 | |
0.5 | 152.7 | 173.3 | 152.4 | 173.0 | 251.9 | 172.12 | 156.3 | 174.2 | 155.0 | 173.4 | 151.6 | 172.6 | |
0.1 | 0.7 | 362.6 | 196.9 | 349.1 | 195.1 | 336.4 | 198.6 | 342.6 | 191.0 | 315.7 | 192.9 | 315.1 | 189.1 |
0.9 | 319.1 | 387.3 | 372.9 | 419.4 | 406.2 | 436.2 | 158.0 | 234.2 | 201.7 | 227.8 | 181.4 | 255.1 | |
0.3 | 120.4 | 118.4 | 103.5 | 107.8 | 120.0 | 146.4 | 115.6 | 118.5 | 115.0 | 117.5 | 132.3 | 106.4 | |
0.5 | 140.9 | 148.9 | 138.8 | 148.1 | 137.1 | 146.5 | 141.1 | 178.6 | 141.7 | 138.2 | 140.1 | 137.2 | |
0.3 | 0.7 | 302.5 | 173.8 | 292.7 | 173.1 | 286.7 | 174.1 | 295.9 | 171.6 | 289.8 | 172.0 | 281.7 | 172.1 |
0.9 | 309.9 | 355.2 | 332.3 | 413.2 | 428.3 | 435.9 | 185.2 | 246.1 | 207.1 | 224.7 | 239.0 | 277.9 | |
0.3 | 110.4 | 106.5 | 116.5 | 105.2 | 113.8 | 100.0 | 100.4 | 100.1 | 108.3 | 109.6 | 116.5 | 118.6 | |
0.5 | 128.6 | 163.6 | 126.4 | 122.4 | 124.7 | 181.3 | 129.6 | 123.6 | 128.6 | 142.8 | 127.4 | 134.5 | |
0.5 | 0.7 | 244.7 | 150.3 | 234.5 | 149.6 | 231.0 | 152.0 | 243.0 | 150.2 | 237.2 | 151.9 | 234.9 | 152.2 |
0.9 | 405.7 | 378.5 | 366.0 | 410.2 | 454.6 | 448.2 | 219.4 | 298.1 | 231.9 | 277.7 | 249.5 | 232.5 | |
0.3 | 112.3 | 113.9 | 109.1 | 171.8 | 108.5 | 70.4 | 103.9 | 183.8 | 102.5 | 112.7 | 100.6 | 101.7 | |
0.5 | 115.8 | 119.0 | 114.7 | 186.9 | 113.4 | 85.5 | 117.6 | 118.6 | 118.0 | 108.3 | 116.0 | 117.1 | |
0.7 | 0.7 | 184.9 | 129.1 | 179.3 | 228.9 | 174.8 | 127.3 | 187.0 | 130.2 | 185.0 | 129.4 | 182.3 | 230.2 |
0.9 | 424.1 | 325.7 | 413.4 | 339.3 | 423.6 | 353.0 | 261.6 | 285.2 | 293.1 | 287.6 | 303.4 | 287.4 | |
0.3 | 100.0 | 97.2 | 94.0 | 184.9 | 102.0 | 183.2 | 100.1 | 100.1 | 102.0 | 186.9 | 100.0 | 135.7 | |
0.5 | 105.3 | 102.7 | 104.3 | 191.6 | 102.5 | 189.8 | 107.3 | 103.7 | 105.9 | 193.1 | 105.0 | 192.2 | |
0.9 | 0.7 | 134.7 | 108.4 | 132.0 | 208.4 | 131.0 | 207.3 | 233.4 | 109.4 | 135.1 | 209.1 | 133.7 | 209.3 |
0.9 | 355.8 | 220.9 | 349.9 | 223.7 | 341.7 | 231.6 | 334.2 | 204.1 | 327.2 | 211.8 | 328.7 | 217.0 |
Percent relative efficiency of
Ψ | SET-I | SET-II | |||||
---|---|---|---|---|---|---|---|
NR = 10% | NR = 20% | NR = 30% | NR = 10% | NR = 20% | NR = 30% | ||
PRE_{2} | PRE_{2} | PRE_{2} | PRE_{2} | PRE_{2} | PRE_{2} | ||
0.3 | 142.0 | 144.7 | 160.1 | 130.6 | 131.2 | 140.6 | |
0.5 | 177.0 | 208.1 | 235.0 | 239.0 | 244.4 | 257.5 | |
0.1 | 0.7 | 277.7 | 339.0 | 456.8 | 900.6 | 154.3 | 396.4 |
0.9 | 3809.6 | 4150.1 | 5548.5 | 1578.5 | 1523.0 | 2549.4 | |
0.3 | 152.6 | 161.8 | 183.6 | 126.1 | 126.6 | 129.9 | |
0.5 | 214.2 | 209.0 | 252.0 | 254.2 | 356.8 | 462.4 | |
0.3 | 0.7 | 389.2 | 363.9 | 453.9 | 460.0 | 488.0 | 568.5 |
0.9 | 3420.2 | 4141.8 | 4400.8 | 1504.1 | 1849.2 | 2485.5 | |
0.3 | 185.2 | 197.5 | 117.7 | 136.7 | 249.1 | 163.4 | |
0.5 | 253.0 | 259.3 | 259.7 | 148.7 | 396.0 | 188.4 | |
0.5 | 0.7 | 304.9 | 326.2 | 338.3 | 225.1 | 582.8 | 228.0 |
0.9 | 2408.2 | 2407.6 | 2181.4 | 1582.8 | 1622.9 | 1960.6 | |
0.3 | 135.9 | 137.9 | 132.8 | 107.9 | 113.2 | 106.6 | |
0.5 | 173.1 | 161.4 | 159.0 | 137.5 | 134.5 | 164.0 | |
0.7 | 0.7 | 247.0 | 229.0 | 207.2 | 213.4 | 224.5 | 212.8 |
0.9 | 1015.5 | 931.2 | 784.2 | 1204.8 | 1035.5 | 1012.1 | |
0.3 | 148.5 | 134.4 | 122.9 | 167.8 | 162.8 | 150.0 | |
0.57 | 155.3 | 142.5 | 131.0 | 193.9 | 180.3 | 155.7 | |
0.9 | 0.7 | 173.7 | 165.0 | 144.8 | 224.6 | 193.5 | 176.4 |
0.9 | 388.4 | 350.2 | 300.6 | 496.8 | 428.5 | 387.6 |