Effects of measurement and response errors on estimation based on centre sampling and two-phase centre sampling for elusive populations

Introduction

In elusive surveys, the population units are not only difficult to capture but response errors can be quite high especially if the study variable is an undesirable characteristic. Elusive populations with undesirable characteristics include populations such as homosexuals, drug addicts and illegal immigrants who are difficult to sample because they are rare in the general population and often not easy to identify. Such populations are generally best sampled and surveyed on specific locations or venues, commonly known as centers of sampling. The process is referred to as Centre Sampling (CS). Centre sampling was developed in Italy to carry out surveys on immigrant population composed of regular people (those with a residence permit according to the law) and irregular people (those illegally residing in the country), (Blangiardo, 2004) [3] . Work on elusive surveys has been done by Mecatti (2002) [16] and Migliorati (2003) [18] , amongst others. Fuller and Burmeister (1972) considered estimators for samples selected from two overlapping frames. Most sampling theory methods are limited to sampling designs in which the selection of the samples is done before the survey, such that none of the decisions about the sampling depends on what is observed when gathering the data. In such typical surveys decisions such as how to sample during the experiment are arrived at and fixed in advance. In repeated surveys, differences in response given by respondents may be caused by lack of understanding of the questions, difficult in determining a true value or due to lack of information required to answer the question correctly. The interviewer may contribute to the differences in responses by giving different interpretations which may lead to the interviewee not understanding the subject matter and the purpose of the survey (Burton and Blair, 1991) [6] . In studies where sampling is carried out instead of a census, two major categories of errors in survey estimates arise, namely; sampling errors and non-sampling errors. http://www.mathsjournal.com Sampling errors occur when only a part of the population is studied to make inferences about a target population. Sampling error would therefore not exist if a census is conducted. Non sampling errors arise in the course of collecting data, through the measuring process, interviewing or during observations. The other source of non-sampling errors is non-response, where respondents fail to respond due to reasons associated with the study variable. The centre sampling relies on the habit of the immigrants to congregate in particular places for social contacts, healthcare, religion or places such as churches, mosques, bus stations, markets, hospitals etc. Such centers which are finite in number are usually spread all over the geographical area of interest and overlap covering the entire population. Due to this, the number of centers is taken to be more than 2. In order to handle the problem of non-response and response errors, a two-phase sampling is necessary since under the general conditions of a survey, the measurement for any individual varies over repeated trials of the survey measurement process. The survey question and answer process is a four-step process that involves comprehension of the question, retrieval of the information from memory, assessment of the correspondence between the retrieved information and the requested information and communication (Sudman et al., 1996). The encoding of the information, a process outside the control of the survey interview, determines whether the information of interest is available for the respondent to retrieve from long term memory.

Estimation in Elusive surveys under non contaminated data

In this section consideration is made on estimation when data is assumed to be free of measurement and response errors under Centre sampling. Suppose the population is finite of size N and the study variable is  . Let the population true (error free) values be  

Profiles are very important practical tools since they can be assumed observable by asking the sampled unit the question; "Which other centers among the L considered and besides the one where you were sampled do you attend?".

According to Mecatti and Migliorati (2003) [17] , the population mean is given by (Birnbaum and Sirken, 1965; Thompson, 2002) [2,22] , then using profiles and multiplicity,  can be expressed as

is unbiased for l  under the second phase sampling design (Mecatti and Migliorati, 2003) [17] .

is an unbiased estimator for  according to standard theory for two phase centre sampling (Hedayat and Sinha,1991) [12] . By using sample totals instead of sample means the estimator (4) agrees with Maiti, Pal and Sinha (1993) [14] estimator for the population size N under two-dimensional sampling. Using standard results from two phase centre sampling, the estimator (4) has variance

with ll   being the joint inclusion probability of the pair of centers l and l at the first phase selection and   var l  conform with the general form (5).

, according to Mecatti and Migliorati (2003) [17] +, an unbiased estimator of the variance ( 6) is given by

Estimation under Centre Sampling (CS) for elusive population when there are measurement and response errors

Estimation is now considered in the presence of measurement and response errors. It is assumed that there exists a true value for each population unit which we denote by lrq  for unit q with profile r in centre l in the population of size N. In the process of making observations, a response error occurs for individual q with profile r in centre l such that rather than

 as the estimator of l  given by 21

Where lrq  is the 1 st order inclusion probability of unit q with profile r in centre l and ˆl  is the estimate of l 

The expectation of ˆl  is considered as a two-phase process where in the 1 st phase a unit is selected into the sample and in the 2 nd phase, an error randomly occurs from the population of possible errors. The expectation is therefore

The bias due to measurement errors is

The variance of the estimate is given by

N lrq lrq lrq lrq lrq lrq r q q q l r lrq lrq lrq

V I I I Nm

rr NN lrq lrq lrr qq r r q q r r lrq lr q mm

We then obtain NN lrq lrq lrrqq lrrqq lrr qq lrr qq r q q q r r q q l r lrq lrq lrq l r r lrq r q

, assuming independence between centres to get,

Simple Random Sampling

Under simple random sampling without replacement,

An estimator of the variance is

The variance of the estimator can be written as

4. Estimation under two phase Centre Sampling (CS) for elusive population when there are measurement and response errors two phase centre sampling a sample say S' is selected at the first phase (phase 1) and some auxiliary variable X is observed. In phase 2, a subsample is selected from the phase 1 sample.

Let as usual lrq  be the inclusion probability of unit q after the two phases. We can write   lrq P(unit q is selected after both phases) = P(unit q is selected at phase 1)  P(unit q is selected at phase 2 given that it was selected at phase 1) The variance under the two-phase sampling and measurement model is given by

The three components are obtained as follows,  

ll rr NN lrq lrq lr qq lrq lr q r r q q r r lrq lrq uu mm

1 } S lrq lr q lrr qq lrq lr q r r q q r r lrq lr q uu mm

NN lrq lrq lrq p lrq lrq rq lrqq lrq lrq lrr qq s r q q q l r lrq lrq lrq NN lrq lr q lrr qq lrq lr q r r q q r r lrq lrq

L ll rr ll rr lrr qq NN lrq lrq lrq lrq lrqq lrqq lrq lrq r q q q l r lrq lrq lrq NN lrq lr q lrr qq lrr qq lrq lr q r q q r r lrq lr q u u u Nm uu mm

L S S S lrq lrqq lrr qq pp r q s q q r r q q l r lrq lrq lrq r r lrq lr q

L SS lrq lrqq lrr qq lrr qq lrr qq s p p m p p lrqs r q s q q r r q q l r lrq lrq lrq r r lrq lr q

L SS lrqq lrqq lrr qq lrr qq lrq p r q s q q r r q q l r lrq lrq lrq lrq r r lrq lr q

N lrq lrqq lrr qq lrr qq lrr qq r q q q r r l r lrq lrq lrq r r lrq lr q N m m m

A numerical example

Considering an example with four centers. This implies that there are 2 L -1=15 possible profiles with multiplicities as shown in Table 1.

A normal population of size N=200 was randomly simulated with mean=60 and variance=10. The population values were then assigned to the profiles randomly. In Table 2 the mean and variance increase as the value of k and p increases. This implies that the more the measurement and response errors the more the mean and variance are overestimated. We deduce that for an increasing error (k) on the mean level, the measurement and response errors consequently increase.

Two phase Centre Sampling

A sample of size n=60 is then randomly selected from the N=200 simulated values as the first phase units to obtain 3. In Table 3 as the values of p and k increases, the mean and variance are overestimated. Compared to the results in Table 2, both the means and variance estimates are higher in Table 3, that represents the second phase sampling. Although two phase sampling is a strategy to reduce cost, when measurement and response errors are present the estimates are compromised. The presence of measurement and response errors cannot be ruled out in any survey.

Conclusion

Carrying out of any survey involves enormous amounts of finances, a lot of time and meticulous planning and coordination of the process. The data obtained is used for important issues such as preparation of budgets, development of strategic plans as well as designing intervention measures for certain undesirable trends in the society. It is therefore imperative that the sources of errors in surveys are identified so as to reduce their effects on the survey data obtained. This paper has shown that the presence of measurement and response errors in elusive surveys greatly affects the results by overestimating and underestimating the estimates. This will seek to improve efficiency on planning, forecasting as well policy formulation processes.

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