Multiple Characters in Multiphase Sampling

Abstract

History of learning about population by using sampling methods could be traced out even to very early stages of primitive life of mankind. It was more than 250 years ago that Bernoulli developed the theory of independent random sampling of elements from a population when the unit of sampling and the unit of analysis was the same (Hansen and Hurvits, 1943). A century later, Polsson indicated the use of stratification in sampling to get possibly more precise estimate of the parameters. Later on the techniques such as cluster sampling, sub-sampling, double sampling and successive sampling were developed with an idea of making the best use of the available resources. Research in the theory of sampling for surveys has been mainly concerned with the development of more efficient sampling systems, the system including both, the sampling design and the method of sampling. One sampling system is said to be more efficient than the other, if the variance or the mean square error of the estimate with the former system is less than that of the later, provided the cost of obtaining the data is the same. The development of stratified, multistage, multiphase, systematic and other sampling designs over simple random sampling, have all resulted in increased efficiency in specific circumstances. With an idea to build up more precise estimators of the population parameters, the information collected on auxiliary variates are used. The auxiliary information is obtained in addition to the character under study and some extra information can be derived from it. The use of auxiliary information can be made on two occasions, one at the time of designing the survey and the other at the time of estimating the parameters. Sometimes the sampling units are selected with probability proportional to some measure of size, size being the value of auxiliary variable. The estimators which make use of auxiliary information include ratio and regression methods of estimation, which are by far the most common, though the estimators are biased. In sample surveys it happens sometimes that several auxiliary variables are available which are highly correlated with the variable under study. It is, therefore, of interest to investigate the effect of using all or some of them in building up an estimate of the population mean or total of the character under study.