Regression Technique in Multi Stage Design

Abstract

In the present work, the regression method of estimation based on large sample theory has been extended to multi-stage designs. The discussion has been confined to a three-stage design and the results are obtained by considering a linear ‘difference estimator’ of the population mean Y bar of a character under study. The formulae for estimates and their variances are all expressed in terms of the regression coefficient B and the correlation coefficient of the sample means of the ancillary character and the character under study. It has been shown that when the unites are highly correlated, the correlation coefficient of the sample means x bar and y bar will be a fairly good approximation to the overall correlation coefficient in the population. The estimates thus obtained are biased though consistent and the bias has been shown to be negligible for large sample. A comparison with the ratio type estimates in multi-stage design is made and it has been shown that the regression estimates are always more officiant than the ratio estimates. The case for stratification in multi-stage design is also considered and the formulae for the two common estimate vis., separate regression estimate and combined regression estimate are obtained. Results for the double sampling scheme when the population mean of the ancillary character is not known, is also considered and it has been shown that the regression estimates when double sampling is adopted, are always officiant than the simple estimates. The question of optimum allocation of sample size for the double sampling scheme are considered under certain limitations and the theory has been illustrated with reference to the data collected in the pilot survey for obtaining block level estimates of wheat crop carried out in the Patna District of Bihar during 1962-63. Finally, generalisation of the results to several auxiliary variables are also taken into consideration by forming a multiple regression estimate and modifications of the formulae when the population means of the ancillary characters are not known, are obtained by considering the technique of double sampling for several auxiliary variables.