ABSTRACT
In the current book, rotatable and slope-rotatable designs with correlated errors are discussed in Chapters 2 to 8. First-and second-order correlated regression models have been discussed under different error structures (intra-class, inter-class, compound symmetry, tridiagonal and autocorrelated). Note that intra-class and inter-class structures are some particular cases of compound symmetry structure (Chapter 2). Using robust designs (controlled setting) or environmental data, one may estimate the regression model parameters.
Different sources of arising correlation in the observations have been discussed in Chapter 2. For example, on the organismal level, there may be measurements on several tumors, both hands, all teeth, and so on. It is usually expected in such cases that the measurements on a given individual are more similar than those on different individuals. One may consider the regression analysis for some sets of observations such that within each set, observations have an autocorrelation structure, and any two observations from any two sets have a constant correlation coefficient, and the variance of all the observations are same. In general, the pattern (or form) of the correlation structure is known for a given situation of data set but the parameter (parameters) involved in the correlation structure is (are) always unknown. The present chapter describes different method of regression analyses on planned and unplanned data under the compound symmetry and compound autocorrelated error structures. We have discussed some robust methods of estimating the best linear unbiased estimators of all the regression parameters except the intercept, which is often unimportant in practice. In this connection, we have also described some testing procedures for any set of linear hypotheses regarding the unknown regression coefficients. Confidence interval (for an estimable linear function) and confidence ellipsoid (for a set of estimable linear functions) of regression parameters have been developed. Index of fit has also been described for the fitted regression models. Two applications of correlated regression analysis in block designs have been described. Some examples (with simulated data) illustrate the theories described in the present chapter.
