= Tj

Of the three models discussed in this chapter, the fixed effects model is by far the most developed. It is interesting that R. A. Fisher (1924) developed a procedure for analyzing the fixed model and called the procedure "the analysis of variance." The procedure included optimum methods for point estimation, confidence intervals, and hypothesis tests. In a classical paperS. Kolodziejczyk (1935) gave a general theory of the linear model that included the fixed effects model. During the 1930s, 1940s, and 1950s a tremendous amount of work extended and generalized this model. The components-of-variance model received much less attention during this period in part because it was not so amenable to optimum procedures and exact confidence intervals and tests. One of the early papers that attempted approximate procedures was by Smith ( 1936). Two of the important early papers were by Yates and Zacopanay (1935) and Daniels (1939). By 1951 enough work on components-of-variance models had been done for Crump (1951) to write a paper entitled, " The present status of variance components." Most of the papers on variance components during the 1930s and 1940s concerned point estimation. During the late 1940s and throughout the 1950s much work was done on procedures for obtaining approximate confidence intervals on linear combinations of variances. This work included contributions by Satterthwaite ( 1941, 1946), Green (1954), Welch (1956), Huitson (1955) , Bulmer (1957), and Moriguti (1955). Most of the work on variance component models during this period was based on the analysis of variance as if the model was a fixed effects model. Graybill and Hultquist ( 1961) gave conditions when this procedure led to optimum point estimation of linear combinations of variance components. During the 1960s and 1970s point estimation of variance components received a great deal of attention. Searle (1971) provides a summary of these and other activities. Papers concerning confidence intervals began appearing in the late 1970s. Burdick and Graybill (1988) reviewed this research and reported the present status of confidence intervals for functions of variance components. Most of the results for confidence intervals on functions of variance components have considered balanced components-of-variance models. More recently results for unbalanced models have been published. For readers who want to study all aspects of components-of-variance models,

fairly complete bibliographies are provided by Sahai ( 1979), Khuri and Sahai (1985), and Sahai, Khuri, and Kapadia (1985). Searle (1988) reviews some of the history and results on unbalanced and mixed models.