| Abstract: | In this dissertation, we develop an extension of the new method (EM-REML-AVE) to estimate variance components including continuous variables in the fixed effect design matrix, suggest a procedure to estimate and make inferences related to the fixed effects, and develop software to estimate variance components with this new procedure. To develop this extension, we first find the REML estimators for variance components including continuous variables in the fixed effect design matrix in the balanced case. Then, we develop estimators for variance components in the unbalanced case applying the EM algorithm to the REML estimation with the inclusion of continuous variables in the design matrix. This development is made under the assumption of orthogonality of the continuous variables to all effects. Finally, we propose a methodology to assure the assumption of orthogonality and extend the EM-REML estimation to AVE method. It is interesting to note that the REML, EM-REML, and classical methods only provide one point estimator for each variance component. However, the AVE algorithm additionally provides several individual estimators for each variance component which provides diagnostic information, a characteristic that, in the classical methodologies is not present. That extension is called EM-REML-AVE method which includes the balanced and unbalanced cases with or without continuous variables in the fixed design matrix. To estimate fixed effects and make inferences on them, we propose a methodology which is a consequence of the imputed balance in the data and the equivalence of the ordinary and generalized least square estimation when the data is balanced. Finally, we present the description of a software, which is written in FORTRAN 77 in the UNIX SYSTEM on the SUN WORK STATION in the Department of Statistics of the Texas A&M University, to compute estimates of variance components using the EM-REML-AVE algorithm. This software has a capacity to handle a maximum of five factors and four continuous variables (covariables). The extension of the method and the use of this software is illustrated with several numerical examples. |
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