APPLICATION OF BAYESIAN METHOD ON GAUSSIAN LINEAR MIXED MODEL - Vantika, Sandy
In general, regression analysis is defined as an analysis of the dependence of a variable on other variables, namely the independent variables (predictor) in order to make estimates or predictions of the value of the dependent variable (response) if the value of the independent variables (predictor) is known. At first, this analysis was widely applied in the fields of medical, animal husbandry, and education. In its development, regression analysis began to be used in other fields, namely: plantations, genetics, hydrogeology, and forestry. In this dissertation, this analysis will be applied to genetics. Linear mixed model (LMM) are generally written as y=X?+Z?+? with y is an n × 1 observation vector, X is a known n × p design matrix, ? is an unknown p × 1 vector of parameter of regression (often called the fixed effect parameter), Z is known n × q design matrix, ? is a q × 1 random-effect parameter vector, and ? is a n × 1 error vector. Compared to the linear regression model (LRM), it is clear that the difference lies in the addition of Z?. Based on the LMM and considering the distribution of the ? parameter, Gaussian LMM was formed using the Bayesian method. The sum of the random effects Z? and error ? is denoted by ?. Therefore, LMM can also be written as y=X?+?. The vector ? is assumed to have a distribution of N_n(0, V=K+R). The K matrix is a n × n covariance function matrix. Meanwhile, matrix R is the covariance matrix of error ?. The characteristics of the covariance matrix V, especially the singularity are expressed in a lemma. Bayesian LMM has three parameters consisting of a fixed effect parameter vector ?, random effect parameter variance ?_?^2, and error variance ?_?^2. In parameter estimation, the method used is maximum likelihood (ML), restricted maximum likelihood (RML), ML-Bayes A, ML-Bayes B, RML-Bayes A, and RML-Bayes B. ML and RML become methods for estimating two variance parameters. ML-Bayes A and RML-Bayes A use the result of the parameter variance estimation to estimate the vector of fixed effect parameters. Meanwhile, ML-Bayes B and RML-Bayes B estimate the value of new observations without having to estimate the vector of fixed effect parameters. As an application, LMM was applied to the data of the weight of corn from the Drought Tolerance Maize for Africa Project of CIMMYT's Global Maize Program. As a novelty, the exponential and quadratic exponential covariance matrices are formed as alternative from the genomic relationship matrix which is a linear covariance matrix. The exponential covariance matrix and quadratic exponential are matrices whose components consist of exponential covariance functions and quadratic exponential with certain hyperparameters. The linear covariance matrix provides a higher estimated value of fixed effect parameter than the exponential and quadratic exponential covariance matrix with a significant difference in the mean value of the squared error (MSE). The advantage of exponential and quadratic exponential covariance matrices is that we can see the different characteristics of random effects on each individual pair and use adjustable parameters. Lemma becomes a significant contribution to the development of algebra in LMM. Gaussian LMM programming algorithm with Bayesian method is also generated. Based on the characteristics of similar data, the algorithm can be used by practitioners in many fields other than genetics.