Guassian linear mixed models

Normalni linearni mešani modeli so modeli oblike $Y = Xbeta + Zalpha + epsilon$ in zajemajo tako fiksne učinke $beta$, kot tudi slučajne učinke $alpha$. Pomembni predpostavki v teh modelih sta predpostavka normalne porazdeljenosti vektorja slučajnih učinkov $alpha sim N(0, sigma^2I_n)$ in vektorja slučajnih odstopanj $epsilon sim N(0, tau^2I_m)$, ki nista nujno enakih razsežnosti ter predpostavka neodvisnosti slučajnih vektorjev $alpha$ in $epsilon$. Variančne komponente v modelih, obravnavanih v diplomskem delu, se lahko ocenjuje po običajni in restringirani metodi največjega verjetja, med drugim pa tudi z uporabo metode iterativnega uteženega povprečja najmanjših kvadratov, z metodo analize varianc in z metodo kvadratičnega nepristranskega ocenjevanja minimalnih norm. V normalnih linearnih mešanih modelih se da konstrurati več različnih tipov intervalov zaupanja, med drugim eksaktne intervale zaupanja za variančne komponente in intervale zaupanja za fiksne učinke.Gaussian linear mixed models can be expressed as $Y = Xbeta + Zalpha + epsilon$, where vector $beta$ represents fixed effects and vector $alpha$ represents random effects. There are two important assumptions in these models. The first is the assumption that both, random effects $alpha$ and errors $epsilon$ are normally distributed, the former with mean zero and variance $sigma^2$ and the latte with mean zero and variance $tau^2$. The second important assumption is that random effects and errors are assumed to be independent. Variance components in Gaussian linear mixed models can be estimated with maximum likelihood method or with restricted maximum likelihood method. Variance components can also be estimated with iterative weighted least squares method, analysis of variance, or minimum norm quadratic unbiased estimation. Confidence intervals in Gaussian linear mixed models include exact confidence intervals for variance components and confidence intervals for fixed effects, among others