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lec1 - STAT 8260 Theory of Linear Models Lecture Notes...

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STAT 8260 — Theory of Linear Models Lecture Notes Classical linear models are at the core of the field of statistics, and are probably the most commonly used set of statistical techniques in practice. For these reasons a large portion of your coursework is devoted to them. The two main subclasses of the classical linear model are (1) linear regression models, and (2) analysis of variance (ANOVA) models. These classes of models are covered from a very applied perspective in the courses STAT 6320 (or STAT 6230) and STAT 8200, respectively. A third subclass, (3) analysis of covariance models (ANCOVA) models, combines elements of regression and ANOVA, and this subclass usually receives some treatment in STAT 6320. Because these models are such important practical tools for data analysis, instruction in the practical aspects of their application is a crucial part of your training as statisticians. It is important to realize, however, that these methods are not a collection of unrelated, specialized techniques. A general theory of estimation and inference in the linear model is available that subsumes the methods of regression, ANOVA, and ANCOVA. This theory is worth studying because it unifies and yields insight into the methods used in many, many important subcases of the linear model; and because its ideas point the way and, in some cases, carry over directly to more general (not-necessarily-linear) modelling of data. In summary, this is a theory course, and as such it is not a complete course in linear models. Very important practical aspects of these models will be omitted; they are covered elsewhere in your course-work. 1
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Introduction to Linear Models Linear models are parametric statistical models that summarize how the probability distribution of a response variable (usually denoted as Y ) depends upon one or more explanatory variables (usually denoted with X ’s: X 0 , X 1 , X 2 , . . . , X k ). They are statistical (or probabilistic) because they specify a (con- ditional) probability distribution of a random variable (or at least some aspects of that distribution, like its mean and variance). They are parametric because the probability distribution is specified up to a finite number of unknown constants, or parameters. They are linear because the mean of the conditional probability dis- tribution of Y , E( Y | X 0 , X 1 , . . . , X k ), is specified to be a linear func- tion of model parameters. This conditional mean, E( Y | X 0 , X 1 , . . . , X k ), is called the regression function for Y on X 0 , . . . , X k . The classical linear model specifies that Y = X 0 β 0 + X 1 β 1 + · · · X k β k + e = x T β + e where x = X 0 . . . X k ( k +1) × 1 , β = β 0 . . . β k ( k +1) × 1 , where e has conditional mean 0, and variance σ 2 . Notice that this model implies that the regression function is linear in the β ’s: E( Y | x ) = X 0 β 0 + X 1 β 1 + · · · X k β k .
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