# general linear model

In statistical modeling of $N$ data observations ($N<\infty$), two types of variables are usually defined. One is the response variable or variate, usually denoted by $Y$, and the other is the explanatory variable or covariate $X$. While there is only one response variable, there may be one or more than one explanatory variables. The response variable is considered random, where as the explanatory variable(s) may or may not be random.

Based on the above setup, a general linear model, or normal linear model, is a statistical model with the following assumptions  :

1. 1.

the response variable $Y$ is a continuous random variable

2. 2.

the response variable $Y$ can be expressed as a linear combination  of functions $z_{i}(\textbf{X})$, of the explanatory variables, plus a random error term $\varepsilon$:

 $Y=\beta_{0}z_{0}(\textbf{X})+\cdots+\beta_{k}z_{k}(\textbf{X})+\varepsilon.$

The portion of $Y$ without the error term is known as the systematic component    of $Y$.

3. 3.
4. 4.

random error variables $\varepsilon_{i}$ for the $N$ observations are iid normal with mean 0 and variance  $\sigma^{2}$

Remarks

 Title general linear model Canonical name GeneralLinearModel Date of creation 2013-03-22 14:31:23 Last modified on 2013-03-22 14:31:23 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 6 Author CWoo (3771) Entry type Definition Classification msc 62J10 Classification msc 62J05 Synonym normal linear model Defines analysis of variance Defines ANOVA Defines analysis of covariance Defines ANCOVA