Editing General linear model (section)

== Comparison to multiple linear regression ==
{{Further|Multiple linear regression}}
Multiple linear regression is a generalization of [[simple linear regression]] to the case of more than one independent variable, and a [[special case]] of general linear models, restricted to one dependent variable. The basic model for multiple linear regression is

:<math> Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \ldots + \beta_p X_{ip} + \epsilon_i</math> or more compactly <math>Y_i = \beta_0 + \sum \limits_{k=1}^{p} {\beta_k X_{ik}} + \epsilon_i</math>

for each observation ''i'' = 1, ... , ''n''.

In the formula above we consider ''n'' observations of one dependent variable and ''p'' independent variables. Thus, ''Y''<sub>''i''</sub> is the ''i''<sup>th</sup> observation of the dependent variable, ''X''<sub>''ik''</sub> is ''k''<sup>th</sup> observation of the ''k''<sup>th</sup> independent variable, ''j'' = 1, 2, ..., ''p''. The values ''β''<sub>''j''</sub> represent parameters to be estimated, and ''ε''<sub>''i''</sub> is the ''i''<sup>th</sup> independent identically distributed normal error.

In the more general multivariate linear regression, there is one equation of the above form for each of ''m'' > 1 dependent variables that share the same set of explanatory variables and hence are estimated simultaneously with each other:

:<math> Y_{ij} = \beta_{0j} + \beta_{1j} X_{i1} + \beta_{2j}X_{i2} + \ldots + \beta_{pj} X_{ip} + \epsilon_{ij}</math> or more compactly <math>Y_{ij} = \beta_{0j} + \sum \limits_{k=1}^{p} { \beta_{kj} X_{ik}} + \epsilon_{ij}</math>

for all observations indexed as ''i'' = 1, ... , ''n'' and for all dependent variables indexed as ''j = 1'', ''...'' , ''m''.

Note that, since each dependent variable has its own set of regression parameters to be fitted, from a computational point of view the general multivariate regression is simply a sequence of standard multiple linear regressions using the same explanatory variables.