Linear model

Linear models and linear comparisons are statistical methods for comparing how well different models match a given set of data. It is usually written as:

<math>Y = X \beta + \varepsilon</math>

In the comparison two different models will be matched to the data, <math>\varepsilon</math> contains the size of the error for how well the model and data match. The larger the <math>\varepsilon</math> the worse the match. However, models must be penalized for the number of free parameters (<math>\beta</math>) that they posses. A theoretical linear model with an infinite number of parameters can perfectly explain any data set, but this is not a valuable model. Usually the liner model a statistician is interested in is compared against the null hypothesis linear model which has fewer free parameters, as such the more complicated model must have a smaller <math>\varepsilon</math> in proportion to the number of free parameters to be statistically significant. The measurement of free parameters is referred to as the degrees of freedom.