# Linear model

Statistics
Major approaches
Frequency probability
Bayesian inference
Non-parametric statistics
Common methods
Analysis of variance
Chi-Square test
Students t-test
Z test
Linear regression
Bayesian model selection
Bootstrapping

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:

$Y = X \beta + \varepsilon$

In the comparison two different models will be matched to the data, $\varepsilon$ contains the size of the error for how well the model and data match. The larger the $\varepsilon$ the worse the match. However, models must be penalized for the number of free parameters (β) 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 linear 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 $\varepsilon$ 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.