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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} ) 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.