Linear model

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Statistics
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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.

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