Various Applications

Abstract

Linear regression analysis represents the criterion variable y by the sum of a linear combination of p predictor variables x ₁,x ₂,...,x _pand an error term ε,

$$ y_j = \alpha + \beta _1 x_{1j} +... + \beta _p x_{pj} + \in _j \left( {j = 1,...,n} \right),$$

where j indexes cases (observation units, subjects, etc.) and n indicates the total number of cases, and where α and β _i(i = 1,...,p)are regres- sion coefficients (parameters) to be estimated. Assume first that the error termsε₁,ε₂,...,ε_nare mutually independent with an equal variance σ².We may obtain the estimates a,b₁,...,b_pof the regression coefficients using the method of least squares (LS) that minimizes

$$ \sum\limits_{j = 1}^n {\left( {y_j - a_1 - b_1 x_1j -\cdots - b_p x_{pj} } \right)^2 }$$

.