Beyond Linear Regression

1. Standardized Linear Regression

Standardize the coefficients in order to make comparisons directly.

Standardize (normalize) the data withing zero mean and one variance (standard deviation)

\[y^*_i = \frac{y_i - \bar{y}}{s_y} \qquad \text{and} \qquad x^*_{i,j} = \frac{x_{i,j} - \bar{x}_j }{ s_{x_j} }, \ \forall j=1,\cdots, m \]

where \(s_y\) and \(s_{x_j}\) represent the standard deviation of respond variable \(y\) and \(j\)th explanatory variable \(x_j\), respectively.

Standardized Linear Regression

\[\frac{y_i - \bar{y}}{s_y} = \beta^*_1 \cdot \frac{x_{i,1} - \bar{x}_1 }{ s_{x_1} } + \beta^*_2 \cdot \frac{x_{i,2} - \bar{x}_2 }{ s_{x_2} } + \cdots + \varepsilon_i \quad \forall i =1,\cdots \]

• Note that no constant in this formula.

Thus, the formula for converting from an unstandardized coefficient to a
standardized one

\[\beta^*_j = \beta_j \frac{s_{x_1}}{s_y}, \quad \forall j=1,2,\cdots \]