e-Statistics

Regression Line

The coefficients $\beta_0$ and $\beta_1$ of the linear regression model

$\displaystyle Y_i = \beta_0 + \beta_1 x_i + \epsilon_i, \quad i=1,\ldots,n,$

are called the intercept and the slope parameters, respectively.

Data is not assigned yet.

The data set consists of

explanatory variable for 's;
dependent variable for 's.

Then the point estimates $\hat{\beta}_0$ and $\hat{\beta}_1$ of the parameters $\beta_0$ and $\beta_1$ are obtained as follows.

$\hat{\beta}_0 = \bar{Y} - \hat{\beta}_1 \bar{x}$ = and $\hat{\beta}_1 = \displaystyle\frac{S_{xy}}{S_{xx}}$ = .

Here the values $\bar{x}$ , $\bar{Y}$ , $S_{xx}$ , and $S_{xy}$ are computed as in the following table.

Variable	Mean	Sum of squares
Explanatory	$\displaystyle \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$	$\displaystyle S_{xx} = \sum_{i=1}^{n} (x_i - \bar{x})^2$
Response	$\displaystyle \bar{Y} = \frac{1}{n} \sum_{i=1}^{n} Y_i$	$\displaystyle S_{xy} = \sum_{i=1}^{n} (x_i - \bar{x})(Y_i - \bar{Y})$

Fitted model. The fitted linear model

$\displaystyle {y} = \hat{\beta}_0 + \hat{\beta}_1 x$

is called the prediction equation (or the regression line). The scatter plot together with regression line (which should appear below when it is produced) suggests how well the line fits along the data.

Correlation. The sample correlation

$\hat{\rho} = \displaystyle\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}$ = .

describes the strength of linear relationship for the pair

of data. Here $S_{yy} = \sum_{i=1}^{n} (Y_i - \bar{Y})^2$ is the sum of squares within the response variable

's. The value $\hat{\rho}$ is always between

and

. The value $\hat{\rho}$ is close to

when the pairs lie close to the straight line with positive slope, and it is close to

when it is aligned with a negative slope.