Simple linear regression from raw data
Let $(x_i, y_i), i=1,2, \cdots , n$
be $n$ pairs of observations.
The simple linear regression model of $Y$ on $X$ is
$$y_i=\beta_0 + \beta_1x_i +e_i$$ where,
- $y$ is a dependent variable,
- $x$ is an independent variable,
- $\beta_0$ is an intercept,
- $\beta_1$ is the slope,
- $e$ is the error term.
Formula
By the method of least square, the model parameters $\beta_0$ and $\beta_1$ can be estimated as
The regression coefficients $\beta_0$ (intercept) and $\beta_1$ (slope) can be estimated as
$\hat{\beta}_1 = \frac{n \sum xy - (\sum x)(\sum y)}{n(\sum x^2) -(\sum x)^2}$
$\hat{\beta}_0=\overline{y}-\hat{\beta}_1\overline{x}$
where,
$\overline{x}=\dfrac{1}{n}\sum_{i=1}^n x_i$
is the sample mean of $X$,$\overline{y}=\dfrac{1}{n}\sum_{i=1}^n y_i$
is the sample mean of $Y$,- $n$ is the number of data points.
Important Results
- Explained variation
$SSR = \sum(\hat{y}-\overline{y})^2$
- Unexplained variation
$SSE = \sum (y-\hat{y})^2$
- Total variation
$SST = \sum (y-\overline{y})^2$
- Coefficient of determination
$R^2 =\dfrac{SSR}{SST}$
- Standard error of estimate
$S_e = \sqrt{\dfrac{\sum(y-\hat{y})^2}{n-2}}=\sqrt{\dfrac{SSE}{n-2}}$
Related Resources
Suggestions and comments will be appreciated.