16.1 Straigh Line Relationship

Let’s use this app Simple Linear Regression to understand what a straight line relationship between two variables is.

Output = Response variable = Dependent Variable = $Y$

Input = Predictor variable = Regressor = Explanatory = Independent Variable = $X$

Response variable = Model function + Random error

$$Y = f(X) +\epsilon$$

$$\begin{equation} Y = \beta_0 + \beta_1 X + \epsilon \end{equation}$$

$$\begin{equation} y_i = \beta_0 + \beta_1 x_i + \epsilon_i \end{equation}$$

The simple linear regression model is:

$$\begin{equation} y_i = \mu_i + \epsilon_i = \beta_0 + \beta_1 x_i + \epsilon_i \end{equation}$$

Here,

1. $\mu_i = \beta_0 + \beta_1 x_i$ is the mean value of the dependent variable when the value of the independent variable $X$ is $x_i$
2. $\epsilon_i$ is an error term that describes the effects on $y_i$ of all factors other than the value $x_i$ of the independent variable $X$
3. $\beta_0$ (the $y$-intercept) is the mean value of the dependent variable when the value of the independent variable $X$ is zero.
4. $\beta_1$ (the slope) is the change in the mean value of the dependent variable.
5. If $\beta_1$ is positive, the mean value of the dependent variable increases as the value of the independent variable increases. See Figure @ref(sales-adv-regression).
6. If $\beta_1$ is negative, the mean value of the dependent variable decreases as the value of the independent variable increases.See Figure @ref(fuel-temperature-regression).