5.1.2 Simple Linear Regression

5.1.2 Simple Linear Regression



The objective of a regression model is to try to explain the relationship that exists between a dependent variable, (response variable) and a set of independent variables, (explanatory variables)
In the simple linear regression model, we try to explain the relationship that exists between the response variable AND a single explanatory variable X.

                                                                     Y = α + βX + ε

Where α is the ordinate at the origin (the value that Y takes when X is 0)

β is the slope of the line, (and indicates how Y changes by increasing X by one unit)
ε is a variable that includes a large set of factors, each of which influences the response only in a small amount to what is called "error".


ESTIMATION OF THE REGRESSION STRAIGHT BY THE MINIMUM SQUARE METHOD

First, we will proceed to represent the scatter diagram, or point cloud. Suppose it is the one obtained in the figure. Although the cloud reveals a large dispersion, we can observe a certain linear tendency by increasing X and Y (a trend that is not entirely accurate, for example, if we assume that X is age and Y is the size, obviously, not only the size it depends on the age, in addition there can also be measurement errors).

The regression line should have a mid-line character, it should fit well with most of the data, that is, it should pass as close as possible to all the points, that you have little of each and every one of them means that we should adopt a particular criterion that is generally known as SQUARE MINIMUM. This criterion means that the sum of the squares of the vertical distances of the points to the line must be as small as possible.