Regression is a statistical technique used to study the relation among quantitative variables. Both, in cases of two variables (simple regression) as in cases of more than two variables (multiple regression), regression analysis can be used for exploring and quantifying the relation among a so-called dependent variable ( y) and one or more so-called independent variables (x1, x2 . . . , xp). In others words, to find out to what degree the dependent variable can be explained by the independent variable(s). Furthermore, we can write an equation that allows us to predict the values of a dependent ( y) variable, knowing the values of one or more independent (x) variables.
The simplest way of expressing a relation among variables is through a mathematical equation, which allows us to describe the form as one variable changes with the other one. The regression equation is as follows:
Y = alpha + beta (X) + epsilon
Y is the dependent variable; the alpha parameter is a constant or value of ''Y'' when ''X'' is equal to 0; the beta coefficient determines the slope of the line; X is the independent variable (in multiple regression the equation is extended to include additional x variables''; and epsilon is the error comprised of variation in ''Y'' not accounted for by the remainder of the equation.
The problem is that in the social sciences it is difficult to find perfect linear associations among variables and therefore it is necessary to find the regression line that better adjusts to the data. The most used regression form is ordinary least squares linear regression. But even this line is not always a good summary of the existing relation in the data and so we usually resort to the coefficient of determination - R squared - to know the goodness of fit. This coefficient takes values between 0 (absence of relation among the variables) and 1 (perfect relation among the variables), and the value of R squared represents the degree of profit that is obtained when we predict a variable from the knowledge that we have of other variable(s). The higher the value of R squared, the better the fit of the equation and therefore the greater is our ability to predict the values of the dependent variable (y) knowing the values of the independent variable(s) (x).
Allison, P. D. (1999) Multiple Regression: A Primer. Pine Forge Press, Thousand Oaks, CA.
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