The regression equation

When studying a phenomenon or process, it is often necessary to find out whether there is a correlation between the factors (variables) and the response function (the dependent quantity), and how close their interaction is. To do this allows regression analysis, which is performed in several stages.

One of the main stages of regression analysis is the calculation of the mathematical relationship between the factors and the response function, which makes it possible to quantify the relationship between them. This dependence is called the regression equation. Formally, the least-squares method is considered to be the basic analytical method for determining this equation , since this method is optimal and allows smoothing out the points of the correlation field. In practice, however, finding such a function can be quite difficult, since it is necessary to rely on theoretical knowledge about the phenomenon being studied, on the experience of its predecessors in this scientific field, or through the trial and error method, to perform a simple search and evaluation of various functions. If successful, a regression equation will be obtained, which allows to adequately evaluate the effect of various factors on the response function, that is, to find the expected value of the response function (dependent variable) for certain values of the factors (dependent variables).

The initial data for the regression analysis are the values of the factor x and the corresponding value of the response function Y, obtained during the experimental part of the work. For clarity and more comfortable perception, these values are presented in tabular form.

The linear regression equation , as a rule, has the following form: Y = a + b ∙ X. It includes a constant coefficient (constant) a, and a regression coefficient (slope) b multiplied by the value of the variable factor X. The coefficient b shows the average change in the response function when the factor value is changed by one unit. When constructing the graph of the regression equation using the coefficient b, one can also determine the slope of the line to the abscissa line. It should be noted that this coefficient has certain properties:

· B can take different values;

· B is not symmetric, that is, it changes its value in the case of studying the influence of Y on X;

· Unit of measurement of the correlation coefficient is the ratio of the unit of measure of the response function Y to the unit of measurement of the variables X;

· If the units of measurement for the X and Y variables change, the value of the regression coefficient also changes.

In most cases, the observed values are rarely located exactly on a straight line. In practice, it is always possible to observe a certain scatter of experimental data on the regression line, which I form the predicted values. The deviation of an individual point from the regression line from its theoretical or predicted value is called the remainder.

Very often in practice, a sample regression equation is determined, the main method of calculating the coefficient values of which is the method of least squares. The coefficients are calculated from the initial data representing the sample of the values of the variable factor and the response function.

At first glance, it may seem that the calculation of the coefficients in the regression equation is quite complex and time-consuming. But this is not so. At the service of researchers are numerous application software packages (the simplest is Microsoft Excel), which according to your input data will not only calculate all the coefficients in the equation, they will be able to establish the degree of interrelation between variables and dependent values, but will present the values in graphical form.