When studying a phenomenon or process, it is often necessary to find out if there is a relationship between factors (variables) and the response function (dependent value), and how close their interaction is. This can be done by regression analysis, which is performed in several stages.
One of the main stages of the regression analysis is the calculation of the mathematical relationship between factors and the response function, which allows you to quantify the relationship between them. This dependence is called the regression equation. Formally, the least squares method is considered the main analytical method for determining this equation , since this method is optimal and allows you to smooth the points of the correlation field. In practice, finding such a function can be quite difficult, since one has to rely on theoretical knowledge about the phenomenon being studied, on the experience of his predecessors in this scientific field, or using the trial and error method, to make simple enumeration and evaluation of various functions. If successful, a regression equation will be obtained that allows one to adequately evaluate the effect of various factors on the response function, that is, find the expected value of the response function (dependent variable) for certain values of factors (dependent variables).
The values of factor x and the corresponding value of the response function Y obtained during the experimental part of the work are used as initial data for the regression analysis. For clarity and more convenient 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 (angular coefficient) 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 changes by one unit. When plotting the regression equation using the coefficient b, you can also determine the angle of inclination of the straight line to the abscissa line. It should be noted that this coefficient has certain properties:
· B can take on different meanings;
· B is not symmetrical, that is, it changes its value in the case of studying the influence of Y on X;
· The unit of measurement of the correlation coefficient is the ratio of the unit of measurement of the response function Y to the unit of measurement of variable factors X;
· If the units of measurement of the variables X and Y change, the value of the regression coefficient also changes.
In most cases, the observed values are rarely located exactly on a straight line. Almost always, one can observe some scatter of experimental data relative to the regression line, which the predicted values form. 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 selective regression equation is determined, the main method of calculating the coefficient values of which is the least squares method. The coefficients are calculated from the source data, representing a sample of the values of the variable factor and the response function.
At first glance, it might seem that calculating the value of the coefficients included in the regression equation is quite complicated and time-consuming. But this is not so. Researchers are offered numerous application packages (the simplest is Microsoft Excel), which according to your source data will not only calculate all the coefficients included in the equation, can establish the degree of relationship between variables and dependent quantities, but present the obtained values in a graphical form.