1 Multiple Regression Two values can be connected by functional dependence, or statistical dependence or to be independent. For example. If you know circle radius, you absolutely exactly can calculate its area. There are no concepts approximately. The radius is equal 3, the area means is equal R 2 =3.14*9. R=3. There are values absolutely independent. For example, temperature behind a window and length of a book. functional dependence independent values

2 Statistical dependence takes an average place between functional dependence and independence of values. Dependence at which change of one of sizes brings distribution change another is called as statistical. Statistical dependence at which change of one of values brings change of average value another, is called as correlation. Imagine, we probe dependence between quality of feed and health of dog. It is a question of correlation dependence. A health depends and from quality of feed, and from the terms of residence, mode, surrounding environment, genes. Correlation dependences are intermediate between functional dependence and full independence of variables. It is impossible to set strict connection between Х and Y. Cause is in that a dependent variable of Y value is determined not only a variable of X value but also other factors. For example, weight of man is related to growth certain dependence. A child grows, with the increase of his growth weight is increased too. But by weight have influence and half man, and genes of ancestors, and condition of life. We do not can say that if growth is equal to 180, and weight is equal to 80.

3 Correlation dependences play an important role in description of processes of production, estimations of dynamics of market development, researches of tendencies of growth of enterprises. And the income of enterprise, and result of execution of technological processes, depend on the great number of factors which it is possible to estimate and forecast only statistically. It means that it is needed to collect experimental information, create a statistical model and use it for description of processes, and for forecasting. positive correlation negative correlation The set of points { Xi, Yi } on the XY plane is called as a correlation field

4 If points of a correlation field form an ellipse which main diagonal has a positive tilt angle, positive correlation takes place. If points of a correlation field form an ellipse which main diagonal has a negative tilt angle, negative correlation takes place. If in an arrangement of points there is no regularity, say that zero correlation in this case is observed. If with increase x value of a dependent variable Y on the average increases, such dependence is called as direct or positive. If average value Y at increase x decreases, negative or return correlation takes place. If with change x values Y on the average don't change, say that correlation – zero.

5 The equation of regression describes dependence between values. It allows to predict value Y on known value X, to estimate extent of influence X on Y. For example, you estimate influence of amount of fertilizers on the size of a crop. You need to know, what amount of fertilizers should be used to receive a certain crop. You besides want to know, whether the amount of fertilizers essentially influences a crop or there are other values which influence. influence of application of phosphoric fertilizers on productivity. with phosphorus without phosphorus For example, study sales volume depending on a number of factors. It and personal characteristics of sellers, both quality of production, and the price, and presence of competitors with the similar goods. There is a question, whether everything characteristics make identical impact? On what it is necessary to pay attention first of all to receive the maximum sales volume. And one more task for an example. For increase in realization of production research influence of different types of advertizing on sales volume. Used three types of advertizing: television, bigboards and newspapers. The task was put to estimate, what type of advertising gives a most benefit?

6 Here x -, - - the sizes mean of variables X and Y. The value R 2 is called as factor of determination. Defines a share of a variation of one of variables which speaks a variation of other variable (RI). Use for check of the general quality of the equation of regression: =Regress/Total The equation of regression looks like: n- amount of datas; a and b - unknown parameters (factors of regression). Y i = a + bx i, Size of influencing of factor X on Y estimated the coefficient of correlation r xy, which characterizes the degree of linear connection between two by variables. Factor determine by a formula (in a package Statistica it is designated as Multiple R):

7 Open the menu of package and choose will point Multiple Regression. Copy file data Excel in a package. The following task is put: Studied, influencing of landing density, age and initial thickness of tree on the annual increment of thickness of larch. A larch is a very costly tree, with a hardwood which is not afraid of water,. In our task У - Thickness of annual layer. X1-Landing density, thousand on hectare ; X2 - Radius of barrel, cм; X3 – Age. You must get such file of basic data Open menu Analysis /Startup Panel

8 We set variables for calculations. Dependent variables - it is Y, independent - Х. Press on the button OK.

9 It is a main window of results of calculation of regression window. R 2 or RI is a coefficient of determination; Numeral expresses the stake of variation of dependency variable, explained by regressive equation. What anymore R2, then the greater stake of values is explained by variables, plugged in a model. Multiple R is a coefficient of plural correlation ; Characterizes the degree of linear connection between dependent and all of independent variables. Can take on values from 0 to 1. Beta coefficients determine the degree of influence a variable Х on the size of Y. Than nearer they by 1, the it influencing stronger. If beta 0, it means that with an increase the X value Y increased. In our example all of coefficients are positive, means all of the chosen variables Х promote in an increase Y

10 Button Regression summary allows to look over the basic results of regressive analysis : BETA are coefficients of equation ; St. Err. of BETA is standard errors of coefficients ; В are coefficients of equation of regression; St. Err. of B is standard errors of coefficients of equation of regression; р-level is a level of significance. If р<0.05, a model is considered good. We can write down equation of regression in a kind : У= *X *X *X2

11 Button Pred. & residuals allows to get a table with the set of basic data, counted up on equation of regression and to estimate rests, difference between basic data and theoretical. You must copy them by a command Copy Raw in Excel and to build two charts. On an ax Х put aside Landing density, on an ax Y is theoretical and initial information. You will compare two charts. you will estimate, how well describes the got equation basic data.

12 For forecasting it is possible to use only two variables : X1-Landing density, thousand on hectare ; X3 - Age. Value Radius of barrel (X2) it is difficult to get, do not saw trees. You will repeat all of the operations described higher, since a sliding seat 6 and will get regressive equation. Coefficients turned out for me Const X X Consequently, equation looks like У= *X *X2. forecast : Let Х1=100; Х3=5, then У= * *5=0.9

13 Results of calculations