# REGRESSION ANALYSIS

REGRESSION ANALYSIS — the section of mathematical statistics studying statistical dependence of one variate on another independent (accidental or not casual).

Methods P. and. together with other methods of probability theory (see. Probabilities theory ) are applied as mathematical methods (see) for detection and creation of mathematical dependences (regressions) between quantitative characters on their experimental values. The purpose of such constructions is studying of any medicobiological patterns, the analysis patol. and other processes, development of techniques and the automated systems of machine diagnosis (see. Diagnosis machine ), professional selection, forecast, control and management. Accepting as an explanatory variable time, and as dependent — an average value, dispersion or other characteristics, study dynamics population, immune, etc. processes.

Function of regression at (x), or regression of Y on X, is any numerical characteristic of conditional distribution of probabilities of a variate of Y set for each value of an explanatory variable (regressor) of X. Most often as regression use a conditional average value. Also the median, fashion and other characteristics are applied to creation of regression. The sizes X and Y can be scalars, vectors, functions of time and space. Regressions of Y on X and X on Y usually do not match and are not the mutually return.

Main practical objective of R. and. — creation of assessment at (x) unknown regression at (x) on set: a) to experimental measurements (x, s) values of a dependent variable of Y and regressor of X; b) to criterion and (or) method of creation; c) to a type of required estimates at (x). The type of required regression at (x) usually is chosen on the basis of the visual analysis of experimental data and the priori information on the studied dependence. Among evaluation criteria the minimum of the sum of squares of deviations at (x) from observed values is most common at (see. Leastsquares method ). Regression in this case is called mean square. The regression received on a method of least squares in a class of all possible functions is a conditional average value (mathematical expectation) at (x) = M (Y/x), and on a minimum of the sum of modules of deviations at (x) from at — a conditional median. In a class of linear functions (and at normal joint distribution of X and Y in a class of all functions) on a method of least squares linear regression turns out:

y (h) = at l (x) = a ooh (x — M x ) + M at , where and ooh — ρ ooh σ at x — coefficient, M at and M x — average values, σ at and σ x — dispersions of Y and X, and ρ ooh — a correlation coefficient between them. Thus, in this case R. and. comes down to correlation analysis.

For creation of linear and nonlinear regressions there are various computing methods and the programs entering software are developed electronic computers (see). Data interpretation at R. and. similar to a method correlation analysis (see) begins with drawing up correlation tables and the subsequent assessment of the corresponding probabilistic characteristics in each cell of the table.

River and. investigates also statistical properties of estimates of regressions (not shift, a solvency, efficiency). In expanded understanding of R. and. includes check of statistical hypotheses about communication of Y and X. At the same time it is important to consider that close regression dependence means not only existence of direct relationship of cause and effect, but can be caused by the mediated interrelations and concurrent factors.

Bibliography: Gubler. B. Computing methods of the analysis and recognition of pathological processes, L., 1978, bibliogr.; Kendall M. D. and Styyuart A. Statistical conclusions and bonds, the lane with English, M., 1973; Kramer G. Mathematical methods of statistics, the lane with English, M., 1975; M and with yu to N. S., Drudgery Kean A. S. and Kuznetsov G. P. The correlation and regression analysis in clinical medicine, M., 1975; R and about S. R. Linear statistical methods and their uses, the lane with English, M., 1968

V. G. Laptev.