regression analysis

Also found in: Dictionary, Thesaurus, Medical, Legal, Financial, Wikipedia.
Related to regression analysis: linear regression, Multiple Regression Analysis

regression analysis

[ri′gresh·ən ə‚nal·ə·səs]
The description of the nature of the relationship between two or more variables; it is concerned with the problem of describing or estimating the value of the dependent variable on the basis of one or more independent variables.

Regression Analysis


the branch of mathematical statistics that encompasses practical methods of studying a regression relation between variables on the basis of statistical data. The purposes of regression analysis include the determination of the general form of a regression equation, the construction of estimates of unknown parameters occurring in a regression equation, and the testing of statistical regression hypotheses.

When the relationship between two variables is studied on the basis of the observed values (x1, y1),…, (xn, yn) in accordance with regression theory, it is assumed that one of the variables, Y, has a certain probability distribution when the value of the other variable is fixed as x. This probability distribution is such that

E(Yǀx) = g(x, β)

D(Yǀx) = σ2h2(x)

where β denotes the set of unknown parameters that determine the function g(x), and h(x) is a known function of x—for example, it may have the constant value one. The choice of a regression model is determined by the assumptions regarding the form of the dependence of g(x, β) on x and β. The most natural model, from the standpoint of a unified method of estimating the unknown parameters β, is the regression model that is linear in β:

g(x, β) = β0g0(x) + … + βkgk(x)

Different assumptions may be made regarding the values of the variable x, depending on the nature of the observations and the aims of the analysis. In order to establish the relationship between the variables in an experiment, a model is used that is based on simplified but plausible assumptions. These assumptions are that the variable x is a controllable variable, whose value is assigned during the design of the experiment, and that the observed values of y are expressed in the form

yi = g(xi, β) + i i = 1,…,k

where the quantities i, describe the errors. The errors are assumed to be independent under different measurements and to be identically distributed with zero mean and constant variance σ2. The case where x is an uncontrollable variable differs in that the observed values (x1, y1),…. (xn, yn) constitute a sample from a bivariate population. In either case, the regression analysis is performed in the same way. The interpretation of the results, however, is done substantially differently. If both the variables are random, the relationship between them is studied by the methods of correlation analysis.

A preliminary idea of the nature of the relation between g(x) and x can be obtained by plotting the points (xi, ȳ(xi) in a scatter diagram, which is also called a correlation field when both variables are random. The (xi) are the arithmetic means of the values of y that correspond to a fixed value xi. For example, if the points fall near a straight line, a linear regression can be used as the approximation.

The standard method of estimating the regression line is based on the polynomial model (m ≥ 1):

y(x, β) = β0 + β1x = … + βmxm

One reason for the choice of this model is that every function continuous over some interval can be approximated by a polynomial to any desired degree of accuracy. The unknown regression coefficients β0,…, βm and the unknown variance σ2 are estimated by the method of least squares. The estimates β0,…, β̂0 of the parameters β0, …, βm obtained by this method are called the sample regression coefficients, and the equation

ŷ(x) = β̂0 + … + βmxm

defines what is called the sample regression line. If the observed values are assumed to be normally distributed, this method leads to estimates of β0,…, βm and of σ2 that coincide with estimates obtained by the maximum likelihood method. The estimates obtained by the least squares method are in some sense best estimates even when the distribution is not normal. Thus, if a linear regression hypothesis is to be tested,

where and are the arithmetic means of the xi and yi. The estimate ĝ(x) = β̂0 + β1(x) is an unbiased estimate of g(x); its variance is less than the variance of any other linear estimate. The assumption that the yi have a normal distribution is the most effective method of checking the accuracy of the constructed sample regression equation and of testing the hypotheses on the parameters of the regression model. In this case, the construction of the confidence intervals for the true regression coefficients β0,…, βm and the testing of the hypothesis that no regression relationship exists (βi = 0, i = 1,…, m) are carried out by means of Student’s distribution.

In a more general situation, the observed values y1,…,yn are regarded as values of independent random variables with identical variances and the mathematical expectations

Eyi = βi xu+ … + βkxki i = 1…,n

where the values of the Xji, j = 1,…, k, are assumed known. This form of linear regression model is general in the sense that higher-order models in the variables x1, …, xk reduce to it. Moreover, certain models that are nonlinear in β can also be reduced to this linear form by a suitable transformation.

Regression analysis is one of the most widespread methods of processing the results of observations made during the study of relationships in such fields as physics, biology, economics, and engineering. Such branches of mathematical statistics as analysis of variance and the design of experiments are also based on regression analysis. Regression analysis models are widely used in multivariate statistical analysis.


Yule, G. U., and M. G. Kendall. Teoriia statisliki, 14th ed. Moscow, 1960. (Translated from English.)
Smirnov, N. V., and I. V. Dunin-Barkovskii. Kurs teorii veroiatnostei i matematicheskoi statisliki dlia tekhnicheskikh prilozhenii, 3rd ed. Moscow, 1969.
Aivazian, S. A. Statislicheskoe issledovanie zavisimostei. Moscow, 1968.
Rao, C. R. Lineinye statisticheskie metody i ikh primeneniia. Moscow, 1968. (Translated from English.)


regression analysis

In statistics, a mathematical method of modeling the relationships among three or more variables. It is used to predict the value of one variable given the values of the others. For example, a model might estimate sales based on age and gender. A regression analysis yields an equation that expresses the relationship. See correlation.
References in periodicals archive ?
After the measurements, linear regression analysis was done using cut perpendicularity as dependent variable and laser power P, cutting speed v and assist gas pressure p as independent variables.
DC is known to have a linear relation [26-28] to the eyeball angle, so we performed a multiple regression analysis with the explanatory variables AC, DC, DC integral value (DC_Int), and DC difference (DC_Dif).
In the logistic regression analysis with 0-1 outcome [2], we assume that
In the Regression analysis A for Sample 2, the indicators of working capital management and liquidity are regressed against the 'Return on Assets' for sample 2.
Material factors which affect on yarn quality characteristic, i.e., yarn count were studied by using regression analysis. Yarn count on 12 independent variables (input parameters) was regressed to select the regression model by stepwise regression.
where [H.sub.t] and [[tau].sub.t] are the target height and duration, respectively; [H.sub.e] and [[tau].sub.e] are, respectively, the height and duration estimated by either regression analysis or BPNN; j is the test number; and n is the total number of data sets in each case.
The interval regression analysis by QP approach unifying the possibility and necessity models subject to the inclusion relations, [Y.sub.*] ([x.sub.j]) [not subset] [Y.sub.j] [not subset] [Y.sup.*] ([x.sub.j]), can be represented as
Variable BL WH RH BG CG WH 0.875 - - - - RH 0.830 0.957 - - - BG 0.748 0.789 0.782 - - CG 0.842 0.907 0.894 0.851 - LW 0.749 0.754 0.750 0.617 0.764 All the correlations were significant (P less than 0.01) Only multiple regression analysis without including factor analysis caused to occur VIF values greater than 10 as recognized in variables, WH (VIF=17) and RH (VIF=13), meaning that there was a strong indication of multicollinearity problem (data not shown).
(2010) also underlined the significant effects of age and body weight on testicular traits with the multiple linear regression analysis, which is used to identify the relationship of body weight with some body measurements, prone to multicollinearity problem.
'Linear regression analysis was also performed to determine whether total amount of physical activity was predicted by revision hip arthroplasty.
According to the prediction equation generated by this linear regression analysis, CVD risk could be predicted as CVD Risk = (-4.48 + (WC * 0.209)).