REGRATION ANALYSIS (V. Imp)The dictionary meaning of term “regression” is the act of returning or goingback how ever it is a statistical device with the help of which we are in position toestimate (or predict) the unknown value of one variable known value of anothervariable. The variable, which is used to predict the variable of interest is called,is called independent variable or explanatory variable. Variable we are trying topredict is called dependent variable or explained variable. The independentvariable is denoted by X and dependent variable by Y.For example, while estimating sales of a product for figures on advertisingexpenditure, sale is generally taken as independent variable & advertisingexpenditure as independent variable. However, there may or may not be casualconnection between these 2 factors in the sense that changes in advertisingexpenditure cause change in sales, in fact, in certain cases cause-effect relationmay be just opposite of what appears to be obvious one.Definition: “Regression is the measure of the average relationship between twoor more variables in terms of original unit of the data.” “Regression analysis attempts to establish the nature of relationshipbetween variables that is to study functional relationship between variables &thereby provide a mechanism for prediction or forecasting.” USES OF REGRESSION ANALYSIS Regression analysis is a branch of statistical theory that is widely inalmost all scientific disciplines. It attempts to accomplish the following: --1. Nature of relationship: -- In economics, it is basic technique for measuring or estimating the relationship economic variables that constitute essence of economic theory & economic life. For example, if we know that 2 variables, price (X) and demand (Y) are closely related we can find out the most probable value of X for a given value of Y or most probable value of Y for a given value of X.2. Useful in economic & business Research: -- Study of regression is of considerable help to economists and businessmen. The uses of regression are not confined to economics & business field only. Its application is extended to almost all natural, physical & social science.3. Prediction: -- Regression analysis provides estimates of value of dependency variables from value of independent variable. The device used to accomplish this estimation procedure is regression line which describes average relationship existing between X & Y variables i.e., it displays mean values of X for given values of Y. For example; if the price of a commodity rises, what will be the probable fall in demand, this can be predicted by regression.
4. Measure of error: -- Regression analysis helps to measure the error as a basis for estimation. For this purpose standard error estimate is calculated. This is a measure of scatter of observed value of Y around the corresponding value estimated from regression line. If the line fits data closely that is if there is little scatter of observations around regression line, good estimates can be made of Y variable. On the other hand, if there is great deal of scatter of observation around regression line, line will not produce accurate estimates of dependent variable. With the help of regression coefficients we can calculate correlation coefficient. The square of correlation coefficient γ called coefficient of determination, measures the degree of association of correlation that exists between 2 variables. In general, greater the value of γ 2 the better is the fit & more useful the regression equations as a predictive device. TYPES OF REGRESSION ANALYSIS1. Simple & Multiple: -- In simple regression analysis, we study only 2 variables at a time, in which one variable is dependent and another is independent. The functional relationship between income & expenditure is an example of simple regression. On the contrary, we study more than 2 variables at a time in multiple regression analysis (i.e., at least 3 variables) in which one is dependent variable & other is independent variable. The study of effect of rain & irrigation on yield of what is an example of multiple regressions.2. Linear and non-linear regression: -- When one variable changes with other variable in some fixed ratio, this is called as linear regression. Such type of relationship of relationship is depicted on a graph by means of straight line or a first-degree equation. On the contrary, when one variable varies with other variable in a changing ratio, than it is referred to as curvi-linear/non- linear regression. This relationship expressed on a graph takes the form of a curve & this is presented by way of 2nd & 3rd degree equation.2. Partial & total regression: -- When 2 or more variables are studied for functional relationship but at a time, relationship between only 2 variables is studied & other variables. DIFFERENCE BETWEEN CORRELATION AND REGRESSION Correlation and regression analysis are constructed under differentassumptions they furnish different types of information & it is not always clearas to which measure should be used in a given problem situation. The followingare the points of difference between the two: -Correlation and regression analysis are constructed under different assumptionthey furnish different types of information & it is not always clear as to whichmeasure should be used in a given problem situation. The following are thepoints of difference between the two: --
1. Degree and nature of relationship: -- Correlation coefficient is a measure of degree of co variability between X & Y, where as the relationship between variables so that we may be able to predict the value of one on the basis of another. The closer the relationship between two variables, the greater the confidence that may be placed in the estimates.2. Cause & effect relationship: -- Correlation is merely a tool of ascertaining the degree of relationship between 2 variables & therefore, we cannot say that one variable is cause & other effect. For example, a high degree of correlation between price & demand for a certain commodity or a particular point of time may not suggest which cause is & which effect is. However, in regression analysis one variable is taken as dependent which other as independent thus making it possible to study the cause & effect relationship.3. Symmetric: -- In correlation γ xy is a measure of direction & degree of linear relationship between 2 variables X & Y, γ xy & γ yx are symmetric (γ xy = γ yx) i.e., it is immaterial which of X & Y is dependent variable & which is independent variable. In regression analysis the regression coefficient bxy & byx are not symmetric i.e., bxy = byx and hence it makes a difference as to which variable is dependent and which is independent.4. There may be nonsense correlation between two variables, which is purely due to chance & has no practical relevance such as increase in income & increase in weight of a group of people. However, there is nothing like nonsense regression.5. Origin & Scale: -- Correlation coefficient is independent of change of scale & origin. Regression coefficients are independent of change of origin but not of scale. There is something common in both regression and correlation analysis. The coefficient of correlation (γ) takes same sign as the regression coefficient (bxy and byx).