Regression

Basic concept of regression analysis

1. Saurav Das MURP 180405 Lecture 5 R E G R E S S I O N Quantitative Methods Analysis and Techniques MURP 5259

2. Linear regression analysis Multi linear regression analysis Standard error of the regression Coefficient of determination Application of regression analysis R E G R E S S I O N Content

3. Regression analysis is a powerful statistical method that allows you to examine the relationship between two or more variables of interest. While there are many types of regression analysis, at their core they all examine the influence of one or more independent variables on a dependent variable . Regression analysis is widely used for  prediction and forecasting such as Advertisement and Product sells R E G R E S S I O N Definition

4. Types of regression Simple: In case of simple relationship only two variables are considered. Multiple: In the case of multiple relationship, more than two variables are involved. On this while one variable is a dependent variable the remaining variables are independent Linear: The linear relationships are based on straight-line trend , the equation of which has no-power higher than one. Non- Linear: In the case of non-linear relationship curved trend lines are derived. Total: In the case of total relationships all the important variables are considered. Partial: In the case of partial relationship one or more variables are considered , but not all.

5. Types of regression

6. What is “Linear”? Linear regression analysis Remember this: Y= mX + B? B m What’s Slope? A slope of 2 means that every 1-unit change in X yields a 2-unit change in Y. I f you know something about x, this knowledge helps you predict something about Y (conditional probabilities) Prediction

7. R E G R E S S I O N Linear regression analysis Linear regression equation Y = a + b X Where, b = slope a = intercept

8. Methods of Regression Analysis Y = a + b X

9. Graphic Method X axis Y axis . . . . . . . . . 1. Points are plotted on a graph paper representing various parts of values 2. These points give a picture of a scatter diagram 3. A regression line may be drawn in between these points 4. Line should be drawn faithfully as the line of best fit leaving equal number of points on both sides

10. Obtain a random sample of n data pairs (X, Y), where X is the explanatory variable and Y is the response variable. Using the data pairs, compute ∑X, ∑Y, ∑X 2 , ∑Y 2 , and ∑XY. Then compute the sample means X and Y. With n = sample size, ∑X, ∑Y, ∑X 2 , ∑Y 2 , ∑XY, X and Y, you are ready to compute the slope b and intercept a using the computation formulas Slope: b = Intercept: a = Y— bX 4. The equation of the least-squares line computed from your sample data is, Y = a + b X   Algebraic Method

11. Calculation Find the regression equations from the following data: Solution: X = = = 6 Y = = = 8 b = = -0.65 a = Y— bX = 11.9 Y = a + b X = 11.9 + 0.65X  

12. Multi linear regression analysis Simple regression considers the relation between a single explanatory variable and response variable Multiple regression simultaneously considers the influence of multiple explanatory variables on a response variable Y A multiple regression equation expresses a linear relationship between a response variable y and two or more predictor variables (x 1 , x 2 , . . ., x k ). The general form of a multiple regression equation obtained from sample data is y = b + b 1 x 1 + b 2 x 2 + ………………. + b k x k

13. Simple linear & Multi linear

14. Standard error of the regression The standard error of the estimate is a measure of the accuracy of predictions made with a regression line. The square root of the average squared error of prediction is used as a measure of the accuracy of prediction. This measure is called the standard error of the estimate and is designated as σ est . where N is the number of pairs of (X,Y) points

15. Calculation The slope and intercept of the regression line are 3.2716 and 7.1526 respectively. Y' = 3.2716X + 7.1526

16. Coefficient of determination The coefficient of determination (denoted by R 2 ) is a key output of regression analysis. It is interpreted as the proportion of the variance in the dependent variable that is predictable from the independent variable. The coefficient of determination is the square of the correlation (r) between predicted y scores and actual y scores; thus, it ranges from 0 to 1. With linear regression, the coefficient of determination is also equal to the square of the correlation between x and y scores.

17. Coefficient of determination An R 2 of 0 means that the dependent variable cannot be predicted from the independent variable. An R 2 of 1 means the dependent variable can be predicted without error from the independent variable. An R 2 between 0 and 1 indicates the extent to which the dependent variable is predictable. An R2 of 0.10 means that 10 percent of the variance in Y is predictable from X; an R2 of 0.20 means that 20 percent is predictable; and so on.

18. Calculation Correlation coefficient, r = = - 0.92 Coefficient of determination = r 2 = (-0.92) 2 = 0.85   Here, n = 5 ∑𝑿 = 30 ∑𝒀 = 40 ∑𝑿𝒀 = 214 ∑ = 220 (∑ = 900 ∑ = 340 (∑ = 1600   Interpretation: 0.85 means that 85 percent of the variance in Y is predictable from X

19. 1. Regression analysis helps in establishing a functional relationship between two or more variables. 2. Since most of the problems of economic analysis are based on cause and effect relationships, the regression analysis is a highly valuable tool in economic and business research . 3. Regression analysis predicts the values of dependent variables from the values of independent variables. 4. We can calculate coefficient of correlation ( r) and coefficient of determination ( r2) with the help of regression coefficients. 5. In statistical analysis of demand curves, supply curves, production function, cost function, consumption function etc., regression analysis is widely used. Application of regression analysis