2Relationship between “Cause” and “Effect”t Regression analysis establishes relationship between a dependent variable and independent variablesRelationship between “Cause” and “Effect”tRelationship between variables
3Usefulness of regression analysis Regression analysis is a vary widely used tool for research.It shows type and magnitude of relationship between two variables.
4Example of Usefulness of Regression Analysis :Shows for example whether there is any relationship between an increase in household income (Y) land an increase in consumption (C ).Whether there is positive or negative relationship between Y and C. Whether if :YC or reverseHow much of an increase in income (Y) is spent on consumption ( C ).
5Example of Usefulness of Regression Analysis Regression is also used for prediction and forecasting,Regression analysis allows to measure confidence or significance level of the findings.
6Example of Usefulness of Regression Analysis Increase in traffic jam (hours of non-movement) depends on Increase in number of cars in Dhaka City. (+ dependency)A decrease in number of School drop-out depends on an increase I income of parents.(-ve dependency)An increase in household income leads to an increase in household consumption.
7Other Logical Examples of Positive and Negative Dependency
8Forms of regression models A regression model relates dependent variable Y to be a function/relation of independent variable X.Symbolically, Y = f (Xi)Where i = 1,2,3,4,…
9Diagrammatic Representation of Regression Model Consumption Expenditure(,000Tk)Each dot represent sample data for Income and Expenditure for each sample household10090130Income of the Household (,000 Tk)120
10Consumption Expenditure ( C ) C = a + byRegression analysis draw a mean /average line with equation C = a + b Y so that difference between sample data and estimated data is minimized.Income of the Household (Y)Does dotted line minimize deviations?
11Deviations between sample value and the mean value Mean value line
12Diagrammatic Representation of Regression Equation In mean or average line, square of the deviation ( C i) for each of thesample from mean ( C )is minimized.Why ?Because simple sum of difference from mean is always zero.
13Example C - C Sum is zero 1 -1 0 (C – C)**2 1 0 (C – C)**2Sum of square is + numberY1089Av Y is 9C67Av C is 7Dependent variable
14Formula for Regression coefficient b when sum of square is minimized , b = (Ci – C) (Yi –Y) (Yi – Y) 2 i = 1,2, ….n
15General FormulaIf Y is dependent variable and X is independent variable e.g. Y = f (x) thenRegression coefficient =Sum of (Xi –X) (Yi – Y)Sum of (Xi –X)**2
16Example : Given the following data C = f (Y), predict Consumption level for a household with annual income of 500 thousand TakaAnnual Income (Y)(,000Tk)100150200250300Annual Expenditure(C )8090110120
17Example : Given the following data, predict Consumption level for a household with annual income of 500 thousand Taka. (Fig in,000Tk)Annual Income (Y)Av Y = 200Yi - Y100-100150-5020025050300Annual Expenditure(C )Av C = 100Ci - C80-2090-101101012020
19Calculated Regression Equation Example C = a + b YOr C = a Y or C = a YOr a = C -0.2 YOr a = x 200 = 100 – 40 = 60Therefore C = Y
20Calculated Regression Equation Example C = YWhat kind of relationship between Y and C ?How much consumption increases for Tk 1000 increase in income ?
21C = 60 +0.2 Y What is consumption, when income is zero? What is predicted consumption, when income is Tk 500,000?
22Correlation : A measure of simple relationship Correlation shows only associanship or relationship between two variables.Whereas Regression analysis shows dependency relationshipCorrelation between two variables ( for example Income and Expenditure) is measured by a formula shown as ;
23Formula of Correlation coefficient r is (Ci – C) (Yi –Y) (Yi – Y) 2 (Ci – C)2
24Formula of Correlation coefficient r in terms of regression coefficient r (Yi – Y)**2r = b(Ci – C )**2
26Class AssignmentGiven the following data, calculate correlation coefficient between Income and Expenditure. Also predict how much Consumption will increase for a 1000 Tk increase in household income?Annual Income (Y)(,000Tk)110160210260310Annual Expenditure(C )758595105115