WELCOME TO THE PRESENTATION ON LINEAR REGRESSION ANALYSIS & CORRELATION (BI-VARIATE) ANALYSIS

LINEAR REGRESSION ANALYSIS

Linear Regression Linear regression analyzes the cause and effect relationship between two variables, X and Y. When both X and Y are known and want to find the best straight line through the data.

Regression or Correlation? When Correlation To measure how well x and y are associated. To Calculate the Pearson (parametric) correlation coefficient if it is assumed that both X and Y are sampled from normally- distributed populations.

Regression or Correlation? When Regression When one of the variables (X) is likely to precede or cause the other variable (Y). When we want to identify the nature of relationship.

Difference CorrelationRegression It measures the degree of relationship between x and y It measures the nature of relationship between the variables It doesn’t indicate the cause and effect relation between the variables. It indicates the cause and effect relation between the variables. For example- there is a strong correlation between rooster’s crows with the rising of the sun but the rooster does not cause the sun to rise. For example – there is a cause and effect relationship between the sales price and demand. If price decreases demand increases.

Uses Forecasting- Sales, Production etc. Effect of one variable to another- Advertisement on sales.

Linear Equation Y = a + bX Here, Y = dependent variable a = Y intercept / fixed value b = Slope X = independent variable

Linear Regression Equation

Linear Regression Least square method This method mathematically determines the best fitting regression line for the observed data.

Least square method xy

Least Square Method If we take the sum of deviation or absolute deviation, it does not stress the magnitude of the error. So, if we want to penalize the large absolute errors so that we can avoid them, we can accomplish this if we square the individual errors before we add them. Squaring each term accomplishes two goals: It magnifies or penalizes the larger errors. It cancels the effect of the positive and negative values. As we are looking for the estimating line that minimizes the sum of the suares of the errors, this is called least square method.

Example The Vice President for research and development of M.M. Ispahani limited, a large tea marketing company, believes that the firm’s annual profits depend on the amount spent on R & D. the new chairman does not agree and has asked for evidence. Here are the data for 6 years. The vice president wants to make a relationship and a prediction of profit for the year 2011 if the R&D expenditure is 9 lakhs. YearR&D expenses (in Lakhs) Annual Profit (in lakhs)

Solution YearXYX2X2 XY ∑X= 30∑Y= 180∑X 2 = 200∑XY = 1000

Solution (Cont…) Now, by putting the value in the equation we find the value of a = 20 and b = 2 so, Y = X Now, to forecast the profit for year 2010 if the R&D expenditure is 9 lakhs. Y = (9) = 38 lakhs So, it is expected that about 38 lakh taka of profit will come it the R&D expense is 9 lakh for year 2011.

Demonstration

CORRELATION (BI-VARIATE) ANALYSIS

Correlation The statistical tool with the help of which relationships between two or more then two variables are studied is called correlation.

Correlation Analysis Correlation analysis refers to the techniques used in measuring the closeness of the relationship between the variables. The measure of correlation called the coefficient of correlation A correlation analysis typically involves one dependent variable and one independent variable

Scatter Diagram

Positive Correlation

Negative Correlation

Zero Correlation

Quantitative Evaluation

The correlation coefficient = Value of the dependent variable obtained from respondent i = Average value of the dependent variable = Value of the independent variable obtained from respondent i = Average value of the independent variable

When r =The correlation between the two variables is 1.0Perfect.90Very Strong Strong Moderate.40-lessWeak

Problem A toy shop of Gulshan wants to give a new television commercial to promote their product. For this the owner of the shop wants to know how much money parents of Gulshan spent on the toy of their children last month and what is their monthly expenditure. Because he was interested in finding out if a relationship exists between toy purchase (the dependent variable ) and monthly expenditure ( the independent variable ). By analyzing this he wanted to know to give a television advertisement is feasible or not. So, to find the relationship the owner performed a correlation analysis.

ParentsExpenditure on toyMonthly expenditure ( Tk.00)( Tk.000) =13.67=24.33

i=

=

Expenditure on toy and Monthly expenditure

The relation between toy expenditure and monthly expenditure is positively correlated Correlation between two variables is strong.

Conclusion In conclusion we can say that correlation analysis attempts to identify patterns of variation common to a dependent variable and an independent variable. When both the dependent and the independent variables are continuous researchers can use correlation analysis to examine the relationship between the variables. This results in an objectively arrived at correlation coefficient, which indicates how strongly the two variables share a common pattern of change and whether the pattern is positive or negative.