Quantitative Business Analysis for Decision Making

1. Quantitative Business Analysis for Decision Making Simple Linear Regression

2. Lecture Outlines • Scatter Plots • Correlation Analysis • Simple Linear Regression Model • Estimation and Significance Testing • Coefficient of Determination • Confidence and Prediction Intervals • Analysis of Residuals 403.7

3. Regression Analysis ? Regression analysis is used for modeling the mean of “response” variable Y as a function of “predictor” variables X1, X2,.., Xk. When K = 1, it is called simple regression analysis. 403.7

4. Random Sample Y: Response Variable, X: Predictor Variable For each unit in a random sample of n, the pair (X, Y) is observed resulting a random sample: (x1, y1), (x2, y2),... (xn, yn) 403.7

5. Scatter Plot Scatter Plot is a graphical displays of the sample (x1, y1), (x2, y2),... (xn, yn) by n points in 2-dimension. It will suggest if there is a relationship between X and Y 403.7

6. A Scatter Plot Showing Linear Trend 403.7

7. A Scatter Plot Showing No Linear Trend 403.7

8. Modeling linear Trend • A perfect linear relationship between Y and X exists if . Coefficient is the slope--quantifying the amount of change in y corresponding to one unit change in x. • There are no perfect linear relationships in practical world. 403.7

9. Simple Linear Regression Model Model: • is linear function (nonrandom) • is random error. It is assumed to be normally distributed mean 0 and standard deviation . So • are parameters of the model 403.7

10. Estimation Simple linear regression analysis estimates the mean of Y (linear trend) by and 403.7

11. Standard deviation Standard deviation (s) of the sample of n points in the scatter plot around the estimated regression line is: 403.7

12. Testing the Slope of Linear Trend For Testing compute t-statistic and its p value: 403.7

13. Coefficient of Determination: R2 • A quantification of the significance of estimated model is denoted by R2. • R2 > 85% = significant model • R2 < 85% = model is perceived as inadequate • Low R2 will suggest a need for additional predictors for modeling the mean of Y 403.7

14. Correlation Coefficient: r The correlation coefficient r is the square root of R2. It is a number between -1 and 1. • Closer r is to -1 or 1, the stronger is the linear trend • Its sign is positive for increasing trend (slope b is positive) • Its sign is negative for decreasing trend (slope b is negative) 403.7

15. Confidence and Prediction Intervals To estimate by a confidence interval, or to predict response Y corresponding to its predictor value x = x0 • 1. Compute: • 2. compute: 403.7

16. What is ? i.e. Standard Error of For estimating , For Predicting Y, 403.7

17. Analysis of Residuals Residuals are defined: • Residual analysis is used to check the normality and homogeneity of variance assumptions of random errors . • Histogram or box plot of residuals will help to ascertain if errors are normally distributed. 403.7

18. Analysis of Residuals (con’t) Plot of residual against observed predictor values xi will help ascertain homogeneity assumption. • random appearance = homogeneity of variance assumption is valid. • non-random appearance =homogeneity assumption is not valid and variance is dependent on predictor values. 403.7