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1、1,What does regression do?,Solve the following problems: Determine whether there has statistical relationship among variables, if has, show the formula. Forecast the value of another variable according to one variable or a group of variables.,2,Linear Regression Assumptions,Normality Every value of
2、X , Y follows the normal distribution The error probability follows the normal distribution Homoscedasticity (Constant Variance) Independence of Errors Linearity,3,Example: X-price，Y-demand for the product We have data: 1. Scatter plot 2. Regression equation(Ordinary Least Square Estimation) 3. Corr
3、elation coefficient r Testing the regression model 4.Forecasting 5.Regression can be linearitied,Simple Linear Regression,4,Linear Regression Model,Variables consist of a linear function.,Y,X,i,i,i,0,1,Slope,Y-Intercept,Independent (Explanatory) Variable,Dependent (Response) Variable,Random Error,5,
4、Population Linear Regression Model,i,= random error,X,YX,i,X,0,1,Y,X,i,i,i,0,1,Observed Value,Observed Value,Y,6,Sample Linear Regression Model,e,i,= random error,Y,X,Y,b,b,X,e,i,i,i,0,1,Y,b,b,X,i,i,0,1,Unsampled Observed Value,Sampled Observed Value,7,Ordinary Least Squares,The least squares method
5、 provides an estimated regression equation that minimizes the sum of squared deviations between the observed values of the dependent variable yi and the estimated values of the dependent variable .,e,2,Y,X,e,1,e,3,e,4,Y,b,b,X,e,i,i,i,0,1,Y,b,b,X,i,i,0,1,OLS Min,e,e,e,e,e,i,i,2,1,1,2,2,2,3,2,4,2,Pred
6、icted Value,8,Coefficient & Equations,Y,b,X,b,X,Y,n,X,Y,X,n,X,b,Y,b,X,i,i,i,i,i,n,i,i,n,0,1,1,1,2,2,1,0,1,Sample regression equation,Slope for the estimated regression equation,Intercept for the estimated regression equation,b,9,Evaluating the Model,Test Coefficient of Determination and Standard Dev
7、iation of Estimation Residual Analysis Test Coefficients of Significance,10,Measures of Variation in Regression,1. Total Sum of Squares (SST) Measure the variation between the observed value Yi and the mean Y. 2. Explained Variation (SSR) Variation caused by the relationship between X and Y. 3. Unex
8、plained Variation (SSE) Variation caused by other factors.,11,Variation Measures,Y,X,Y,X,i,SST (Yi - Y)2,SSE (Yi -Yi)2,SSR (Yi - Y)2,Yi,Y,b,b,X,i,i,0,1,12,Coefficient of Determination,0 r2 1,r,b,Y,b,X,Y,n,Y,Y,n,Y,i,i,i,i,n,i,n,i,i,n,2,0,1,2,1,1,2,1,2,Explained variation,Total variation,SSR,SST,A mea
9、sure of the goodness of fit of the estimated regression equation. It can be interpreted as the proportion of the variation in the dependent variable y that is explained by the estimated regression equation.,13,Correlation Coefficient,A numerical measure of linear association between two variables th
10、at takes values between 1 and +1. Values near +1 indicate a strong positive linear relationship, values near 1 indicate a strong negative linear relationship, and values near zero indicate lack of a linear relationship.,14,Coefficients of Determination (r2) and Correlation (r),15,Test of Slope Coeff
11、icient for Significance,1. Tests a Linear Relationship Between X & Y 2.Hypotheses H0: 1 = 0 (No Linear Relationship) H1: 1 0 (Linear Relationship) 3.Test Statistic,16,Example Test of Slope Coefficient,H0: 1 = 0 H1: 1 0 .05 df 5 - 2 = 3 Critical value:,Statistic: Determine: Conclusion:,Reject at = 0.
12、05,There is evidence of a relationship.,17,Multiple Regression Model,There exists linear relationship among an dependent variable and two or more than two independent variables.,Y,X,X,X,i,i,i,P,Pi,i,0,1,1,2,2,slope of population,intercept of population Y,random error,Dependent Variable,Independent V
13、ariables,18,Example: New York Times,You work in the advertisement department of New York Times(NYT). You will find to what extent do ads size(square inch ) and publishing volume (thousand) influence the response to ads(hundred).,You have collected the following data: response size volume 11248813135
14、72644106,19,Example (NYT) Computer Output,Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Param=0 Prob|T| INTERCEP 1 0.0640 0.2599 0.246 0.8214 ADSIZE 1 0.2049 0.0588 3.656 0.0399 CIRC 1 0.2805 0.0686 4.089 0.0264,20,Interpretation of Coefficients,1.Slope (b1) If the publ
15、ishing volume remains unchanged,when ads size increases one square inch, the response is expected to increase 0.2049 hundred times. 2.Slope (b2) If ads size remains unchanged, when publishing volume increases one thousand, the response is expected to in- crease 0.2805 hundred times.,21,Evaluating the Model,1.How does the model describe the relationship among variables? 2.Closeness of Best Fit 3.Ass
