MBA统计学第10章

nullnullRelationship among variables Functional relationship Statistical relationship(correlation) Y depends on X, but isn’t merely determined by X. Example: price and sales daily high temperature—the demand for air-conditioning Regression—According to observed data, establish regression equation and make statistical reference (predict) .Chapter 10 (P 227) Correlation and Regression AnalysisnullWhat does regression do?Solve the following problems: Determine whether there is statistical relationship among variables, if does, give the regression equation. Forecast the value of another variable (dependent) according to one variable or a group of variables (independent).nullExample: X-price，Y-sales for a kind of product We collect data: 1. Scatter plot 2. Regression equation(the Least Square Estimation) 3. Correlation coefficient (Testing the regression model) 4.Forecasting (point and interval forecasting ) Simple Linear RegressionnullLinear Regression ModelVariables consist of a linear function. YXiii01 Slope Y-InterceptIndependent (Explanatory) Variable Dependent (Response) Variable Random ErrornullSample Linear Regression Modelei = random errorYXYbbXeiii01^YbbXii01Sampled Observed ValuenullSample Linear Regression ModelThe least squares method 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 . nullLeast Squares estimatione2YXe1e3e4YbbXeiii01^YbbXii01OLS Min eeeeeii2112223242Predicted Value nullCoefficient & EquationYbXbXYnXYXnXbYbXiiiiiniin011122101Sample regression equationSlope for the estimated regression equation P 238 (10.17)Intercept for the estimated regression equationbnullSignificance Test Test Coefficient of Determination and Standard Deviation of Estimation Residual Analysis Evaluating the ModelnullMeasures of Variation in Regression SST = SSR + SSE 1. Total Sum of Squares (SST) P 239 (10.20) Measure the variation between the observed value Yi and the mean Y. 2. Sum of Squares due to Regression (SSR) Variation caused by the relationship between X and Y. 3. Sum of Squares due to Error (SSE) Variation caused by other factors.nullVariation MeasuresYXYXiSST (Yi - Y)2 SSE (Yi -Yi)2 ^ SSR (Yi - Y)2 ^Yi ^YbbXii01null A measure 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.Coefficient of Determination 0  r2  1rbYbXYnYYnYiiiininiin201211212Explained variation Total variationSSRSSTnullCorrelation CoefficientA numerical measure of linear association between two variables that 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.nullCoefficients of Determination (r2) and Correlation (r)null 1. Tests a Linear Relationship Between X & Y 2. Hypotheses H0: 1 = 0 (No Linear Relationship) H1: 1  0 (Linear Relationship) 3. Test StatisticTest of Slope Coefficient for SignificancenullH0: 1 = 0 H1: 1  0   .05 df  5 - 2 = 3 Critical value: Example Test of Slope CoefficientStatistic: Determine: Conclusion: Reject at  = 0.05There is evidence of a relationship.null There exists linear relationship among an dependent variable and two or more than two independent variables.Multiple Regression ModelYXXXiiiPPii01122slope of populationintercept of population Yrandom errorDependent Variable Independent Variables null 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). Example: New York Times You have collected the following data: response size volume 1 1 2 4 8 8 1 3 1 3 5 7 2 6 4 4 10 6nullExample (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 null 1.Slope (b1) If the publishing 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. Interpretation of Coefficientsnull1. How does the model describe the relationship among variables? 2. Closeness of ‘Best Fit’ 3. Assumptions met 4. Significance of estimates 5. Correlation among variables 6. Outliers (unusual observations)Evaluating the ModelnullTesting Overall SignificanceTest whether there is linear relationship between Y and all the independent variables. 2. Use F statistic. Hypothesis H0:  1 =  2 = ... =  P = 0 There is no linear relationship between Y and independent variables. H1: At least there is a coefficient isn’t equal to 0. At least there is an independent variable influences YnullTesting Overall Significance Computer OutputAnalysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 2 9.2497 4.6249 55.440 0.0043 Error 3 0.2503 0.0834 C Total 5 9.5000 Pn - P -1n - 1MSR / MSEp ValuenullNon-linear models that can be transformed into linear models (convenient to carry out OLS). Data Transformation Multiplicative Model Example Transformations in Regression ModelsnullSquare-Root TransformationnullLogarithmic TransformationnullExponential Transformation