Quantitative Methods Simple Regression.

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Presentation transcript:

Quantitative Methods Simple Regression

Multiple Regression Simple Regression Model Least Squares Method Coefficient of Determination Model Assumptions Testing for Significance Using the Estimated Regression Equation for Estimation and Prediction

The Simple Regression Model y = 0 + 1x1 +  The Estimated Simple Regression Equation y = b0 + b1x1 ^

The Least Squares Method Least Squares Criterion Computation of Coefficients’ Values The formulas for the regression coefficients b0, b1 b1 = n Σ x y - Σx Σy /nΣx2 – (Σx)2 b0 = ¯y - b1 ¯x. or b1 = COV(X,Y)/S2x b1 = r (Sy/Sx) A Note on Interpretation of Coefficients b1 represents an estimate of the change in y corresponding to a one-unit change in x. It is known as regression coefficient ^

The Coefficient of Determination Relationship Among SST, SSR, SSE SST = SSR + SSE Coefficient of Determination R 2 = SSR/SST It measures extent of variation in Y explained by the regression equation ^ ^

The Coefficient of Determination Coefficient of Determination R2:It is the square of correlation coefficient r & it measures strength of association in regression.

Model Assumptions Assumptions About the Error Term  The error  is a random variable with mean of zero. The variance of  , denoted by 2 The values of  are independent. The error  is a normally distributed random variable

Testing for Significance: F Test Hypotheses H0: 1 = 0 Ha: 1 ≠0 Test Statistic F = MSR/MSE Rejection Rule Reject H0 if F > F where F is based on an F distribution with 1d.f. in the numerator and n – 2 d.f. in the denominator.

Testing for Significance: t Test Hypotheses H0: 1= 0 Ha: 1 = 0 Test Statistic Rejection Rule Reject H0 if t < -tor t > t where t is based on a t distribution with n - p - 1 degrees of freedom.

Using the Estimated Regression Equation for Estimation and Prediction The procedures for estimating and predicting an individual value of y in simple regression We substitute the given values of x into the estimated regression equation and use the corresponding value of y as the point estimate. . ^