Quantitative Methods Simple Regression Multiple Regression Simple Regression

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 • The Simple Regression Model y = 0 + 1 x 1 + • The Estimated Simple Regression Equation ^ y = b 0 + b 1 x 1

The Least Squares Method • Least Squares Criterion ^ • Computation of Coefficients’ Values The formulas for the regression coefficients b 0, b 1 • b 1 = n Σ x y - Σx Σy /nΣx 2 – (Σx)2 • b 0 = ¯y - b 1 ¯x. or • b 1 = COV(X, Y)/S 2 x • b 1 = r (Sy/Sx) • A Note on Interpretation of Coefficients b 1 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 R 2: 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 H 0: 1 = 0 Ha: 1 ≠ 0 Test Statistic F = MSR/MSE • Rejection Rule Reject H 0 if F > F where F is based on an F distribution with 1 d. f. in the numerator and n – 2 d. f. in the denominator.

Testing for Significance: t Test • Hypotheses H 0: 1= 0 Ha : 1 = 0 • Test Statistic • Rejection Rule Reject H 0 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. .