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Simple Linear Regression and Correlation (Part II)

Simple Linear Regression and Correlation (Part II). By Asst. Prof. Dr. Min Aung. Regression equation. = The best-fitting line along which all sample points are scattering. Regression equation = Least-squares equation.

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Simple Linear Regression and Correlation (Part II)

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  1. Simple Linear Regression and Correlation(Part II) By Asst. Prof. Dr. Min Aung

  2. Regression equation • = The best-fitting line along which all sample points are scattering • Regression equation = Least-squares equation • = the straight line whose total squared vertical distance from all scatter points is minimum(Least-squared) • Ŷ = a + bX • Ŷ is the point estimate for Y given by the regression equation • Find A in calculator and press it. • Find B in calculator and press it. • B = Formula 1 (the first one) and A = Formula 1 (the second one)

  3. Regression Line (1) • Substitute the smallest Xvalue in the regression equation Ŷ = a + bX and compute Ŷ. • Regression line = Least-squares line • Then, you get a pair (smallest X, corresponding Ŷ). • Substitute the largest Xvalue in the regression equation Ŷ = a + bX and compute Ŷ. • Then, you get a pair (largest X, corresponding Ŷ). • Plot the two points (smallest X, corresponding Ŷ) and(largest X, corresponding Ŷ). • Connect the two points by a straight line segment.

  4. Regression Line (2) Y (4, 3) (2, 1) X 2 4

  5. Y (4, 3) (2, 1) X 2 4 Constant or Y-intercept • A is the value of Ŷwhen X = 0. • In the regression equation, A is called constant or slope. • Interpretation of A: If X is 0 unit, the estimated Y is A units. (0, 1) 1is called the y-intercept of the lineŶ = 1 + 0.5X. 1is called the constant of theequationŶ = 1 + 0.5X.

  6. Regression Coefficient or Slope • B is the value by which Ŷ increases when X increases by 1 unit. • In the regression equation, B is called Regression Coefficient or Slope. • Interpretation of B: If X increases by 1 unit, the estimated Y will increase by B units. Y (4, 3) (2, 2) 0.5 1 0.5 1 X 2 4 0.5is called the regression coefficient of theequationŶ = 1 + 0.5Xand slope of the line. 0.5is called the slope of thelinewith the equation Ŷ = 1 + 0.5X.

  7. Interval Estimates • : b  tSb, where t is found at t-table, Df = n – 2, two-tailed • Compute Se by Formula 7, then use Se and Formula 4 to computeSb.

  8. ANOVA Table • SST = (Total variation of Y values from Ῡ) =  (Y - Ῡ)2 • An = Analysis, O = of, V = Variance ANOVA • SSR = (Total variation of Ŷ values from Ῡ) =  (Ŷ - Ῡ)2 • SSE = (Total variation of Y values from Ŷ) =  (Y -Ŷ)2 • Table Structure F SS Df MS Formula 9: numerator R = SSR MSR  MSE 1 n-2 E = SSE  (n-2) SST - SSR Formula 9: Denominator T n-1

  9. Three Statistics from ANOVA Table • F = MSR  MSE: The larger F is, the more likely is the regression model significant • R2 = SSR  SST : The larger R2 is, the better can the regression model predict Y values • Se = MSE : The smaller Se is, the more precise is the interval estimates for Y values

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