This document discusses linear regression analysis. It defines the linear regression equation as Y = a + bX, or Y = β0 + β1X + ε, where Y is the dependent variable, X is the independent variable, β0 is the intercept, β1 is the slope, and ε is the error term. It provides formulas for estimating the slope and intercept using summation notation. It also explains how to interpret the regression results by decomposing the total sum of squares into the regression sum of squares and error sum of squares, with a better fit indicated by a larger ratio of SSR to SST and a smaller error sum of squares.

1. Biostatistics
Instructor: H M Nawaz Mehar

2. Regression

3. Regression

4. Regression

5. Regression Equation
𝑌 = 𝑎 + 𝑏𝑋
𝑌 = 𝛽0 + 𝛽1𝑋 + 𝜀
Dependent Variable Independent Variable
Intercept Slope
Error
Order 1 equation

6. Dependent & Independent Variables

7. Linear Regression

8. Linear Model

9. Slope & Intercept

10. Error Term

11. Estimation

12. Estimation
𝑌 = 𝛽0 + 𝛽1𝑋 + 𝜀
𝛽0=
𝑌 𝑋2− 𝑋 𝑋𝑌
𝑛 𝑋2− 𝑋 2
𝛽1=
𝑛 𝑋𝑌− 𝑋 𝑌
𝑛 𝑋2− 𝑋 2
𝑆𝑆𝐸 =
𝑖=1
𝑛
𝑌𝑖 − 𝑌𝑖
2

13. Interpretation

14. Interpretation

15. Interpretation
• SST = SSR + SSE
• SST = Total sum of square
• SSR = Regression sum of squares
• SSE = Error sum of squares
• The fit of the estimated regression line would be best if every value of the
dependent variable y falls on the regression line.
• If SSE = 0 i.e. e = (y – ŷ) = 0 then SST = SSR.
• For the perfect fit of the regression model, the ratio of SSR to SST must be equal
to unity.
• If SSE = 0 then the model would be perfect.
• If SSE would be larger, the fit of the regression line would be poor.

16. Interpretation