2. Medical Statistics (full English class) Ji-Qian Fang School of Public Health Sun Yat-Sen University

3. Chapter 12 Linear Correlation and Linear Regression

6. Initial meaning of “regression”: Galdon noted that if father is tall, his son will be relatively tall; if father is short, his son will be relative short. But, if father is very tall, his son will not taller than his father usually; if father is very short, his son will not shorter than his father usually. Otherwise, ……?! Galdon called this phenomenon “regression to the mean” 12.3 Linear regression

7. What is regression in statistics? To find out the track of the means

8.  Given the value of chest circumference (X), the vital capacity (Y) vary around a center (  y|x )  All the centers locate on a line -- regression line. The relationship between the center  y|x and X – regression equation

9. Linear regression Try to estimate  and , getting Where a -- estimate of , intercept b -- estimate of , slop -- estimate of  y|x 1. Linear regression equation

10. Least square method To find suitable a and b such that By calculus,

11. Slop b Slop b Intercept a Regression Equation

12. 2. t test for regression coefficient b is sample regression coefficient, change from sample to sample There is a population regression coefficient, denoted by  Question : Whether  =0 or not? H 0 :  =0, H 1 :  ≠0 α=0.05

13. Statistic Standard deviation of regression coefficient Standard deviation of residual Sum of squared residuals

14. Back to Example 12-3

15. 3. Application of regression 1) To describe how the value of Y depending on X 2) To estimate or predict the value of Y through a value of X (known) -- based on the regression of Y on X. 3) To control the value of X through a value of Y (known) -- If X is not a random variable, based on the regression of Y on X. -- If X is also a random variable, based on the regression of X on Y.

16. 12.4 The relationship between Regression and Correlation 1. Distinguish and connection Distinguish: Correlation: Both X and Y are random Regression: Y is random X is not random – Type  regression X is also random – Type  regression

17. Connection: When both X and Y are random 1) Same sign for correlation coefficient and regression coefficient 2) t tests are equivalent t r = t b

18. 3) Coefficient of determination Without regression, given the value of X i we can only predict, the sum of squared residuals is After regression, given the value of X i we can predict, the sum of squared residuals is Contribution of regression It can be proved

19. 2. Caution -- for regression and correlation 1)Don’t put any two variables together for correlation and regression – They must have some relation in subject matter; 2)Correlation does not necessary mean causality -- sometimes may be indirect relation or even no any real relation;

20. 3) A big value of r does not necessary mean a big regression coefficient b; 4) To reject H 0 : ρ=0 does not necessary mean the correlation is strong -- ρ≠0; 5) Scatter diagram is useful before working with linear correlation and linear regression; 6) The regression equation is not allowed to be applied beyond the range of the data set.