1. 1 Linear Regression Model

2. 2 Types of Regression Models

3. EPI 809/Spring 20083 Linear Equations © 1984-1994 T/Maker Co.

4. YX iii  01 Linear Regression Model  1.Relationship Between Variables In a Linear Function Dependent (Response) Variable (e.g., income.) Independent (Explanatory) Variable (e.g., age) Population Slope Population Y-Intercept Random Error

5. EPI 809/Spring 20085 Population & Sample Regression Models

6. EPI 809/Spring 20086 Population & Sample Regression Models Population

7. EPI 809/Spring 20087 Population & Sample Regression Models Unknown Relationship Population

8. EPI 809/Spring 20088 Population & Sample Regression Models Unknown Relationship Population Random Sample

9. EPI 809/Spring 20089 Population & Sample Regression Models Unknown Relationship Population Random Sample

10. EPI 809/Spring 200810 Population Linear Regression Model Observed value  i = Random error

11. EPI 809/Spring 200811 Sample Linear Regression Model Unsampled observation  i = Random error Observed value ^

12. EPI 809/Spring 200812 Estimating Parameters: Least Squares Method

13. EPI 809/Spring 200813 0 20 40 60 0204060 X Y Scatter plot  1.Plot of All (X i, Y i ) Pairs  2.Suggests How Well Model Will Fit

14. EPI 809/Spring 200814 Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’? 0 20 40 60 0204060 X Y

15. EPI 809/Spring 200815 Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’? 0 20 40 60 0204060 X Y Slope changed Intercept unchanged

16. EPI 809/Spring 200816 Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’? 0 20 40 60 0204060 X Y Slope unchanged Intercept changed

17. EPI 809/Spring 200817 Thinking Challenge How would you draw a line through the points? How do you determine which line ‘fits best’? 0 20 40 60 0204060 X Y Slope changed Intercept changed

18. EPI 809/Spring 200818 Least Squares Least Squares  1.‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum. But Positive Differences Off- Set Negative ones

19. EPI 809/Spring 200819 Least Squares Least Squares  1.‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values is a Minimum. But Positive Differences Off-Set Negative ones. So square errors!

20. EPI 809/Spring 200820 Least Squares Least Squares  1.‘Best Fit’ Means Difference Between Actual Y Values & Predicted Y Values Are a Minimum. But Positive Differences Off- Set Negative. So square errors!  2.LS Minimizes the Sum of the Squared Differences (errors) (SSE)

21. EPI 809/Spring 200821 Least Squares Graphically

22. EPI 809/Spring 200822 Coefficient Equations  Prediction equation  Sample slope  Sample Y - intercept

23. EPI 809/Spring 200823 Derivation of Parameters (1)  Least Squares (L-S): Minimize squared error

24. EPI 809/Spring 200824 Derivation of Parameters (1)  Least Squares (L-S): Minimize squared error MULTIPLY BOTH SIDES BY (x –xbar)

25. EPI 809/Spring 200825 Computation Table

26. EPI 809/Spring 200826 Interpretation of Coefficients

27. EPI 809/Spring 200827 Interpretation of Coefficients  1.Slope (  1 ) Estimated Y Changes by  1 for Each 1 Unit Increase in X Estimated Y Changes by  1 for Each 1 Unit Increase in X If  1 = 2, then Y Is Expected to Increase by 2 for Each 1 Unit Increase in XIf  1 = 2, then Y Is Expected to Increase by 2 for Each 1 Unit Increase in X ^ ^ ^

28. EPI 809/Spring 200828 Interpretation of Coefficients  1.Slope (  1 ) Estimated Y Changes by  1 for Each 1 Unit Increase in X Estimated Y Changes by  1 for Each 1 Unit Increase in X If  1 = 2, then Y Is Expected to Increase by 2 for Each 1 Unit Increase in XIf  1 = 2, then Y Is Expected to Increase by 2 for Each 1 Unit Increase in X  2.Y-Intercept (  0 ) Average Value of Y When X = 0 Average Value of Y When X = 0 If  0 = 4, then Average Y Is Expected to Be 4 When X Is 0If  0 = 4, then Average Y Is Expected to Be 4 When X Is 0 ^ ^ ^ ^ ^

29. EPI 809/Spring 200829 Parameter Estimation Example  Obstetrics: What is the relationship between Mother’s Estriol level & Birthweight using the following data? Estriol Birthweight (mg/24h)(g/1000) (mg/24h)(g/1000) 11 21 32 42 54 11 21 32 42 54

30. EPI 809/Spring 200830 Scatterplot Birthweight vs. Estriol level Birthweight Estriol level

31. EPI 809/Spring 200831 Parameter Estimation Solution Table

32. EPI 809/Spring 200832 Parameter Estimation Solution Follows from slide 24.

33. EPI 809/Spring 200833 Coefficient Interpretation Solution

34. EPI 809/Spring 200834 Coefficient Interpretation Solution  1.Slope (  1 ) Birthweight (Y) Is Expected to Increase by.7 Units for Each 1 unit Increase in Estriol (X) Birthweight (Y) Is Expected to Increase by.7 Units for Each 1 unit Increase in Estriol (X) ^

35. EPI 809/Spring 200835 Coefficient Interpretation Solution  1.Slope (  1 ) Birthweight (Y) Is Expected to Increase by.7 Units for Each 1 unit Increase in Estriol (X) Birthweight (Y) Is Expected to Increase by.7 Units for Each 1 unit Increase in Estriol (X)  2.Intercept (  0 ) Average Birthweight (Y) Is -.10 Units When Estriol level (X) Is 0 Average Birthweight (Y) Is -.10 Units When Estriol level (X) Is 0 Difficult to explainDifficult to explain The birthweight should always be positiveThe birthweight should always be positive ^ ^

36. EPI 809/Spring 200836 SAS codes for fitting a simple linear regression  Data BW; /*Reading data in SAS*/  input estriol birthw@@;  cards;  11 21 32 42 54  ;  run;  PROC REG data=BW; /*Fitting linear regression models*/  model birthw=estriol;  run;

37. EPI 809/Spring 200837 Parameter Estimates Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 -0.10000 0.63509 -0.16 0.8849 Estriol 1 0.70000 0.19149 3.66 0.0354 Parameter Estimation SAS Computer Output 00 ^ 11 ^

38. EPI 809/Spring 200838 Parameter Estimation Thinking Challenge  You’re a Vet epidemiologist for the county cooperative. You gather the following data:  Food (lb.) Milk yield (lb.) 43.0 65.5 106.5 129.0  What is the relationship between cows’ food intake and milk yield? © 1984-1994 T/Maker Co.

39. EPI 809/Spring 200839 Scattergram Milk Yield vs. Food intake* M. Yield (lb.) Food intake (lb.)

40. EPI 809/Spring 200840 Parameter Estimation Solution Table*

41. EPI 809/Spring 200841 Parameter Estimation Solution*

42. EPI 809/Spring 200842 Coefficient Interpretation Solution*

43. EPI 809/Spring 200843 Coefficient Interpretation Solution*  1.Slope (  1 ) Milk Yield (Y) Is Expected to Increase by.65 lb. for Each 1 lb. Increase in Food intake (X) Milk Yield (Y) Is Expected to Increase by.65 lb. for Each 1 lb. Increase in Food intake (X) ^

44. EPI 809/Spring 200844 Coefficient Interpretation Solution*  1.Slope (  1 ) Milk Yield (Y) Is Expected to Increase by.65 lb. for Each 1 lb. Increase in Food intake (X) Milk Yield (Y) Is Expected to Increase by.65 lb. for Each 1 lb. Increase in Food intake (X)  2.Y-Intercept (  0 ) Average Milk yield (Y) Is Expected to Be 0.8 lb. When Food intake (X) Is 0 Average Milk yield (Y) Is Expected to Be 0.8 lb. When Food intake (X) Is 0 ^ ^