1. BIOSTATISTICS Linear regression

2. Copyright ©2011, Joanna Szyda INTRODUCTION 1.Linear regression equation 2.Estimation of linear regression coefficients Least squares Weighted least squares 3.Examples of regression equations Multiple regression Polynomials Logarithmic regression 4.Results interpretation

3. LINEAR REGRESSION EQUATION

4. Copyright ©2011, Joanna Szyda REGRESSION EQUATION 1.Dependent variable 2.Independent variable BODY WEIGHT ADIPOSE TISSUE 8928 8827 6624 5923 9329 7325 8229 7725 10030 6723 DATA SET

5. Copyright ©2011, Joanna Szyda REGRESSION EQUATION INTERCEPT

6. Copyright ©2011, Joanna Szyda REGRESSION EQUATION x i -x i-1 y i -y i-1 slope

7. Copyright ©2011, Joanna Szyda no dependence gradient = 0 changes in the same direction gradient > 0 changes in opposite directions gradient < 0 EXAMPLES how much will the dependent variable change when the independent variable increases by 1 unit REGRESSION EQUATION SLOPE

8. Copyright ©2011, Joanna Szyda ERROR error intercept REGRESSION EQUATION

9. Copyright ©2011, Joanna Szyda small errorlarge error REGRESSION EQUATION ERROR

10. Copyright ©2011, Joanna Szyda observed value (y) predicted value (ŷ) REGRESSION EQUATION

11. Copyright ©2011, Joanna Szyda Dependent variableIndependent variable intercept slope REGRESSION EQUATION

12. Copyright ©2011, Joanna Szyda dependent variableindependent variable intercept slope HOW TO INTERPRET THOSE VALUES ??? REGRESSION EQUATION

13. ESTIMATION OF REGRESSION COEFFICIENTS

14. Copyright ©2011, Joanna Szyda LEAST SQUARES HOW TO ESTIMATE REGRESSION COEFFICIENTS ? SO THAT DISTANCES BETWEEN OBSERVED AND PREDICTED VALUES WERE AS SMALL AS POSSIBLE - least squares method

15. Copyright ©2011, Joanna Szyda  ( ) 2 → minimum LEAST SQUARES

16. Copyright ©2011, Joanna Szyda LEAST SQUARES

17. Copyright ©2011, Joanna Szyda LEAST SQUARES

18. Copyright ©2011, Joanna Szyda WEIGHTING OF OBSERVATIONS WEIGHTED LEAST SQUARES HOW TO ESTIMATE REGRESSION COEFFICIENTS WHEN y ARE MEASURED WITH DIFFERENT ACCURACY?

19. Copyright ©2011, Joanna Szyda WEIGHTED LEAST SQUARES

20. EXAMPLES OF REGRESSION EQUATIONS

21. Copyright ©2011, Joanna Szyda EXAMPLES OF REGRESSION EQUATIONS MULTIPLE REGRESSION BODY WEIGHTAGE ADIPOSE TISSUE 893528 884327 6624 595623 934829 733125 826129 772725 1003130 674723

22. Copyright ©2011, Joanna Szyda POLYNOMIALS 1st grade polynomial 2nd grade polynomial 3rd grade polynomial EXAMPLES OF REGRESSION EQUATIONS

23. Copyright ©2011, Joanna Szyda LOGHARITMIC REGRESSION EXAMPLES OF REGRESSION EQUATIONS

24. RESULT INTERPRETATION

25. Copyright ©2011, Joanna Szyda RESULT INTERPRETATION

26. Copyright ©2011, Joanna Szyda 1.Time between calls diminishes with temparature rise 2.Temperature increase by 1°C results in call interval shorter by 0.21 s. 3.At 10°C frogs call every 6.26 s. on average: RESULT INTERPRETATION

27. Copyright ©2011, Joanna Szyda RESULT INTERPRETATION

28. Copyright ©2011, Joanna Szyda 1.Model: 2.Partners who in 2003 were on average 40 years old and where a husband was 15 years older than his wife, have on average 2.42 children: 3.Partners who in 2003 were on average 25 years old and where a husband was 4 years older than his wife, have on average 2.66 children: RESULT INTERPRETATION

29. LINEAR REGRESSION