1. Copyright ©2012, Joanna Szyda Nonlinear regression BIOSTATISTICS

2. 1.Binomial distribution - example 2.Transformations of binomial data logit probit log-log 3.Logistic regression equation equation parameter estimation example INTRODUCTION Copyright ©2012, Joanna Szyda

3. BINOMIAL DISTRIBUTION

4. Examples ??? Copyright ©2012, Joanna Szyda

5. DATA SAMPLE 1. 1.Testing the strength of joints in aircrafts 2.Various pressure forces tested Copyright ©2012, Joanna Szyda pressure Total number defect 25005010 27007017... 43006551

6. DATA SAMPLE 1. Copyright ©2012, Joanna Szyda

7. sick k healthy n-k case212 control1913 1.Case - a room filled with cigarette smoke every hour for 12 hr/day 2.Control – identical room without smoke 3.Individuals killed after one year of experiment - lung cancer DATA SAMPLE 2. Copyright ©2012, Joanna Szyda

8. 1.Data is not continuous 2.Data originate from a binomial distribution 3.Y  { sick, healthy } 4.Probability of cancer health status [y] group [x] 10= case 10 00... 11= cont 11 01 DATA SAMPLE 2. Copyright ©2012, Joanna Szyda

9. DATA TRANSFORMATIONS

10. DATA TRANSFORMATION 0 probability 1 -  function of probability + transformation Copyright ©2012, Joanna Szyda

11. 1.Logistic transformation p→0 logit(p)→-  p→1 logit(p)→+  p=0.5 logit(p)=1 DATA TRANSFORMATION Copyright ©2012, Joanna Szyda

12. 1.Logistic transformation DATA TRANSFORMATION Copyright ©2012, Joanna Szyda nonlinear dependence linear dependence

13. 2.Probit transformation DATA TRANSFORMATION Copyright ©2012, Joanna Szyda

14. 3.Log-log transformation DATA TRANSFORMATION Copyright ©2012, Joanna Szyda

15. 1.Logit transformation – most common 2.Probit transformation – similar to logit, but more difficult to calculate 3.Log-log transformation – used when data modeling requires nonsymmetrical distribution with respect to p=0.5 DATA TRANSFORMATION Copyright ©2012, Joanna Szyda

16. LOGISTIC REGRESSION EQUATION

17. Copyright ©2012, Joanna Szyda

18. PARAMETER ESTIMATION Iterative least squares method Fisher's method of scoring Copyright ©2012, Joanna Szyda

19. EXAMPLES

20. EXAMPLE – SAMPLE 1 1.Regression equation Copyright ©2012, Joanna Szyda

21. 2.E.g. Probability of damage by pressure = 5 000 1.Probability of damage increases with pressure EXAMPLE – SAMPLE 1 Copyright ©2012, Joanna Szyda

22. 1.Regression equation choroba k=0/1 n=1 grupa x=0/1 10=b. 10 00... 11=k. 11 01 3.Probability of cancer in case group: 2.Probability of cancer is lower in a control group 4.Probability of cancer in control group: EXAMPLE – SAMPLE 2 Copyright ©2012, Joanna Szyda

23. EXAMPLE Copyright ©2012, Joanna Szyda

24. NONLINEAR REGRESSION example datadata transformation regression equation results interpretation