1. Statistical Fridays J C Horrow, MD, MS STAT Clinical Professor, Anesthesiology Drexel University College of Medicine

2. Previous Session Review Student’s t test. Frequency Data. Chi-square contingency tables.

3. Session Outline: Regression Regression v. Correlation The regression model Types of regression How to do linear regression Features of well-performed regression How to examine regression data

4. Regression v. Correlation Correlation: observational data Regression: cause-effect (experimental) –Do not imply a cause-effect relationship with observational data

5. The regression model SAMPLE: (x i,y i ). –Dependent variable: x. May have >1 –Response variable: y. May have >1 MODEL: y = x +  –where  describes the error  ~ N(0,  2 ). –Models can be very complicated

6. Types of Regression SIMPLE: one dependent variable MULTIPLE: several dependent variables LINEAR: x variables appear to 1 st power y =  0 +  1 x QUADRATIC: y =  0 +  1 x + +  2 x 2 LOGISTIC: outcome is dichotamous (0,1)

7. Simple Linear Regression Obtain data pairs (x i,y i ). Plot your data: should look linear. X i measured without error Y i measured with common error  ~ N(0,  2 ). Minimize: S(  0,  1 ) =   2 w.r.t.  0,  1

8. Simple Linear Regression

10. Obtain data pairs (x i,y i ). Plot your data: should look linear. X i measured without error Y i measured with common error  ~ N(0,  2 ). Minimize: S(  0,  1 ) =   2 w.r.t.  0,  1

11. Simple Linear Regression Find minimum by taking derivitives:  S/  0 =0 and  S/  1 =0. Get 2 equations, 2 unknowns. Solve  1 = S xy /S xx where S xy =  (x i y i ) – (  x i )(  y i )/n and S xx =  (x i 2 ) – (  x i ) 2 /n Then  0 = ybar -  1 (xbar)

12. No relationship !!

13. Features of well-performed regression Test  1 against 0 (no relationship) Can do this because we know its variance Test assumptions of: –Linearity –Homoschedasticity: Var(  i )=  2 for all i –  i ~ N(0,  2 ) “Plot residuals” (y i – yhat i )

14. How to Examine Regression Data Check r 2 value –if >0.70, then fit is good –if <0.60, very suspicious Look for “influential” points –Usually at extremes of dependent range

15. Example of an influential point Slope from –9.9 to –9.0Omit Point

16. Example of an influential point Slope from –9.9 to –5.2Move Point

17. Multiple Regression Lots of explanatory variables: –Y = X 1 + X 2 + X 3 + … + X k +  Art as well as science: –All possible regressions (2 k possibilities) –Forward selection –Backward elimination –Stepwise

18. Multiple Regression Fewer explanatory variables are better Stepwise > Backward > Forward Check final model for common error  2 Best model has smallest error  2 Beware multi-collinearity –Age as surrogate for decr renal function –Weight as surrogate for diabetes mellitus

19. Logistic Regression Results Outcome variable is an event (yes/no) –Measured as “incidence” Can be simple or multiple Results as p-value and as odds-ratio –O.R.: point estimate and confidence interval –C.I. Includes 1.0  not significant (p=NS)

20. Odds Ratios v. Hazard Ratios Odds Ratios –Relate to event incidences (%) –Measured variable is occurrence of event (y/n) Hazard Ratios –Relate to event rates (% per time) –Measured variable is time to event: “survival analysis”

21. Session Review: Regression Regression v. Correlation The regression model Types of regression How to do linear regression Features of well-performed regression How to examine regression data