1. Exploring the Shape of the Dose-Response Function

2.  Traditional approach to dose-response analysis  The “step function”  Alternative: “Flexible” regression line  Spline regression  Examples: logistic/linear/Cox Outline

3. Example: Sleep-Disordered Breathing and Stroke  Study: the Sleep Heart Health Study  Data set: cross-sectional  Exposure variable: apnea-hypopnea index (AHI)  Dependent variable: self-reported stroke  Potential confounders: known stroke risk factors

4. Data set  Observations: N=5,192  Self-reported stroke: N=204 Mean Percentile Distribution 5th 25th 50th 75th 95th 8.9 0.2 1.4 4.5 11.3 34.1 Apnea- Hypopnea Index (AHI)

5. Traditional Approach: Categorical Analysis  Categorization dummy coding AHI Q2 Q3 Q4 0 - 1.4 0 0 0 1.5 - 4.5 1 0 0 4.6 - 11.3 0 1 0 >11.3 0 0 1

6. Traditional Approach: Step Function Model: Log odds (stroke) =  1 +  2 Q2 +  3 Q3 +  4 Q4 +  Z Maximum Likelihood Estimates: Log odds (stroke) = (-9.924) + (0.301)Q2 + (0.344)Q3 + (0.454)Q4 +  Z

7. Adjusted Odds Ratios of Prevalent STROKE by Quartile of the Apnea-Hypopnea Index AHI Quartile 1.0 (ref.)1.35 (0.84 - 2.18) 1.41 (0.88 - 2.26) 1.57 (0.98 - 2.53) IIIIII IV

8. Traditional Approach: Step Function

9. Traditional Approach: “Step Function” Log odds (stroke) =  1 +  2 Q2 +  3 Q3 +  4 Q4 +  Z AHI Fitted Model 0 - 1.4 Log (odds of stroke) =  1 +  Z 1.5 - 4.5 Log (odds of stroke) =  1 +  2 +  Z 4.6 - 11.3 Log (odds of stroke) =  1 +  3 +  Z > 11.3Log (odds of stroke) =  1 +  4 +  Z

10. Traditional Approach: “Step Function” Log odds (stroke) =  1 +  2 Q2 +  3 Q3 +  4 Q4 +  Z AHI Fitted Model 0 - 1.4 Log (odds of stroke) = -9.924 +  Z 1.5 - 4.5 Log (odds of stroke) = -9.623 +  Z 4.6 - 11.3 Log (odds of stroke) = -9.580 +  Z > 11.3 Log (odds of stroke) = -9.470 +  Z

11. Traditional Approach: Step Function -9.470 +  Z -9.580 +  Z -9.924 +  Z Log odds (stroke) -9.623 +  Z 0 1.4 4.5 11.3 AHI

12.  Unrealistic assumptions  A “step function”  We actually don’t believe it; our mind tries to draw an imaginary smooth line through the step  Choice of categories could influence the shape  Test for trend  Not a test for monotonic dose-response  Statistical hypothesis testing Step Function: Problems

13. Alternative: “Flexible” Regression Line  Spline Regression  Categorize (specify cutoff points) (as in categorical analysis)  Fit the regression line in segments (as in categorical analysis)  Enforce continuity at the junctions (knots) (new)

14. EXAMPLE: Linear Spline Regression Log odds (stroke) 0 1.4 4.5 11.3 AHI

15. Linear Spline Regression Log odds (stroke) 0 1.4 4.5 11.3

16. Linear Spline Regression  Fit two straight regression lines  Ensure continuity at the knot (AHI=1.4) Method:  Define a new variable, S S=0, if AHI<1.4 S=AHI-1.4, if AHI>1.4

17. Log odds (stroke) =  0 +  1 (AHI)+  2 (S)+  Z To the left of the knot: S=0 Log odds (stroke) =  0 +  1 (AHI) +  Z To the right of the knot: S=AHI-1.4 Log odds (stroke) =  0 +  1 (AHI) +  2 (AHI-1.4) +  Z =  0 -1.4  2 + (  1 +  2 )AHI +  Z  Different slopes  Identical predicted value at the knot (AHI=1.4) Linear Spline Regression

18. More Flexible Spline Regression  Quadratic spline AHI + AHI 2  Cubic spline AHI + AHI 2 + AHI 3

19. Basic quadratic spline: Step #1  Determine cutpoints (C1, C2, C3) on the exposure scale (4 categories)  These are either percentiles or some other values. That is, decide on the values of C1, C2, C3 of your choice C1=?; C2=?; C3=?;

20. Step #2 S1 = EXP 2 ; S2 = 0; S3 = 0; S4 = 0; IF EXP > C1 THEN S2 = (EXP-C1) 2 ; IF EXP > C2 then S3 = (EXP-C2) 2 ; IF EXP > C3 then S4 = (EXP-C3) 2 ;

21. Step #3 Step #4 Regress the dependent variable on EXP S1 S2 S3 S4 covariates And find the four regression equations: one per exposure category (together they form a continuous dose-response function) Compute and display the dose-response function

22. C1=14; C2=29;Example: pack-years of smoking and CHD C3=43;EXP = pack-years S1 = EXP**2; S2=0; S3=0; S4=0; IF EXP > C1 THEN S2 = (EXP-C1)**2; IF EXP > C2 then S3 = (EXP-C2)**2; IF EXP > C3 then S4 = (EXP-C3)**2;

23. PROC LOGISTIC; MODEL DIS = EXP S1 S2 S3 S4;

24. Maximum Likelihood Estimates Parameter DF Estimate Intercept 1 -1.7022(α) EXP 1 -0.0203 (β 0 ) S1 1 0.00252(β 1 ) S2 1 -0.00265(β 2 ) S3 1 -0.00047(β 3 ) S4 1 0.000305(β 4 )

25. Log odds (CHD) = α +  0 (EXP)+  1 (S1) +  2 (S2) +  3 (S3) +  4 (S4) EXPFour regression equations < 14 Log odds (CHD) = S1=EXP 2, S2=0, S3=0, S4=0 15-29 Log odds (CHD) = S1=EXP 2, S2=(EXP-14) 2, S3=0, S4=0 30-43 Log odds (CHD) = S1=EXP 2, S2=(EXP-14) 2, S3=(EXP-29) 2, S4=0 >43 Log odds (CHD) = S1=EXP 2, S2=(EXP-14) 2, S3=(EXP-29) 2, S4=(EXP-43) 2

28. Cubic Spline Regression Log odds (stroke) vs. AHI 3 Knots: 0.2, 4.5, 34.1

29. Cubic Spline Regression Log odds (stroke) vs. AHI 4 knots: 0.2, 1.4, 11.3, 34.1

30. Spline Regression: Applications Regression Dependent SAS Procedure Model Variable Logistic log odds (Y=1) PROC LOGISTIC Linear mean Y PROC REG Cox log (hazard) PROC PHREG All models are linear functions of the predictors

31. Spline Regression (within PROC REG) Systolic BP vs. AHI 3 knots: 0.1, 3.6, 29.1

32. Spline Regression (within PROC REG) Systolic BP vs. AHI 4 knots: 0.1, 1.1, 9.5, 29.1

33. Spline Regression (within PROC REG) Systolic BP vs. AHI 5 knots: 0.1, 1.1, 3.6, 9.5, 29.1

34. Spline Regression Key Advantages  Less restrictive assumptions  More regional flexibility  Does not rely on statistical hypothesis testing  Not as sensitive to the choice of cutoff points  Visual inspection of the dose-response pattern  Might be used to guide the choice of categories for traditional categorical analysis

35. Spline Regression Key Issues  Moderately sensitive to the number of knots (especially if only 3 are specified)  What do the “bumps and valleys” really mean?  Visual (subjective) interpretation  Consider the scale of the Y-axis  Consider the amount of data at the tail(s)  Straight line at the outermost segments