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Exploring the Shape of the Dose-Response Function

Exploring the Shape of the Dose-Response Function. Outline. Traditional approach to dose-response analysis The “step function” Alternative: “Flexible” regression line Spline regression Examples: logistic/linear/Cox. Example: Sleep-Disordered Breathing and Stroke.

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Exploring the Shape of the Dose-Response Function

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  1. Exploring the Shape of the Dose-Response Function

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

  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 Apnea- Hypopnea Index (AHI) Mean Percentile Distribution 5th 25th 50th 75th 95th 8.9 0.2 1.4 4.5 11.3 34.1

  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 + 2Q2 + 3Q3 + 4Q4 + Z Maximum Likelihood Estimates: Log odds (stroke) = (-9.924) + (0.301)Q2 + (0.344)Q3 + (0.454)Q4 + Z

  7. Adjusted Odds Ratios of PrevalentSTROKEby Quartile of the Apnea-Hypopnea Index AHI Quartile I II III IV 1.0 (ref.) 1.35 (0.84 - 2.18) 1.41 (0.88 - 2.26) 1.57 (0.98 - 2.53)

  8. Traditional Approach: Step Function

  9. Traditional Approach: “Step Function” Log odds (stroke) = 1 + 2Q2 + 3Q3 + 4Q4 + 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 > 11.3 Log (odds of stroke) = 1 + 4 + Z

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

  12. Step Function: Problems • 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

  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. Linear Spline Regression 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)

  18. More Flexible Spline Regression • Quadratic spline AHI + AHI2 • Cubic spline AHI + AHI2 + AHI3

  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 = EXP2; 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 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) Step #4 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. PROCLOGISTIC; MODEL DIS = EXP S1 S2 S3 S4;

  24. Maximum Likelihood Estimates Parameter DF Estimate Intercept 1 -1.7022 (α) EXP 1 -0.0203 (β0) S1 1 0.00252 (β1) S2 1 -0.00265 (β2) S3 1 -0.00047 (β3) S4 1 0.000305 (β4)

  25. Log odds (CHD) =α + 0(EXP)+ 1(S1) + 2(S2) + 3(S3) + 4(S4) EXP Four regression equations < 14 Log odds (CHD) = S1=EXP2, S2=0, S3=0, S4=0 15-29 Log odds (CHD) = S1=EXP2, S2=(EXP-14)2, S3=0, S4=0 30-43 Log odds (CHD) = S1=EXP2, S2=(EXP-14)2, S3=(EXP-29)2, S4=0 >43 Log odds (CHD) = S1=EXP2, S2=(EXP-14)2, S3=(EXP-29)2, S4=(EXP-43)2

  26. Cubic Spline RegressionLog odds (stroke) vs. AHI3 Knots: 0.2, 4.5, 34.1

  27. Cubic Spline RegressionLog odds (stroke) vs. AHI4 knots: 0.2, 1.4, 11.3, 34.1

  28. 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

  29. Spline Regression (within PROC REG) Systolic BP vs. AHI3 knots: 0.1, 3.6, 29.1

  30. Spline Regression (within PROC REG) Systolic BP vs. AHI4 knots: 0.1, 1.1, 9.5, 29.1

  31. Spline Regression (within PROC REG) Systolic BP vs. AHI5 knots: 0.1, 1.1, 3.6, 9.5, 29.1

  32. 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

  33. 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

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