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Exploring the Shape of the Dose-Response Function

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

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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

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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)

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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

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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

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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

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Traditional Approach: Step Function

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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

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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

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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

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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

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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)

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

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

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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

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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

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More Flexible Spline Regression Quadratic spline AHI + AHI 2 Cubic spline AHI + AHI 2 + AHI 3

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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=?;

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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 ;

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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

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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;

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

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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 )

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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

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

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

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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

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

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

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

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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

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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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