1. 1997 Summer Public Health Research Institute on Minority Health, UNC-CH Living Beyond Our “Means”: New Methods for Comparing Distributions Camara Phyllis Jones, MD, MPH, PhD Department of Health and Social Behavior Department of Epidemiology Harvard School of Public Health

2. Why Study Distributions? Preserve information on location, spread, shape Describe populations The Strategy of Preventive Medicine probability density systolic blood pressure

3. Overview Critique of current approaches Proposed methods –Projection plot –Projection spline –Iter-1 test Systolic blood pressure by “race”

4. Critique of Current Approaches Location Spread Shape Stable Test Proportions - - - yes yes Moments: Mean yes - - yes yes Std deviation - yes - yes yes Skewness - - symm yes yes Kurtosis - - peak yes yes Boxplots yes yes symm yes - Histograms yes yes yes - yes Kernel densities yes yes yes - - Cum distrib fns yes yes - yes yes Q-Q plot yes yes yes yes -

6. Fitting a Line to the Quantile-Quantile Plot Covariance of quantiles Asymmetry of modeling

7. Covariance of Sample Quantiles For i < j, the covariance is p i * (1 - p j ) n * dens i * dens j

8. Asymmetry of Modeling

9. Modeling B as dependent on A: quant B = 19.28 + 0.781 quant A SE of intercept = 3.78 SE of slope = 0.031

10. Asymmetry of Modeling (cont.) Modeling A as dependent on B: quant A = -22.64 +1.263 quant B SE of intercept = 7.17 SE of slope = 0.064 Noteworthy: Slopes are not exact reciprocals Standard errors differ in magnitude

11. Symmetry with respect to the line y=x x quantiles y quantiles line y=x (A,B) (B,A)

12. Perpendicular distance of the point (A,B) from the line y=x A (A,B) B-A B A sqrt(2)

13. Projection of the point (A,B) on the line y=x (A,B) B AB B-A 2 A+B 2

14. Rotating the plot 45 degrees x quantiles y quantiles line y=x B-A sqrt(2) (A,B) (A+B, A+B) 2 2

15. Projection plot difference between corresponding quantiles B-A A+B 2 average of corresponding quantiles line dif=0

19. Example of a Knotted Linear Spline Y = b0 + b1 x + b2 xplus (knot1) + b3 xplus (knot2) slope = b1 + b2 + b3 slope = b1 + b2 slope = b1 knot 1 knot 2

20. Iterative Fit of a Knotted Linear Spline Iteration 6 Iteration 7 Iteration 5 Candidate knots Initial knots Iteration 1 Iteration 4 Iteration 3Iteration 2 ^ ^ ^ ^ ^ ^...

22. Highest Level of Difference Shape? If not shape: Spread? If neither shape nor spread: Location? None? Shape Spread Location None Segments Slope Intercept >1 1 1 1 0 0 0 00 00

23. Interpretation of Projection Splines Number of segments – More than one SHAPE – One evaluate slope Slope – Differs from zero SPREAD – Zero evaluate intercept Intercept – Differs from zero LOCATION – Zero NONE

24. Synopsis Two groups Continuous outcome Graphical Comparison –Location –Spread –Shape Statistical test –Global –Level of difference

25. Design of NHANES 1 Probability survey All 50 states 1971-1975 Direct examination Systolic blood pressure 2,178 “black” females 9,778 “white” females

29. P-values from the iter-1 test, with projection spline inferences Crude distributions p-value Level of difference 5-14 15-24 25-34 35-44 45-54 55-64 65-74 0.443 0.229 0.000 0.003 0.000 -- Shape Spread

30. P-values from the iter-1 test, with projection spline inferences (cont.) Adjusted for body mass index p-value Level of difference 5-14 15-24 25-34 35-44 45-54 55-64 65-74 0.087 0.004 0.011 0.000 0.027 0.000 -- Shape Spread Shape

31. P-values from the iter-1 test, with projection spline inferences (cont.) Age-shifted analysis p-value Level of difference bf 5-14 / wf 15-24 bf 15-24 / wf 25-34 bf 25-34 / wf 35-44 bf 35-44 / wf 45-54 bf 45-54 / wf 55-64 bf 55-64 / wf 65-74 0.000 0.022 0.664 0.955 0.001 0.503 Shape Location -- Shape --

32. Systolic Blood Pressure by “Race” Same-age comparisons – No differences in childhood – Shape differences in middle age Age-shifted comparisons – Acceleration of age-dependence – Shift of entire distributions

33. Significance of Age-Shifting Blood pressure and age Social meaning of “race” Other health conditions

34. Hypotheses “Race” - associated differences in health outcomes in the U.S. are due to accelerated aging of the black population. Accelerated aging of the black population in the U.S. is due to racism.