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

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

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

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 -

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

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

Asymmetry of Modeling

Modeling B as dependent on A: quant B = quant A SE of intercept = 3.78 SE of slope = 0.031

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

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

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

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

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

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

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

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

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

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

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

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

P-values from the iter-1 test, with projection spline inferences Crude distributions p-value Level of difference Shape Spread

P-values from the iter-1 test, with projection spline inferences (cont.) Adjusted for body mass index p-value Level of difference Shape Spread Shape

P-values from the iter-1 test, with projection spline inferences (cont.) Age-shifted analysis p-value Level of difference bf 5-14 / wf bf / wf bf / wf bf / wf bf / wf bf / wf Shape Location -- Shape --

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

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

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.