In this project, Jessica Rudd analyzed longitudinal data from a study of African American and Caucasian girls between ages 9-19 to build time-variant parametric models for predicting systolic and diastolic blood pressure. She used normalization, smoothing, and statistical methods like local polynomial and kernel smoothing to estimate blood pressure curves over time for the entire cohort and separately by race. The results showed local polynomial smoothing provided smoother estimates and slightly higher blood pressure for African American girls, though significant differences between races existed at only 22% of time points for systolic and 18% for diastolic blood pressure.

1. Faculty Advisor: Dr. Mohammed Chowdhury
Predicting Systolic and Diastolic Blood Pressure from Time Variant
Parametric Models With Longitudinal Data
Jessica M. Rudd, PhD Student, Analytics and Data Science
METHODS
• The data for this study comes from the National Heart,
Lung, and Blood Institute Growth and Health Study
(NGHS). NGHS was an observational study started in
1985 with a cohort of 1,213 African American and 1,166
Caucasian girls followed from the ages of 9 thru 19.
• Samples were binned from ages 9.1 years through 19.9
years by rounding months up, creating 100 time points.
• I used normal Q-Q plots and Shapiro-Wilk tests to test for
normality of the blood pressure data. Neither SBP or DBP
are normally distributed in this dataset. I normalized blood
pressure within time points using log transformation for
SBP and square transformation for DBP in order to apply
the parametric models.
• Mean transformed SBP and DBP was found for each time
point for the entire sample, as well as separately for
African Americans and White girls.
• Local polynomial and kernel smoothing procedures were
both applied to provide smoothing estimates as a function
of time.
• Properties of the statistical methods were evaluated by
simulation.
In this project I used longitudinal data to build time variant
parametric models for the prediction of systolic (SBP) and
diastolic (DBP) blood pressure in a cohort of African
American and Caucasian girls with annual study visits
between the ages of 9 and 19.
Considering repeated measures over time with variable
sample sizes at each time point, and increasing blood
pressure with age, raw estimates of blood pressure can only
be applied at each time point. The application of smoothing
methods as a function of time allows curve estimates over
the entire time range.
INTRODUCTION
DISCUSSION
• Box-Cox transformation were first used to normalize the
blood pressure data for each time point. While this
transformation was better at normalizing the data within
each time point, it created too much variation between
time points to allow for effective smoothing.
• I used a bandwidth of h=5 and Epinechnikov kernel in the
smoothing procedures. In this case, the bandwidth was
chosen subjectively. In the future of this study, I plan to
calculate a data-driven bandwidth using a cross-validation
approach.
R CODE SAMPLES
RESULTS
• Local polynomial smoothing method provides a smoother estimate of blood pressure
over time in this cohort.
• While it appears from the curves that African-American girls have a slightly higher
blood pressure over time than Caucasian girls, one-sided ttests only show significant
differences at 22% of the time points for SBP and 18% for DBP.
• Bootstrap confidence bands demonstrate that the correct bandwidth was selected for
the smoothing methods since the smoothing bands do not cross the confidence
bands.
• In a simulation of sample size 100 across 100 time points, replicated 500 times,
coverage probabilities are higher for local polynomial smoothing method
• The ratio of simulated local polynomial smoothing MSE to kernel smoothing MSE
demonstrates local polynomial has smaller error at 80% of the time points.
STATISTICAL METHODS
Local Polynomial Smoothing Estimators:
Suppose that q(t) is (p + 1) times continuously differentiable
with respect to t ϵ τ. Let q(q)(t) be the qth derivative of q(t), 1
≤ q ≤ and βq (t) = q(q) (t) /q!. By the Taylor expansion of q(t),
for t in some neighborhood of a0. With simplification, the pth
order local polynomial estimator of q(q)(t) based on q(tj)
which minimizes QG [β(t)], is
REFERENCES
1. Fan, K. and Zhang, J.T. (2000). “Two-step estimation of
functional linear models with applications to longitudinal
data”. J.R. Statist. Soc. Ser. B, 62, 303-332.
2. Chowdhury, M. (2014). Nonparametric Smoothing
Estimation of Conditional Distribution Function with
Longitudinal Data and Time-Varying Parametric Models.
PhD Thesis.
STATISTICAL METHODS
Kernel Smoothing Estimators:
Suppose a random sample of bivariate data N(t1,q (t1))…(tJ,
q(tJ)) from a joint pdf f(t, q(t)). Let m(t) be an unknown
regression function. Then the nonparametric regression
model is
With transformation we have