||5R01CA075097-03 Interpret this number
||University Of Wisconsin Madison
||Nonparametric Analysis of Multiple Curves in Cancer
Self Modeling Regression (SEMOR) is a flexible, semiparametric approach
to fitting multiple curves of the type generated in tumor re-growth
experiments. SEMOR is based on the assumption that each of a group of
growth curves is a simple parametric transformation of some smooth shape
function. This proposal concentrates on incorporating a free-knot
spline into SEMOR as the estimate of the shape function and in
developing confidence regions and testing procedures for the resulting
Free-knot splines have not enjoyed popularity as nonparametric
estimators because of the computational difficulty of finding least
squares estimates of the knots. We have recently developed a penalized
estimate of the knot locations which improves the computational
properties of the estimation problem without sacrificing significant
flexibility in the resulting curve. We propose further work on both
penalized and unpenalized free-knot splines including the development
of starting value algorithms for the knot locations, confidence regions
for the fitted curve, and tests for differences between two fitted
curves. In addition, the reversible jump Markov chain Monte Carlo
(MCMC) algorithm will be applied to the problem of estimating the number
and locations of the knots in a Bayesian setting and its performance
compared to (restricted) maximum likelihood estimation.
The second group of issues addressed in this proposal relate to the
incorporation of penalized free-knot splines into a SEMOR model and the
application of the SEMOR model to tumor regrowth data. In previous work
we have proposed a nonlinear mixed effects SEMOR model where the
parameters that define the individual curve transformations are assumed
to follow a distribution in the population of curves. Adding penalized
estimation of the shape function to this setting requires careful
consideration of the definition of the estimator as well as the
optimization algorithm. A number of approaches are proposed including
two penalized maximum likelihood estimates and reversible jump MCMC
estimation of a Bayesian approach. Bootstrap based confidence bands and
testing procedures are proposed.
The applicability of theoretical results to small samples will be
checked by simulation and all successful algorithms will be implemented
in the S statistical language and made publicly available.
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, Lindstrom M.J.
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