semiparametric methods for colonic crypt signaling
DESCRIPTION
Semiparametric Methods for Colonic Crypt Signaling. Raymond J. Carroll Department of Statistics Faculty of Nutrition and Toxicology Texas A&M University http://stat.tamu.edu/~carroll. Outline. Problem : Modeling correlations among functions and apply to a colon carcinogenesis experiment - PowerPoint PPT PresentationTRANSCRIPT
Semiparametric Methods for Colonic Crypt Signaling
Raymond J. CarrollDepartment of Statistics
Faculty of Nutrition and Toxicology
Texas A&M Universityhttp://stat.tamu.edu/~carroll
Outline
• Problem: Modeling correlations among functions and apply to a colon carcinogenesis experiment
• Biological background• Semiparametric framework• Nonparametric Methods• Asymptotic summary• Analysis• Summary
Biologist Who Did The Work
Meeyoung Hong
Postdoc in the lab of Joanne Lupton and Nancy Turner
Data collection represents a year of work
Basic Background
• Apoptosis: Programmed cell death• Cell Proliferation: Effectively the
opposite• p27: Differences in this marker are
thought to stimulate and be predictive of apoptosis and cell proliferation
• Our experiment: understand some of the structure of p27 in the colon when animals are exposed to a carcinogen
Data Collection
• Structure of Colon• Note the finger-like
projections• These are colonic
crypts• We measure
expression of cells within colonic crypts
Data Collection
• p27 expression: Measured by staining techniques
• Brighter intensity = higher expression
• Done on a cell by cell basis within selected colonic crypts
• Very time intensive
Data Collection
• Animals sacrificed at 4 times: 0 = control, 12hr, 24hr and 48hr after exposure
• Rats: 12 at each time period• Crypts: 20 are selected• Cells: all cells collected, about 30 per crypt• p27: measured on each cell, with
logarithmic transformation
Nominal Cell Position
• X = nominal cell position
• Differentiated cells: at top, X = 1.0
• Proliferating cells: in middle, X=0.5
• Stem cells: at bottom, X=0
Standard Model
• Hierarchical structure: cells within crypts within rats within times
rcrc
rc
r rc
r
Y (x) = η(x)+γ (x)+ +ε (x)
η(x)+γ (x)=
= crypt-level functions, typ
rat-lev
ically
θ (
as
el fun
sumed
x)
θ (inde
cti
penx
o
)
n
dent
Standard Model
• Hierarchical structure: cell locations of cells within crypts within rats within times
• In our experiment, the residuals from fits at the crypt level are essentially white noise
• However, we also measured the location of the colonic crypts
Crypt Locations at 24 Hours, Nominal zero
Scale: 1000’s of microns
Our interest: relationships at between 25-200 microns
Standard Model
• Hypothesis: it is biologically plausible that the nearer the crypts to one another, the greater the relationship of overall p27 expression.
• Expectation: The effect of the carcinogen might well alter the relationship over time
• Technically: Functional data where the functions are themselves correlated
Basic Model
• Two-Levels: Rat and crypt level functions
• Rat-Level: Modeled either nonparametrically or semiparametrically
• Semiparametrically: low-order regression spline
• Nonparametrically: some version of a kernel fit
Basic Model
• Crypt-Level: A regression spline, with few knots, in a parametric mixed-model formulation
rc,I rc,L rc,S
prc S 2
p
rc = + x +C(x) ;
C(x) spline basis functions
0cov( )
0
θ (x)
I
Linear spline:
In practice, we used a quadratic spline.
The covariance matrix of the quadratic part used the Pourahmadi-Daniels construction
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Basic Model
• Crypt-Level: We modeled the functions across crypts semiparametrically as regression splines
• The covariance matrix of the parameters of the spline modeled as separable
• Matern family used to model correlations
m
mm
m
1
Matern correlation : (d, , )
1 2d 2d(d, , ) K2 ( )
K Modified Bessel functio
m
n
m
m mm
Basic Model
• Crypt-Level: regression spline, few knots• Separable covariance structure
rc,I rc,L rc,Sr
j S
c
ri r
= + x +C(x) ;
1 d(i, j), ,cov( , )
d(i, j), , 1
d(i, j) dis tance between crypts (
m
θ (x
m
i, j
)
)
Xihong Lin
Theory for Parametric Version
• The semiparametric approach we used fits formally into a new general theory
• Recall that we fit the marginal function at the rat level “nonparametrically”.
• At the crypt level, we used a parametric mixed-model representation of low-order regression splines
General Formulation
• Yij = Response• Xij, Zij = cell and crypt locations• Likelihood (or criterion function)
• The key is that the function is evaluated multiple times for each rat
• This distinguishes it from standard semiparametric models
• The goal is to estimate and efficiently
θ( )
i i i1 imY ,Z , , (X ),..., (θ )θβ X
θ( ) β
General Formulation: Overview
• Likelihood (or criterion function)
• For iid versions of these problems, we have developed constructive kernel-based methods of estimation with• Asymptotic expansions and inference available
• If the criterion function is a likelihood function, then the methods are semiparametric efficient.• Methods avoid solving integral equations
i i i1 imY ,Z , , (X ),..., (θ )θβ X
General Formulation: Overview
• We also show • The equivalence of profiling and backfitting• Pseudo-likelihood calculations
• The results were submitted 7 months ago, and we confidently expect 1st reviews within the next year or two , depending on when the referees open their mail
General Formulation: Overview
• In the application, modifications necessary both for theoretical and practical purposes
• Techniques described in the talk of Tanya Apanasovich can be used here as well to cut down on the computational burden due to the correlated functions.
Nonparametric Fits
• Equal Spacing: Assume cell locations are equally spaced
• Define = covariance between crypt-level functions that are apart, one at location x1 and the other at location x2
• Assume separable covariance structure
1 2 1 2V(x ,x ,Δ)=G(x ,x ) ρ(Δ)
1 2V(x ,x ,Δ)
Nonparametric Fits
• Discrete Version: Pretend x1 and x2 take on a small discrete set of values (we actually use a kernel-version of this idea)
• Form the sample covariance matrix per rat at x1 and x2 , then average across rats.
• Call this estimate
1 2V̂(x ,x ,Δ)
Nonparametric Fits
• Separability: Now use the separability to get a rough estimate of the correlation surface.
1 2
1 2
1 2 1 2
1 2x ,x
1 2x ,x
V(x ,x ,Δ)=G(x ,x )
V̂(x ,x ,Δ)=
V̂(x ,x ,0)
ρ(Δ)
ρ(Δ)
Nonparametric Fits
• The estimate is not a proper correlation function
• We fixed it up using a trick due to Peter Hall (1994, Annals), thus forming , a real correlation function
• Basic idea is to do a Fourier transform, force it to be non-negative, then invert
• Slower rates of convergence than the parametric fit, more variability, etc.
• Asymptotics worked out (non-trivial)
ρ(Δ)
ρ̂(Δ)
Semiparametric Method Details
• In the example, a cubic polynomial actually suffices for the rat-level functions
• Maximize in the covariance structure of the crypt-level splines, including Matern-order
• The covariance structure includes• Matern order m (gridded)• Matern parameter • Spline smoothing parameter• Quadratic part’s covariance matrix
(Pourahmadi-Daniels)• AR(1) for residuals
Results: Part I
• Matern Order: Matern order of 0.5 is the classic autoregressive model
• Our Finding: For all times, an order of about 0.15 was the maximizer
• Simulations: Can distinguish from autocorrelation
• Different Matern orders lead to different interpretations about the extent of the correlations
Marginal over time Spline Fits
Note• time
effects• Location
effects
24-Hour Fits: The Matern order matters
Nonparametric and Semiparametric Fits at 24 Hours
Comparison of Fits
• The semiparametric and nonparametric fits are roughly similar
• At 200 microns, quite far apart, the correlations in the functions are approximately 0.40 for both
• Surprising degree of correlation
Summary
• We have studied the problem of crypt-signaling in colon carcinogenesis experiments
• Technically, this is a problem of hierarchical functional data where the functions are not independent in the standard manner
• We developed (efficient, constructive) semiparametric and nonparametric methods, with asymptotic theory
• The correlations we see in the functions are surprisingly large.
Statistical Collaborators
Yehua Li Naisyin Wang
Summary
Insiration for this work:Water goanna, Kimberley Region, Australia