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