Semiparametric Methods for Colonic Crypt Signaling Raymond J. Carroll Department of Statistics Faculty of Nutrition and Toxicology Texas A&M University

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

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 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 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

Basic Model Crypt-Level: regression spline, few knots Separable covariance structure

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 Y ij = Response X ij, Z ij = 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

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

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 1 st 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 x 1 and the other at location x 2 Assume separable covariance structure

Nonparametric Fits Discrete Version: Pretend  x 1 and x 2 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  x 1 and x 2, then average across rats. Call this estimate

Nonparametric Fits Separability: Now use the separability to get a rough estimate of the correlation surface.

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)

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