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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 • 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 Yrc (x) = η(x)+γ r (x)+θrc (x)+ε rc (x) η(x)+γ r (x) = rat-level function θrc (x) = crypt-level functions, typically assumed independent 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 θrc (x) = rc,I + x rc,L + C(x) rc,S ; C(x) spline basis functions p cov( rc ) S 0 0 2 p I Linear spline: In practice, we used a quadratic spline. The covariance matrix of the quadratic part used the PourahmadiDaniels construction Advertisement David Ruppert http://stat.tamu.edu/~carroll/semiregbook/ Please buy our book! Or steal it Matt Wand 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 Matern correlation : (d, , m ) m 2d m 2d m 1 (d, , m ) m 1 K m 2 (m ) K m Modified Bessel function Basic Model • Crypt-Level: regression spline, few knots • Separable covariance structure θrc (x) = rc,I + rc,L x + C(x) rc,S ; 1 d(i, j), , m cov( ri , rj ) S 1 d(i, j), , m d(i, j) dis tan ce between crypts ( i, j) 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 Xihong Lin General Formulation • Yij = Response • Xij, Zij = cell and crypt locations • Likelihood (or criterion function) Yi ,Zi ,β , θ(X i1 ),...,θ( X im ) • The key is that the function θ() is evaluated multiple times for each rat • This distinguishes it from standard semiparametric models • The goal is to estimate θ() and β efficiently General Formulation: Overview • Likelihood (or criterion function) Yi ,Zi ,β , θ(X i1 ),...,θ( X im ) • 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 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 V(x1 ,x2 ,Δ) = covariance between cryptlevel functions that are D apart, one at location x1 and the other at location x2 • Assume separable covariance structure V(x1 ,x2 ,Δ) =G(x1 ,x2 ) ρ(Δ) Nonparametric Fits • Discrete Version: Pretend D, 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 D, x1 and x2 , then average across rats. • Call this estimate V̂(x1 ,x2 ,Δ) Nonparametric Fits • Separability: Now use the separability to get a rough estimate of the correlation surface. V(x1 ,x 2 ,Δ) = G(x1 ,x 2 ) ρ(Δ) V̂(x ,x ,Δ) 1 ρ(Δ) = 2 x1 ,x 2 V̂(x ,x ,0) 1 x1 ,x 2 2 Nonparametric Fits ρ(Δ) • The estimate is not a proper correlation function • We fixed it up using a trick due to Peter Hall (1994, Annals), thus forming ρ̂(Δ) , 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 (PourahmadiDaniels) • 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