Functional mixed effects models

In this article, a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced. The linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects, which are modeled as realizations of zero-mean stochastic processes. The fixed functional effects and the random functional effects are modeled in the same functional space, which guarantee the population-average and subject-specific curves have the same smoothness property. These models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures, can include continuous covariates as well as dummy factors in both the fixed or random design matrices, and include the nested curves models as special cases. Two estimation procedures are proposed. The first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software. The second procedure is a sequential estimation procedure using Kalman filtering. This algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets. A generalized maximum likelihood (GML) ratio test is proposed for inference and model selection. An application to comparison of cortisol profiles is used as an illustration.     

Semiparametric regression splines in matched case-control studies

We develop semiparametric methods for matched case-control studies using regression splines. Three methods are developed: 1) an approximate cross-validation scheme to estimate the smoothing parameter inherent in regression splines, as well as 2) Monte Carlo expectation maximization (MCEM) and 3) Bayesian methods to fit the regression spline model. We compare the approximate cross-validation approach, MCEM, and Bayesian approaches using simulation, showing that they appear approximately equally efficient; the approximate cross-validation method is computationally the most convenient. An example from equine epidemiology that motivated the work is used to demonstrate our approaches.     

Testing for cubic smoothing splines under dependent data

Summary In most research on smoothing splines the focus has been on estimation, while inference, especially hypothesis testing, has received less attention. By defining design matrices for fixed and random effects and the structure of the covariance matrices of random errors in an appropriate way, the cubic smoothing spline admits a mixed model formulation, which places this nonparametric smoother firmly in a parametric setting. Thus nonlinear curves can be included with random effects and random coefficients. The smoothing parameter is the ratio of the random‐coefficient and error variances and tests for linear regression reduce to tests for zero random‐coefficient variances. We propose an exact F‐test for the situation and investigate its performance in a real pine stem data set and by simulation experiments. Under certain conditions the suggested methods can also be applied when the data are dependent.     

Determination of the Activity of Immobilized Biocatalysts in the Case of Poorly Soluble Substrates: The Example of D-5Phenylhydantoinase

The present communication describes the determination of activity of immobilized biocatalysts, from progress curves, in the case of a poorly soluble substrate. Computer simulation is used for the comparison of five well known methods for the determination of initial velocity of the reaction with the modified method of smoothing experimental data by cubic spline. The computer simulation data show that, in the case of poorly soluble substrate, it is expedient to use the method of data linearization in the co-ordinate system p/t versus p and method of splines.     

Simple Incorporation of Interactions into Additive Models

Often, the functional form of covariate effects in an additive model varies across groups defined by levels of a categorical variable. This structure represents a factor-by-curve interaction. This article presents penalized spline models that incorporate factor-by-curve interactions into additive models. A mixed model formulation for penalized splines allows for straightforward model fitting and smoothing parameter selection. We illustrate the proposed model by applying it to pollen ragweed data in which seasonal trends vary by year.     

The effect of end-point adjustments on smoothing splines used for tree-ring standardization

The cubic smoothing spline has been a popular method for detrending tree-ring data since the 1980s. The common implementation of this procedure (e.g., ARSTAN, dplR) uses a unique method for determining the smoothing parameter that is widely known as the %n criterion. However, this smoothing parameter selection method carries the assumption that end point effects are ignorable. In this paper, we complete the mathematical derivation and show how the original method differs from the complete version, both in the interpretation of the smoothing parameter and in the spline fit. Frequency response curves (FRC) demonstrate how the smoothing parameter is affected by the original assumption. For example, the FRC results indicate that a tree core of 250-year length has a 14% difference in the cut-off frequency when looking at the 67%n criterion. The FRC analysis shows that the existing approach produces a more flexible fit than anticipated, i.e., it is removing more variance than previously thought. For example, a 67%n spline under the existing approach corresponds to a 53%n spline fit. By using both simulated tree-core sequences and a dataset from a Midwest forest, we discuss which conditions result in greater differences between the spline fits and which conditions will have small differences. Tree-core sequences that have more curvature, such as a large-amplitude growth release, will lead to greater differences. Finally, we provide approximations to the end-point effect procedure. For example, using an 83%n criterion under the original approach produces a spline fit approximating the 67%n fit under the complete approach. These approximations could be easily implemented within existing programs like ARSTAN.     

Determination of the Activity of Immobilized Biocatalysts in the Case of Poorly Soluble Substrates: The Example of D-5Phenylhydantoinase

The present communication describes the determination of activity of immobilized biocatalysts, from progress curves, in the case of a poorly soluble substrate. Computer simulation is used for the comparison of five well known methods for the determination of initial velocity of the reaction with the modified method of smoothing experimental data by cubic spline. The computer simulation data show that, in the case of poorly soluble substrate, it is expedient to use the method of data linearization in the co-ordinate system p/t versus p and method of splines.     

Exploration of long-term growth changes using the tree-ring detrending program “Spotty”

A new tree-ring detrending program (Spotty) performs analyses of changing growth trends and environmental signals with tree age. Spotty is particularly useful in understanding the nature of long-term, centennial scale trends in tree-ring data but has a variety of dendrochronological applications. The program permits the user to define up to fifty age classes from a file of increments data. Age classes can be defined as ranges (e.g., 25–50, 100–150 years) or as single years (e.g., 25, 40, 55 years). The program will select the data corresponding to these classes and fit spline functions to the respective age class, data clouds. The user is allowed to choose a certain spline bandwidth, and can decide to fit splines to either the raw or Regional Curve Standardization (RCS) detrended data – with the RCS routine being included in Spotty – thus permitting the analysis of different underlying frequencies for each selected age class. The individual, age class splines may also be compared for common (or deviating) variance. Spotty facilitates color graphics and allows saving results for further use in other programs. PC and Mac versions of the program are available at the WSL Dendro Sciences (www.wsl.ch/forschung/forschungsunits/dendro) and LDEO Tree-Ring Lab (www.ldeo.columbia.edu/res/fac/trl/public/publicSoftware.html) web pages.     

Flexibility versus “rigidity” of the functional architecture of AChE active center

Functional architecture of the AChE active center appears to be characterized by both structural “rigidity”, necessary to stabilize the catalytic triad as well as by flexibility in accommodating the different, high affinity AChE ligands. These seemingly conflicting structural properties of the active center are demonstrated through combination of structural methods with kinetic studies of the enzyme and its mutant derivatives with plethora of structurally diverse ligands and in particular with series of stereoselective covalent and noncovalent AChE ligands. Thus, steric perturbation of the acyl pocket precipitates in a pronounced stereoselectivity toward methylphosphonates by disrupting the stabilizing environment of the catalytic histidine rather than through steric exclusion demonstrating the functional importance of the “rigid” environment of the catalytic machinery. The acyl pocket, the cation-binding subsite (Trp86) and the peripheral anionic subsite were also found to be directly involved in HuAChE stereoselectivity toward charged chiral phosphonates, operating through differential positioning of the ligand cationic moiety within the active center. Residue Trp86 is also a part of the “hydrophobic patch” which seems flexible enough to accommodate the structurally diverse ligands like tacrine, galanthamine and the two diastereomers of huperzine A. Also, we have recently discovered further aspects of the role of both the unique structure and the flexibility of the “hydrophobic patch” in determining the reactivity and stereoselectivity of HuAChE toward certain carbamates including analogs of physostigmine. In these cases the ligands are accommodated mostly through hydrophobic interactions and their stereoselectivity delineates precisely the steric limits of the pocket. Hence, the HuAChE stereoselectivity provides a sensitive tool in the in depth exploration of the functional architecture of the active center. These studies suggest that the combination of “rigidity” and flexibility within the HuAChE gorge are an essential element of its molecular design.     

Detecting pulsatile hormone secretions using nonlinear mixed effects partial spline models

Neuroendocrine ensembles communicate with their remote and proximal target cells via an intermittent pattern of chemical signaling. The identification of episodic releases of hormonal pulse signals constitutes a major emphasis of endocrine investigation. Estimating the number, temporal locations, secretion rate, and elimination rate from hormone concentration measurements is of critical importance in endocrinology. In this article, we propose a new flexible statistical method for pulse detection based on nonlinear mixed effects partial spline models. We model pulsatile secretions using biophysical models and investigate biological variation between pulses using random effects. Pooling information from different pulses provides more efficient and stable estimation for parameters of interest. We combine all nuisance parameters including a nonconstant basal secretion rate and biological variations into a baseline function that is modeled nonparametrically using smoothing splines. We develop model selection and parameter estimation methods for the general nonlinear mixed effects partial spline models and an R package for pulse detection and estimation. We evaluate performance and the benefit of shrinkage by simulations and apply our methods to data from a medical experiment.     

Generalized additive modeling with implicit variable selection by likelihood-based boosting

The use of generalized additive models in statistical data analysis suffers from the restriction to few explanatory variables and the problems of selection of smoothing parameters. Generalized additive model boosting circumvents these problems by means of stagewise fitting of weak learners. A fitting procedure is derived which works for all simple exponential family distributions, including binomial, Poisson, and normal response variables. The procedure combines the selection of variables and the determination of the appropriate amount of smoothing. Penalized regression splines and the newly introduced penalized stumps are considered as weak learners. Estimates of standard deviations and stopping criteria, which are notorious problems in iterative procedures, are based on an approximate hat matrix. The method is shown to be a strong competitor to common procedures for the fitting of generalized additive models. In particular, in high-dimensional settings with many nuisance predictor variables it performs very well.     

Shape-space smoothing splines for planar landmark data

A method is developed for fitting smooth curves through a seriesof shapes of landmarks in two dimensions using unrolling andunwrapping procedures in Riemannian manifolds. An explicit methodof calculation is given which is analogous to that of Jupp &Kent (1987) for spherical data. The resulting splines are calledshape-space smoothing splines. The method resembles that offitting smoothing splines in real spaces in that, if the smoothingparameter is zero, the resulting curve interpolates the datapoints, and if it is infinitely large the curve is a geodesicline. The fitted path to the data is defined such that its unrolledversion at the tangent space of the starting point is a cubicspline fitted to the unwrapped data with respect to that path.Computation of the fitted path consists of an iterative procedurewhich converges quickly, and the resulting path is given ina discretised form in terms of a piecewise geodesic path. Theprocedure is applied to the analysis of some human movementdata, and a test for the appropriateness of a mean geodesiccurve is given.     

Semiparametric frailty models for clustered failure time data

We consider frailty models with additive semiparametric covariate effects for clustered failure time data. We propose a doubly penalized partial likelihood (DPPL) procedure to estimate the nonparametric functions using smoothing splines. We show that the DPPL estimators could be obtained from fitting an augmented working frailty model with parametric covariate effects, whereas the nonparametric functions being estimated as linear combinations of fixed and random effects, and the smoothing parameters being estimated as extra variance components. This approach allows us to conveniently estimate all model components within a unified frailty model framework. We evaluate the finite sample performance of the proposed method via a simulation study, and apply the method to analyze data from a study of sexually transmitted infections (STI).     

Varying coefficient model with unknown within-subject covariance for analysis of tumor growth curves

SUMMARY: In this article we develop a nonparametric estimation procedure for the varying coefficient model when the within-subject covariance is unknown. Extending the idea of iterative reweighted least squares to the functional setting, we iterate between estimating the coefficients conditional on the covariance and estimating the functional covariance conditional on the coefficients. Smoothing splines for correlated errors are used to estimate the functional coefficients with smoothing parameters selected via the generalized maximum likelihood. The covariance is nonparametrically estimated using a penalized estimator with smoothing parameters chosen via a Kullback-Leibler criterion. Empirical properties of the proposed method are demonstrated in simulations and the method is applied to the data collected from an ovarian tumor study in mice to analyze the effects of different chemotherapy treatments on the volumes of two classes of tumors.     

Global conformation analysis of irradiated xyloglucans

Xyloglucan isolated and purified from tamarind seed was subjected to various degrees of γ-irradiation treatments, from 10 to 70 kGy, monitored for radiation damage and then studied using a new combined hydrodynamic approach with regards to conformation and flexibility. Radiation products were analysed with regard to molecular weight (weight average) M_w from size exclusion chromatography coupled to multi-angle laser light scattering (SEC–MALLs), intrinsic viscosity [η] and sedimentation coefficient s^o_20,w. Sedimentation coefficient distributions and elution profiles from SEC–MALLs confirmed the unimodal nature of the molecular weight distribution for each sample in solution. The chain flexibility was then investigated in terms of the persistence length, L_p of the equivalent worm-like chain model. The traditional Bushin–Bohdanecky (intrinsic viscosity) and Yamakawa–Fujii (sedimentation coefficient) relations were used separately then combined together by minimisation of a target function according to a recently published procedure [Ortega, A., & García de la Torre, J. (2007). Equivalent radii and ratios of radii from solution properties as indicators of macromolecular conformation, shape, and flexibility. Biomacromolecules, 8, 2464–2475 [see also Ortega, A. Metodologías computacionales para propiedades en disolución de macromoléculas rígidas y flexibles. Ph.D. Dissertation, Universidad de Murcia, 2005]] and yielded an estimate for L_p in the range 4–9 nm using floated and fixed mass per unit length analysis protocols and “point” global analysis: irradiated xyloglucans behave as flexible structures in common with pressure/heat treated materials.     

Method for an automatic analysis of the ECG

A method is presented for an analysis of the ECG. Using cubic spline techniques we proceed first to a smoothing of the signal and then to the elimination of baseline drift. The properties of the calculated derivatives are used to establish criteria for the identification of the ECG waves and the measurement of their essential parameters. The complete procedure can be carried out by a computer, without human intervention. The results of this fully automatic procedure can be used directly as a means of classifying the ECG.     

On the asymptotics of penalized splines

We study the asymptotic behaviour of penalized spline estimatorsin the univariate case. We use B-splines and a penalty is placedon mth-order differences of the coefficients. The number ofknots is assumed to converge to infinity as the sample sizeincreases. We show that penalized splines behave similarly toNadaraya--Watson kernel estimators with ‘equivalent’kernels depending upon m. The equivalent kernels we obtain forpenalized splines are the same as those found by Silverman forsmoothing splines. The asymptotic distribution of the penalizedspline estimator is Gaussian and we give simple expressionsfor the asymptotic mean and variance. Provided that it is fastenough, the rate at which the number of knots converges to infinitydoes not affect the asymptotic distribution. The optimal rateof convergence of the penalty parameter is given. Penalizedsplines are not design-adaptive.     

Smoothing splines of apex predator movement: Functional modeling strategies for exploring animal behavior and social interactions

The collection of animal position data via GPS tracking devices has increased in quality and usage in recent years. Animal position and movement, although measured discretely, follows the same principles of kinematic motion, and as such, the process is inherently continuous and differentiable. I demonstrate the functionality and visual elegance of smoothing spline models. I discuss the challenges and benefits of implementing such an approach, and I provide an analysis of movement and social interaction of seven jaguars inhabiting the Taiamã Ecological Station, Pantanal, Brazil, a region with the highest known density of jaguars. In the analysis, I derive measures for pairwise distance, cooccurrence, and spatiotemporal association between jaguars, borrowing ideas from density estimation and information theory. These measures are feasible as a result of spline model estimation, and they provide a critical tool for a deeper investigation of cooccurrence duration, frequency, and localized spatio‐temporal relationships between animals. In this work, I characterize a variety of interactive relationships between pairs of jaguars, and I particularly emphasize the relationships in movement of two male–female and two male–male jaguar pairs exhibiting highly associative relationships.