Self-modeling for two-dimensional response curves

Two-dimensional response curves are an important experimental outcome in speech kinematics and other areas of research. These parameterized curves are usually obtained by recording the two-dimensional location of an object over time. In this setting, time is the independent variable and the x and y locations on specified coordinate axes define the multivariate response. Collections of such parameterized curves can be obtained either from one subject or from a number of different subjects, each producing one or several realizations of the response curve. When only one dependent variable is observed over time and no parametric model is specified, self-modeling regression (SEMOR) is an attractive modeling approach. SEMOR assumes that each of a collection of curves differs from a smooth, average shape function by some simple parametric transformation of the coordinate axes (usually linear). We will describe the extension of SEMOR to two-dimensional parameterized curves using affine transformations of a two-dimensional, time-parameterized shape function.     

Flexible modeling via a hybrid estimation scheme in generalized mixed models for longitudinal data

To circumvent the computational complexity of likelihood inference in generalized mixed models that assume linear or more general additive regression models of covariate effects, Laplace's approximations to multiple integrals in the likelihood have been commonly used without addressing the issue of adequacy of the approximations for individuals with sparse observations. In this article, we propose a hybrid estimation scheme to address this issue. The likelihoods for subjects with sparse observations use Monte Carlo approximations involving importance sampling, while Laplace's approximation is used for the likelihoods of other subjects that satisfy a certain diagnostic check on the adequacy of Laplace's approximation. Because of its computational tractability, the proposed approach allows flexible modeling of covariate effects by using regression splines and model selection procedures for knot and variable selection. Its computational and statistical advantages are illustrated by simulation and by application to longitudinal data from a fecundity study of fruit flies, for which overdispersion is modeled via a double exponential family.     

An alternative approach to normalization and evaluation for gait patterns: Procrustes analysis applied to the cyclograms of sprinters and middle-distance runners

In order to compare gait patterns, a common procedure is to normalize strides both in time and magnitude. The stride duration is usually normalized to a time percentage before averaging curves. As the timing of event occurrences may shift across strides, the shape of the averaged curves is distorted and therefore the standard deviation is overvalued. Stride magnitude normalization is performed by means of dimensionless numbers. However, there is little agreement on which body size correction methods should be used. The Procrustes method describes curve shape and shape change in a mathematical and statistical framework, independently of time and size factors. The present study aims to explore how this technique may be used for time- and magnitude-stride normalization to reflect individual and group mean responses. The Procrustes method, which combines quantitative and visual features, is applied to the shape of the ankle and knee cyclograms. Superimposition of 25 cyclograms (10 for sprinters (SP) and 15 for middle-distance runners (MDR)) was supplemented by statistical procedures (principal component analysis, discriminant function) to extract the main key events, which vary according to the athletic specialities. In comparison with the MDR (poulaine-shaped cyclogram), the ovoid cyclogram of SP reveals the following gait indicators: a short braking phase, a rapid initial lower limb swing in the forward direction, a fast upward movement of the knee and ankle, and an active foot contact. The Procrustes approach could be used to describe other quasi-periodic movements through relative motion plots (e.g., cyclograms, angle-angle diagrams, phase plane portraits).     

Semiparametric regression of multidimensional genetic pathway data: least-squares kernel machines and linear mixed models

We consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway, where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least-squares kernel machines (LSKMs). This unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way. This semiparametric model also makes it possible to test for the overall genetic pathway effect. We show that the LSKM semiparametric regression can be formulated using a linear mixed model. Estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software. Both the regression coefficients of the covariate effects and the LSKM estimator of the genetic pathway effect can be obtained using the best linear unbiased predictor in the corresponding linear mixed model formulation. The smoothing parameter and the kernel parameter can be estimated as variance components using restricted maximum likelihood. A score test is developed to test for the genetic pathway effect. Model/variable selection within the LSKM framework is discussed. The methods are illustrated using a prostate cancer data set and evaluated using simulations.     

A Bayesian model for sparse functional data

Summary.   We propose a method for analyzing data which consist of curves on multiple individuals, i.e., longitudinal or functional data. We use a Bayesian model where curves are expressed as linear combinations of B-splines with random coefficients. The curves are estimated as posterior means obtained via Markov chain Monte Carlo (MCMC) methods, which automatically select the local level of smoothing. The method is applicable to situations where curves are sampled sparsely and/or at irregular time points. We construct posterior credible intervals for the mean curve and for the individual curves. This methodology provides unified, efficient, and flexible means for smoothing functional data.     

Effect of registration on cyclical kinematic data

Given growing interest in functional data analysis (FDA) as a useful method for analyzing human movement data, it is critical to understand the effects of standard FDA procedures, including registration, on biomechanical analyses. Registration is used to reduce phase variability between curves while preserving the individual curve's shape and amplitude. The application of three methods available to assess registration could benefit those in the biomechanics community using FDA techniques: comparison of mean curves, comparison of average RMS values, and assessment of time-warping functions. Therefore, the present study has two purposes. First, the necessity of registration applied to cyclical data after time normalization is assessed. Second, we illustrate the three methods for evaluating registration effects. Masticatory jaw movements of 22 healthy adults (2 males, 21 females) were tracked while subjects chewed a gum-based pellet for 20 s. Motion data were captured at 60 Hz with two gen-locked video cameras. Individual chewing cycles were time normalized and then transformed into functional observations. Registration did not affect mean curves and warping functions were linear. Although registration decreased the RMS, indicating a decrease in inter-subject variability, the difference was not statistically significant. Together these results indicate that registration may not always be necessary for cyclical chewing data. An important contribution of this paper is the illustration of three methods for evaluating registration that are easy to apply and useful for judging whether the extra data manipulation is necessary.     

Bayesian inference for stationary points in Gaussian process regression models for event-related potentials analysis

Stationary points embedded in the derivatives are often critical for a model to be interpretable and may be considered as key features of interest in many applications. We propose a semiparametric Bayesian model to efficiently infer the locations of stationary points of a nonparametric function, which also produces an estimate of the function. We use Gaussian processes as a flexible prior for the underlying function and impose derivative constraints to control the function's shape via conditioning. We develop an inferential strategy that intentionally restricts estimation to the case of at least one stationary point, bypassing possible mis-specifications in the number of stationary points and avoiding the varying dimension problem that often brings in computational complexity. We illustrate the proposed methods using simulations and then apply the method to the estimation of event-related potentials derived from electroencephalography (EEG) signals. We show how the proposed method automatically identifies characteristic components and their latencies at the individual level, which avoids the excessive averaging across subjects that is routinely done in the field to obtain smooth curves. By applying this approach to EEG data collected from younger and older adults during a speech perception task, we are able to demonstrate how the time course of speech perception processes changes with age.     

Application of the time-dependent ROC curves for prognostic accuracy with multiple biomarkers

The rapid advancement in molecule technology has led to the discovery of many markers that have potential applications in disease diagnosis and prognosis. In a prospective cohort study, information on a panel of biomarkers as well as the disease status for a patient are routinely collected over time. Such information is useful to predict patients' prognosis and select patients for targeted therapy. In this article, we develop procedures for constructing a composite test with optimal discrimination power when there are multiple markers available to assist in prediction and characterize the accuracy of the resulting test by extending the time-dependent receiver operating characteristic (ROC) curve methodology. We employ a modified logistic regression model to derive optimal linear composite scores such that their corresponding ROC curves are maximized at every false positive rate. We provide theoretical justification for using such a model for prognostic accuracy. The proposed method allows for time-varying marker effects and accommodates censored failure time outcome. When the effects of markers are approximately constant over time, we propose a more efficient estimating procedure under such models. We conduct numerical studies to evaluate the performance of the proposed procedures. Our results indicate the proposed methods are both flexible and efficient. We contrast these methods with an application concerning the prognostic accuracies of expression levels of six genes.     

Sensitivity of Fibroblasts and Their Cytoskeletons to Substratum Topographies: Topographic Guidance and Topographic Compensation by Micromachined Grooves of Different Dimensions

Fibroblasts alter their shape, orientation, and direction of movement to align with the direction of micromachined grooves, exhibiting a phenomenon termed topographic guidance. In this study we examined the ability of the microtubule and actin microfilament bundle systems, either in combination with or independently from each other, to affect alignment of human gingival fibroblasts on sets of micromachined grooves of different dimensions. To assess specifically the role of microtubules and actin microfilament bundles, we examined cell alignment, over time, in the presence or absence of specific inhibitors of microtubules (colcemid) and actin microfilament bundles (cytochalasin B). Using time-lapse videomicroscopy, computer-assisted morphometry and confocal microscopy of the cytoskeleton we found that the dimensions of the grooves influenced the kinetics of cell alignment irrespective of whether cytoskeletons were intact or disturbed. Either an intact microtubule or an intact actin microfilament-bundle system could produce cell alignment with an appropriate substratum. Cells with intact microtubules aligned to smaller topographic features than cells deficient in microtubules. Moreover, cells deficient in microtubules required significantly more time to become aligned. An unexpected finding was that very narrow 0.5-μm-wide and 0.5-μm-deep grooves aligned cells deficient in actin microfilament bundles (cytochalasin B-treated) better than untreated control cells but failed to align cells deficient in microtubules yet containing microfilament bundles (colcemid treated). Thus, the microtubule system appeared to be the principal but not sole cytoskeletal substratum-response mechanism affecting topographic guidance of human gingival fibroblasts. This study also demonstrated that micromachined substrata can be useful in dissecting the role of microtubules and actin microfilament bundles in cell behaviors such as contact guidance and cell migration without the use of drugs such as cytochalasin and colcemid.     

Inherited resistance to N- and B-tropic murine leukemia viruses in vitro: titration patterns in strains SIM and SIM.R congenic at the Fv-1 locus.

We have investigated the titration patterns of murine leukemia viruses on mouse embryo cultures derived from a pair of congenic strains differing at the Fv-1 locus. XC plaque and infectious center assays carried out with N- and B-tropic viruses on both SIM (Fv-1nn) and SIM.R(Fv-1bb) host cells yielded results that were best approximated by Poisson one-hit curves. Titration curves of N-tropic virus by direct XC plaque assay were linear and parallel on the different hosts, with titers 1.8 to 2.7 log10 lower on SIM.R and on (SIM X SIM.R)F1 than on SIM cells; similar linear and parallel curves were found for B-tropic virus, with titers 1.4 to 2.0 log10 lower on SIM and (SIM XSIM-R)F1 than on SIM-R cells. In the infectious center assays, the proportion of infected cells was linearly related to multiplicity of infection on both permissive (N- on SIM and B- on SIM.R) restrictive (B- on SIM and N- on SIM.R) genotypes at multiplicities of infection below 0.5; the line relating the variables was about 1 log10 lower in the restrictive than in the permissive situations. At multiplicities of infection where the proportion of infected cells reached a plateau, differences between the results on permissive and restrictive genotypes were considerably reduced. This appeared to be due to the action of non-Fv-1 factors in permissive host. We conclude that the major action of the restrictive allele at the Fv-1 locus in this system is to reduce the probability of successful murine leukemia virus infection without a change in hitness.     

A Dynamical Model of Hierarchical Selection and Coordination in Speech Planning

Studies of the control of complex sequential movements have dissociated two aspects of movement planning: control over the sequential selection of movement plans, and control over the precise timing of movement execution. This distinction is particularly relevant in the production of speech: utterances contain sequentially ordered words and syllables, but articulatory movements are often executed in a non-sequential, overlapping manner with precisely coordinated relative timing. This study presents a hybrid dynamical model in which competitive activation controls selection of movement plans and coupled oscillatory systems govern coordination. The model departs from previous approaches by ascribing an important role to competitive selection of articulatory plans within a syllable. Numerical simulations show that the model reproduces a variety of speech production phenomena, such as effects of preparation and utterance composition on reaction time, and asymmetries in patterns of articulatory timing associated with onsets and codas. The model furthermore provides a unified understanding of a diverse group of phonetic and phonological phenomena which have not previously been related.     

A statistical method for chromatographic alignment of LC-MS data

Integrated liquid-chromatography mass-spectrometry (LC-MS) is becoming a widely used approach for quantifying the protein composition of complex samples. The output of the LC-MS system measures the intensity of a peptide with a specific mass-charge ratio and retention time. In the last few years, this technology has been used to compare complex biological samples across multiple conditions. One challenge for comparative proteomic profiling with LC-MS is to match corresponding peptide features from different experiments. In this paper, we propose a new method--Peptide Element Alignment (PETAL) that uses raw spectrum data and detected peak to simultaneously align features from multiple LC-MS experiments. PETAL creates spectrum elements, each of which represents the mass spectrum of a single peptide in a single scan. Peptides detected in different LC-MS data are aligned if they can be represented by the same elements. By considering each peptide separately, PETAL enjoys greater flexibility than time warping methods. While most existing methods process multiple data sets by sequentially aligning each data set to an arbitrarily chosen template data set, PETAL treats all experiments symmetrically and can analyze all experiments simultaneously. We illustrate the performance of PETAL on example data sets.     

Nonparametric mixed effects models for unequally sampled noisy curves

We propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients. The method is applicable when the individual curves are sampled at variable and irregularly spaced points. This produces a low-rank, low-frequency approximation to the covariance structure, which can be estimated naturally by the EM algorithm. Smooth curves for individual trajectories are constructed as best linear unbiased predictor (BLUP) estimates, combining data from that individual and the entire collection. This framework leads naturally to methods for examining the effects of covariates on the shapes of the curves. We use model selection techniques--Akaike information criterion (AIC), Bayesian information criterion (BIC), and cross-validation--to select the number of breakpoints for the spline approximation. We believe that the methodology we propose provides a simple, flexible, and computationally efficient means of functional data analysis.     

Flexible Regression Model Selection for Survival Probabilities: With Application to AIDS

Summary Clinicians are often interested in the effect of covariates on survival probabilities at prespecified study times. Because different factors can be associated with the risk of short‐ and long‐term failure, a flexible modeling strategy is pursued. Given a set of multiple candidate working models, an objective methodology is proposed that aims to construct consistent and asymptotically normal estimators of regression coefficients and average prediction error for each working model, that are free from the nuisance censoring variable. It requires the conditional distribution of censoring given covariates to be modeled. The model selection strategy uses stepup or stepdown multiple hypothesis testing procedures that control either the proportion of false positives or generalized familywise error rate when comparing models based on estimates of average prediction error. The context can actually be cast as a missing data problem, where augmented inverse probability weighted complete case estimators of regression coefficients and prediction error can be used ( Tsiatis, 2006 , Semiparametric Theory and Missing Data). A simulation study and an interesting analysis of a recent AIDS trial are provided.     

A flexible estimating equations approach for mapping function-valued traits

In genetic studies, many interesting traits, including growth curves and skeletal shape, have temporal or spatial structure. They are better treated as curves or function-valued traits. Identification of genetic loci contributing to such traits is facilitated by specialized methods that explicitly address the function-valued nature of the data. Current methods for mapping function-valued traits are mostly likelihood-based, requiring specification of the distribution and error structure. However, such specification is difficult or impractical in many scenarios. We propose a general functional regression approach based on estimating equations that is robust to misspecification of the covariance structure. Estimation is based on a two-step least-squares algorithm, which is fast and applicable even when the number of time points exceeds the number of samples. It is also flexible due to a general linear functional model; changing the number of covariates does not necessitate a new set of formulas and programs. In addition, many meaningful extensions are straightforward. For example, we can accommodate incomplete genotype data, and the algorithm can be trivially parallelized. The framework is an attractive alternative to likelihood-based methods when the covariance structure of the data is not known. It provides a good compromise between model simplicity, statistical efficiency, and computational speed. We illustrate our method and its advantages using circadian mouse behavioral data.     

Shape-invariant modeling of circadian rhythms with random effects and smoothing spline ANOVA decompositions

Medical studies often collect physiological and/or psychological measurements over time from multiple subjects, to study dynamics such as circadian rhythms. Under the assumption that the expected response functions of all subjects are the same after shift and scale transformations, shape-invariant models have been applied to analyze this kind of data. The shift and scale parameters provide efficient and interpretable data summaries, while the common shape function is usually modeled nonparametrically, to provide flexibility. However, due to the deterministic nature of the shift and scale parameters, potential correlations within a subject are ignored. Furthermore, the shape of the common function may depend on other factors, such as disease. In this article, we propose shape-invariant mixed effects models. A second-stage model with fixed and random effects is used to model individual shift and scale parameters. A second-stage smoothing spline ANOVA model is used to study potential covariate effects on the common shape function. We apply our methods to a real data set to investigate disease effects on circadian rhythms of cortisol, a hormone that is affected by stress. We find that patients with Cushing's syndrome lost circadian rhythms and their 24-hour means were elevated to very high levels. Patients with major depression had the same circadian shape and phases as normal subjects. However, their 24-hour mean levels were elevated and amplitudes were dampened for some patients.     

Determining alternative models for vegetation response analysis: a non-parametric approach

Abstract. Vegetation models based on multiple logistic regression are of growing interest in environmental studies and decision making. The relatively simple sigmoid Gaussian optimum curves are most common in current vegetation models, although several different other response shapes are known. However, improvements in the technical means for handling statistical data now facilitate fast and interactive calculation of alternative complex, more data-related, non-parametric models. The aim in this study was to determine whether, and if so how often, a complex response shape could be more adequate than a linear or quadratic one. Using the framework of Generalized Additive Models, both parametric (linear and quadratic) and non-parametric (smoothed) stepwise multiple logistic regression techniques were applied to a large data set on wetlands and water plants and to six environmental variables: pH, chloride, orthophosphate, inorganic nitrogen, thickness of the sapropelium layer and depth of the water-body. All models were tested for their goodness-of-fit and significance. Of all 156 generalized additive models calculated, 77 % were found to contain at least one smoothed predictor variable, i.e. an environmental variable with a response better fitted by a complex, non-parametric, than by a linear or quadratic parametric curve. Chloride was the variable with the highest incidence of smoothed responses (48 %). Generally, a smoothed curve was preferable in 23 % of all species-variable correlations calculated, compared to 25 % and 18 % for sigmoid and Gaussian shaped curves, respectively. Regression models of two plant species are presented in detail to illustrate the potential of smoothers to produce good fitting and biologically sound response models in comparison to linear and polynomial regression models. We found Generalized Additive Modelling a useful and practical technique for improving current regression-based vegetation models by allowing for alternative, complex response shapes.