Using a surrogate marker for early testing of a treatment effect

The development of methods to identify, validate, and use surrogate markers to test for a treatment effect has been an area of intense research interest given the potential for valid surrogate markers to reduce the required costs and follow‐up times of future studies. Several quantities and procedures have been proposed to assess the utility of a surrogate marker. However, few methods have been proposed to address how one might use the surrogate marker information to test for a treatment effect at an earlier time point, especially in settings where the primary outcome and the surrogate marker are subject to censoring. In this paper, we propose a novel test statistic to test for a treatment effect using surrogate marker information measured prior to the end of the study in a time‐to‐event outcome setting. We propose a robust nonparametric estimation procedure and propose inference procedures. In addition, we evaluate the power for the design of a future study based on surrogate marker information. We illustrate the proposed procedure and relative power of the proposed test compared to a test performed at the end of the study using simulation studies and an application to data from the Diabetes Prevention Program.     

Numerical discretization-based estimation methods for ordinary differential equation models via penalized spline smoothing with applications in biomedical research

Differential equations are extensively used for modeling dynamics of physical processes in many scientific fields such as engineering, physics, and biomedical sciences. Parameter estimation of differential equation models is a challenging problem because of high computational cost and high-dimensional parameter space. In this article, we propose a novel class of methods for estimating parameters in ordinary differential equation (ODE) models, which is motivated by HIV dynamics modeling. The new methods exploit the form of numerical discretization algorithms for an ODE solver to formulate estimating equations. First, a penalized-spline approach is employed to estimate the state variables and the estimated state variables are then plugged in a discretization formula of an ODE solver to obtain the ODE parameter estimates via a regression approach. We consider three different order of discretization methods, Euler's method, trapezoidal rule, and Runge-Kutta method. A higher-order numerical algorithm reduces numerical error in the approximation of the derivative, which produces a more accurate estimate, but its computational cost is higher. To balance the computational cost and estimation accuracy, we demonstrate, via simulation studies, that the trapezoidal discretization-based estimate is the best and is recommended for practical use. The asymptotic properties for the proposed numerical discretization-based estimators are established. Comparisons between the proposed methods and existing methods show a clear benefit of the proposed methods in regards to the trade-off between computational cost and estimation accuracy. We apply the proposed methods t an HIV study to further illustrate the usefulness of the proposed approaches.     

Estimation of the size of an open population using local estimating equations II: a partially parametric approach

Kernel smoothing methods are applied to extend a modification of the closed population approach of Lloyd and Yip (1991, in Estimating Equations, 65-88) to open populations with frequent capture occasions. The method complements previous nonparametric methods and, when the parametric assumptions are met, simulations show the new method has a smaller integrated mean squared error than the previous fully nonparametric method. The method is applied to capture-recapture data on short-tailed shearwaters collected annually for 48 years.     

Smoothed quantile regression for partially functional linear models in high dimensions

Practitioners of current data analysis are regularly confronted with the situation where the heavy-tailed skewed response is related to both multiple functional predictors and high-dimensional scalar covariates. We propose a new class of partially functional penalized convolution-type smoothed quantile regression to characterize the conditional quantile level between a scalar response and predictors of both functional and scalar types. The new approach overcomes the lack of smoothness and severe convexity of the standard quantile empirical loss, considerably improving the computing efficiency of partially functional quantile regression. We investigate a folded concave penalized estimator for simultaneous variable selection and estimation by the modified local adaptive majorize-minimization (LAMM) algorithm. The functional predictors can be dense or sparse and are approximated by the principal component basis. Under mild conditions, the consistency and oracle properties of the resulting estimators are established. Simulation studies demonstrate a competitive performance against the partially functional standard penalized quantile regression. A real application using Alzheimer's Disease Neuroimaging Initiative data is utilized to illustrate the practicality of the proposed model.     

Rapid cladogenesis in marine fishes revisited

Lineages that underwent rapid cladogenesis are attractive systems for the study of mechanisms underlying taxonomic, ecological, morphological, and behavioral diversification. Recently developed statistical methods provide insights into historical patterns of diversity and allow distinguishing bursts of cladogenesis from stochastic background rates in the presence of confounding factors such as extinction and incomplete taxon sampling. Here, we compare the dynamics of speciation in several marine fish lineages some of which were previously proposed to have undergone significant changes of cladogenesis through time. We tested for evidence of episodes of rapid cladogenesis using the constant rate and Monte Carlo constant rate tests that are robust to incomplete taxon sampling. These tests employ the statistic gamma to measure the relative position of internal node in a chronogram. For the first time, we conducted a comparative analysis to address the behavior of the statistic under different chronogram-constructing methods (Langley-Fitch, nonparametric rate smoothing, and penalized likelihood). Although estimates of gamma sometimes differ widely among methods, acceptance or rejection of the constant rate model within a particular clade appears to be robust to the choice of method. Bursts of cladogenesis were detected in 14 of 34 studied datasets. Some of these were previously proposed to represent marine fish "radiations," whereas others are identified anew. Our results indicate that the wider application of tree shape methods that are able to detect significantly elevated rates of speciation is useful to more precisely define clades that underwent episodes of rapid cladogenesis in marine fish clades. Contrasting the patterns of phylogenetic diversification in marine fish lineages may facilitate the identification of common evolutionary trajectories versus idiosyncrasies, and ultimately help towards a better understanding of the factors and processes underlying speciation in the marine realm.     

Robust estimation for ordinary differential equation models

Applied scientists often like to use ordinary differential equations (ODEs) to model complex dynamic processes that arise in biology, engineering, medicine, and many other areas. It is interesting but challenging to estimate ODE parameters from noisy data, especially when the data have some outliers. We propose a robust method to address this problem. The dynamic process is represented with a nonparametric function, which is a linear combination of basis functions. The nonparametric function is estimated by a robust penalized smoothing method. The penalty term is defined with the parametric ODE model, which controls the roughness of the nonparametric function and maintains the fidelity of the nonparametric function to the ODE model. The basis coefficients and ODE parameters are estimated in two nested levels of optimization. The coefficient estimates are treated as an implicit function of ODE parameters, which enables one to derive the analytic gradients for optimization using the implicit function theorem. Simulation studies show that the robust method gives satisfactory estimates for the ODE parameters from noisy data with outliers. The robust method is demonstrated by estimating a predator-prey ODE model from real ecological data.     

Poisson Regression with Change‐Point Prior in the Modelling of Disease Risk around a Point Source

Bayesian estimation of the risk of a disease around a known point source of exposure is considered. The minimal requirements for data are that cases and populations at risk are known for a fixed set of concentric annuli around the point source, and each annulus has a uniquely defined distance from the source. The conventional Poisson likelihood is assumed for the counts of disease cases in each annular zone with zone‐specific relative risk and parameters and, conditional on the risks, the counts are considered to be independent. The prior for the relative risk parameters is assumed to be piecewise constant at the distance having a known number of components. This prior is the well‐known change‐point model. Monte Carlo sampling from the posterior results in zone‐specific posterior summaries, which can be applied for the calculation of a smooth curve describing the variation in disease risk as a function of the distance from the putative source. In addition, the posterior can be used in the calculation of posterior probabilities for interesting hypothesis. The suggested model is suitable for use in geographical information systems (GIS) aimed for monitoring disease risks. As an application, a case study on the incidence of lung cancer around a former asbestos mine in eastern Finland is presented. Further extensions of the model are discussed.     

Locally adaptive function estimation for binary regression models

In this paper we present a nonparametric Bayesian approach for fitting unsmooth or highly oscillating functions in regression models with binary responses. The approach extends previous work by Lang et al. for Gaussian responses. Nonlinear functions are modelled by first or second order random walk priors with locally varying variances or smoothing parameters. Estimation is fully Bayesian and uses latent utility representations of binary regression models for efficient block sampling from the full conditionals of nonlinear functions.     

Application Fields of the Empirical Regression - A Case Study

Empirical regression is defined as conditional expected value based on an estimation of a twodimensional density. It is a modelfree mathematical means for a first evaluation of measured data of an unknown stochastical relation between two quantities. The numerical procedures may be applied for calculation of the mean course of an unknown relation hidden in the measured data. Disregarding the statistical background an other aspect of application is the analyzing of time series, especially smoothing of time series and modelfree recording of the trend component in non-stationary time series. The calculated regression curve provides an objective basis for comparing of different measured courses as well as for a further evaluation, e. g. in respect of a suitable choice of an analytical expression. The possibility of interpolation and the smoothing properties of empirical regression give essential advantages for internal regression as one step in the process of model construction.