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.     

Modeling and testing treated tumor growth using cubic smoothing splines

Human tumor xenograft models are often used in preclinical study to evaluate the therapeutic efficacy of a certain compound or a combination of certain compounds. In a typical human tumor xenograft model, human carcinoma cells are implanted to subjects such as severe combined immunodeficient (SCID) mice. Treatment with test compounds is initiated after tumor nodule has appeared, and continued for a certain time period. Tumor volumes are measured over the duration of the experiment. It is well known that untreated tumor growth may follow certain patterns, which can be described by certain mathematical models. However, the growth patterns of the treated tumors with multiple treatment episodes are quite complex, and the usage of parametric models is limited. We propose using cubic smoothing splines to describe tumor growth for each treatment group and for each subject, respectively. The proposed smoothing splines are quite flexible in modeling different growth patterns. In addition, using this procedure, we can obtain tumor growth and growth rate over time for each treatment group and for each subject, and examine whether tumor growth follows certain growth pattern. To examine the overall treatment effect and group differences, the scaled chi-squared test statistics based on the fitted group-level growth curves are proposed. A case study is provided to illustrate the application of this method, and simulations are carried out to examine the performances of the scaled chi-squared tests.     

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.     

Time-varying-response smoothing

Cubic smoothing splines with a fixed-period response are used widely in producing “expected” growth curves for ring-width and density data in dendroclimatology. A simple modification to the procedure which generates these splines enables the use of a smoothing spline with a user-specified, time-varying flexibility and hence time-varying-response characteristics. The revised procedure is presented here, along with different examples of its application in the context of Regional Curve Standardisation (RCS). The ability to generate a smoothing spline with time-dependent flexibility may have wider application in tree-ring studies.     

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.     

Penalized partial likelihood for frailties and smoothing splines in time to first insemination models for dairy cows

In many epidemiological studies time to event data are clustered and the physiological relationship between (time-dependent) covariates and the log hazard is often not linear as assumed in the Cox model. Introducing frailties in the Cox model can account for the clustering of the data and smoothing splines can be used to describe nonlinear relations. These two extensions of the Cox model are introduced jointly and it is shown how penalized partial likelihood techniques can be used to fit the extended model. We demonstrate the need for such a model to study the relation between the physiological covariates milk ureum and protein concentration and the log hazard of first insemination in dairy cows, with the farms as clusters.     

Spatially constrained adaptive rewiring in cortical networks creates spatially modular small world architectures

A modular small-world topology in functional and anatomical networks of the cortex is eminently suitable as an information processing architecture. This structure was shown in model studies to arise adaptively; it emerges through rewiring of network connections according to patterns of synchrony in ongoing oscillatory neural activity. However, in order to improve the applicability of such models to the cortex, spatial characteristics of cortical connectivity need to be respected, which were previously neglected. For this purpose we consider networks endowed with a metric by embedding them into a physical space. We provide an adaptive rewiring model with a spatial distance function and a corresponding spatially local rewiring bias. The spatially constrained adaptive rewiring principle is able to steer the evolving network topology to small world status, even more consistently so than without spatial constraints. Locally biased adaptive rewiring results in a spatial layout of the connectivity structure, in which topologically segregated modules correspond to spatially segregated regions, and these regions are linked by long-range connections. The principle of locally biased adaptive rewiring, thus, may explain both the topological connectivity structure and spatial distribution of connections between neuronal units in a large-scale cortical architecture.