We study the asymptotic behaviour of penalized spline estimators in the univariate case. We use B-splines and a penalty is placed on mth-order differences of the coefficients. The number of knots is assumed to converge to infinity as the sample size increases. We show that penalized splines behave similarly to Nadaraya--Watson kernel estimators with ‘equivalent’ kernels depending upon m. The equivalent kernels we obtain for penalized splines are the same as those found by Silverman for smoothing splines. The asymptotic distribution of the penalized spline estimator is Gaussian and we give simple expressions for the asymptotic mean and variance. Provided that it is fast enough, the rate at which the number of knots converges to infinity does not affect the asymptotic distribution. The optimal rate of convergence of the penalty parameter is given. Penalized splines are not design-adaptive.

We introduce the functional generalized additive model (FGAM), a novel regression model for association studies between a scalar response and a functional predictor. We model the link-transformed mean response as the integral with respect to t of F{X(t), t} where F( ·, ·) is an unknown regression function and X(t) is a functional covariate. Rather than having an additive model in a finite number of principal components as by Müller and Yao ( 2008 Müller, H.G. , and Yao, F. (2008), “Functional Additive Models,” Journal of the American Statistical Association, 103, 1534–1544. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] ), our model incorporates the functional predictor directly and thus our model can be viewed as the natural functional extension of generalized additive models. We estimate F( ·, ·) using tensor-product B-splines with roughness penalties. A pointwise quantile transformation of the functional predictor is also considered to ensure each tensor-product B-spline has observed data on its support. The methods are evaluated using simulated data and their predictive performance is compared with other competing scalar-on-function regression alternatives. We illustrate the usefulness of our approach through an application to brain tractography, where X(t) is a signal from diffusion tensor imaging at position, t, along a tract in the brain. In one example, the response is disease-status (case or control) and in a second example, it is the score on a cognitive test. The FGAM is implemented in R in the refund package. There are additional supplementary materials available online.

In high dimensions, the classical Hotelling’s T² test tends to have low power or becomes undefined due to singularity of the sample covariance matrix. In this article, this problem is overcome by projecting the data matrix onto lower dimensional subspaces through multiplication by random matrices. We propose RAPTT (RAndom Projection T²-Test), an exact test for equality of means of two normal populations based on projected lower dimensional data. RAPTT does not require any constraints on the dimension of the data or the sample size. A simulation study indicates that in high dimensions the power of this test is often greater than that of competing tests. The advantages of RAPTT are illustrated on a high-dimensional gene expression dataset involving the discrimination of tumor and normal colon tissues.

We consider statistical inference for additive partial linear models when the linear covariate is measured with error. We propose attenuation-to-correction and simulation-extrapolation, simex, estimators of the parameter of interest. It is shown that the first resulting estimator is asymptotically normal and requires no undersmoothing. This is an advantage of our estimator over existing backfitting-based estimators for semiparametric additive models which require undersmoothing of the nonparametric component in order for the estimator of the parametric component to be root-n consistent. This feature stems from a decrease of the bias of the resulting estimator, which is appropriately derived using a profile procedure. A similar characteristic in semiparametric partially linear models was obtained by Wang et al. (2005). We also discuss the asymptotics of the proposed simex approach. Finite-sample performance of the proposed estimators is assessed by simulation experiments. The proposed methods are applied to a dataset from a semen study.

Myalgic Encephalomyelitis/Chronic Fatigue Syndrome (ME/CFS) is a disease of enigmatic origin with no established cure. Its constellation of symptoms has silently ruined the lives of millions of people around the world. A plethora of hypotheses have been vainly investigated over the past few decades, so that the biological basis of this debilitating condition remains a mystery. In this study, we investigate whether there is a disturbance in homeostasis of metabolic networks in the plasma of a female 32-patient cohort compared to 19 healthy female controls. Extensive analysis of the 832-metabolite dataset generated by Metabolon^®, covering eight biological classes, generated important insight into metabolic disruptions that occur in ME/CFS. We report on 14 metabolites with differences in abundance, allowing us to develop a theory of broad redox imbalance in ME/CFS patients, which is consistent with findings of prior work in the ME/CFS field. Moreover, exploration of enrichment analysis using www.MetaboAnalyst.ca provides information concerning similarities between metabolite disruptions in ME/CFS and those that occur in other diseases, while its biomarker analysis unit yielded prospective plasma biomarkers for ME/CFS. This work contributes key elements to the development of ME/CFS diagnostics, a crucial step required for discovering a therapy for any disease of unknown origin.