Thaddeus Tarpey

Clustering Functional Data

The problem of clustering functional data is addressed. Results on principal points (cluster means for probability distributions) are given for functional Gaussian distributions. Examples and simulations are provided to illustrate results.

Linear Transformations and the k-Means Clustering Algorithm: Applications to Clustering Curves

Functional data can be clustered by plugging estimated regression coefficients from individual curves into the k-means algorithm. Clustering results can differ depending on how the curves are fit to the data. Estimating curves using different sets of basis functions corresponds to different linear transformations of the data. k-means clustering is not invariant to linear transformations of the data. The optimal linear transformation for clustering will stretch the distribution so that the primary direction of variability aligns with actual differences in the clusters. It is shown that clustering the raw data will often give results similar to clustering regression coefficients obtained using an orthogonal design matrix. Clustering functional data using an L2 metric on function space can be achieved by clustering a suitable linear transformation of the regression coefficients. An example where depressed individuals are treated with an antidepressant is used for illustration.

Self-Consistency and Principal Component Analysis

Abstract I examine the self-consistency of a principal component axis; that is, when a distribution is centered about a principal component axis. A principal component axis of a random vector X is self-consistent if each point on the axis corresponds to the mean of X given that X projects orthogonally onto that point. A large class of symmetric multivariate distributions are examined in terms of self-consistency of principal component subspaces. Elliptical distributions are characterized by the preservation of self-consistency of principal component axes after arbitrary linear transformations. A “lack-of-fit” test is proposed that tests for self-consistency of a principal axis. The test is applied to two real datasets.

A Note on the Prediction Sum of Squares Statistic for Restricted Least Squares

Abstract There is a well-known simple formula for computing prediction sum of squares (PRESS) residuals in a regression problem without having to refit the curve for each observation. This note shows that the same basic result holds for fitting a regression function when the regression coefficients are subject to linear constraints.

Profiling Placebo Responders by Self-Consistent Partitioning of Functional Data

Identification of placebo responders among subjects treated with active drug has significant clinical and research implications. In clinical practice, when a patient treated with medication improves, this improvement may be attributed to the chemical component of the drug itself, a “placebo effect,” or some combination of these. Determining the proper subsequent treatment and maintenance of the patient may be greatly aided by understanding the mechanism of patient improvement. In a research context, classification of patient response has bearing on how efficacy and effectiveness clinical trials are designed and conducted. This article presents a framework for studying placebo response in diverse areas of medicine. To identify placebo responders among drug-treated patients, a profile of the clinical status over time (outcome profile) is estimated for each subject. Self-consistent partitioning techniques are used to group subjects based on the amount of curvature in the profile as well as the overall trend in the profile. The resulting partitions determine representative profiles for subjects in the drug group that subsequently can be used to classify patients. The proposed method is applied to data from a clinical trial for treatment of depression involving placebo and the active drug phenelzine. Data from the placebo arm of the study is used to help validate the procedure, because the drug-treated and placebo-treated subjects should share common profiles.

Massively Parallel Nonparametric Regression, With an Application to Developmental Brain Mapping

A penalized approach is proposed for performing large numbers of parallel nonparametric analyses of either of two types: restricted likelihood ratio tests of a parametric regression model versus a general smooth alternative, and nonparametric regression. Compared with naïvely performing each analysis in turn, our techniques reduce computation time dramatically. Viewing the large collection of scatterplot smooths produced by our methods as functional data, we develop a clustering approach to summarize and visualize these results. Our approach is applicable to ultra-high-dimensional data, particularly data acquired by neuroimaging; we illustrate it with an analysis of developmental trajectories of functional connectivity at each of approximately 70,000 brain locations. Supplementary materials, including an appendix and an R package, are available online.

Latent Regression Analysis

Finite mixture models have come to play a very prominent role in modelling data. The finite mixture model is predicated on the assumption that distinct latent groups exist in the population. The finite mixture model therefore is based on a categorical latent variable that distinguishes the different groups. Often in practice, distinct sub-populations do not actually exist. For example, disease severity (e.g., depression) may vary continuously and therefore, a distinction of diseased and non-diseased may not be based on the existence of distinct sub-populations. Thus, what is needed is a generalization of the finite mixture’s discrete latent predictor to a continuous latent predictor. We cast the finite mixture model as a regression model with a latent Bernoulli predictor. A latent regression model is proposed by replacing the discrete Bernoulli predictor by a continuous latent predictor with a beta distribution. Motivation for the latent regression model arises from applications where distinct latent classes do not exist, but instead individuals vary according to a continuous latent variable. The shapes of the beta density are very flexible and can approximate the discrete Bernoulli distribution. Examples and a simulation are provided to illustrate the latent regression model. In particular, the latent regression model is used to model placebo effect among drug-treated subjects in a depression study.

Interpreting Meta-Regression: Application to Recent Controversies in Antidepressants' Efficacy

A recent meta-regression of antidepressant efficacy on baseline depression severity has caused considerable controversy in the popular media. A central source of the controversy is a lack of clarity about the relation of meta-regression parameters to corresponding parameters in models for subject-level data. This paper focuses on a linear regression with continuous outcome and predictor, a case that is often considered less problematic. We frame meta-regression in a general mixture setting that encompasses both finite and infinite mixture models. In many applications of meta-analysis, the goal is to evaluate the efficacy of a treatment from several studies, and authors use meta-regression on grouped data to explain variations in the treatment efficacy by study features. When the study feature is a characteristic that has been averaged over subjects, it is difficult not to interpret the meta-regression results on a subject level, a practice that is still widespread in medical research. Although much of the attention in the literature is on methods of estimating meta-regression model parameters, our results illustrate that estimation methods cannot protect against erroneous interpretations of meta-regression on grouped data. We derive relations between meta-regression parameters and within-study model parameters and show that the conditions under which slopes from these models are equal cannot be verified on the basis of group-level information only. The effects of these model violations cannot be known without subject-level data. We conclude that interpretations of meta-regression results are highly problematic when the predictor is a subject-level characteristic that has been averaged over study subjects.

Oviposition Preferences of Agrilus planipennis (Coleoptera: Buprestidae) for Different Ash Species Support the Mother Knows Best Hypothesis

ABSTRACT The “mother knows best” hypothesis states that adults should choose hosts for oviposition on which their offspring will best perform, maximizing their own fitness. It has been hypothesized that this preference—performance relationship for wood-boring insects is especially important because larvae are not able to switch hosts, although no study has examined oviposition choices for these insects. We examined oviposition preferences of the emerald ash borer, Agrilus planipennis Fairmaire (Coleoptera: Buprestidae), in two common gardens, one on the campus of Wright State University in Dayton, OH, and the other at the Michigan State University Tollgate Research Farm in Novi, MI, by wrapping cheesecloth around ash trunks to assess passive oviposition patterns.Wefound that in both gardens, ash species native to North America, which are highly susceptible to the emerald ash borer, consistently received more ova than Manchurian ash, which is indigenous to Asia and more resistant to the emerald ash borer. Susceptible trees in the Novi garden received 93 times the number of ova and susceptible trees at the Wright State garden received up to 25 times the number of ova that were received by Manchurian ash in each of their respective gardens. Neither tree size nor vigor affected oviposition choice. There were also higher numbers of adult exit holes on North American than Manchurian ash in both common gardens. The observed oviposition preferences in this study align with patterns of adult feeding preference, ash host mortality, and exit hole numbers from other studies. These observations also suggest that oviposition preferences may contribute to interspecific patterns of host resistance and mortality. Collectively, our results demonstrate that the emerald ash borer prefers to oviposit on species on which its offspring will best perform, suggesting that there is strong selection for the ability to recognize host cues that predict better larval survival and performance.

The Effects of Antipsychotic Medication on Factor and Cluster Structure of Neurologic Examination Abnormalities in Schizophrenia

This study extends a previous study of the factor structure of the neurologic examination in unmedicated schizophrenia, utilizing cluster analysis and adding a medicated condition. We administered a modified version of the Neurologic Evaluation Scale (NES) on two occasions to 80 patients with schizophrenia or schizoaffective disorder, once while on antipsychotic medications and once while off medication. Data were distilled by combining right- and left-side scores, and by excluding rarely abnormal and unreliable items from the analysis. Principal components analysis yielded an intuitive four-factor solution in the unmedicated condition, but an inscrutable five-factor solution during medication. Cluster analysis revealed three groups: normal, cognitively impaired, and diffusely impaired. These results were also less interpretable with data from the medicated condition. Neurologic performance was better in the medicated than in the unmedicated condition. As is the case with other domains of symptoms and performance in schizophrenia, relationships among neurologic exam variables are altered by the presence of antipsychotic medication.