Continuous-time causal mediation analysis.

While causal mediation analysis has seen considerable recent development for a single measured mediator (M) and final outcome (Y), less attention has been given to repeatedly measured M and Y. Previous methods have typically involved discrete-time models that limit inference to the particular measurement times used and do not recognize the continuous nature of the mediation process over time. To overcome such limitations, we present a new continuous-time approach to causal mediation analysis that uses a differential equations model in a potential outcomes framework to describe the causal relationships among model variables over time. A connection between the differential equation models and standard repeated measures models is made to provide convenient model formulation and fitting. A continuous-time extension of the sequential ignorability assumption allows for identifiable natural direct and indirect effects as functions of time, with estimation based on a two-step approach to model fitting in conjunction with a continuous-time mediation formula. Novel features include a measure of an overall mediation effect based on the "area between the curves," and an approach for predicting the effects of new interventions. Simulation studies show good properties of estimators and the new methodology is applied to data from a cohort study to investigate sugary drink consumption as a mediator of the effect of socioeconomic status on dental caries in children.

Random local temporal structure of category fluency responses.

The Category Fluency Test (CFT) provides a sensitive measurement of cognitive capabilities in humans related to retrieval from semantic memory. In particular, it is widely used to assess progress of cognitive impairment in patients with dementia. Previous research shows that, in the first approximation, the intensity of tested individuals' responses within a standard 60-s test period decays exponentially with time, with faster decay rates for more cognitively impaired patients. Such decay rate can then be viewed as a global (macro) diagnostic parameter of each test. In the present paper we focus on the statistical properties of the properly de-trended time intervals between consecutive responses (inter-call times) in the Category Fluency Test. In a sense, those properties reflect the local (micro) structure of the response generation process. We find that a good approximation for the distribution of the de-trended inter-call times is provided by the Weibull Distribution, a probability distribution that appears naturally in this context as a distribution of a minimum of independent random quantities and is the standard tool in industrial reliability theory. This insight leads us to a new interpretation of the concept of "navigating a semantic space" via patient responses.

Lévy flights in evolutionary ecology.

We are interested in modeling Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation, and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a random mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. In the case we are interested in, the probability distribution of mutations has a heavy tail and belongs to the domain of attraction of a stable law and the corresponding diffusion admits jumps. This could be seen as an alternative to Gould and Eldredge's model of evolutionary punctuated equilibria. We investigate the large-population limit with allometric demographies: larger populations made up of smaller individuals which reproduce and die faster, as is typical for micro-organisms. We show that depending on the allometry coefficient the limit behavior of the population process can be approximated by nonlinear Lévy flights of different nature: either deterministic, in the form of non-local fractional reaction-diffusion equations, or stochastic, as nonlinear super-processes with the underlying reaction and a fractional diffusion operator. These approximation results demonstrate the existence of such non-trivial fractional objects; their uniqueness is also proved.

Detrending Changes the Temporal Dynamics of a Semantic Fluency Task.

Objective: To study the dynamics of clustering semantic fluency responses and switching between clusters. Methods: We conducted a cross-sectional study of participants (N = 60) in a study of patient reported outcomes who were given the Saint Louis University Mental Status test. Sixty-second animal naming tests were scored for the timing of responses as well as the clustering of responses into semantic categories. Time scores were detrended to correct for exponential exhaustion and normalize the time scale across individuals. Results: Grouped by number of responses given, low performers (LP; Carter et al., 2012) switched between clusters fewer times than medium performers (MP) and high performers (HP). Prior to detrending, LP showed increased intracluster response times when compared to the other groups, but no differences were shown in intercluster response times. After detrending, however, the difference in intracluster response times disappeared and LP showed significantly faster detrended intercluster response times compared to both MP and HP. Conclusion: Prior to detrending, slower intracluster response times appear to be driving poorer performance. When time scores are detrended, our findings suggest that LP participants have quicker intercluster response times but exhaust more quickly as well. Detrending can help describe the interplay between the structure-loss and retrieval-slowing models of declining semantic fluency by isolating the component mechanisms involved in each.

Optimal channel selection for analysis of EEG-sleep patterns of neonates.

This paper extends our previous work on automated detection and classification of neonate EEG sleep stages. In [19] we adapted and integrated a range of computational, mathematical and statistical tools for the analysis of neonatal electroencephalogram (EEG) sleep recordings with the aim of facilitating the assessment of neonatal brain maturation and dismaturity by studying the structure and temporal patterns of their sleep. That work relied on algorithms using a single channel of EEG. The present paper builds on our previous work by incorporating a larger selection of EEG channels that capture both the spatial distribution and temporal patterns of EEG during sleep. Using a multivariate analysis approach, we obtain the "optimal" selection of the EEG channels and characteristics that are most suitable for EEG sleep state separation.

Automated detection of neonate EEG sleep stages.

The paper integrates and adapts a range of advanced computational, mathematical and statistical tools for the purpose of analysis of neonate sleep stages based on extensive electroencephalogram (EEG) recordings. The level of brain dysmaturity of a neonate is difficult to assess by direct physical or cognitive examination, but dysmaturity is known to be directly related to the structure of neonatal sleep as reflected in the nonstationary time series produced by EEG signals which, importantly, can be collected trough a noninvasive procedure. In the past, the assessment of sleep EEG structure has often been done manually by experienced clinicians. The goal of this paper is to develop rigorous algorithmic tools for the same purpose by providing a formal scheme to separate different sleep stages corresponding to different stationary segments of the EEG signal based on statistical analysis of the spectral and nonlinear characteristics of the sleep EEG recordings. The methods developed in this paper can, potentially, be translated to other areas of biomedical research.