A generalization of the Leibniz rule for derivatives

The Leibniz rule of derivatives gives the n-th derivative of a product of two functions. We extend the rule to products of m functions using a combinatorial analysis of labelled balls in boxes

Classification of incomplete feature vectors by radial basis function networks

The paper describes the use of radial basis function neural networks with Gaussian basis functions to classify incomplete feature vectors. The method uses the fact that any marginal distribution of a Gaussian distribution can be determined from the mean vector and covariance matrix of the joint distribution

Neural computation in medicine: Perspectives and prospects

In 1998, over 400 papers on artificial neural networks (ANNs) were published in the context of medicine, but why is there this interest in ANNs? And how do ANNs compare with traditional statistical methods? We propose some answers to these questions, and go on to consider the ‘black box’ issue. Finally, we briefly look at two directions in which ANNs are likely to develop, namely the use of Bayesian statistics and knowledge data fusion

Prediction regions for the visualization of incomplete datasets

A complication in the visualization of biomedical datasets is that they are often incomplete. A response to this is to multiply impute each missing datum prior to visualization in order to convey the uncertainty of the imputations. In our approach, the initially complete cases in a real-valued dataset are represented as points in a principal components plot and, for each initially incomplete case in the dataset, we use an associated prediction region or interval displayed on the same plot to indicate the probable location of the case. When a case has only one missing datum, a prediction interval is used in place of a region. The prediction region or interval associated with an incomplete case is determined from the dispersion of the multiple imputations of the case mapped onto the plot. We illustrate this approach with two incomplete datasets: the first is based on two multivariate normal distributions; the second on a published, simulated health survey

Confidence Intervals and Prediction Intervals for Feed-Forward Neural Networks

The chapter opens with an introduction to regression and its implementation within the maximum-likelihood framework. This is followed by a general introduction to classical confidence intervals and prediction intervals. We set the scene by first considering confidence and prediction intervals based on univariate samples, and then we progress to regarding these intervals in the context of linear regression and logistic regression. Since a feed-forward neural network is a type of regression model, the concepts of confidence and prediction intervals are applicable to these networks, and we look at several techniques for doing this via maximum-likelihood estimation. An alternative to the maximum-likelihood framework is Bayesian statistics, and we examine the notions of Bayesian confidence and predictions intervals as applied to feed-forward networks. This includes a critique on Bayesian confidence intervals and classification

Robust outcome prediction for intensive care patients

Missing data are a major plague of medical databases in general, and of Intensive Care Units databases in particular. The time pressure of work in an Intensive Care Unit pushes the physicians to omit randomly or selectively record data. These different omission strategies give rise to different patterns of missing data and the recommended approach of completing the database using median imputation and fitting a logistic regression model can lead to significant biases. This paper applies a new classification method, called robust Bayes classifier, that does not rely on any particular assumption about the pattern of missing data and compares it to the traditional median imputation approach using a database of 324 Intensive Care Unit patients

Highly epidemic strains of methicillin-resistant Staphylococcus aureus (MRSA) do not differ from other MRSA or methicillin-sensitive strains in capsule formation, Protein A content or adherence to HEp-2 cells

During the 1990s, two strains of epidemic methicillin-resistant Staphylococcus aureus, designated ‘phage types EMRSA-15 and EMRSA-16, have emerged as significant hospital pathogens. They have resisted standard methods of control and spread widely amongst in the UK, often becoming endemic, while the incidence of other epidemic types of MRSA has either declined or not changed. This suggests that EMRSA-15 and EMRSA-16 possess special properties that favour their dissemination and survival. In order to investigate this hypothesis, a study was undertaken that examined methicillin-sensitive and methicillinresistant strains of Staphylococcus aureus, including EMRSA types 1, 2, 3, 15 and 16, for capsule formation, the amount of bound protein A produced, and quantitative adherence to the human continuous epithelial cell line HEp-2. Although all these properties varied amongst the strains examined, there was no relationship between any of them and methicillin resistance or epidemic type, and, incidentally, no relationship between cell-wall bound protein A content and adherence

Nested sampling for Bayesian model comparison in the context of Salmonella disease dynamics.

Understanding the mechanisms underlying the observed dynamics of complex biological systems requires the statistical assessment and comparison of multiple alternative models. Although this has traditionally been done using maximum likelihood-based methods such as Akaike's Information Criterion (AIC), Bayesian methods have gained in popularity because they provide more informative output in the form of posterior probability distributions. However, comparison between multiple models in a Bayesian framework is made difficult by the computational cost of numerical integration over large parameter spaces. A new, efficient method for the computation of posterior probabilities has recently been proposed and applied to complex problems from the physical sciences. Here we demonstrate how nested sampling can be used for inference and model comparison in biological sciences. We present a reanalysis of data from experimental infection of mice with Salmonella enterica showing the distribution of bacteria in liver cells. In addition to confirming the main finding of the original analysis, which relied on AIC, our approach provides: (a) integration across the parameter space, (b) estimation of the posterior parameter distributions (with visualisations of parameter correlations), and (c) estimation of the posterior predictive distributions for goodness-of-fit assessments of the models. The goodness-of-fit results suggest that alternative mechanistic models and a relaxation of the quasi-stationary assumption should be considered.RD was funded by the Biotechnology and Biological Sciences Research Council (BBSRC) (grant number BB/I002189/1). TJM was funded by the Biotechnology and Biological Sciences Research Council (BBSRC) (grant number BB/I012192/1). OR was funded by the Royal Society.This paper was originally published in PLOS ONE (Dybowski R, McKinley TJ, Mastroeni P, Restif O, PLoS ONE 2013, 8(12): e82317. doi:10.1371/journal.pone.0082317)

Single passage in mouse organs enhances the survival and spread of Salmonella enterica.

Intravenous inoculation of Salmonella enterica serovar Typhimurium into mice is a prime experimental model of invasive salmonellosis. The use of wild-type isogenic tagged strains (WITS) in this system has revealed that bacteria undergo independent bottlenecks in the liver and spleen before establishing a systemic infection. We recently showed that those bacteria that survived the bottleneck exhibited enhanced growth when transferred to naive mice. In this study, we set out to disentangle the components of this in vivo adaptation by inoculating mice with WITS grown either in vitro or in vivo. We developed an original method to estimate the replication and killing rates of bacteria from experimental data, which involved solving the probability-generating function of a non-homogeneous birth-death-immigration process. This revealed a low initial mortality in bacteria obtained from a donor animal. Next, an analysis of WITS distributions in the livers and spleens of recipient animals indicated that in vivo-passaged bacteria started spreading between organs earlier than in vitro-grown bacteria. These results further our understanding of the influence of passage in a host on the fitness and virulence of Salmonella enterica and represent an advance in the power of investigation on the patterns and mechanisms of host-pathogen interactions.This work was funded by a Medical Research Council (MRC) grant (G0801161) awarded to AJG, PM and DJM. RD was supported by BBSRC grant BB/I002189/1 awarded to PM. OR is supported by a University Research Fellowship from the Royal Society.This is the final version of the article. It was first available from Royal Society Publishing via http://dx.doi.org/10.1098/rsif.2015.070

An efficient moments-based inference method for within-host bacterial infection dynamics.

Over the last ten years, isogenic tagging (IT) has revolutionised the study of bacterial infection dynamics in laboratory animal models. However, quantitative analysis of IT data has been hindered by the piecemeal development of relevant statistical models. The most promising approach relies on stochastic Markovian models of bacterial population dynamics within and among organs. Here we present an efficient numerical method to fit such stochastic dynamic models to in vivo experimental IT data. A common approach to statistical inference with stochastic dynamic models relies on producing large numbers of simulations, but this remains a slow and inefficient method for all but simple problems, especially when tracking bacteria in multiple locations simultaneously. Instead, we derive and solve the systems of ordinary differential equations for the two lower-order moments of the stochastic variables (mean, variance and covariance). For any given model structure, and assuming linear dynamic rates, we demonstrate how the model parameters can be efficiently and accurately estimated by divergence minimisation. We then apply our method to an experimental dataset and compare the estimates and goodness-of-fit to those obtained by maximum likelihood estimation. While both sets of parameter estimates had overlapping confidence regions, the new method produced lower values for the division and death rates of bacteria: these improved the goodness-of-fit at the second time point at the expense of that of the first time point. This flexible framework can easily be applied to a range of experimental systems. Its computational efficiency paves the way for model comparison and optimal experimental design.Biotechnology and Biological Sciences Research Council grant BB/M020193/1 awarded to OR, and to support DJP. Biotechnology and Biological Sciences Research Council grant BB/I002189/1 awarded to PM, and to support RD