Statistics

There is a growing interest in leveraging the prevalence of mobile technology to improve health by delivering momentary, contextualized interventions to individuals' smartphones. A just-in-time adaptive intervention (JITAI) adjusts to an individual's changing state and/or context to provide the right treatment, at the right time, in the right place. Micro-randomized trials (MRTs) allow for the collection of data which aid in the construction of an optimized JITAI by sequentially randomizing participants to different treatment options at each of many decision points throughout the study. Often, this data is collected passively using a mobile phone. To assess the causal effect of treatment on a near-term outcome, care must be taken when designing the data collection system to ensure it is of appropriately high quality. Here, we make several recommendations for collecting and managing data from an MRT. We provide advice on selecting which features to collect and when, choosing between "agents" to implement randomization, identifying sources of missing data, and overcoming other novel challenges. The recommendations are informed by our experience with HeartSteps, an MRT designed to test the effects of an intervention aimed at increasing physical activity in sedentary adults. We also provide a checklist which can be used in designing a data collection system so that scientists can focus more on their questions of interest, and less on cleaning data.

This paper introduces pragmatic hypotheses and relates this concept to the spiral of scientific evolution. Previous works determined a characterization of logically consistent statistical hypothesis tests and showed that the modal operators obtained from this test can be represented in the hexagon of oppositions. However, despite the importance of precise hypothesis in science, they cannot be accepted by logically consistent tests. Here, we show that this dilemma can be overcome by the use of pragmatic versions of precise hypotheses. These pragmatic versions allow a level of imprecision in the hypothesis that is small relative to other experimental conditions. The introduction of pragmatic hypotheses allows the evolution of scientific theories based on statistical hypothesis testing to be interpreted using the narratological structure of hexagonal spirals, as defined by Pierre Gallais.

Examination of precinct level data in US presidential elections reveals a correlation of large precincts and increased fraction of Republican votes. The large precinct bias is analyzed with respect to voter heterogeneity and voter inconvenience as precinct size increases. The analysis shows that voter inconvenience is a significant factor in election outcomes in certain states, and may significantly disadvantage Democratic candidates.

Product risk assessment is the overall process of determining whether a product, which could be anything from a type of washing machine to a type of teddy bear, is judged safe for consumers to use. There are several methods used for product risk assessment, including RAPEX, which is the primary method used by regulators in the UK and EU. However, despite its widespread use, we identify several limitations of RAPEX including a limited approach to handling uncertainty and the inability to incorporate causal explanations for using and interpreting test data. In contrast, Bayesian Networks (BNs) are a rigorous, normative method for modelling uncertainty and causality which are already used for risk assessment in domains such as medicine and finance, as well as critical systems generally. This article proposes a BN model that provides an improved systematic method for product risk assessment that resolves the identified limitations with RAPEX. We use our proposed method to demonstrate risk assessments for a teddy bear and a new uncertified kettle for which there is no testing data and the number of product instances is unknown. We show that, while we can replicate the results of the RAPEX method, the BN approach is more powerful and flexible.

Triangular distributions are a well-known class of distributions that are often used as an elementary example of a probability model. Maximum likelihood estimation of the mode parameter of the triangular distribution over the unit interval can be performed via an order statistics-based method. It had been conjectured that such a method can be conducted using only a constant number of likelihood function evaluations, on average, as the sample size becomes large. We prove two theorems that validate this conjecture. Graphical and numerical results are presented to supplement our proofs.

A data matrix may be seen simply as a means of organizing observations into rows ( e.g., by measured object) and into columns ( e.g., by measured variable) so that the observations can be analyzed with mathematical tools. As a mathematical object, a matrix defines a linear mapping between points representing weighted combinations of its rows (the row vector space) and points representing weighted combinations of its columns (the column vector space). From this perspective, a data matrix defines a relationship between the information that labels its rows and the information that labels its columns, and numerical methods are used to analyze this relationship. A first step is to normalize the data, transforming each observation from scales convenient for measurement to a common scale, on which addition and multiplication can meaningfully combine the different observations. For example, z-transformation rescales every variable to the same scale, standardized variation from an expected value, but ignores scale differences between measured objects. Here we develop the concepts and properties of projective decomposition, which applies the same normalization strategy to both rows and columns by separating the matrix into row- and column-scaling factors and a scale-normalized matrix. We show that different scalings of the same scale-normalized matrix form an equivalence class, and call the scale-normalized, canonical member of the class its scale-invariant form that preserves all pairwise relative ratios. Projective decomposition therefore provides a means of normalizing the broad class of ratio-scale data, in which relative ratios are of primary interest, onto a common scale without altering the ratios of interest, and simultaneously accounting for scale effects for both organizations of the matrix values. Both of these properties distinguish it from z-transformation.

In many risk analyses the results are only given as mean values and often the input data are also mean values. However the required accuracy of the result is often an interval of values e. g. for the derivation of a Safety Integrity Level (SIL). In this paper we reason what should be the accuracy of the input data of risk analyses if a particular certainty of the result is demanded. Also the backside of the coin, the SIL composition is discussed. The results show that common methods for risk analysis are faulty and that SIL allocation by a kind of SIL calculus seems infeasible without additional requirements on the composed components. A justification of a common practice for parameter scaling in well-constructed semi-quantitative risk analysis is also provided.

For linear and Gaussian state space models parametrized by θ 0 ∈Θ⊂ R r ,r≥1 corresponding to the vector of parameters of the model, the Kalman filter gives exactly the solution for the optimal filtering under weak assumptions. This result supposes that θ 0 is perfectly known. In most real applications, this assumption is not realistic since θ 0 is unknown and has to be estimated. In this paper, we analysis the Kalman filter for a biased estimator of θ 0 . We show the propagation of this bias on the estimation of the hidden state. We give an expression of this propagation for linear and Gaussian state space models and we extend this result for almost linear models estimated by the Extended Kalman filter. An illustration is given for the autoregressive process with measurement noises widely studied in econometrics to model economic and financial data.

In this paper, we explore the relationships that individuals have with their communities. This work was prepared as part of the ASA Data Expo '13 sponsored by the Graphics Section and the Computing Section, using data provided by the Knight Foundation Soul of the Community survey. The Knight Foundation in cooperation with Gallup surveyed 43,000 people over three years in 26 communities across the United States with the intention of understanding the association between community attributes and the degree of attachment people feel towards their community. These include the different facets of both urban and rural communities, the impact of quality education, and the trend in the perceived economic conditions of a community over time. The goal of our work is to facilitate understanding of why people feel attachment to their communities through the use of an interactive and web-based visualization. We will explain the development and use of web-based interactive graphics, including an overview of the R package Shiny and the JavaScript library D3, focusing on the choices made in producing the visualizations and technical aspects of how they were created. Then we describe the stories about community attachment that unfolded from our analysis.

International verification of nuclear warheads is a practical problem in which the protection of secret warhead information is of paramount importance. We propose a measure that would enable a weapon owner to evaluate the privacy of a proposed protocol in a technology-neutral fashion. We show the problem is reducible to `natural' and `corrective' learning. The natural learning can be computed without assumptions about the inspector, while the corrective learning accounts for the inspector's prior knowledge. The natural learning provides the warhead owner a useful lower bound on the information leaked by the proposed protocol. Using numerical examples, we demonstrate that the proposed measure correlates better with the accuracy of a maximum a posteriori probability estimate than alternative measures.