The Friedman rank sum test is a widely-used nonparametric method in computational biology. In addition to examining the overall null hypothesis of no significant difference among any of the rank sums, it is typically of interest to conduct pairwise comparison tests. Current approaches to such tests rely on large-sample approximations, due to the numerical complexity of computing the exact distribution. These approximate methods lead to inaccurate estimates in the tail of the distribution, which is most relevant for p-value calculation. We propose an efficient, combinatorial exact approach for calculating the probability mass distribution of the rank sum difference statistic for pairwise comparison of Friedman rank sums, and compare exact results with recommended asymptotic approximations. Whereas the chi-squared approximation performs inferiorly to exact computation overall, others, particularly the normal, perform well, except for the extreme tail. Hence exact calculation offers an improvement when small p-values occur following multiple testing correction. Exact inference also enhances the identification of significant differences whenever the observed values are close to the approximate critical value. We illustrate the proposed method in the context of biological machine learning, were Friedman rank sum difference tests are commonly used for the comparison of classifiers over multiple datasets. We provide a computationally fast method to determine the exact p-value of the absolute rank sum difference of a pair of Friedman rank sums, making asymptotic tests obsolete. Calculation of exact p-values is easy to implement in statistical software and the implementation in R is provided in one of the Additional files and is also available at http://www.ru.nl/publish/pages/726696/friedmanrsd.zip .

Computational modeling plays an important role in modern neuroscience research. Much previous research has relied on statistical methods, separately, to address two problems that are actually interdependent. First, given a particular computational model, Bayesian hierarchical techniques have been used to estimate individual variation in parameters over a population of subjects, leveraging their population-level distributions. Second, candidate models are themselves compared, and individual variation in the expressed model estimated, according to the fits of the models to each subject. The interdependence between these two problems arises because the relevant population for estimating parameters of a model depends on which other subjects express the model. Here, we propose a hierarchical Bayesian inference (HBI) framework for concurrent model comparison, parameter estimation and inference at the population level, combining previous approaches. We show that this framework has important advantages for both parameter estimation and model comparison theoretically and experimentally. The parameters estimated by the HBI show smaller errors compared to other methods. Model comparison by HBI is robust against outliers and is not biased towards overly simplistic models. Furthermore, the fully Bayesian approach of our theory enables researchers to make inference on group-level parameters by performing HBI t-test. Computational modeling of brain and behavior plays an important role in modern neuroscience research. By deconstructing mechanisms of behavior and quantifying parameters of interest, computational modeling helps researchers to study brain-behavior mechanisms. In neuroscience studies, a dataset includes a number of samples, and often the question of interest is to characterize parameters of interest in a population: Do patients with attention-deficit hyperactive disorders exhibit lower learning rate than the general population? Do cognitive enhancers, such as Ritalin, enhance parameters influencing decision making? The success of these efforts heavily depends on statistical methods making inference about validity and robustness of estimated parameters, as well as generalizability of computational models. In this work, we present a novel method, hierarchical Bayesian inference, for concurrent model comparison, parameter estimation and inference at the population level. We show, both theoretically and experimentally, that our approach has important advantages over previous methods. The proposed method has implications for computational modeling research in group studies across many areas of psychology, neuroscience, and psychiatry.

High-throughput techniques allow for massive screening of drug combinations. To find combinations that exhibit an interaction effect, one filters for promising compound combinations by comparing to a response without interaction. A common principle for no interaction is Loewe Additivity which is based on the assumption that no compound interacts with itself and that two doses from different compounds having the same effect are equivalent. It then should not matter whether a component is replaced by the other or vice versa. We call this assumption the Loewe Additivity Consistency Condition (LACC). We derive explicit and implicit null reference models from the Loewe Additivity principle that are equivalent when the LACC holds. Of these two formulations, the implicit formulation is the known General Isobole Equation (Loewe, 1928), whereas the explicit one is the novel contribution. The LACC is violated in a significant number of cases. In this scenario the models make different predictions. We analyze two data sets of drug screening that are non-interactive (Cokol et al., 2011; Yadav et al., 2015) and show that the LACC is mostly violated and Loewe Additivity not defined. Further, we compare the measurements of the non-interactive cases of both data sets to the theoretical null reference models in terms of bias and mean squared error. We demonstrate that the explicit formulation of the null reference model leads to smaller mean squared errors than the implicit one and is much faster to compute.

Background: Wearable sensors have been used successfully to characterize bradykinetic gait in patients with Parkinson disease (PD), but most studies to date have been conducted in highly controlled laboratory environments. Objective: This paper aims to assess whether sensor-based analysis of real-life gait can be used to objectively and remotely monitor motor fluctuations in PD. Methods: The Parkinson@Home validation study provides a new reference data set for the development of digital biomarkers to monitor persons with PD in daily life. Specifically, a group of 25 patients with PD with motor fluctuations and 25 age-matched controls performed unscripted daily activities in and around their homes for at least one hour while being recorded on video. Patients with PD did this twice: once after overnight withdrawal of dopaminergic medication and again 1 hour after medication intake. Participants wore sensors on both wrists and ankles, on the lower back, and in the front pants pocket, capturing movement and contextual data. Gait segments of 25 seconds were extracted from accelerometer signals based on manual video annotations. The power spectral density of each segment and device was estimated using Welch’s method, from which the total power in the 0.5- to 10-Hz band, width of the dominant frequency, and cadence were derived. The ability to discriminate between before and after medication intake and between patients with PD and controls was evaluated using leave-one-subject-out nested cross-validation. Results: From 18 patients with PD (11 men; median age 65 years) and 24 controls (13 men; median age 68 years), ≥10 gait segments were available. Using logistic LASSO (least absolute shrinkage and selection operator) regression, we classified whether the unscripted gait segments occurred before or after medication intake, with mean area under the receiver operator curves (AUCs) varying between 0.70 (ankle of least affected side, 95% CI 0.60-0.81) and 0.82 (ankle of most affected side, 95% CI 0.72-0.92) across sensor locations. Combining all sensor locations did not significantly improve classification (AUC 0.84, 95% CI 0.75-0.93). Of all signal properties, the total power in the 0.5- to 10-Hz band was most responsive to dopaminergic medication. Discriminating between patients with PD and controls was generally more difficult (AUC of all sensor locations combined: 0.76, 95% CI 0.62-0.90). The video recordings revealed that the positioning of the hands during real-life gait had a substantial impact on the power spectral density of both the wrist and pants pocket sensor. Conclusions: We present a new video-referenced data set that includes unscripted activities in and around the participants’ homes. Using this data set, we show the feasibility of using sensor-based analysis of real-life gait to monitor motor fluctuations with a single sensor location. Future work may assess the value of contextual sensors to control for real-world confounders.

This article discusses inference problems in probabilistic graphical models that often oc- cur in a machine learning setting. In particular it presents a unified view of several recently proposed approximation schemes. Expectation consistent approximations and expectation propagation are both shown to be related to Bethe free energies with weak consistency con- straints, i.e. free energies where local approximations are only required to agree on certain statistics instead of full marginals.

As neural networks become more popular, the need for accompanying uncertainty estimates increases. There are currently two main approaches to test the quality of these estimates. Most methods output a density. They can be compared by evaluating their loglikelihood on a test set. Other methods output a prediction interval directly. These methods are often tested by examining the fraction of test points that fall inside the corresponding prediction intervals. Intuitively, both approaches seem logical. However, we demonstrate through both theoretical arguments and simulations that both ways of evaluating the quality of uncertainty estimates have serious flaws. Firstly, both approaches cannot disentangle the separate components that jointly create the predictive uncertainty, making it difficult to evaluate the quality of the estimates of these components. Specifically, the quality of a confidence interval cannot reliably be tested by estimating the performance of a prediction interval. Secondly, the loglikelihood does not allow a comparison between methods that output a prediction interval directly and methods that output a density. A better loglikelihood also does not necessarily guarantee better prediction intervals, which is what the methods are often used for in practice. Moreover, the current approach to test prediction intervals directly has additional flaws. We show why testing a prediction or confidence interval on a single test set is fundamentally flawed. At best, marginal coverage is measured, implicitly averaging out overconfident and underconfident predictions. A much more desirable property is pointwise coverage, requiring the correct coverage for each prediction. We demonstrate through practical examples that these effects can result in favouring a method, based on the predictive uncertainty, that has undesirable behaviour of the confidence or prediction intervals. Finally, we propose a simulation-based testing approach that addresses these problems while still allowing easy comparison between different methods. This approach can be used for the development of new uncertainty quantification methods.