Unraveling implicit human behavioral effects on dynamic characteristics of Covid-19 daily infection rates in Taiwan.

We investigate the dynamic characteristics of Covid-19 daily infection rates in Taiwan during its initial surge period, focusing on 79 districts within the seven largest cities. By employing computational techniques, we extract 18 features from each district-specific curve, transforming unstructured data into structured data. Our analysis reveals distinct patterns of asymmetric growth and decline among the curves. Utilizing theoretical information measurements such as conditional entropy and mutual information, we identify major factors of order-1 and order-2 that influence the peak value and curvature at the peak of the curves, crucial features characterizing the infection rates. Additionally, we examine the impact of geographic and socioeconomic factors on the curves by encoding each of the 79 districts with two binary characteristics: North-vs-South and Urban-vs-Suburban. Furthermore, leveraging this data-driven understanding at the district level, we explore the fine-scale behavioral effects on disease spread by examining the similarity among 96 age-group-specific curves within urban districts of Taipei and suburban districts of New Taipei City, which collectively represent a substantial portion of the nation's population. Our findings highlight the implicit influence of human behaviors related to living, traveling, and working on the dynamics of Covid-19 transmission in Taiwan.

Ordinal Conditional Entropy Displays Reveal Intrinsic Characteristics of the Rosenberg Self-Esteem Scale.

Individual subjects' ratings neither are metric nor have homogeneous meanings, consequently digital- labeled collections of subjects' ratings are intrinsically ordinal and categorical. However, in these situations, the literature privileges the use of measures conceived for numerical data. In this paper, we discuss the exploratory theme of employing conditional entropy to measure degrees of uncertainty in responding to self-rating questions and that of displaying the computed entropies along the ordinal axis for visible pattern recognition. We apply this theme to the study of an online dataset, which contains responses to the Rosenberg Self-Esteem Scale. We report three major findings. First, at the fine scale level, the resultant multiple ordinal-display of response-vs-covariate entropy measures reveals that the subjects on both extreme labels (high self-esteem and low self-esteem) show distinct degrees of uncertainty. Secondly, at the global scale level, in responding to positively posed questions, the degree of uncertainty decreases for increasing levels of self-esteem, while, in responding to negative questions, the degree of uncertainty increases. Thirdly, such entropy-based computed patterns are preserved across age groups. We provide a set of tools developed in R that are ready to implement for the analysis of rating data and for exploring pattern-based knowledge in related research.

Measuring dominance certainty and assessing its impact on individual and societal health in a nonhuman primate model: a network approach.

The notion of dominance is ubiquitous across the animal kingdom, wherein some species/groups such relationships are strictly hierarchical and others are not. Modern approaches for measuring dominance have emerged in recent years taking advantage of increased computational power. One such technique, named Percolation and Conductance (Perc), uses both direct and indirect information about the flow of dominance relationships to generate hierarchical rank order that makes no assumptions about the linearity of these relationships. It also provides a new metric, known as 'dominance certainty', which is a complimentary measure to dominance rank that assesses the degree of ambiguity of rank relationships at the individual, dyadic and group levels. In this focused review, we will (i) describe how Perc measures dominance rank while accounting for both nonlinear hierarchical structure as well as sparsity in data-here we also provide a metric of dominance certainty estimated by Perc, which can be used to compliment the information dominance rank supplies; (ii) summarize a series of studies by our research team reflecting the importance of 'dominance certainty' on individual and societal health in large captive rhesus macaque breeding groups; and (iii) provide some concluding remarks and suggestions for future directions for dominance hierarchy research. This article is part of the theme issue 'The centennial of the pecking order: current state and future prospects for the study of dominance hierarchies'.

Time-Frequency Analysis of Scalp EEG With Hilbert-Huang Transform and Deep Learning.

Electroencephalography (EEG) is a brain imaging approach that has been widely used in neuroscience and clinical settings. The conventional EEG analyses usually require pre-defined frequency bands when characterizing neural oscillations and extracting features for classifying EEG signals. However, neural responses are naturally heterogeneous by showing variations in frequency bands of brainwaves and peak frequencies of oscillatory modes across individuals. Fail to account for such variations might result in information loss and classifiers with low accuracy but high variation across individuals. To address these issues, we present a systematic time-frequency analysis approach for analyzing scalp EEG signals. In particular, we propose a data-driven method to compute the subject-specific frequency bands for brain oscillations via Hilbert-Huang Transform, lifting the restriction of using fixed frequency bands for all subjects. Then, we propose two novel metrics to quantify the power and frequency aspects of brainwaves represented by sub-signals decomposed from the EEG signals. The effectiveness of the proposed metrics are tested on two scalp EEG datasets and compared with four commonly used features sets extracted from wavelet and Hilbert-Huang Transform. The validation results show that the proposed metrics are more discriminatory than other features leading to accuracies in the range of 94.93% to 99.84%. Besides classification, the proposed metrics show great potential in quantification of neural oscillations and serving as biomarkers in the neuroscience research.

Using Clustering to Examine Inter-individual Variability in Topography of Auditory Event-Related Potentials in Autism and Typical Development.

Although prior studies have compared sensory event-related potential (ERP) responses between groups of autistic and typically-developing participants, it is unclear how heterogeneity contributes to the results of these studies. The present study used examined individual differences in these responses. 130 autistic children and 81 typically-developing children, aged between 2 and 5 years, listened to tones at four identity levels while 61-channel electroencephalography was recorded. Hierarchical clustering was used to group participants based on rescaled ERP topographies between 51 and 350 ms. The hierarchical clustering analysis revealed substantial heterogeneity. Some of the seven clusters defined in this analysis were characterized by prolonged fronto-central positivities and/or weak or absent N2 negativities. However, many other participants fell into clusters in which N2 responses were present at varying latencies. Atypical response morphologies such as absent N2 responses and/or prolonged positive-going responses found in some autistic participants may account for prior research findings of attenuated N2 amplitudes in autism. However, there was also considerable overlap between groups, with participants of both groups appearing in all clusters. These results emphasize the utility of using clustering to explore individual differences in brain responses, which can expand on and clarify the results of analyses of group mean differences.

Categorical Exploratory Data Analysis: From Multiclass Classification and Response Manifold Analytics Perspectives of Baseball Pitching Dynamics.

All features of any data type are universally equipped with categorical nature revealed through histograms. A contingency table framed by two histograms affords directional and mutual associations based on rescaled conditional Shannon entropies for any feature-pair. The heatmap of the mutual association matrix of all features becomes a roadmap showing which features are highly associative with which features. We develop our data analysis paradigm called categorical exploratory data analysis (CEDA) with this heatmap as a foundation. CEDA is demonstrated to provide new resolutions for two topics: multiclass classification (MCC) with one single categorical response variable and response manifold analytics (RMA) with multiple response variables. We compute visible and explainable information contents with multiscale and heterogeneous deterministic and stochastic structures in both topics. MCC involves all feature-group specific mixing geometries of labeled high-dimensional point-clouds. Upon each identified feature-group, we devise an indirect distance measure, a robust label embedding tree (LET), and a series of tree-based binary competitions to discover and present asymmetric mixing geometries. Then, a chain of complementary feature-groups offers a collection of mixing geometric pattern-categories with multiple perspective views. RMA studies a system's regulating principles via multiple dimensional manifolds jointly constituted by targeted multiple response features and selected major covariate features. This manifold is marked with categorical localities reflecting major effects. Diverse minor effects are checked and identified across all localities for heterogeneity. Both MCC and RMA information contents are computed for data's information content with predictive inferences as by-products. We illustrate CEDA developments via Iris data and demonstrate its applications on data taken from the PITCHf/x database.

Mimicking Complexity of Structured Data Matrix's Information Content: Categorical Exploratory Data Analysis.

We develop Categorical Exploratory Data Analysis (CEDA) with mimicking to explore and exhibit the complexity of information content that is contained within any data matrix: categorical, discrete, or continuous. Such complexity is shown through visible and explainable serial multiscale structural dependency with heterogeneity. CEDA is developed upon all features' categorical nature via histogram and it is guided by all features' associative patterns (order-2 dependence) in a mutual conditional entropy matrix. Higher-order structural dependency of k(≥3) features is exhibited through block patterns within heatmaps that are constructed by permuting contingency-kD-lattices of counts. By growing k, the resultant heatmap series contains global and large scales of structural dependency that constitute the data matrix's information content. When involving continuous features, the principal component analysis (PCA) extracts fine-scale information content from each block in the final heatmap. Our mimicking protocol coherently simulates this heatmap series by preserving global-to-fine scales structural dependency. Upon every step of mimicking process, each accepted simulated heatmap is subject to constraints with respect to all of the reliable observed categorical patterns. For reliability and robustness in sciences, CEDA with mimicking enhances data visualization by revealing deterministic and stochastic structures within each scale-specific structural dependency. For inferences in Machine Learning (ML) and Statistics, it clarifies, upon which scales, which covariate feature-groups have major-vs.-minor predictive powers on response features. For the social justice of Artificial Intelligence (AI) products, it checks whether a data matrix incompletely prescribes the targeted system.

Color-complexity enabled exhaustive color-dots identification and spatial patterns testing in images.

Our computational developments and analyses on experimental images are designed to evaluate the effectiveness of chemical spraying via unmanned aerial vehicle (UAV). Our evaluations are in accord with the two perspectives of color-complexity: color variety within a color system and color distributional geometry on an image. First, by working within RGB and HSV color systems, we develop a new color-identification algorithm relying on highly associative relations among three color-coordinates to lead us to exhaustively identify all targeted color-pixels. A color-dot is then identified as one isolated network of connected color-pixel. All identified color-dots vary in shapes and sizes within each image. Such a pixel-based computing algorithm is shown robustly and efficiently accommodating heterogeneity due to shaded regions and lighting conditions. Secondly, all color-dots with varying sizes are categorized into three categories. Since the number of small color-dot is rather large, we spatially divide the entire image into a 2D lattice of rectangular. As such, each rectangle becomes a collective of color-dots of various sizes and is classified with respect to its color-dots intensity. We progressively construct a series of minimum spanning trees (MST) as multiscale 2D distributional spatial geometries in a decreasing-intensity fashion. We extract the distributions of distances among connected rectangle-nodes in the observed MST and simulated MSTs generated under the spatial uniformness assumption. We devise a new algorithm for testing 2D spatial uniformness based on a Hierarchical clustering tree upon all involving MSTs. This new tree-based p-value evaluation has the capacity to become exact.

Livestock Informatics Toolkit: A Case Study in Visually Characterizing Complex Behavioral Patterns across Multiple Sensor Platforms, Using Novel Unsupervised Machine Learning and Information Theoretic Approaches.

Large and densely sampled sensor datasets can contain a range of complex stochastic structures that are difficult to accommodate in conventional linear models. This can confound attempts to build a more complete picture of an animal's behavior by aggregating information across multiple asynchronous sensor platforms. The Livestock Informatics Toolkit (LIT) has been developed in R to better facilitate knowledge discovery of complex behavioral patterns across Precision Livestock Farming (PLF) data streams using novel unsupervised machine learning and information theoretic approaches. The utility of this analytical pipeline is demonstrated using data from a 6-month feed trial conducted on a closed herd of 185 mix-parity organic dairy cows. Insights into the tradeoffs between behaviors in time budgets acquired from ear tag accelerometer records were improved by augmenting conventional hierarchical clustering techniques with a novel simulation-based approach designed to mimic the complex error structures of sensor data. These simulations were then repurposed to compress the information in this data stream into robust empirically-determined encodings using a novel pruning algorithm. Nonparametric and semiparametric tests using mutual and pointwise information subsequently revealed complex nonlinear associations between encodings of overall time budgets and the order that cows entered the parlor to be milked.

Mind the Queue: A Case Study in Visualizing Heterogeneous Behavioral Patterns in Livestock Sensor Data Using Unsupervised Machine Learning Techniques.

Sensor technologies allow ethologists to continuously monitor the behaviors of large numbers of animals over extended periods of time. This creates new opportunities to study livestock behavior in commercial settings, but also new methodological challenges. Densely sampled behavioral data from large heterogeneous groups can contain a range of complex patterns and stochastic structures that may be difficult to visualize using conventional exploratory data analysis techniques. The goal of this research was to assess the efficacy of unsupervised machine learning tools in recovering complex behavioral patterns from such datasets to better inform subsequent statistical modeling. This methodological case study was carried out using records on milking order, or the sequence in which cows arrange themselves as they enter the milking parlor. Data was collected over a 6-month period from a closed group of 200 mixed-parity Holstein cattle on an organic dairy. Cows at the front and rear of the queue proved more consistent in their entry position than animals at the center of the queue, a systematic pattern of heterogeneity more clearly visualized using entropy estimates, a scale and distribution-free alternative to variance robust to outliers. Dimension reduction techniques were then used to visualize relationships between cows. No evidence of social cohesion was recovered, but Diffusion Map embeddings proved more adept than PCA at revealing the underlying linear geometry of this data. Median parlor entry positions from the pre- and post-pasture subperiods were highly correlated (R = 0.91), suggesting a surprising degree of temporal stationarity. Data Mechanics visualizations, however, revealed heterogeneous non-stationary among subgroups of animals in the center of the group and herd-level temporal outliers. A repeated measures model recovered inconsistent evidence of a relationships between entry position and cow attributes. Mutual conditional entropy tests, a permutation-based approach to assessing bivariate correlations robust to non-independence, confirmed a significant but non-linear association with peak milk yield, but revealed the age effect to be potentially confounded by health status. Finally, queueing records were related back to behaviors recorded via ear tag accelerometers using linear models and mutual conditional entropy tests. Both approaches recovered consistent evidence of differences in home pen behaviors across subsections of the queue.

Defining clusters of young autistic and typically developing children based on loudness-dependent auditory electrophysiological responses.

BACKGROUND: Autistic individuals exhibit atypical patterns of sensory processing that are known to be related to quality of life, but which are also highly heterogeneous. Previous investigations of this heterogeneity have ordinarily used questionnaires and have rarely investigated sensory processing in typical development (TD) alongside autism spectrum development (ASD). METHODS: The present study used hierarchical clustering in a large sample to identify subgroups of young autistic and typically developing children based on the normalized global field power (GFP) of their event-related potentials (ERPs) to auditory stimuli of four different loudness intensities (50, 60, 70, 80 dB SPL): that is, based on an index of the relative strengths of their neural responses across these loudness conditions. RESULTS: Four clusters of participants were defined. Normalized GFP responses to sounds of different intensities differed strongly across clusters. There was considerable overlap in cluster assignments of autistic and typically developing participants, but autistic participants were more likely to display a pattern of relatively linear increases in response strength accompanied by a disproportionately strong response to 70 dB stimuli. Autistic participants displaying this pattern trended towards obtaining higher scores on assessments of cognitive abilities. There was also a trend for typically developing participants to disproportionately fall into a cluster characterized by disproportionately/nonlinearly strong 60 dB responses. Greater auditory distractibility was reported among autistic participants in a cluster characterized by disproportionately strong responses to the loudest (80 dB) sounds, and furthermore, relatively strong responses to loud sounds were correlated with auditory distractibility. This appears to provide evidence of coinciding behavioral and neural sensory atypicalities. LIMITATIONS: Replication may be needed to verify exploratory results. This analysis does not address variability related to classical ERP latencies and topographies. The sensory questionnaire employed was not specifically designed for use in autism. Hearing acuity was not measured. Variability in sensory responses unrelated to loudness is not addressed, leaving room for additional research. CONCLUSIONS: Taken together, these data demonstrate the broader benefits of using electrophysiology to explore individual differences. They illuminate different neural response patterns and suggest relationships between sensory neural responses and sensory behaviors, cognitive abilities, and autism diagnostic status.

A Data-Driven Approach to Predict and Classify Epileptic Seizures from Brain-Wide Calcium Imaging Video Data.

The prediction of epileptic seizures has been an essential problem of epilepsy study. The calcium imaging video data images the whole brain-wide neurons activities with electrical discharge recorded by calcium fluorescence intensity (CFI). In this paper, using the zebrafish's brain-wide calcium image video data, we propose a data-driven approach to effectively detect the systemic change-point, and further predict the epileptic seizures. Our approach includes two phases: offline training and online testing. Specifically, during offline training, we extract features and confirm the existence of systemic change-point, then estimate the ratio of unchanged system duration to interictal period duration. For online testing, we implement a statistical model to estimate the change-point, and then predict the onset of epileptic seizure. The testing results show that our proposed approach could effectively predict the time range of future epileptic seizure. Furthermore, we explore the macroscopic patterns of epileptic and control cases, and extract features based on the pattern difference, then implement and compare the classification performance from four machine learning models. Based on the data structure, we also propose a new method to discretize related features, and combine with hierarchical clustering to better visualize and explain the pattern difference between epileptic and control cases.

DCG++: A data-driven metric for geometric pattern recognition.

Clustering large and complex data sets whose partitions may adopt arbitrary shapes remains a difficult challenge. Part of this challenge comes from the difficulty in defining a similarity measure between the data points that captures the underlying geometry of those data points. In this paper, we propose an algorithm, DCG++ that generates such a similarity measure that is data-driven and ultrametric. DCG++ uses Markov Chain Random Walks to capture the intrinsic geometry of data, scans possible scales, and combines all this information using a simple procedure that is shown to generate an ultrametric. We validate the effectiveness of this similarity measure within the context of clustering on synthetic data with complex geometry, on a real-world data set containing segmented audio records of frog calls described by mel-frequency cepstral coefficients, as well as on an image segmentation problem. The experimental results show a significant improvement on performance with the DCG-based ultrametric compared to using an empirical distance measure.

Decoupling social status and status certainty effects on health in macaques: a network approach.

BACKGROUND: Although a wealth of literature points to the importance of social factors on health, a detailed understanding of the complex interplay between social and biological systems is lacking. Social status is one aspect of social life that is made up of multiple structural (humans: income, education; animals: mating system, dominance rank) and relational components (perceived social status, dominance interactions). In a nonhuman primate model we use novel network techniques to decouple two components of social status, dominance rank (a commonly used measure of social status in animal models) and dominance certainty (the relative certainty vs. ambiguity of an individual's status), allowing for a more complex examination of how social status impacts health. METHODS: Behavioral observations were conducted on three outdoor captive groups of rhesus macaques (N = 252 subjects). Subjects' general physical health (diarrhea) was assessed twice weekly, and blood was drawn once to assess biomarkers of inflammation (interleukin-6 (IL-6), tumor necrosis factor-alpha (TNF-α), and C-reactive protein (CRP)). RESULTS: Dominance rank alone did not fully account for the complex way that social status exerted its effect on health. Instead, dominance certainty modified the impact of rank on biomarkers of inflammation. Specifically, high-ranked animals with more ambiguous status relationships had higher levels of inflammation than low-ranked animals, whereas little effect of rank was seen for animals with more certain status relationships. The impact of status on physical health was more straightforward: individuals with more ambiguous status relationships had more frequent diarrhea; there was marginal evidence that high-ranked animals had less frequent diarrhea. DISCUSSION: Social status has a complex and multi-faceted impact on individual health. Our work suggests an important role of uncertainty in one's social status in status-health research. This work also suggests that in order to fully explore the mechanisms for how social life influences health, more complex metrics of social systems and their dynamics are needed.

Integrative Inferences on Pattern Geometries of Grapes Grown under Water Stress and Their Resulting Wines.

Multiple datasets of two consecutive vintages of replicated grape and wines from six different deficit irrigation regimes are characterized and compared. The process consists of four temporal-ordered signature phases: harvest field data, juice composition, wine composition before bottling and bottled wine. A new computing paradigm and an integrative inferential platform are developed for discovering phase-to-phase pattern geometries for such characterization and comparison purposes. Each phase is manifested by a distinct set of features, which are measurable upon phase-specific entities subject to the common set of irrigation regimes. Throughout the four phases, this compilation of data from irrigation regimes with subsamples is termed a space of media-nodes, on which measurements of phase-specific features were recoded. All of these collectively constitute a bipartite network of data, which is then normalized and binary coded. For these serial bipartite networks, we first quantify patterns that characterize individual phases by means of a new computing paradigm called "Data Mechanics". This computational technique extracts a coupling geometry which captures and reveals interacting dependence among and between media-nodes and feature-nodes in forms of hierarchical block sub-matrices. As one of the principal discoveries, the holistic year-factor persistently surfaces as the most inferential factor in classifying all media-nodes throughout all phases. This could be deemed either surprising in its over-arching dominance or obvious based on popular belief. We formulate and test pattern-based hypotheses that confirm such fundamental patterns. We also attempt to elucidate the driving force underlying the phase-evolution in winemaking via a newly developed partial coupling geometry, which is designed to integrate two coupling geometries. Such partial coupling geometries are confirmed to bear causal and predictive implications. All pattern inferences are performed with respect to a profile of energy distributions sampled from network bootstrapping ensembles conforming to block-structures specified by corresponding hypotheses.

Connections Matter: Social Networks and Lifespan Health in Primate Translational Models.

Humans live in societies full of rich and complex relationships that influence health. The ability to improve human health requires a detailed understanding of the complex interplay of biological systems that contribute to disease processes, including the mechanisms underlying the influence of social contexts on these biological systems. A longitudinal computational systems science approach provides methods uniquely suited to elucidate the mechanisms by which social systems influence health and well-being by investigating how they modulate the interplay among biological systems across the lifespan. In the present report, we argue that nonhuman primate social systems are sufficiently complex to serve as model systems allowing for the development and refinement of both analytical and theoretical frameworks linking social life to health. Ultimately, developing systems science frameworks in nonhuman primate models will speed discovery of the mechanisms that subserve the relationship between social life and human health.

Systemic testing on Bradley-Terry model against nonlinear ranking hierarchy.

We take a system point of view toward constructing any power or ranking hierarchy onto a society of human or animal players. The most common hierarchy is the linear ranking, which is habitually used in nearly all real-world problems. A stronger version of linear ranking via increasing and unvarying winning potentials, known as Bradley-Terry model, is particularly popular. Only recently non-linear ranking hierarchy is discussed and developed through recognition of dominance information contents beyond direct dyadic win-and-loss. We take this development further by rigorously arguing for the necessity of accommodating system's global pattern information contents, and then introducing a systemic testing on Bradley-Terry model. Our test statistic with an ensemble based empirical distribution favorably compares with the Deviance test equipped with a Chi-squared asymptotic approximation. Several simulated and real data sets are analyzed throughout our development.

Invariance in the recurrence of large returns and the validation of models of price dynamics.

Starting from a robust, nonparametric definition of large returns ("excursions"), we study the statistics of their occurrences, focusing on the recurrence process. The empirical waiting-time distribution between excursions is remarkably invariant to year, stock, and scale (return interval). This invariance is related to self-similarity of the marginal distributions of returns, but the excursion waiting-time distribution is a function of the entire return process and not just its univariate probabilities. Generalized autoregressive conditional heteroskedasticity (GARCH) models, market-time transformations based on volume or trades, and generalized (Lévy) random-walk models all fail to fit the statistical structure of excursions.

Phylogenetic analysis, expression patterns, and transcriptional regulation of human CTEN gene.

Cten is a focal adhesion molecule and a member of the tensin family. Its expression is highly enriched in the prostate and placenta, suggesting that cten gene might be closely associated with mammalian species. Recent studies have reported that cten expression is frequently up-regulated in a variety of cancers and its levels appear to correlate with tumorigenicity. Here, we have (1) analyzed cten sequences of various species to build a phylogenetic tree, (2) examined cten mRNA levels in human and mouse tissues to establish its expression profiles, and (3) determined the promoter region of human CTEN gene in cell lines and in a mouse model to understand its transcriptional regulation. Our analyses indicate that all currently known cten genes are present in mammals. The prostate and placenta are the two most cten abundant tissues in human and mouse, meanwhile brain and lung also express low levels of cten. Results from cell culture reporter assays demonstrate that a 327-bp fragment is the shortest functional promoter. All functional promoter constructs produce 40- to 160-fold increases in luciferase reporter activities in normal prostate cells, whereas lower activities (<40-fold) are detected in non-prostatic cell lines. To evaluate CTEN promoter activity in mice and develop a new tissue specific Cre recombinase mouse model, we have established pCTEN-Cre:R26R mice by crossing R26R ß-galactosidase reporter mice with pCTEN-Cre transgenic mice, in which the 327-bp cten promoter drives the expression of Cre recombinase. X-gal analysis has shown strong ß-galactosidase activities in the prostate, brain, and few other tissues in pCTEN-Cre:R26R mice. Altogether, we have identified the promoter region of human cten gene and provided a useful tool for investigating cell lineages and generating tissue-specific knockout or knockin mice.

Using Markov chain Monte Carlo (MCMC) to visualize and test the linearity assumption of the Bradley-Terry class of models.

The construction of dominance hierarchies for animal societies is an important aspect of understanding the nature of social relationships, and the models to calculate dominance ranks are many. However, choosing the appropriate model for a given data set may appear daunting to the average behaviourist, especially when many of these models assume linearity of dominance. Here, we present a method to test whether or not a data set fits the assumption of linearity using the Bradley-Terry model as a representative of the class of models that assume linearity. Our method uses the geometry of a posterior distribution of possible rankings given the data by using a random walk on this distribution. This test is intuitive, efficient, particularly for large number of individuals, and represents an improvement over previous linearity tests because it takes into account all information (i.e. both linear and apparently circular or nonlinear information) from the data with few restrictions due to high dimensionality. Such a test is not only useful in determining whether a linear hierarchy is relevant to a given animal society, but is necessary in justifying the results of any analysis for which the assumption of linearity is made, such as the Bradley-Terry model. If the assumption of linearity is not met, other methods for ranking, such as the beta random field method proposed by Fushing et al. (2011, PLoS One, 6, e17817) should be considered.