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... The non-linear constraint represented by the elliptope has likewise been investigated by probability theorists and statisticians before in contexts far removed from physics. As we will see in Section 2.6, it can be found in a paper by Udny Yule (1897) on what are now called Pearson correlation coefficients as well as in papers by Ronald A. Fisher (1924) and Bruno de Finetti (1937). Yule, like Pearson, was especially interested in applications in evolutionary biology (see notes 38 and 39). ...

... discuss the application of this general result both to our quantum banana peeling and tasting experiment and to the raffles designed to simulate them (Sections 2.6.2-2.6.3) and provide a geometrical perspective on the general result (Section 2.6.3), following some remarkable papers by two famous statisticians a generation after Pearson and Yule, Ronald A. Fisher (1915, 1924) and Bruno De Finetti (1937. 39 38 Yule was an associate of Karl Pearson and is remembered by historians of biology today for his role in bridging the divide between Mendelians and Darwinian biometrists, which would eventually result in the modern synthesis (Bowler, 2003, p. 329). ...

... In this subsection, we indicate how Yule (1897) found the general constraint on correlation coefficients in Eq. (2.64) in the context of regression theory (i.e., finding the straight lines best approximating correlations between variables) and how Fisher (1915, 1924) and De Finetti (1937 recovered the result Yule found algebraically by treating random variables as vectors and correlations in terms of angles between those vectors. The importance of this geometric approach was emphasized by Pearson (1916): ...

We use Bub's (2016) correlation arrays and Pitowksy's (1989b) correlation polytopes to analyze an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-$\frac12$ particles. The class of correlations allowed by quantum mechanics in this setup is represented by an elliptope inscribed in a non-signaling cube. The class of correlations allowed by local hidden-variable theories is represented by a tetrahedron inscribed in this elliptope. We extend this analysis to pairs of particles of arbitrary spin. The class of correlations allowed by quantum mechanics is still represented by the elliptope; the subclass of those allowed by local hidden-variable theories by polyhedra with increasing numbers of vertices and facets that get closer and closer to the elliptope. We use these results to advocate for an interpretation of quantum mechanics like Bub's. Probabilities and expectation values are primary in this interpretation. They are determined by inner products of vectors in Hilbert space. Such vectors do not themselves represent what is real in the quantum world. They encode families of probability distributions over values of different sets of observables. As in classical theory, these values ultimately represent what is real in the quantum world. Hilbert space puts constraints on possible combinations of such values, just as Minkowski space-time puts constraints on possible spatio-temporal constellations of events. Illustrating how generic such constraints are, the equation for the elliptope derived in this paper is a general constraint on correlation coefficients that can be found in older literature on statistics and probability theory. Yule (1896) already stated the constraint. De Finetti (1937) already gave it a geometrical interpretation.

... Both of these conditions affect the confidence intervals, which should be asymmetrical and based on a non-distorted estimate of the standard error. Fisher (1915Fisher ( , 1921Fisher ( , 1924 first demonstrated these two problems, noted they are greatest in small samples (N < 30), and suggested solutions. He also noted one other oddity: the greatest underestimate of the ρ is not, as one might expect, with high absolute values but actually is worse for mid-range correlations. ...

... The Figures illustrate the mean bias that occurs due to the distribution of observed correlations. Fisher (1915Fisher ( , 1924 considered this problem and introduced two formulas for correcting r's so that means will most closely approximate ρ. The first was so complex it has not been used. ...

... The jth most significant among the significant ones enters the model. 3. Examine all conditional associations (y, xi) |CS. ...

... where r(y; x) denotes the correlation between y & x, and r(y; z), r(x; z) are the correlation coefficients between y & z and between x & z respectively. The general method of calculating the partial correlation, for | | = ≥ 1 is given by (Fisher et al., 1924) ...

... It is worth mentioning that the recent work of Shah and Peters (2018) shows that testing conditional independence without restricting the form of conditional independence is impossible in general. For testing whether a specific marginal or conditional independence holds, it is common to use the correlation coefficient or partial correlation coefficient under Fisher's z-transformation (Fisher, 1924) as the test statistic. Under independence, the transformed correlation coefficient is approximately distributed as a normal distribution with zero mean and variance determined by the sample size and the number of variables being conditioned on (Hotelling, 1953;Anderson, 1984). ...

... This method is adapted from Drton and Perlman (2004). We construct (1 − α)-level non-simultaneous confidence intervals on correlation coefficients ρ 12 and ρ 12·3 with Fisher's z-transform (Fisher, 1924). The decision rule is ...

We consider testing marginal independence versus conditional independence in a trivariate Gaussian setting. The two models are non-nested and their intersection is a union of two marginal independences. We consider two sequences of such models, one from each type of independence, that are closest to each other in the Kullback-Leibler sense as they approach the intersection. They become indistinguishable if the signal strength, as measured by the product of two correlation parameters, decreases faster than the standard parametric rate. Under local alternatives at such rate, we show that the asymptotic distribution of the likelihood ratio depends on where and how the local alternatives approach the intersection. To deal with this non-uniformity, we study a class of "envelope" distributions by taking pointwise suprema over asymptotic cumulative distribution functions. We show that these envelope distributions are well-behaved and lead to model selection procedures with uniform error guarantees and near-optimal power. To control the error even when the two models are indistinguishable, rather than insist on a dichotomous choice, the proposed procedure will choose either or both models.

... If the vector (X, Y, Z) is Gaussian, then the population partial correlation is zero if and only if X ⊥ ⊥ Y | Z hence the empirical estimate can form the basis of a test. Fisher (1924) derived the distribution of the sample partial correlation under this assumption, which can be used to construct an exact test. If the regression functions for the X on Z and Y on Z regressions are linear (as is the case when the vector (X, Y, Z) is Gaussian), the aforementioned partial correlation test remains an asymptotically valid test, although it is not consistent against all alternatives (Arnold, 1984;Huber, 1973). ...

This thesis concerns the ubiquitous statistical problem of variable significance testing. The first chapter contains an account of classical approaches to variable significance testing including different perspectives on how to formalise the notion of `variable significance'. The historical development is contrasted with more recent methods that are adapted to both the scale of modern datasets but also the power of advanced machine learning techniques. This chapter also includes a description of and motivation for the theoretical framework that permeates the rest of the thesis: providing theoretical guarantees that hold uniformly over large classes of distributions. The second chapter deals with testing the null that Y ⊥ X | Z where X and Y take values in separable Hilbert spaces with a focus on applications to functional data. The first main result of the chapter shows that for functional data it is impossible to construct a non-trivial test for conditional independence even when assuming that the data are jointly Gaussian. A novel regression-based test, called the Generalised Hilbertian Covariance Measure (GHCM), is presented and theoretical guarantees for uniform asymptotic Type I error control are provided with the key assumption requiring that the product of the mean squared errors of regressing Y on Z and X on Z converges faster than n$^{-1}$, where n is the sample size. A power analysis is conducted under the same assumptions to illustrate that the test has uniform power over local alternatives where the expected conditional covariance operator has a Hilbert--Schmidt norm going to 0 at a $\sqrt[n]{n}$-rate. The chapter also contains extensive empirical evidence in the form of simulations demonstrating the validity and power properties of the test. The usefulness of the test is demonstrated by using the GHCM to construct confidence intervals for the boundary point in a truncated functional linear model and to detect edges in a graphical model for an EEG dataset. The third and final chapter analyses the problem of nonparametric variable significance testing by testing for conditional mean independence, that is, testing the null that E(Y | X, Z) = E(Y | Z) for real-valued Y. A test, called the Projected Covariance Measure (PCM), is derived by considering a family of studentised test statistics and choosing a member of this family in a data-driven way that balances robustness and power properties of the resulting test. The test is regression-based and is computed by splitting a set of observations of (X, Y, Z) into two sets of equal size, where one half is used to learn a projection of Y onto X and Z (nonparametrically) and the second half is used to test for vanishing expected conditional correlation given Z between the projection and Y. The chapter contains general conditions that ensure uniform asymptotic Type I control of the resulting test by imposing conditions on the mean-squared error of the involved regressions. A modification of the PCM using additional sample splitting and employing spline regression is shown to achieve the minimax optimal separation rate between null and alternative under Hölder smoothness assumptions on the regression functions and the conditional density of X given Z=z. The chapter also shows through simulation studies that the test maintains the strong type I error control of methods like the Generalised Covariance Measure (GCM) but has power against a broader class of alternatives.

... Moreover, the human disturbance effect needed to be excluded to the greatest extent when assessing vegetation changes caused by climatic factors; here, only those pixels classified as forestland throughout the study period were selected for correlation analysis. The second-order partial correlation coefficient of the NDVI and any MF (taking MF1 as an example) after removing the effect of MF2 and MF3 is expressed as [53,54]: ...

Vegetation degeneration has become a serious ecological problem for karst regions in the Anthropocene. According to the deficiency of long serial and high-resolution analysis of karst vegetation, this paper reconstructed the variation of vegetation landscape changes from 1987 to 2020 in a typical karst region of China. Using Landsat time series data, the dynamic changes and driving factors of natural karst vegetation were identified at the landscape scale. On the premise of considering the time-lag effect, the main climatic factors that influence vegetation growth were presented at the interannual timescale. Then, the approach of residual analysis was adopted to distinguish the dominant factors affecting vegetation growth. Results of trend analysis revealed that 21.5% of the forestland showed an overall significant decline in vegetation growth, while only 1.5% showed an increase in vegetation growth during the study period. Precipitation and radiation were the dominant meteorological factors influencing vegetation at the interannual timescale, as opposed to temperature. More than 70% of the natural vegetation growth was dominated by climatic factors. The area percentage of negative human impact has increased gradually since 2009 and reached 18.5% in 2020, indicating the currently serious situation of vegetation protection; fortunately, in recent years, human disturbances on vegetation have been mitigated in karst areas with the promotion of ecological conservation and restoration projects.

... The network model used was the Gaussian Graphical Model (GGM) [60]. The GGM models the precision matrix (i.e., inverse of the variance covariance matrix) such that (after standardizing the precision matrix and reversing the sign) nodes represent items and edges represent partial correlation coefficients [61] between items. Thus, an edge indicates conditional dependence (after conditioning on the entire set of variables) between two items, while the absence of an edge indicates conditional independence [60]. ...

Over the past decades, increasing research interest has been directed towards the psychosocial factors that impact Aboriginal health, including stress, coping and social support. However, there has been no study that examined whether the behaviours, cognitions and emotions related to stress, coping and social support constitute a psychological network in an Aboriginal population and that examined its properties. To address this gap, the current study employed a new methodology, network psychometrics, to evaluate stress, coping and social support in an Aboriginal Australian population. This study conducted a secondary analysis of the South Australian Aboriginal Birth Cohort (SAABC) study, a randomised controlled trial in South Australia, which included 367 pregnant Aboriginal women at study baseline. The Gaussian Graphical Model was estimated with least absolute shrinkage and selection operator (LASSO). Node centrality was evaluated with eigencentrality, strength and bridge centrality. Network communities were investigated with the walktrap algorithm. The findings indicated that stress, coping and social support constituted a connected psychological network in an Aboriginal population. Furthermore, at the centre of the network were the troubles experienced by the Aboriginal pregnant women, bridging their perceptions of stress and coping and constituting a potential target for future interventions.

... A partial correlation coefficient (PCC) was used to analyse the relationship between the continuous random variables (material properties and initial MC) and the calculated mould growth index. A PCC determines the 'clean' correlation between random variables Y and X, while eliminating the linear impact of the third variable Z [60,61]. A PCC, r(Y, X|Z), between X and Y is expressed by the conventional linear correlation coefficients of X on Z 1 , …, Z n and Y on Z 1 , …, Z n , see Eq. (3). ...

In terms of hygrothermal performance, solid wood panels composed of cross-laminated timber (CLT) are sensitive to moisture. However, to the best of our knowledge, no clear indications of critical initial moisture conditions for CLT envelopes regarding the dry-out of built-in moisture have been provided. Therefore, our main objective was to set critical limit values for CLT external wall design in terms of their initial moisture content (built-in moisture) and dry-out capacity (vapour resistance of wall layers) using a stochastic approach. The focus is on five types of CLT external walls that differ in their dry-out capacity. The key factors in the hygrothermally safe design of the CLT external envelope are sufficient dry-out capacity and control of the CLT initial moisture content level during the construction phase. The results of our stochastic analysis confirmed that a high initial moisture content in CLT causes a high mould growth risk when CLT is covered with vapour tight layers. Therefore, the limit values for critical initial moisture content of CLT and the water vapour resistance of the wall layers that ensure dry-out capacity of different external walls to prevent the risk of mould growth on the CLT surface in a cold and humid climate were determined. These limit values take into consideration the weather conditions during the construction phase and classes of material sensitivity to mould growth.

... In the classical scenario, the Pearson's correlation coefficient r and the partial correlation coefficient q follow the same probability distribution f 0 , differing only in their degrees of freedom k (Fisher, 1924). Under the null hypothesis H 0 : q ¼ 0, the density of the partial correlation is ...

Background:
Gaussian graphical models (GGMs) are network representations of random variables (as nodes) and their partial correlations (as edges). GGMs overcome the challenges of high-dimensional data analysis by using shrinkage methodologies. Therefore, they have become useful to reconstruct gene regulatory networks from gene expression profiles. However, it is often ignored that the partial correlations are 'shrunk' and that they cannot be compared/assessed directly. Therefore, accurate (differential) network analyses need to account for the number of variables, the sample size, and also the shrinkage value, otherwise, the analysis and its biological interpretation would turn biased. To date, there are no appropriate methods to account for these factors and address these issues.
Results:
We derive the statistical properties of the partial correlation obtained with the Ledoit-Wolf shrinkage. Our result provides a toolbox for (differential) network analyses as i) confidence intervals, ii) a test for zero partial correlation (null-effects), and iii) a test to compare partial correlations. Our novel (parametric) methods account for the number of variables, the sample size, and the shrinkage values. Additionally, they are computationally fast, simple to implement, and require only basic statistical knowledge. Our simulations show that the novel tests perform better than DiffNetFDR -a recently published alternative-, in terms of the trade-off between true and false positives. The methods are demonstrated on synthetic data and two gene expression datasets from Escherichia coli and Mus musculus.
Availability:
The R package with the methods and the R script with the analysis are be available in https://github.com/V-Bernal/GeneNetTools.

... Our specific aims are to present measures designed to disentangle individual functional relations between triplets of variables in the presence of redundancy, to computationally test whether these measures are robust to noise in source and target variables, and to propose and discuss potential improvements. We focus on three existing measures: partial correlation (PCorr) (Fisher, 1924), variance partitioning ( VP) (Borcard, Legendre, & Drapeau, 1992), and partial information decomposition (PID) (P. L. Williams & Beer, 2010). ...

An important goal in systems neuroscience is to understand the structure of neuronal interactions, frequently approached by studying functional relations between recorded neuronal signals. Commonly used pairwise measures (e.g. correlation coefficient) offer limited insight, neither addressing the specificity of estimated neuronal relations nor potential synergistic coupling between neuronal signals. Tripartite measures, such as partial correlation, variance partitioning, and partial information decomposition, address these questions by disentangling functional relations into interpretable information atoms (unique, redundant and synergistic). Here, we apply these tripartite measures to simulated neuronal recordings to investigate their sensitivity to noise. We find that the considered measures are mostly accurate and specific for signals with noiseless sources but experience significant bias for noisy sources. We show that permutation-testing of such measures results in high false positive rates even for small noise fractions and large data sizes. We present a conservative null hypothesis for significance testing of tripartite measures, which significantly decreases false positive rate at a tolerable expense of increasing false negative rate. We hope our study raises awareness about the potential pitfalls of significance testing and of interpretation of functional relations, offering both conceptual and practical advice.

... Our specific aims are to present metrics designed to disentangle individual functional relations between triplets of variables in the presence of redundancy, to computationally test whether these metrics are robust to impurities in source and target variables, and to propose and discuss potential improvements. We focus on three existing metrics: Partial Correlation (PCorr) [30], Variance Partitioning (VP) [12], and Partial Information Decomposition (PID) [80]. Precise definitions of these metrics are given in the methods section. ...

An important goal in systems neuroscience is to understand the structure of neuronal interactions, frequently approached by studying functional relations between recorded neuronal signals. Commonly used pairwise metrics (e.g. correlation coefficient) offer limited insight, neither addressing the specificity of estimated neuronal interactions nor potential synergistic coupling between neuronal signals. Tripartite metrics, such as partial correlation, variance partitioning, and partial information decomposition, address these questions by disentangling functional relations into interpretable information atoms (unique, redundant and synergistic). Here, we apply these tripartite metrics to simulated neuronal recordings to investigate their sensitivity to impurities (like noise or other unexplained variance) in the data. We find that all considered metrics are accurate and specific for pure signals but experience significant bias for impure signals. We show that permutation-testing of such metrics results in high false positive rates even for small impurities and large data sizes. We present a conservative null hypothesis for significance testing of tripartite metrics, which significantly decreases false positive rate at a tolerable expense of increasing false negative rate. We hope our study raises awareness about the potential pitfalls of significance testing and of interpretation of functional relations, offering both conceptual and practical advice.
Author Summary
Tripartite functional relation metrics enable the study of interesting effects in neural recordings, such as redundancy, functional connection specificity and synergistic coupling. However, common estimators of such relations are designed for pure (e.g. non-noisy) signals rare for such recordings. We study the performance of tripartite estimators using simulated impure neural signals. We demonstrate that permutation-testing is not a robust procedure for inferring ground truth interactions from studied estimators. We develop an adjusted conservative testing procedure, reducing false positive rate of studied estimators for impure data. Besides addressing significance testing, our results should aid in accurate interpretation of tripartite functional relations and functional connectivity.

... x and lag t. Forecast skill significance was determined following Stock et al. (2015), which uses a Fisher's Z transformation (Fisher, 1915;Fisher, 1924;Lund et al., 2000) to determine whether the dynamical forecast R 2 is: (a) Significantly above 0, and (b) Significantly greater than the persistence forecast R 2 . For our comparisons, we use a 90% confidence interval to denote significance. ...

Accurate dynamical forecasts of ocean variables in the California Current System (CCS) are essential decision support tools for advancing ecosystem-based marine resource management. However, model and dynamical uncertainties present a significant challenge when attempting to incorporate these forecasts into a formal decision making process. To provide guidance on the reliability of dynamical forecasts, previous studies have suggested that deterministic climate processes associated with atmospheric or oceanic teleconnections may provide opportunities for enhanced forecast skill. Recent computational advances have led to the availability of subseasonal-to-seasonal (S2S) forecasts of key oceanic variables such as sea surface height (SSH), which may be leveraged to identify such “forecast opportunities”. In this study, we conduct a S2S forecast skill assessment of SSH anomalies in the CCS using an ensemble of 46-day reforecasts made by the European Center for Medium-range Weather Forecasting (ECMWF) model for the period 2000-2018. We find that the ECMWF model consistently produces skillful dynamical forecasts of SSH, particularly in both the southern and northern CCS at leads of 4-7 weeks. Using a high-resolution ocean reanalysis, we develop a new index designed to characterize the location and intensity of coastally trapped waves propagating through the CCS. We then show that the S2S dynamical forecasts have enhanced skill in forecasts of SSH in weeks 4-7 when initialized with strong or extreme coastally trapped wave conditions, explaining 30-40% more SSH variance than the corresponding persistence forecast.

... We then concatenated these three EEG predictions into one 223 matrix, Z. And, finally, we used the MATLAB built-in function partialcorr (X, Y, Z) where X = 224 the actual recorded EEG, Y = the predicted EEG in response to the feature of interest (the feature 225 whose unique contribution is to be identified, e.g., f), and Z = the concatenated predicted EEGs 226 in response to the other features (features that are to be partialled out). This function computes 227 the partial correlation coefficients between X and Y, while controlling for the variables in Z 228 (Fisher 1924). ...

Humans have the remarkable ability to selectively focus on a single talker in the midst of other competing talkers. The neural mechanisms that underlie this phenomenon remain incompletely understood. In particular, there has been longstanding debate over whether attention operates at an early or late stage in the speech processing hierarchy. One way to better understand this is to examine how attention might differentially affect neurophysiological indices of hierarchical acoustic and linguistic speech representations. In this study, we do this by using encoding models to identify neural correlates of speech processing at various levels of representation. Specifically, we recorded EEG from fourteen human subjects (nine female and five male) during a “cocktail party” attention experiment. Model comparisons based on these data revealed phonetic feature processing for attended, but not unattended speech. Furthermore, we show that attention specifically enhances isolated indices of phonetic feature processing, but that such attention effects are not apparent for isolated measures of acoustic processing. These results provide new insights into the effects of attention on different prelexical representations of speech, insights that complement recent anatomic accounts of the hierarchical encoding of attended speech. Furthermore, our findings support the notion that, for attended speech, phonetic features are processed as a distinct stage, separate from the processing of the speech acoustics.

... For Fisher's z-test (Fisher, 1924), it is assumed that (X, Y, Z) T follows a multivariate normal distribution. Then ρ XY.Z = 0 if and only if X ⊥ ⊥ Y | Z; this is the null hypothesis of Fisher's z-test. ...

Causal discovery algorithms estimate causal graphs from observational data. This can provide a valuable complement to analyses focussing on the causal relation between individual treatment-outcome pairs. Constraint-based causal discovery algorithms rely on conditional independence testing when building the graph. Until recently, these algorithms have been unable to handle missing values. In this paper, we investigate two alternative solutions: Test-wise deletion and multiple imputation. We establish necessary and sufficient conditions for the recoverability of causal structures under test-wise deletion, and argue that multiple imputation is more challenging in the context of causal discovery than for estimation. We conduct an extensive comparison by simulating from benchmark causal graphs: As one might expect, we find that test-wise deletion and multiple imputation both clearly outperform list-wise deletion and single imputation. Crucially, our results further suggest that multiple imputation is especially useful in settings with a small number of either Gaussian or discrete variables, but when the dataset contains a mix of both neither method is uniformly best. The methods we compare include random forest imputation and a hybrid procedure combining test-wise deletion and multiple imputation. An application to data from the IDEFICS cohort study on diet- and lifestyle-related diseases in European children serves as an illustrating example.

... The test statistic is computed as the difference in deviances, and is distributed as a χ 2 with degrees of freedom equal to the difference in number of parameters between the models (Wilks 1938). For example, the G-test for conditional independence (Agresti 2002) is a likelihood-ratio test for multinomial data, while the partial correlation test (Fisher 1924) is a test for multivariate Gaussian data, which is asymptotically equivalent to a likelihood-ratio test using linear regression (Christensen 2011). Interested readers may refer to Sections 2.2 and 6.2 in Tsamardinos et al. (2019) for a more detailed description of likelihood-ratio tests and how to implement them. ...

Most feature selection methods identify only a single solution. This is acceptable for predictive purposes, but is not sufficient for knowledge discovery if multiple solutions exist. We propose a strategy to extend a class of greedy methods to efficiently identify multiple solutions, and show under which conditions it identifies all solutions. We also introduce a taxonomy of features that takes the existence of multiple solutions into account. Furthermore, we explore different definitions of statistical equivalence of solutions, as well as methods for testing equivalence. A novel algorithm for compactly representing and visualizing multiple solutions is also introduced. In experiments we show that (a) the proposed algorithm is significantly more computationally efficient than the TIE* algorithm, the only alternative approach with similar theoretical guarantees, while identifying similar solutions to it, and (b) that the identified solutions have similar predictive performance.

... Partial correlation [37] is a reliable coefficient for preventing the correlation results of two variables being contaminated by other correlation relationships. It is used in this study to avoid the interference of green-blue infrastructure coverage factors. ...

The urban heat island (UHI) effect in cities and its driving factors have long been investigated. 3D buildings are key components of urban structures and have notable effect on UHI effect. However, due to the incomplete 3D building information in urban database, only a few studies investigated the impact of 3D building morphology factors on the land surface temperature (LST). In this study, a total set of 14 2D and 3D building morphology factors were selected to investigate the correlation between UHI and building morphology across 3 megacities in China (Beijing, Wuhan, Shanghai) with Landsat-8 LST scenarios in summer and winter of 2018 and 2019, respectively. Both Spearman correlation and partial correlation analysis were applied at block scale after urban functional zone (UFZ) mapping. A number of significant observations were noted based on the multi-spatial and multi-temporal experimental results: (1) both 2D and 3D building morphology factors influence LST, among them building coverage ratio (BCR), building ground area divided by facade area (GA2FA) and sky view factor (SVF) yield stronger correlations with LST; (2) UFZ mapping scheme and partial correlation method help with controlling the anthropogenic heat release, greenspace, and water bodies coverage variables when targeting at the influence of building morphology factors on LST. This study provides insights for building morphology design in future urban planning and management.

... Correlations between the different salivary biomarkers were evaluated. Since salivary biomarkers usually correlate with the protein concentration in saliva, partial correlations were calculated following the procedure of Fisher (1924), in which TP was used as the control variable in order to minimize spurious correlations. Statistical analyses were performed using the SPSS statistics package (IBM SPSS Statistics for Windows, Version 26.0. ...

Salivary biomarkers were studied in 17 healthy Large White sows from early gestation to the end of lactation. Saliva samples were obtained at 34 ± 3 days from insemination (G30), 24 ± 4 days before farrowing (G90), within the first 24 h after farrowing (L1) and at the end of a lactation period of 21 days (L21). The measurements in saliva included stress-related biomarkers (cortisol, chromogranin A, α-amylase, cholinesterase [BChE] and lipase [Lip]), inflammatory biomarkers (adenosine deaminase isoenzymes 1 [ADA1] and 2 [ADA2], and haptoglobin [Hp]) and oxidative stress biomarkers (cupric reducing antioxidant capacity, trolox equivalent antioxidant capacity, ferric reducing ability, uric acid, advanced oxidation protein products [AOPP] and hydrogen peroxide [H2O2]), as well as routine biochemistry analytes (aspartate aminotransferase [AST], alkaline phosphatase [ALP], γ-glutamine transferase [GGT], lactate dehydrogenase [LDH], creatine kinase [CK], urea, creatinine, triglycerides, lactate, calcium and phosphorus).
The main changes were observed at farrowing, with increases in biomarkers of stress (cortisol and BChE), inflammation (ADA isoenzymes and Hp) and oxidative stress (AOPP and H2O2), as well as muscle and hepatic enzymes (CK, AST, ALP, GGT and LDH). Lactate and triglycerides increased at the end of gestation and remained at high concentrations until the end of lactation. Lip was higher in gestation than at lactation. Thus, changes in biomarkers of stress, immune function, oxidative stress, hepatic and muscle integrity, and energy mobilization occur in sow saliva during pregnancy, farrowing and lactation. These changes, caused by physiological conditions, should be taken into consideration when these biomarkers are used for the evaluation of sow health and welfare.

... In case X is one-dimensional, the F-statistic is also equivalent to the square of the partial correlation [29,30], which is used in Anderson and Robinson [18]. The partial correlation is the sample Pearson correlation of RY and RX, ...

Permutation testing in linear models, where the number of nuisance coefficients is smaller than the sample size, is a well-studied topic. The common approach of such tests is to permute residuals after regressing on the nuisance covariates. Permutation-based tests are valuable in particular because they can be highly robust to violations of the standard linear model, such as non-normality and heteroscedasticity. Moreover, in some cases they can be combined with existing, powerful permutation-based multiple testing methods. Here, we propose permutation tests for models where the number of nuisance coefficients exceeds the sample size. The performance of the novel tests is investigated with simulations. In a wide range of simulation scenarios our proposed permutation methods provided appropriate type I error rate control, unlike some competing tests, while having good power.

... In a geometrical sense, Pearson's pair-wise correlation is a cosine value of the angle formed by two vectors, and is thus not additive. As a result, the variance stabilization methods of Fisher's transformation (Fisher (1915) and Fisher (1924)) should be used to estimate the average value. We accordingly define an averaging operator as avgðr 1 ; r 2 ; …; r n Þ : ¼ ðexpð2zÞ À 1Þ=ðexpð2zÞ þ 1Þ, where r 1 ; r 2 ; …; r n are the sample correlation elements and.z ¼ 1 n P n i¼1 0:5 log 1þri 1Àri groups A and B are allowed to have overlapping elements (or can be even identical) due to the condition, i 6 ¼ j, which excludes trivial self-correlations for the overlapping stocks. ...

The return on assets of the investment universe tends to form a cluster structure. This study quantifies this strength of the clustering tendency as a single econometric measure, referred to as modularity. Through an empirical study of the US equity market, we demonstrate that the strength of the clustering tendency changes over time with market fluctuations. That is, normal markets tend to have a clear cluster structure (high modularity), while stressed markets tend to have a blurry cluster structure (low modularity). Modularity assesses the quality of an investment opportunity set in terms of potential diversification benefits. Modularity is an important pricing variable in the cross-sectional returns of US stocks. From 1992 to 2015, the average return of the stocks with the lowest sensitivity to modularity (low modularity beta) exceeds that of the stocks with the highest sensitivity (high modularity beta) by approximately 10.49% annually, adjusted for the Fama-French five-factor exposures. The inclusion of modularity as an asset pricing factor, therefore, expands the investment opportunity set for factor-based investors.

... We used the paired sample t-test to study within-subjects' statistics for craving score and within-subjects' EEG power while they watched NFG and FG videos, and ANCOVA, a technique for analyzing grouped data having covariate (in this study, EEG power while participants watched neutral videos) to study between-subjects' statistics for EEG power in the HC and IGD groups while they watched NFG and FG videos (Keselman et al., 1998). Partial correlation was used to analyze the correlation between self-reported craving score and EEG features on game-play videos controlling two covariates (self-reported craving score and EEG features on neutral videos) (Fisher, 1924). We adjusted our statistical significance by performing the Bonferroni correction because we had 12 indicators [areas of interest (prefrontal, central and parietooccipital areas) × frequency band of interest (relative delta, theta, alpha and beta power)]. ...

Recently, the World Health Organization included "gaming disorder" in its latest revision of the international classification of diseases (ICD-11). Despite extensive research on internet gaming disorder (IGD), few studies have addressed game-related stimuli eliciting craving, which plays an important role in addiction. Particularly, most previous studies did not consider personal preferences in games presented to subjects as stimuli. In this study, we compared neurophysiological responses elicited for favorite game (FG) videos and non-favorite game (NFG) videos. We aimed to demonstrate neurophysiological characteristics according to the game preference in the IGD group. We measured participants' EEG while they watched FG, NFG and neutral videos. For FG videos, the parieto-occipital theta power (TPPO) were significantly increased compared with those for NFG videos (p<0.05, paired t-test). TPPO also differed significantly between the healthy control and IGD groups only on FG videos controlling covariate (TPPO on neutral videos) (p<0.05, ANCOVA). And TPPO was significantly correlated to self-reported craving score only on FG videos (r = 0.334, p<0.05). In the present study, we demonstrate that FG videos induce higher TPPO than that induced by NFG videos in the IGD group and TPPO is a reliable EEG feature associated with craving for gaming.

... Nonetheless, many tests exist and are commonly used in practice, despite their various theoretical deficiencies. A classic approach is to form a test statistic from the partial correlation coefficient [8]. This vanishes if X ⊥ ⊥ Y | Z, but only under the strong assumptions that all variables are Gaussian and all dependences linear. ...

... For interpretation purposes, rs from treatment outcome studies were reversed so that positive correlations represent positive scores on supervision process associating with better client outcomes. To correct for non-normality of the correlation effect size and stabilize variance estimation, rs were converted to z scores for the meta-analysis and then back-transformed to rs for interpretation (Fisher, 1924). Variability in effect sizes across studies was examined using the I 2 statistic, which quantifies the amount of "true" between-study heterogeneity in effect sizes not due to chance (Huedo-Medina, Sánchez-Meca, Marín-Martinez, & Botella, 2006). ...

Clinical supervision is deemed an essential element in the development of therapist competence and provision of psychotherapy to clients. However, the association between supervision and psychotherapy process and outcome has been mixed, unclear, and presumed to vary widely given the idiosyncratic features of the supervision and therapy process. Thus, to provide an up-to-date (articles published until May, 2019) quantitative summary, we conducted a meta-analytic review to examine the associations between supervision variables and psychotherapy process and outcome variables including: therapeutic relationship, client satisfaction, and treatment outcomes. Using a random effects
model, the pooled Pearson’s correlation between supervision and psychotherapy process and outcome variables was .21 across 12 studies (32 effects) that were included. Thus, supervision accounted for 4% of the variance in client outcomes. Approximately 54% of the total variance between studies was due to heterogeneity and not to chance. An additional meta-analysis without the 4 studies that assessed client outcomes using supervisor/therapist ratings yielded a slightly higher correlation (r .24), accounting for 6% of the variance in client outcomes. Effect sizes regarding the therapeutic relationship and client satisfaction varied widely while effect sizes for treatment outcomes were less varied with consistently small positive effects. Supervisory working alliance was most frequently examined in assessing supervision and accounted for wider variance in effect sizes. There seemed to be less variance among specific supervision factors (e.g., style, satisfaction, structure) with consistent small to medium positive effects. Implications for future research are discussed.

... A widely used strategy for adjusting for covariates is to use regression models (Fisher, 1924;Baba et al., 2004;Li and Shepherd, 2010). Specifically, we assume that the variable of interest Y k (k = 1, 2, . . . ...

Partial association refers to the relationship between variables Y1,Y2,…,YK while adjusting for a set of covariates X={X1,…,Xp}. To assess such an association when Yk’s are recorded on ordinal scales, a classical approach is to use partial correlation between the latent continuous variables. This so-called polychoric correlation is inadequate, as it requires multivariate normality and it only reflects a linear association. We propose a new framework for studying ordinal-ordinal partial association by using surrogate residuals (Liu and Zhang, JASA, 2018). We justify that conditional on X, Yk and Yl are independent if and only if their corresponding surrogate residual variables are independent. Based on this result, we develop a general measure ϕ to quantify association strength. As opposed to polychoric correlation, ϕ does not rely on normality or models with the probit link, but instead it broadly applies to models with any link functions. It can capture a non-linear or even non-monotonic association. Moreover, the measure ϕ gives rise to a general procedure for testing the hypothesis of partial independence. Our framework also permits visualization tools, such as partial regression plots and 3-D P-P plots, to examine the association structure, which is otherwise unfeasible for ordinal data. We stress that the whole set of tools (measures, p-values, and graphics) is developed within a single unified framework, which allows a coherent inference. The analyses of the National Election Study (K = 5) and Big Five Personality Traits (K = 50) demonstrate that our framework leads to a much fuller assessment of partial association and yields deeper insights for domain researchers.

... We tested for associations between phylogenetic turnover, geographical distance, and environmental distance using partial correlations, with the association between phylogenetic turnover and geographical distance being conditioned on the environmental distance matrix (Fisher 1924). We structured our analysis to provide a test of stated hypotheses in response to earlier criticisms which regarded biogeographical approaches as merely "a narrative addition to phylogenetic studies" (Crisp et al. 2011). ...

Speciation is thought to be predominantly driven by the geographical separation of populations of the ancestral species. Yet, in the marine realm, there is substantial biological diversity despite a lack of pronounced geographical barriers. Here, we investigate this paradox by considering the biogeography of marine mammals: cetaceans (whales and dolphins) and pinnipeds (seals and sea lions). We test for associations between past evolutionary diversification and current geographical distributions, after accounting for the potential effects of current environmental conditions. In general, cetacean lineages are widely dispersed and show few signs of geographically driven speciation, albeit with some notable exceptions. Pinnipeds, by contrast, show a more mixed pattern, with true seals (phocids) tending to be dispersed, whereas eared seals (otariids) are more geographically clustered. Both cetaceans and pinnipeds show strong evidence for environmental clustering of their phylogenetic lineages in relation to factors such as sea temperature, the extent of sea ice, and nitrate concentrations. Overall, current marine mammal biogeography is not indicative of geographical speciation mechanisms, with environmental factors being more important determinants of current species distributions. However, geographical isolation appears to have played a role in some important taxa, with evidence from the fossil record showing good support for these cases.

... A similar segregation pattern is also evident in the spatial distribution of the gender mobility gap: the gap increases significantly when moving from the wealthiest to the most deprived comunas of Santiago. In particular, we measure the semipartial Pearson correlation coefficient (Fisher, 1924) between R S , RN l and the GSE ratio, by controlling for the variations in the call activity by gender and the differences in the sex ratio across comunas. Both R S and RN l are strongly and negatively correlated with the GSE ratio, with correlation coefficients r = −0.59 ...

Mobile phone data have been extensively used to study urban mobility. However, studies based on gender-disaggregated large-scale data are still lacking, limiting our understanding of gendered aspects of urban mobility and our ability to design policies for gender equality. Here we study urban mobility from a gendered perspective, combining commercial and open datasets for the city of Santiago, Chile. We analyze call detail records for a large cohort of anonymized mobile phone users and reveal a gender gap in mobility: women visit fewer unique locations than men, and distribute their time less equally among such locations. Mapping this mobility gap over administrative divisions, we observe that a wider gap is associated with lower income and lack of public and private transportation options. Our results uncover a complex interplay between gendered mobility patterns, socio-economic factors and urban affordances, calling for further research and providing insights for policymakers and urban planners.

... Although the empirical partial correlation coefficientρ ij|S l would be an obvious test statistic, it has the drawback that it is not normally distributed under the null hypothesis (Hotelling, 1953). Fisher (1924) suggested transforming the partial correlation coefficient into the z-statistic ...

Causal discovery algorithms aim to identify causal relations from observational data and have become a popular tool for analysing genetic regulatory systems. In this work, we applied causal discovery to obtain novel insights into the genetic regulation underlying head‐and‐neck squamous cell carcinoma. Some methodological challenges needed to be resolved first. The available data contained missing values, but most approaches to causal discovery require complete data. Hence, we propose a new procedure combining constraint‐based causal discovery with multiple imputation. This is based on using Rubin's rules for pooling tests of conditional independence. A second challenge was that causal discovery relies on strong assumptions and can be rather unstable. To assess the robustness of our results, we supplemented our investigation with sensitivity analyses, including a non‐parametric bootstrap to quantify the variability of the estimated causal structures. We applied these methods to investigate how the high mobility group AT‐Hook 2 (HMGA2) gene is incorporated in the protein 53 signalling pathway playing an important role in head‐and‐neck squamous cell carcinoma. Our results were quite stable and found direct associations between HMGA2 and other relevant proteins, but they did not provide clear support for the claim that HMGA2 itself is a key regulator gene.

... After a while and with increasing volume of submissions from all over the world, the Journal stabilized as quarterly with issues appearing in April, August and December (at least approximately). During the first two decades, many well known papers appeared in the journal, Fisher [5,6], detailing the application of the results previously appeared in Fisher [4] on the distribution of the sample correlation coefficient, and the partial correlation coefficient, Tschuprow [10,11] on optimal allocation in stratified sampling, anticipating some of the ideas in Neyman [8], De Finetti [1], on the probability law of extremes, and Wilks [12], who chose the Journal as the appropriate outlet for some of the main results of his PhD thesis. This is just to mention a few, and it clearly gives a proof of the scientific reputation that the Editor and the Journal had at the time. ...

... In particular, strength-based metrics (Jones et al., 2019;Newman, 2010) are useful for developing hypotheses related to the edge weights of central nodes. This is perhaps unsurprising, given that they can be calculated using a population parameter with a known distribution (e.g., a partial correlation, Fisher, 1924;Yule, 1897). Together, centrality indices provide untapped sources of information that can be used to narrow the focus on to particular aspects of an estimated network. ...

Network psychometrics is undergoing a time of methodological reflection. In part, this was spurred by the revelation that l1-regularization does not reduce spurious associations in partial correlation networks. In this work, we address another motivation for the widespread use of regularized estimation: the thought that it is needed to mitigate overfitting. We first clarify important aspects of overfitting and the bias-variance tradeoff that are especially relevant for the network literature, where the number of nodes or items in a psychometric scale are not largecompared to the number of observations (i.e., a low p/n ratio). This revealed that bias and especially variance are most problematic in p=n ratios rarely encountered. We then introduce a nonregularized method, based on classical hypothesis testing, that fulfills two desiderata: (1) reducing or controlling the false positives rate and (2) quelling concerns of overfitting by providing accurate predictions. These were the primary motivations for initially adopting the graphical lasso (glasso). In several simulation studies, our nonregularized method provided more than competitive predictive performance, and, in many cases, outperformed glasso. Itappears to be nonregularized, as opposed to regularized estimation, that best satisfies these desiderata. We then provide insights into using our methodology. Here we discuss the multiple comparisons problem in relation to prediction: stringent alpha levels, resulting in a sparse network, can deteriorate predictive accuracy. We end by emphasizing key advantages of our approach that make it ideal for both inference and prediction in network analysis.

... LHS, or stratified sampling without replacement, is an efficient implementation of Monte Carlo simulation that requires fewer samples than random sampling. PRCC is partial correlation coefficient (PCC) (Fisher, 1924;Marino et al., 2008) calculated on the ranks instead of values. PCC is the correlation between a Fig. 2. Graph of optimal control u * ðtÞ on the left and modified optimal control u * s ðtÞ on the right. ...

Wolbachia is a bacterium that is present in 60% of insects but it is not generally found in Aedes aegypti, the primary vector responsible for the transmission of dengue virus, Zika virus, and other human diseases caused by RNA viruses. Wolbachia has been shown to stop the growth of a variety of RNA viruses in Drosophila and in mosquitoes. Wolbachia-infected Ae. aegypti have both reproductive advantages and disadvantages over wild types. If Wolbachia-infected females are fertilized by either normal or infected males, the offspring are healthy and Wolbachia-positive. On the other hand, if Wolbachia-negative females are fertilized by Wolbachia-positive males, the offspring do not hatch. This phenomenon is called cytoplasmic incompatibility. Thus, Wolbachia-positive females have a reproductive advantage, and the Wolbachia is expanded in the population. On the other hand, Wolbachia-infected mosquitoes lay fewer eggs and generally have a shorter lifespan. In recent years, scientists have successfully released these Wolbachia-adapted mosquitoes into the wild in several countries and have achieved a high level of replacement with Wolbachia-positive mosquitoes. Here, we propose a minimal mathematical model to investigate the feasibility of such a release method. The model has five steady-states two of which are locally asymptotically stable. One of these stable steady-states has no Wolbachia-infected mosquitoes while for the other steady-state, all mosquitoes are infected with Wolbachia. We apply optimal control theory to find a release method that will drive the mosquito population close to the steady-state with only Wolbachia-infected mosquitoes in a two-year time period. Because some of the model parameters cannot be accurately measured or predicted, we also perform uncertainty and sensitivity analysis to quantify how variations in our model parameters affect our results.

... Student's t-test was applied against the null hypothesis of no correlation for the assessment of statistical significance [59]. Estimation of the statistical significance of a variable in time series has been explained in different previous studies [58,60], whereas the significance of lag-1 correlation coefficient was firstly introduced by Anderson in 1942 [61], followed by Kendall and Stuart in 1948 [62]. Higher-order lags were used when the first-order correlation insufficiently depicted the serial dependence [63]. ...

Investigating the influence of sea surface temperatures (SSTs) on seasonal rainfall is a crucial factor for managing Ethiopian water resources. For this purpose, SST and rainfall data were used to study a wide range of inhomogeneous areas in Ethiopia with uneven distribution of rainfall for both summer (1951-2015) and spring (1951-2000) seasons. Firstly, a preliminary subdivision of rainfall grid points into zones was applied depending on spatial homogeneity and seasonality of rainfall. This introduced new clusters, including nine zones for summer rainfall peak (July/August) and five zones for spring rainfall peak (April/May). Afterward, the time series for each zone was derived by calculating the rainfall averaged over grid points within the zone. Secondly, the oceanic regions that significantly correlated with the Ethiopian rainfall were identified through cross-correlations between rainfalls averaged over every homogeneous zone and the monthly averaged SST. For summer rainfall as a main rainy season, the results indicated that the Gulf of Guinea and southern Pacific Ocean had a significant influence on rainfall zones at a lag time of 5-6 and 6-7 months. Besides, for summer rainfall zones 8 and 9 at lag time 5-6 months, the common SST regions of the southern Pacific Ocean showed the opposite sense of positive and negative correlations. Thus, the difference in SSTs between the two regions was more strongly correlated (r ≥ 0.46) with summer rainfall in both zones than others. For the spring season, the results indicated that SST of the northern Atlantic Ocean had a strong influence on spring rainfall zones (3 and 5) at a lag time 6-7 months, as indicated by a significant correlation (r ≥ −0.40). Therefore, this study suggests that SSTs of southern Pacific and northern Atlantic oceans can be used as effective inputs for prediction models of Ethiopian summer and spring rainfalls, respectively.

... Nonetheless, many tests exist and are commonly used in practice, despite their various theoretical deficiencies. A classic approach is to form a test statistic from the partial correlation coefficient (Fisher, 1924). This vanishes if X ⊥ ⊥ Y | Z, but only under the strong assumptions that all variables are Gaussian and all dependences linear. ...

This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Polya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of the underlying distributions in a nonparametric way. The Bayesian perspective provides an inherently symmetric probability measure of conditional dependence or independence, a feature particularly advantageous in causal discovery and not employed by any previous procedure of this type.

... The distribution of the sample partial correlation for a Gaussian distribution was described by Fisher [1924] and we would reject H 0 if the absolute value of a transformed test statistic exceeded the critical value from the Student table evaluated at δ/2. The computational complexity of the partial correlation is O(np 2 + p 3 ) which simplifies to O(np 2 ) as n ≥ p. ...

This dissertation presents novel structured sparse learning methods on graphs that address commonly found problems in the analysis of neuroimaging data as well as other high dimensional and few sample data. The first part of the thesis focuses on developing and utilizing convex relaxations of discrete and combinatorial penalties. These are developed with the aim of learning an interpretable predictive linear model satisfying sparse and graph based constraints. In the context of neuroimaging these models can leverage implicit structured sparsity properties to learn predictive and interpretable models that can inform analysis of neuro-imaging data. In particular we study the problem of statistical estimation with a signal known to be sparse, spatially contiguous, and containing many highly correlated variables. We take inspiration from the k-support norm, which has been successfully applied to sparse prediction problems with correlated features, but lacks any explicit structural constraints commonly found in machine learning and image processing. We address this problem by incorporating a total variation penalty in the k-support framework. We introduce the (k, s) support total variation norm as the tightest convex relaxation of the intersection of a set of discrete sparsity and total variation penalties. We show that this norm leads to an intractable combinatorial graph optimization problem, which we prove to be NP-hard. We then introduce a tractable relaxation with approximation guarantees. We demonstrate the effectiveness of this penalty on classification in the low-sample regime with M/EEG neuroimaging data and fMRI data, as well as image recovery with synthetic and real data background subtracted image recovery tasks. We show that our method is particularly useful compared to existing methods in terms of accuracy, interpretability, andstability. We consider structure discovery of undirected graphical models from observational data. We then consider the problem of learning the structure of graphical models with structured sparse constraints. Functional brain networks are well described and estimated from data with Gaussian Graphical Models (GGMs), e.g. using sparse inverse covariance estimators. In this thesis we make two contributions for estimating Gaussian Graphical Models under various constraints. Our first contribution is to identify differences in GGMs known to have similar structure. We characterize the uncertainty of differences with confidence intervals obtained using a parametric distribution on parameters of a sparse estimator. Sparse penalties enable statistical guarantees and interpretable models even in high-dimensional and low-sample settings. Characterizing the distributions of sparse models is inherently challenging as the penalties produce a biased estimator. Recent work invokes the sparsity assumptions to effectively remove the bias from a sparse estimator such as the lasso. These distributions can be used to give confidence intervals on edges in GGMs, and by extension their differences. However, in the case of comparing GGMs, these estimators do not make use of any assumed joint structure among the GGMs. Inspired by priors from brain functional connectivity we derive the distribution of parameter differences under a joint penalty when parameters are known to be sparse in the difference. This leads us to introduce the debiased multi-task fused lasso, whose distribution can be characterized in an efficient manner. We show how thedebiased lasso and multi-task fused lasso can be used to obtain confidence intervals on edge differences in GGMs. We validate the techniques proposed on a set of synthetic examples as well as neuro-imaging dataset created for the study of autism. Finally, we consider a novel approach to the structure discovery of undirected graphical models from observational data. Although, popular methods rely on estimating a penalized maximum likelihood of the precision matrix, in these approaches structure recovery is an indirect consequence of the data-fit term, the penalty can be difficult to adapt for domain-specific knowledge, and the inference is computationally demanding. By contrast, it may be easier to generate training samples of data that arise from graphs with the desired structure properties. We propose to leverage this latter source of information as training data to learn a function, parametrized by a neural network that maps empirical covariance matrices to estimated graph structures. Learning this function brings two benefits: it implicitly models the desired structure or sparsity properties to form suitable priors, and it can be tailored to the specific problem of edge structure discovery, rather than maximizing data likelihood. Applying this framework, we find our learnable graph-discovery method trained on synthetic data generalizes well: identifying relevant edges in both synthetic and real data, completely unknown at training time. We find that on genetics, brain imaging, and simulation data we obtain performance generally superior to analytical methods.

... Ces données sont moyennées temporellement, et il en résulte une carte moyennant chacun des phénostades sur douze années (1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010). La méthode des coefficients de corrélation partielle permet de calculer la corrélation entre plusieurs paires de variables en prenant en compte la dépendance entre chacune des variables (Fisher, 1923). ...

Climate change occuring during last decade will deeply modify climate and environnemental interaction. Inthis PhD, two sites of European Lon-Term Ecosystem Research Network (French Alps and Britany), specializedin human/nature relationships and environnmental studies were selected for analyzing phenological impact withclimate change constraint. The aim of this work is to determine the spatio-temporal variability of climate andphenology over both sites and to predict the response of forestry phenology under climate change constraint.Meteorological spatio-temporal variability is explained from Météo-France time series analyses and also withclimate reanalysis on the 1959-2009 period. Temperature and precipitation appeared to be best climatologicalvariable to discrimine the impact on vegetation. Since 1987, a significant temperature increase of about 1°C appears(less in the Alps and more in Armorics). Precipitation temporal variability appeared to change near year 1990with a quasi-biennial periods before and with a 6-8 year period after 1990. Phenology is used to monitor naturalforestry dynamics and the feedback on climate on vegetation. Bioclimatics data from « Observatoire des Saisons »and determined from remote sensing data (SPOT-VGT and MODIS datasets) are used to follow spatio-temporalvariability of the phenology. Relationships between climate and phenology are determined by statistical modelisation(degree-day model). SAFRAN-France data and phenological remote sensing data are combined to calibrate and tovalidate of the model in present time. Climate forecast from ALADIN model are used to execute the model on afutur period (2021-2050) with an thermal increase condition of around 1°C and a diminution of precipitation inArmorics (100 mm). Degree-day model predict an advance of growth phase over both site according to climateforcing.

... The relationships between these variables and the sediment and soil organic carbon storage were studied using partial correlation analysis based on the Spearman rank correlation coefficient, which indicates the degree of association between the variables, controlling for effects from all other variables (Fisher, 1924). For the categorical variables, the non-parametric Kruskal-Wallis test was applied to test for differences in the distribution of the mineral sediment and soil organic carbon storage per categorical class, followed by pairwise Wilcoxon rank sum tests to study the significance of the differences in storage between the categorical classes. ...

River floodplains constitute an important element in the terrestrial sediment and organic carbon cycle and store variable amounts of carbon and sediment depending on a complex interplay of internal and external driving forces. Quantifying the storage in floodplains is crucial to understand their role in the sediment and carbon cascades. Unfortunately, quantitative data on floodplain storage are limited, especially at larger spatial scales. Rivers in the Scottish Highlands can provide a special case to study alluvial sediment and carbon dynamics because of the dominance of peatlands throughout the landscape, but the alluvial history of the region remains poorly understood. In this study, the floodplain sediment and soil organic carbon storage is quantified for the mountainous headwaters of the River Dee in eastern Scotland (663 km2), based on a coring dataset of 78 floodplain cross‐sections. Whereas the mineral sediment storage is dominated by wandering gravel‐bed river sections, most of the soil organic carbon storage can be found in anastomosing and meandering sections. The total storage for the Upper Dee catchment can be estimated at 5.2 Mt or 2306.5 Mg ha‐1 of mineral sediment and 0.7 Mt or 323.3 Mg C ha‐1 of soil organic carbon, which is in line with other studies on temperate river systems. Statistical analysis indicates that the storage is mostly related to the floodplain slope and the geomorphic floodplain type, which incorporates the characteristic stream power, channel morphology and the deposit type. Mapping of the geomorphic floodplain type using a simple classification scheme shows to be a powerful tool in studying the total storage and local variability of mineral sediment and soil organic carbon in floodplains.

... Хельмертом (1876) [11], К. Пірсоном (1900) [12], В. Госсетом (1908) [13], Р.А. Фішером (1924) [14]. Достатньо детально математичний апарат статистики описаний у ряді фундаментальних праць, зокрема у книзі Г. Крамера (1946) «Математичні методи статистики». ...

... In this paper, rather than building upon relatively recently introduced statistical procedures (e.g., 1 -based methods), we propose a statistical technique that directly builds upon work from a century ago (Fisher, 1915(Fisher, , 1924Yule, 1907), and thus has a closed form solution. We rst introduce Gaussian graphical models. ...

The Gaussian graphical model (GGM) is an increasingly popular technique used in psychology to characterize relationships among observed variables. These relationships are represented as elements in the precision matrix. Standardizing the precision matrix and reversing the sign yields corresponding partial correlations that imply pairwise dependencies in which the effects of all other variables have been controlled for. The graphical lasso (glasso) has emerged as the default estimation method, which uses ℓ1‐based regularization. The glasso was developed and optimized for high‐dimensional settings where the number of variables (p) exceeds the number of observations (n), which is uncommon in psychological applications. Here we propose to go ‘back to the basics’, wherein the precision matrix is first estimated with non‐regularized maximum likelihood and then Fisher Z transformed confidence intervals are used to determine non‐zero relationships. We first show the exact correspondence between the confidence level and specificity, which is due to 1 minus specificity denoting the false positive rate (i.e., α). With simulations in low‐dimensional settings (p ≪ n), we then demonstrate superior performance compared to the glasso for detecting the non‐zero effects. Further, our results indicate that the glasso is inconsistent for the purpose of model selection and does not control the false discovery rate, whereas the proposed method converges on the true model and directly controls error rates. We end by discussing implications for estimating GGMs in psychology.

... In MRPC, we use the standard Fisher's z transformation for Pearson correlation in all the marginal (Fisher, 1915) tests and for the partial correlation in all the conditional tests. Consider testing conditional (Fisher, 1924) independence between variables x and y conditioned on a set S of other variables. From the correlation matrix, one may estimate the partial correlations using an iterative approach (Kalisch et al., 2012). ...

Although large amounts of genomic data are available, it remains a challenge to reliably infer causal (i. e., regulatory) relationships among molecular phenotypes (such as gene expression), especially when multiple phenotypes are involved. We extend the interpretation of the Principle of Mendelian randomization (PMR) and present MRPC, a novel machine learning algorithm that incorporates the PMR in the PC algorithm, a classical algorithm for learning causal graphs in computer science. MRPC learns a causal biological network efficiently and robustly from integrating individual-level genotype and molecular phenotype data, in which directed edges indicate causal directions. We demonstrate through simulation that MRPC outperforms several popular general-purpose network inference methods and PMR-based methods. We apply MRPC to distinguish direct and indirect targets among multiple genes associated with expression quantitative trait loci. Our method is implemented in the R package MRPC, available on CRAN (https://cran.r-project.org/web/packages/MRPC/index.html).

The partial correlation coefficient quantifies the relationship between two variables while taking into account the effect of one or multiple control variables. Researchers often want to synthesize partial correlation coefficients in a meta-analysis since these can be readily computed based on the reported results of a linear regression analysis. The default inverse variance weights in standard meta-analysis models require researchers to compute not only the partial correlation coefficients of each study but also its corresponding sampling variance. The existing literature is diffuse on how to estimate this sampling variance, because two estimators exist that are both widely used. We critically reflect on both estimators, study their statistical properties, and provide recommendations for applied researchers. We also compute the sampling variances of studies using both estimators in a meta-analysis on the partial correlation between self-confidence and sports performance. This article is protected by copyright. All rights reserved.

Gaussian graphical models (GGMs) provide a framework for modeling conditional dependencies in multivariate data. In this tutorial, we provide an overview of GGM theory and a demonstration of various GGM tools in R. The mathematical foundations of GGMs are introduced with the goal of enabling the researcher to draw practical conclusions by interpreting model results. Background literature is presented, emphasizing methods recently developed for high‐dimensional applications such as genomics, proteomics, or metabolomics. The application of these methods is illustrated using a publicly available dataset of gene expression profiles from 578 participants with ovarian cancer in The Cancer Genome Atlas. Stand‐alone code for the demonstration is available as an RMarkdown file at https://github.com/katehoffshutta/ggmTutorial.

Motivated by the mechanistic model of the resting energy expenditure, we present a new multiple hypothesis testing approach to evaluate organ/tissue-specific resting metabolic rates. The approach is based on generalized marginal regression estimates for a subset of coefficients along with a stepwise multiple testing procedure with a minimization–maximization of the normalized estimates (maximization over all its components and minimization over all possible choices of the subset). The approach offers a valid way to address challenges in multiple hypothesis testing on regression coefficients in linear regression analysis especially when covariates are highly correlated. Importantly, the approach yields estimates that are conditionally unbiased. In addition, the approach controls a family-wise error rate in the strong sense. The approach was used to analyze a real study on resting energy expenditure in 131 healthy adults, which yielded an interesting and surprising result of age-related decrease in resting metabolic rate of kidneys. Simulation studies were also presented with various strengths of multi-collinearity induced by pre-specified correlation in covariates.

Floodplain sediment storage is an important component of a catchment’s sediment budget. Here, we present the first quantification of floodplain sediment storage for tropical river catchments draining to Lake Tana, NW Ethiopia. The catchments are characterized by relatively gently sloping to flat lowlands towards the lake (1788 m a.s.l.) and steeper river reaches in the uplands towards Mount Guna (4120 m a.s.l.). Sediment storage for 65 homogeneous floodplain segments was estimated by combining information on floodplain spatial extent obtained through the field- and remote sensing-based approaches, with information on sediment thickness obtained through sediment coring and the analysis of river cut-banks. Extrapolation to the entire catchment was done making use of floodplain typology and following the classification scheme of Nanson and Croke (1992) and other floodplain properties and catchment-wide variables. The results showed average sediment storage of 284 Mg m⁻¹ river length or 21,760 Mg ha⁻¹ floodplain area for the Gumara and approx. 227 Mg m⁻¹ river length or 16,909 Mg ha⁻¹ floodplain area for the Rib River catchments. Total floodplain sediment storage at the catchments scale, above the upper gauging stations, amounts to 92.9 Mt and 35.5 Mt for the Gumara and Rib River catchments, respectively. At the scale of homogeneous floodplain segments, sediment storage is related to floodplain width, floodplain slope, upstream catchment area, the average slope of the upstream catchment area, sinuosity, and floodplain type. Overall, floodplain geomorphology largely controls sediment storage. Approx. 70 % of the sediment is stored in low-energy cohesive floodplains which only take up 11 % of the total river length in both catchments. For most of the total river and valley extent, where rivers are incised and have more energy, storage is limited. Comparing our findings with similar studies in temperate catchments of Europe and N-America showed that sediment storage in the Gumara and Rib Rivers is lower by approx. 1 order of magnitude, for similar-sized catchments.

Causal discovery algorithms estimate causal graphs from observational data. This can provide a valuable complement to analyses focusing on the causal relation between individual treatment‐outcome pairs. Constraint‐based causal discovery algorithms rely on conditional independence testing when building the graph. Until recently, these algorithms have been unable to handle missing values. In this article, we investigate two alternative solutions: test‐wise deletion and multiple imputation. We establish necessary and sufficient conditions for the recoverability of causal structures under test‐wise deletion, and argue that multiple imputation is more challenging in the context of causal discovery than for estimation. We conduct an extensive comparison by simulating from benchmark causal graphs: as one might expect, we find that test‐wise deletion and multiple imputation both clearly outperform list‐wise deletion and single imputation. Crucially, our results further suggest that multiple imputation is especially useful in settings with a small number of either Gaussian or discrete variables, but when the dataset contains a mix of both neither method is uniformly best. The methods we compare include random forest imputation and a hybrid procedure combining test‐wise deletion and multiple imputation. An application to data from the IDEFICS cohort study on diet‐ and lifestyle‐related diseases in European children serves as an illustrating example.

It is anticipated that a large number of voltage source converters (VSCs) will be integrated into future power systems, which can potentially be detrimental to system stability. Previous work has utilized an impedance-based approach to analyze the stability boundaries of power systems with multiple VSCs. However, impedance-based modeling limits the analysis of the system feasibility region to just two or three dimensions. Hence, this paper develops a methodology based on bifurcation theory that enables the system multi-dimensional feasibility region to be identified efficiently with consideration of multiple different varying parameters. The developed methodology is generalized so that it can be applied to identify system stability boundaries in other power systems with multiple varying parameters. The partial Spearman correlation coefficient is adopted in this paper to identify the key parameters that affect the stability boundary. Additionally, the calculated correlation indices quantify the interactions between the control loops VSCs in the system. The proposed methodology is verified against other analytical methods and the impact of key parameter variation on the system stability boundaries is discussed.

In this paper, we analyse the dynamic partial correlation network of the constituent stocks of S&P Europe 350. We focus on global parameters such as radius, which is
rarely used in financial networks literature, and also the diameter and distance parameters.
The first two parameters are useful for deducing the force that economic instability should
exert to trigger a cascade effect on the network. With these global parameters, we hone the
boundaries of the strength that a shock should exert to trigger a cascade effect. In addition,
we analysed the homophilic profiles, which is quite new in financial networks literature. We
found highly homophilic relationships among companies, considering firms by country and
industry. We also calculate the local parameters such as degree, closeness, betweenness,
eigenvector, and harmonic centralities to gauge the importance of the companies regarding
different aspects, such as the strength of the relationships with their neighbourhood and
their location in the network. Finally, we analysed a network substructure by introducing
the skeleton concept of a dynamic network. This subnetwork allowed us to study the stability
of relations among constituents and detect a significant increase in these stable connections
during the Covid-19 pandemic.

This chapter explores how Fisher, and then Neyman and Pearson, tried to build a theory of statistical inference from the frequency definition of probability. Fisher, despite his rejection of Laplacean inverse probability, remained with the epistemic tradition, and the fundamental tension in his theory was never resolved. Neyman and Pearson developed a more consistent theory, but Neyman’s commitment to the frequency theory led him inexorably to a theory of inductive behavior, to a theory of statistical decision-making. This approach found paradigmatic application to problems of quality control in manufacturing but had little relevance for basic research in psychology.

We consider testing marginal independence versus conditional independence in a trivariate Gaussian setting. The two models are nonnested, and their intersection is a union of two marginal independences. We consider two sequences of such models, one from each type of independence, that are closest to each other in the Kullback–Leibler sense as they approach the intersection. They become indistinguishable if the signal strength, as measured by the product of two correlation parameters, decreases faster than the standard parametric rate. Under local alternatives at such a rate, we show that the asymptotic distribution of the likelihood ratio depends on where and how the local alternatives approach the intersection. To deal with this nonuniformity, we study a class of envelope distributions by taking pointwise suprema over asymptotic cumulative distribution functions. We show that these envelope distributions are well behaved and lead to model selection procedures with rate-free uniform error guarantees and near-optimal power. To control the error even when the two models are indistinguishable, rather than insist on a dichotomous choice, the proposed procedure will choose either or both models.

This study examined the long-term (1951-2015) spatio-temporal trends, variability, and teleconnections of rainfall of 15 districts in the Terai region of Uttar Pradesh, India. Gridded rainfall data of the India Meteorological Department (IMD) were analyzed using both parametric and non-parametric approaches, and teleconnections of seasonal and annual rainfall with Indian Ocean Dipole (IOD) and El Niño/Southern Oscillation (ENSO) were investigated. Lag-1 autocorrelation coefficient was calculated and tested at 5% level of significance. Our analysis revealed significantly declining trends in monthly rainfall for most of the districts in all the months, except February, April, May, and December which had increasing trends. Monthly rainfall values of the region as a whole had significantly decreasing trends in January, July, August, and October, while February and April had significantly increasing trends. In seasonal and annual rainfall data, only decreasing trends were significant. Monsoon, post-monsoon, and annual rainfall were decreasing in 6, 9, and 7 districts, respectively. The study area as a whole had a significant decrease in monsoon, post-monsoon, and annual rainfall with significantly negative Sen's slope (− 2.7, − 0.39, and − 3.75), Spearman's rho (− 0.25, − 0.21, and − 0.30), and slope of simple linear regression (− 2.67, − 0.98, and − 3.49). CV for annual rainfall of the whole region was 19% with maximum variability recorded in post-monsoon rainfall (CV = 99.81%). Our results also revealed that the monsoon, post-monsoon, and annual rainfall of the whole region had significant teleconnections with both IOD and ENSO events. The results herein suggest decreasing rainfall trends in the Terai region of India with monsoon and annual rainfall having higher ENSO teleconnections while the post-monsoon rainfall teleconnection, dominated by IOD.

We use correlation arrays, the workhorse of Bub's (2016) Bananaworld, to analyze the correlations found in an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-1/2 particles. Adopting an approach pioneered by Pitowsky (1989b) and promoted in Bananaworld, we geometrically represent the class of correlations allowed by quantum mechanics in this setup as an elliptope in a non-signaling cube. To determine which of these quantum correlations are allowed by local hidden-variable theories, we investigate which ones we can simulate using raffles with baskets of tickets that have the outcomes for all combinations of measurement settings printed on them. The class of correlations found this way can be represented geometrically by a tetrahedron contained within the elliptope. We use the same Bub-Pitowsky framework to analyze a generalization of the Mermin setup for measurements on the singlet state of two particles with higher spin. The class of correlations allowed by quantum mechanics in this case is still represented by the elliptope; the subclass of those whose main features can be simulated with our raffles can be represented by polyhedra that, with increasing spin, have more and more vertices and facets and get closer and closer to the elliptope. We use these results to advocate for Bubism (not to be confused with QBism), an interpretation of quantum mechanics along the lines of Bananaworld. Probabilities and expectation values are primary in this interpretation. They are determined by inner products of vectors in Hilbert space. Such vectors do not themselves represent what is real in the quantum world. They encode families of probability distributions over values of different sets of observables. As in classical theory, these values ultimately represent what is real in the quantum world. Hilbert space puts constraints on possible combinations of such values, just as Minkowski space-time puts constraints on possible spatio-temporal constellations of events. Illustrating how generic such constraints are, the constraint derived in this paper, the equation for the elliptope, is a general constraint on correlation coefficients that can be found in older literature on statistics and probability theory. Yule (1896) already stated the constraint. De Finetti (1937) already gave it a geometrical interpretation sharing important features with its interpretation in Hilbert space.

Quantitative researchers often discuss research ethics as if specific ethical problems can be reduced to abstract normative logics (e.g., virtue ethics, utilitarianism, deontology). Such approaches overlook how values are embedded in every aspect of quantitative methods, including ‘observations,’ ‘facts,’ and notions of ‘objectivity.’ We describe how quantitative research practices, concepts, discourses, and their objects/subjects of study have always been value-laden, from the invention of statistics and probability in the 1600s to their subsequent adoption as a logic made to appear as if it exists prior to, and separate from, ethics and values. This logic, which was embraced in the Academy of Management from the 1960s, casts management researchers as ethical agents who ought to know about a reality conceptualized as naturally existing in the image of statistics and probability (replete with ‘constructs’), while overlooking that S&P logic and practices, which researchers made for themselves, have an appreciable role in making the world appear this way. We introduce a different way to conceptualize reality and ethics, wherein the process of scientific inquiry itself requires an examination of its own practices and commitments. Instead of resorting to decontextualized notions of ‘rigor’ and its ‘best practices,’ quantitative researchers can adopt more purposeful ways to reason about the ethics and relevance of their methods and their science. We end by considering implications for addressing ‘post truth’ and ‘alternative facts’ problems as collective concerns, wherein it is actually the pluralistic nature of description that makes defending a collectively valuable version of reality so important and urgent.

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