Three-way Tables Research Articles

Evaluation of Some Uniformity Coefficients for Pressurized Irrigation Systems under Field Conditions in Duhok Governorate (iraqi Kurdistan)

The study was conducted in May–October 2021 within the Duhok University, Iraq, the study was carried out on the land adjacent to the College of Agricultural Engineering Sciences, in the fields of the “Seajay” village latitude 360 55’ 28’’N, longitude 420 52’ 36’’E and 640 (m.a.s.l), the objective of the study was focus on evaluating different uniformity coefficients proposed on investigating the effects of run time and field conditions on the results obtained utilizing those coefficients, a layout of 144 m2 was designed according to the length of lateral lines, 36 catch cans were positioned in a grid of 2 m apart. determining water distribution efficiency affected by different weather conditions during two- daily runs (morning &evening), and two time periods (May & June-July 2021) under operating pressures of (1.0 - 2.0) bar and riser heights of (1.0 - 2 m), solid site sprinkler with a single nozzle type (5mm) was used. The water caught in Cans was subjected to a three-way (ANOVA) table using (LSD) test, the study concluded that riser heights of 1 m had the optimum (CU & DU%) at 2 bar (operating pressure) to reduce water losses, and the results showed that highly significant differences between the means of (CU & DU%) coefficients in (May 2021) at various run times (morning & evening), in time it was shown highly significant differences between the means of all parameters with (PELQ %) coefficients in the (June –July 2021) period. It is not suggested to run the sprinkler irrigation system at wind speeds of (2.6 - 3.6 ms-1)

Assessing the prevalence and association between physical, emotional, and sexual of intimate partner violence against women in Nigeria

BackgroundSeveral studies were carried out on prevalence and associated factors of physical, emotional, and sexual violence against women. However, little attention was given to a comprehensive study that assesses the association between physical, emotional, and sexual violence against women. Thus, this study aimed to assess the association between physical, emotional, and sexual violence against women and their prevalence.MethodsA retrospective cross-sectional design was implemented based on the 2018 Nigeria Demographic and Health Survey involving 8061 married women aged 15–49. A log-linear statistical model for the three-way table was used to assess the association between emotional, physical, and sexual violence. SAS statistical software was used for data management and parameter estimation.ResultsAmong a total of 8061 women considered in the study 3022 (37.49%), 4216 (52.3%) and 1186 (14.71%) women have experienced physical, emotional, and sexual violence, respectively. The estimated odds of the interaction between emotional and physical violence (e1.9281 = 6.876); physical and sexual violence (e−2.0529 = 0.128) were significantly differ from 1.0 with p-values < 0:0001 and 0.0201, respectively.ConclusionOver 33 percent of women experienced at least one incident of physical, emotional, or sexual violence in their lifetime. Physical violence against women has a significant association with emotional and sexual violence. However, it does not imply physical violence causes the other violence since cross-sectional data used for the analysis and other factors were not taken into consideration. The lack of a three-way association between emotional, physical, and sexual violence was also perceived. Therefore, as the prevalence of intimate partner violence against women s high, Nigeria as a country needs to strive to reduce it with the collaboration of other nations in the world to achieve Sustainable Development Goal (SDG). Design and apply guidelines to aware of the community about intimate partner violence against women and besides, take appropriate sentencing on those who commit the violence are the better approaches to prevent violence. Traditional habits that might be the cause of violence should be avoided to reduce or prevent the burden of women due to violence.

The quantification of Simpson’s paradox and other contributions to contingency table theory

The analysis of contingency tables is a powerful statistical tool used in experiments with categorical variables. This study improves parts of the theory underlying the use of contingency tables. Specifically, the linkage disequilibrium parameter as a measure of two-way interactions applied to three-way tables makes it possible to quantify Simpson's paradox by a simple formula. With tests on three-way interactions, there is only one that determines whether the partial interactions of all variables agree or whether there is at least one variable whose partial interactions disagree. To date, there has been no test available that determines whether the partial interactions of a certain variable agree or disagree, and the presented work closes this gap. This work reveals the relation of the multiplicative and the additive measure of a three-way interaction. Another contribution addresses the question of which cells in a contingency table are fixed when the first- and second-order marginal totals are given. The proposed procedure not only detects fixed zero counts but also fixed positive counts. This impacts the determination of the degrees of freedom. Furthermore, limitations of methods that simulate contingency tables with given pairwise associations are addressed.

Phytoplankton Genera Structure Revealed from the Multispectral Vertical Diffuse Attenuation Coefficient

The composition of phytoplankton and the concentration of pigments in their cells make their absorption and specific absorption coefficients key parameters for bio-optical modeling. This study investigated whether the multispectral vertical diffuse attenuation coefficient of downward irradiance (Kd) gradients could be a good framework for accessing phytoplankton genera. In situ measurements of remote sensing reflectance (Rrs), obtained in an Amazon Floodplain Lake (Lago Grande do Curuai), were used to invert Kd, focusing on Sentinel-3/Ocean and Land Color Instrument (OLCI) sensor bands. After that, an analysis based on the organization of three-way tables (STATICO) was applied to evaluate the relationships between phytoplankton genera and Kd at different OLCI bands. Our results indicate that phytoplankton genera are organized according to their ability to use light intensity and different spectral ranges of visible light (400 to 700 nm). As the light availability changes seasonally, the structure of phytoplankton changes as well. Some genera, such as Microcystis, are adapted to low light intensity at 550–650 nm, therefore high values of Kd in this range would indicate the dominance of Microcysts. Other genera, such as Aulacoseira, are highly adapted to harvesting blue-green light with higher intensity and probably grow in lakes with lower concentrations of colored dissolved organic matter that highly absorbs blue light (405–498). These findings are an important step to describing phytoplankton communities using orbital data in tropical freshwater floodplains. Furthermore, this approach can be used with biodiversity indexes to access phytoplankton diversity in these environments.

A New Algorithm for Computing Disjoint Orthogonal Components in the Parallel Factor Analysis Model with Simulations and Applications to Real-World Data

In this paper, we extend the use of disjoint orthogonal components to three-way table analysis with the parallel factor analysis model. Traditional methods, such as scaling, orthogonality constraints, non-negativity constraints, and sparse techniques, do not guarantee that interpretable loading matrices are obtained in this model. We propose a novel heuristic algorithm that allows simple structure loading matrices to be obtained by calculating disjoint orthogonal components. This algorithm is also an alternative approach for solving the well-known degeneracy problem. We carry out computational experiments by utilizing simulated and real-world data to illustrate the benefits of the proposed algorithm.

Dynamics of HDI Index: Temporal Dependence Based on D-vine Copulas Model for Three-Way Data

The HDI (Human Development Index) is a widely used index based on the average of measures of health, education, and income. It assesses the progress of countries worldwide. The publicly available data set associated with the HDI can be seen as a table with 3 dimensions (three-way table): countries, indexes regarding progress, and years (from 2010 to 2018). Thus, modeling the serial dependence structure of this type of intricate three-way tables is a challenge. D-vine copulas are a special class of multivariate copulas that are particularly suited for modeling serial dependence. This work aims to assess the evolution of the dependence relationship between the indexes of the HDI data set over time through D-vine copulas, which has not been fully used before in the area, as far as we are concerned. We tested our approach to European and African countries and compare their results.

Assessing the Italian tax courts system by weighted three-way log-ratio analysis

Deciding on what should be the most suitable reforms of a national judicial system, whether this means reorganising a courts' structure or offering incetives to judges, requires more empirical evidence. The efficiency of the public judicial system is important to any functioning democracy as it affects several aspects of citizens’ life. As a consequence, new statistical methodologies are required for further valuable insights into the issues that affect the effectiveness of a judicial system.The current study attempts to fill this gap by introducing a new statistical method to investigate the heterogeneity among Italian judicial courts. Here we propose a generalisation of log-ratio analysis for the analysis of the association between the variables of a three-way contingency table. Our focus of this investigation will be on the tax court system of the Italian judicial system.Log-ratio analysis presents many advantages, two of which are the computation and the visual representation of the odds ratios on which the analysis is based. For three-way tables, a conditional odds ratio reflects the likelihood that an event (say tax disputes) will take place in a particular geographical area (say North Italy) instead of an another (say South Italy) given a specific condition (type of justice court).Interestingly, the three-way log-ratio analysis presented in this paper allows one to visually describe these conditional odds-ratios in terms of point distances in a biplot between the types of justice court located in different geographical areas of Italy.

Graphical depiction of three-way association in contingency table using higher-order singular value decomposition Tucker3

The anaysis of the association between the variables of a three-way table requires a very different approach to the conventional correspondence analysis techniques. The conventional technique involves by collapsing three-way table into two-way table form. By doing so, no information about three-way interaction among variables. To included interaction among three variables at once, we consider higher-order singular value decomposition (HOSVD) Tucker3. Through the HOSVD Tucker3 decomposition, we can derive marginal of row, column, annd tube categories. Using those marginals, we can easily visualize dependence of the three categories. As a case study, Tucker3 is applied to household data which containing information on race, educational attainment and employment status.

Transformed statistics for tests of conditional independence in [formula omitted] contingency tables

Tests of the hypothesis of conditional independence in J×K×L contingency tables are considered. An expression to approximate the null distribution of the test statistics is derived. Using this expression, transformed statistics are obtained which converge to a chi-square limiting distribution faster than the original statistics do. Simulations are used to compare the transformed statistics with the original ones, and transformed statistics are proposed based on a Bartlett-type adjustment. Through this work and earlier ones, we cover testing hierarchical loglinear models in three-way tables except for a model having three interaction terms.

Estimating Cross-Classified Population Counts of Multidimensional Tables: An Application to Regional Australia to Obtain Pseudo-Census Counts

Abstract Estimating population counts for multidimensional tables based on a representative sample subject to known marginal population counts is not only important in survey sampling but is also an integral part of standard methods for simulating area-specific synthetic populations. In this article several estimation methods are reviewed, with particular focus on the iterative proportional fitting procedure and the maximum likelihood method. The performance of these methods is investigated in a simulation study for multidimensional tables, as previous studies are limited to 2 by 2 tables. The data are generated under random sampling but also under misspecification models, for which sample and target populations differ systematically. The empirical results show that simple adjustments can lead to more efficient estimators, but generally, at the expense of increased bias. The adjustments also generally improve coverage of the confidence intervals. The methods discussed in this article along with standard error estimators, are made freely available in the R package mipfp. As an illustration, the methods are applied to the 2011 Australian census data available for the Illawarra Region in order to obtain estimates for the desired three-way table for age by sex by family type with known marginal tables for age by sex and for family type.

Analysis on the Association Among Stunting, Wasting and Underweight in Malawi: An Application of a Log-Linear Model for the Three-Way Table.

Childhood undernutrition is an important public health problem. Many studies have investigated the factors of childhood undernutrition, but not the association between the undernutrition indicators. This study aimed at investigating the association between the childhood undernutrition indicators. A loglinear model of cell counts of a three way table of stunting, wasting, and underweight was fitted based on the 2010 Malawi demographic health survey data. Interaction terms in the model depicted deviations from independence. A multiple correspondence analysis of undernutrition indicators was also plotted to have a visual impression of association of the undernutrition variables. A loglinear model showed that underweight was associated with both stunting (P<0.001), and wasting (P<0.001). There was no association between stunting and wasting (P=1). Furthermore there was no three way association of stunting, wasting and underweight (P=1). Lack of three way interaction of stunting, wasting and underweight means that childhood undernutrition multidimensional nature is still valid, and no each indicator can represent the other.

Visualising interactions in bi- and triadditive models for three-way tables

This paper concerns the visualisation of interaction in three-way arrays. It extends some standard ways of visualising biadditive modelling for two-way data to the case of three-way data. Three-way interaction is modelled by the Parafac method as applied to interaction arrays that have main effects and biadditive terms removed. These interactions are visualised in three and two dimensions. We introduce some ideas to reduce visual overload that can occur when the data array has many entries. Details are given on the interpretation of a novel way of representing rank-three interactions accurately in two dimensions. The discussion has implications regarding interpreting the concept of interaction in three-way arrays.

Huge tables and multicommodity flows are fixed-parameter tractable via unimodular integer Carathéodory

The three-way table problem is to decide if there exists an l×m×n table satisfying given line sums, and find a table if yes. Recently, it was shown to be fixed-parameter tractable with parameters l,m. Here we extend this and show that the huge version of the problem, where the variable side n is encoded in binary, is also fixed-parameter tractable with parameters l,m. We also conclude that the huge multicommodity flow problem with a huge number of consumers is fixed-parameter tractable. One of our tools is a theorem about unimodular monoids which is of interest on its own right.

Estimates for cell counts and common odds ratio in three-way contingency tables by homogeneous log-linear models with missing data

Missing observations often occur in cross-classified data collected during observational, clinical, and public health studies. Inappropriate treatment of missing data can reduce statistical power and give biased results. This work extends the Baker, Rosenberger and Dersimonian modeling approach to compute maximum likelihood estimates for cell counts in three-way tables with missing data, and studies the association between two dichotomous variables while controlling for a third variable in \( 2\times 2 \times K \) tables. This approach is applied to the Behavioral Risk Factor Surveillance System data. Simulation studies are used to investigate the efficiency of estimation of the common odds ratio.

Dissecting the space-time structure of tree-ring datasets using the partial triadic analysis.

Tree-ring datasets are used in a variety of circumstances, including archeology, climatology, forest ecology, and wood technology. These data are based on microdensity profiles and consist of a set of tree-ring descriptors, such as ring width or early/latewood density, measured for a set of individual trees. Because successive rings correspond to successive years, the resulting dataset is a ring variables × trees × time datacube. Multivariate statistical analyses, such as principal component analysis, have been widely used for extracting worthwhile information from ring datasets, but they typically address two-way matrices, such as ring variables × trees or ring variables × time. Here, we explore the potential of the partial triadic analysis (PTA), a multivariate method dedicated to the analysis of three-way datasets, to apprehend the space-time structure of tree-ring datasets. We analyzed a set of 11 tree-ring descriptors measured in 149 georeferenced individuals of European larch (Larix decidua Miller) during the period of 1967–2007. The processing of densitometry profiles led to a set of ring descriptors for each tree and for each year from 1967–2007. The resulting three-way data table was subjected to two distinct analyses in order to explore i) the temporal evolution of spatial structures and ii) the spatial structure of temporal dynamics. We report the presence of a spatial structure common to the different years, highlighting the inter-individual variability of the ring descriptors at the stand scale. We found a temporal trajectory common to the trees that could be separated into a high and low frequency signal, corresponding to inter-annual variations possibly related to defoliation events and a long-term trend possibly related to climate change. We conclude that PTA is a powerful tool to unravel and hierarchize the different sources of variation within tree-ring datasets.

Huge multiway table problems

Optimization over l×m×n integer threeway tables is NP-hard already for fixed l=3, but solvable in polynomial time with both l,m fixed. Here we consider huge tables, where the variable dimension n is encoded in binary. Combining recent results on Graver bases and recent results on integer cones, we show how to handle such problems in polynomial time. We also show that a harder variant of the problem lies in both NP and coNP. Our treatment goes through the more general class of n-fold integer programming problems.

Estimating the number of zero-one multi-way tables via sequential importance sampling

In 2005, Chen et al. introduced a sequential importance sampling (SIS) procedure to analyze zero-one two-way tables with given fixed marginal sums (row and column sums) via the conditional Poisson (CP) distribution. They showed that compared with Monte Carlo Markov chain (MCMC)-based approaches, their importance sampling method is more efficient in terms of running time and also provides an easy and accurate estimate of the total number of contingency tables with fixed marginal sums. In this paper, we extend their result to zero-one multi-way ( $$d$$ -way, $$d \ge 2$$ ) contingency tables under the no $$d$$ -way interaction model, i.e., with fixed $$d-1$$ marginal sums. Also, we show by simulations that the SIS procedure with CP distribution to estimate the number of zero-one three-way tables under the no three-way interaction model given marginal sums works very well even with some rejections. We also applied our method to Samson’s monks data set.

Application of direct calibration in multivariate image analysis of heterogeneous materials

Many scientific instruments produce multivariate images characterized by three-way tables, an element of which represents the intensity value at a spatial location for a given spectral channel. A problem frequently encountered is to attempt estimating the contributions of some compounds at each location of these images. Usual regression methods of calibration, such as PLS, require having a matrix of calibration X (n×p) and the corresponding vector y of the dependent variable (n×1). X can be built up by sampling pixel-vectors in the images, but y is sometimes difficult to obtain, if the surface of the samples is formed by chemically heterogeneous regions. In this case, the quantitative analyses related to y may be difficult, if the pixels represent very small areas (for example on microscopic images) or very large ones (satellite images). This is for example the case when dealing with biological solid samples representing different tissues. Direct Calibration (DC), sometimes referred to as “spectral unmixing”, do not require having such a calibration set. However, it is indeed needed to have both a matrix of “perturbing” pixel-vectors (noted K) and a vector of the “pure” component spectrum to be analyzed (p), which are more easily obtainable. For estimating the contribution, the unknown pixel vector x and the pure spectrum p are first projected orthogonally onto K giving the vectors x⊥ onto p⊥, respectively. The contribution is then estimated by a second projection of x⊥ onto p⊥. A method, based on principal component analysis, for determining the optimal dimensions of K is proposed. DC was applied on a collection of multivariate images of kernel of wheat to estimate the proportion of three tissues, namely out-layers, “waxy”endosperm and normal endosperm. The eventual results are presented as images of wheat kernels in false colors associated to the estimated proportions of the tissues. It is shown that DC is appropriate for estimating contributions in situations in which the more usual methods of calibration cannot be applied.

Dynamic Markov Bases

This article presents a computational approach for generating Markov bases for multiway contingency tables whose cell counts might be constrained by fixed marginals and by lower and upper bounds. Our framework includes tables with structural zeros as a particular case. Instead of computing the entire Markov bases in an initial step, our framework finds sets of local moves that connect each table in the reference set with a set of neighbor tables. We construct a Markov chain on the reference set of tables that requires only a set of local moves at each iteration. The union of these sets of local moves forms a dynamic Markov basis. We illustrate the practicality of our algorithms in the estimation of exact p-values for a three-way table with structural zeros and a sparse eight-way table. This article has online supplementary materials.

Conditional symmetry models for three-way contingency tables

We generalize the classical conditional or triangular symmetry model for I×I contingency tables to three-way I×I×I tables with commensurable ordinal classification variables. The construction of the new family of models is such that the desirable property that connects conditional symmetry to complete symmetry and marginal homogeneity models in two-way tables is retained in three-way tables. Furthermore, connections between our proposed models obey a coherent structure. We provide maximum likelihood estimation for the new models which is illustrated with a real data example.