Multi-way Tables Research Articles

Using Exact Tests from Algebraic Statistics in Sparse Multi-Way Analyses: An Application to Analyzing Differential Item Functioning

Asymptotic goodness-of-fit methods in contingency table analysis can struggle with sparse data, especially in multi-way tables where it can be infeasible to meet sample size requirements for a robust application of distributional assumptions. However, algebraic statistics provides exact alternatives to these classical asymptotic methods that remain viable even with sparse data. We apply these methods to a context in psychometrics and education research that leads naturally to multi-way contingency tables: the analysis of differential item functioning (DIF). We explain concretely how to apply the exact methods of algebraic statistics to DIF analysis using the R package algstat, and we compare their performance to that of classical asymptotic methods.

Likelihood Ratio Test and the Evidential Approach for 2 × 2 Tables.

Categorical data analysis of 2 × 2 contingency tables is extremely common, not least because they provide risk difference, risk ratio, odds ratio, and log odds statistics in medical research. A χ2 test analysis is most often used, although some researchers use likelihood ratio test (LRT) analysis. Does it matter which test is used? A review of the literature, examination of the theoretical foundations, and analyses of simulations and empirical data are used by this paper to argue that only the LRT should be used when we are interested in testing whether the binomial proportions are equal. This so-called test of independence is by far the most popular, meaning the χ2 test is widely misused. By contrast, the χ2 test should be reserved for where the data appear to match too closely a particular hypothesis (e.g., the null hypothesis), where the variance is of interest, and is less than expected. Low variance can be of interest in various scenarios, particularly in investigations of data integrity. Finally, it is argued that the evidential approach provides a consistent and coherent method that avoids the difficulties posed by significance testing. The approach facilitates the calculation of appropriate log likelihood ratios to suit our research aims, whether this is to test the proportions or to test the variance. The conclusions from this paper apply to larger contingency tables, including multi-way tables.

Awareness of the Cittaslow Brand among Polish Urban Dwellers and Its Impact on the Sustainable Development of Cities

New ways of promoting sustainable urban development are currently being sought. One of them is the slow city model, which has been implemented by some municipalities. This model forms the basis for cooperation between the cities within the international Cittaslow network. It has been assumed that the Cittaslow brand could generate numerous benefits for cities. It can be a significant regional growth and socio-economic development factor. However, city dwellers have to be familiar with the brand, and they have to understand the philosophy behind the Cittaslow movement. The aim of the present study is to assess the awareness of the Cittaslow brand among the residents of the cities that are members of the Cittaslow network. Data from the questionnaires have been classified into multiway tables. The chi-square test is used to examine the relationship between the attributes. The surveys conducted demonstrate that the residents are not sufficiently familiar with the Cittaslow brand to derive any potential benefits from their network membership. This observation suggests that city dwellers should be educated on the main tenets of the slow city concept. The present study demonstrates that urban residents’ awareness of the Cittaslow brand, the slow city philosophy and the Cittaslow logo is influenced by their gender and age. These results show not only the need for education but also the need to promote the desirable forms of urban development; the respondents were familiar with some activities pertaining to the slow city concept, but they were not familiar with the other activities and did not link them to the Cittaslow movement. The most desirable ways of promoting the territorial Cittaslow brand were also identified. There is a general scarcity of research on the Cittaslow brand in Poland and in the world, and thus further studies are needed to formulate meaningful conclusions with the aim of promoting the development of the Cittaslow network.

An Analysis of Health Issues and Challenges Among Urban Migrant Population in Kerala

Abstract: Background: Urbanisation and migration are interconnected. As per Census 2011, 31.16% of the Indian population was urban, whereas in Kerala, 47.72% of the total population was urban. A study conducted by Gulati Institute of Finance and Taxation showed that there were 25 million migrant labourers in Kerala. The migrant labourers were facing various problems both health related and non health related. The objective of the study was to analyze the health problems, disease pattern, health care seeking behaviour and health care utilization amongst the urban in-migrant population of Kerala. Methods: It was a cross sectional descriptive study conducted among a sample of 1217 migrant labourers from the Ernakulam district of Kerala. The study followed a 30 cluster sampling method. Only those respondents having length of stay in Kerala at least six months were included in the study. The Crosstabs procedure forms two-way and multi-way tables and provided a variety of tests and measures of association for two-way tables. A binary logistic regression was done to outline the predictors of health seeking in facilities for migrant workers Findings: The study found that majority of migrant workers were belonged to backward communities, males married and engaged in construction work. The state of origin of the migrant workers were, West Bengal (35%), Odisha (19%), Tamil Nadu (16%), Assam (14%) and Uttar Pradesh, Bihar. Majority of the respondents had been staying in Kerala for duration between one to six years. Accidents and injuries, skin problems and Non communicable diseases were major health problems among the migrant workers. The logistic regression analysis showed possession of health insurance, monthly income, state of origin and easy connectivity to health facilities are positive predictors for health care seeking for migrant labourers. Conclusion: The present study revealed migrant workers were suffering from many illness like Accidents and injuries, NCD, skin problems etc. Improve the awareness of

ENVIRONMENTAL, SOCIAL AND GOVERNANCE FACTORS IN COMPANIES’ BUSINESS MODELS AND THE MOTIVES OF INCORPORATED THEM IN THE CORE BUSINESS

The aim of the article is to study ESG factors and the motives of including the ESG perspective by companies in their business models (BM). The 2-stage research procedure was used to analyse the problem: bibliographic research was carried out with the use of VOSviewer software. In the paper, 99 from 3000 publications were selected for in-depth analyses, and various types of correlation measures were selected, examining the strength and direction of the relationship between the variables included in two-way and multi-way frequency tables. Fourteen binary variables were constructed with two categories, “yes” and “no”, which were assigned ranks 1 and 0, respectively. It turned out that there is a moderate (0.379) significant correlation between ESG factors and companies’ business models. The environmental factor is most often emphasized in these models (0.308). In the SME sector, CSR (1.000 – full correlation) and ESG factors (0.471) are taken into account, with the strongest relationships between this sector and Government (0.615) and Social (0.549). The role of the financial market in supporting sustainability in companies’ business models is growing and taking on importance.

Quasi point-symmetry models based on f-divergence and decomposition of point-symmetry for multi-way contingency tables

For two-way contingency tables, Tomizawa (1985) considered the quasi point-symmetry (QP) model and the marginal point-symmetry (MP) model, and gave the theorem that the point-symmetry (PS) model holds if and only if both the QP and MP models hold. Tahata and Tomizawa (2008) provided similar theorems for multi-way tables. For multi-way tables, the present paper proposes the quasi point-symmetry (QP[f]) model based on f-divergence. The QP[f] model includes the QP model in a special case. It also gives the theorem that the PS model holds if and only if both the QP[f] and MP models hold, and the theorem that the test statistic for goodness-of-fit of the PS model is asymptotically equivalent to the sum of those for the decomposed models under the PS model.

Moment Symmetry Models and Decompositions of Symmetry for Multi-way Contingency Tables

For square contingency tables, Caussinus[ 1 ] demonstrated that the symmetry model holds if and only if both the quasi-symmetry and marginal homogeneity models hold. Bishop, Fienberg, and Holland[ 2 , p. 307] and Bhapkar and Darroch[ 3 ] provided similar theorems for multi-way tables. The present article proposes a moment symmetry model and unique decompositions of the symmetry model. For two-way tables, the second-order moment symmetry model decomposes the symmetry model into the second-order moment symmetry and marginal homogeneity models, while the [Formula: see text]th-order moment symmetry model decomposes the [Formula: see text]th-order marginal symmetry model using the [Formula: see text]th-order marginal symmetry model for multi-way tables. AMS 2000 subject classification: 62H17

Extended marginal homogeneity models based on complementary log-log transform for multi-way contingency tables

For square contingency tables with ordered categories, Saigusa et al. (2018) proposed the marginal cumulative complementary log-log model being an extension of the marginal homogeneity model. The present paper considers the marginal cumulative complementary log-log and conditional marginal cumulative complementary log-log models for multi-way tables. It also gives the decompositions of the marginal homogeneity model into the proposed model and a model of the equality of marginal means for multi-way tables. An example is given.

Estimation of multi-way tables subject to coherence constraints

Nowadays, traditional population censuses based on total enumeration of the population are being accompanied by sample surveys. Sampling within censuses allows to reduce costs and workload of authorities involved in censuses operations, along with the statistical burden for the people involved in the enumeration. In this paper, we deal with estimation of multi-way contingency tables involving variables measured both via census and sampling. In this framework, two main issues need to be addressed: first of all, sample size for estimating some of the entries of the contingency tables may be too small, delivering estimates prone to huge sampling variability. On the other hand, since estimates of the joint distribution need to be coherent with the marginal distribution of the variable collected via a census, estimation methods need to be coherent with the constraint imposed by marginal distribution of variables measured via census. The problem is tackled via a model-based approach that allows to comply with all coherence constraints following a fairly simple procedure. The merit of the proposed methodology is illustrated by means of a simulation study.

A measure of departure from second-order marginal symmetry for multi-way tables with nominal categories

For multi-way tables, Bhapkar and Darroch (1990) gave the second-order marginal symmetry model. The present paper proposes a measure to represent the degree of departure from the second-order marginal symmetry model. The measure is expressed as the weighted sum of the Shannon entropy. The paper also gives the approximate confidence interval of the measure, and shows relationship between the measure and the trivariate normal distribution. Examples are given.

Decomposition of Symmetry into Ordinal Quasi-Symmetry and Marginal Equimoment for Multi-way Tables

For the analysis of square contingency tables with ordered categories, Agresti (1983) introduced the linear diagonals-parameter symmetry (LDPS) model. Tomizawa (1991) considered an extended LDPS (ELDPS) model, which has one more parameter than the LDPS model. These models are special cases of Caussinus (1965) quasi-symmetry (QS) model. Caussinus showed that the symmetry (S) model is equivalent to the QS model and the marginal homogeneity (MH) model holding simultaneously. For square tables with ordered categories, Agresti (2002, p.430) gave a decomposition for the S model into the ordinal quasi-symmetry and MH models. This paper proposes some decompositions which are different from Caussinus’ and Agresti’s decompositions. It gives (i) two kinds of decomposition theorems of the S model for two-way tables, (ii) extended models corresponding to the LDPS and ELDPS, and the generalized model further for multi-way tables, and (iii) three kinds of decomposition theorems of the S model into their models and marginal equimoment models for multi-way tables. The proposed decompositions may be useful if it is reasonable to assume the underlying multivariate normal distribution.

Orthogonality of Decompositions of Symmetry into Extended Symmetry and Marginal Equimoment for Multi-way Tables with Ordered Categories

Yamamoto, Iwashita, and Tomizawa (2007) gave the decompositions of the symmetry (S) model into the extended S and two or three marginal equimoment models for multi-way tables with ordered categories. However the goodness-of-fit test statistic for the S model is not asymptotically equivalent to the sum of those for the decomposed three or four models. This paper gives, for multi-way tables, the modification of their decompositions such that the test statistic for the S model is asymptotically equivalent to the sum of those for the extended S model and one modified marginal equimoment model, which is the constraints combined the marginal means equality and variances equality or combined the marginal means equality, variances equality and correlations equality. Examples are given.

Decomposition of Point-Symmetry Using Ordinal Quasi Point-Symmetry for Ordinal Multi-Way Tables

For multi-way tables with ordered categories, the present paper gives a decomposition of the point-symmetry model into the ordinal quasi point-symmetry and equality of point-symmetric marginal moments. The ordinal quasi point-symmetry model indicates asymmetry for cell probabilities with respect to the center point in the table.

The Effect of Changing Scores for Multi-way Tables with Open-ended Ordered Categories

Log-linear models are used to analyze the contingency tables. If the variables are ordinal or interval, because the score values affect both the model significance and parameter estimates, selection of score values has importance. Sometimes an interval variable contains open-ended categories as the first or last category. While the variable has openended classes, estimates of the lowermost and/or uppermost values of distribution must be handled carefully. In that case, the unknown values of the first and last classes can be estimated firstly, and then the score values can be calculated. In the previous studies, the unknown boundaries were estimated by using interquartile range (IQR). In this study, we suggested interdecile range (IDR), interpercentile range (IPR), and mid-distance range (MDR) as alternative to IQR to detect the effects of score values on model parameters.

Robust integer programming

We provide a complexity classification of four variants of robust integer programming when the underlying Graver basis is given. We discuss applications to robust multicommodity flows and multiway statistical table problems, and describe an effective parametrization of robust integer programming.

Rapid calculation of exact cell bounds for contingency tables from conditional frequencies

We present a fast method for determining the tightest possible bounds, as well as all feasible values, for the underlying cell counts in a two-way contingency table based on knowledge of the corresponding unrounded conditional probabilities, the sample size, and (optionally) bounds on cells and certain sums of cells. This information can be used in statistical inference procedures and also has potential uses in statistical disclosure control, which deals with protecting privacy and confidentiality when data summaries are released to the public. The problem formally consists of a large number of integer linear knapsack optimizations (two per cell). Here we identify special common structure that allows for efficient reuse among cells of intermediate results within a dynamic programming framework. The method runs very quickly on practical examples, thereby enabling a real-time interactive exploration of disclosure risk for two-way rearrangements of large multi-way tables.

Symmetry and asymmetry models and decompositions of models for contingency tables

For analyzing square contingency tables, Bowker [14] proposed the symmetry model. Caussinus [16] proposed the quasi-symmetry model and gave a decomposition of model such that the symmetry model holds if and only if both the quasi-symmetry and the marginal homogeneity models hold. Bhapkar and Darroch [13] gave the similar theorem for multi-way contingency tables. For square tables and for multi-way tables, the present paper (1) reviews various models of symmetry and asymmetry, (2) reviews the decompositions of models, (3) gives some figures which indicate the relationships among various models, and (4) gives a new decomposition of symmetry model.

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.

Measure of departure from marginal homogeneity using marginal odds for multi-way tables with ordered categories

For square contingency tables with ordered categories, this paper proposes a measure to represent the degree of departure from the marginal homogeneity model. It is expressed as the weighted sum of the power-divergence or Patil–Taillie diversity index, and is a function of marginal log odds ratios. The measure represents the degree of departure from the equality of the log odds that the row variable is i or below instead of i+1 or above and the log odds that the column variable is i or below instead of i+1 or above for every i. The measure is also extended to multi-way tables. Examples are given.

Double symmetry model and its orthogonal decomposition for multi-way tables

For two-way contingency tables, Tomizawa (1985b) gave a theorem that the double symmetry (DS) model holds if and only if both the quasi DS and the marginal DS models hold. The present paper proposes, for multiway tables, the DS, some quasi DS and marginal DS models, and extends Tomizawa℉s theorem into multi-way tables. It also shows that for multi-way tables the likelihood ratio statistic for testing goodness of fit of the DS model is asymptotically equivalent to the sum of those for testing the quasi DS model with some order and the marginal DS with the corresponding order. An example is given.