Summary For square contingency tables with ordered categories, Iki, Tahata and Tomizawa (2012) considered a measure to represent the degree of departure from marginal homogeneity. However, the maximum value of this measure cannot distinguish two kinds of marginal inhomogeneity. The present paper proposes a measure which can distinguish two kinds of marginal inhomogeneity. In particular, the proposed measure is useful for representing the degree of departure from marginal homogeneity when the marginal cumulative logistic model holds.
{"title":"A directional measure for marginal homogeneity in square contingency tables with ordered categories","authors":"Kiyotaka Iki, Hiroshi Nakano, S. Tomizawa","doi":"10.2478/bile-2019-0001","DOIUrl":"https://doi.org/10.2478/bile-2019-0001","url":null,"abstract":"Summary For square contingency tables with ordered categories, Iki, Tahata and Tomizawa (2012) considered a measure to represent the degree of departure from marginal homogeneity. However, the maximum value of this measure cannot distinguish two kinds of marginal inhomogeneity. The present paper proposes a measure which can distinguish two kinds of marginal inhomogeneity. In particular, the proposed measure is useful for representing the degree of departure from marginal homogeneity when the marginal cumulative logistic model holds.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"36 1","pages":"1 - 11"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79270393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary The usefulness of combining methods is examined using the example of microarray cancer data sets, where expression levels of huge numbers of genes are reported. Problems of discrimination into two groups are examined on three data sets relating to the expression of huge numbers of genes. For the three examined microarray data sets, the cross-validation errors evaluated on the remaining half of the whole data set, not used earlier for the selection of genes, were used as measures of classifier performance. Common single procedures for the selection of genes—Prediction Analysis of Microarrays (PAM) and Significance Analysis of Microarrays (SAM)—were compared with the fusion of eight selection procedures, or of a smaller subset of five of them, excluding SAM or PAM. Merging five or eight selection methods gave similar results. Based on the misclassification rates for the three examined microarray data sets, for any examined ensemble of classifiers, the combining of gene selection methods was not superior to single PAM or SAM selection for two of the examined data sets. Additionally, the procedure of heterogeneous combining of five base classifiers—k-nearest neighbors, SVM linear and SVM radial with parameter c=1, shrunken centroids regularized classifier (SCRDA) and nearest mean classifier—proved to significantly outperform resampling classifiers such as bagging decision trees. Heterogeneously combined classifiers also outperformed double bagging for some ranges of gene numbers and data sets, but merging is generally not superior to random forests. The preliminary step of combining gene rankings was generally not essential for the performance for either heterogeneously or homogeneously combined classifiers.
{"title":"Gene selection ensembles and classifier ensembles for medical diagnosis","authors":"M. Ćwiklińska-Jurkowska","doi":"10.2478/bile-2019-0007","DOIUrl":"https://doi.org/10.2478/bile-2019-0007","url":null,"abstract":"Summary The usefulness of combining methods is examined using the example of microarray cancer data sets, where expression levels of huge numbers of genes are reported. Problems of discrimination into two groups are examined on three data sets relating to the expression of huge numbers of genes. For the three examined microarray data sets, the cross-validation errors evaluated on the remaining half of the whole data set, not used earlier for the selection of genes, were used as measures of classifier performance. Common single procedures for the selection of genes—Prediction Analysis of Microarrays (PAM) and Significance Analysis of Microarrays (SAM)—were compared with the fusion of eight selection procedures, or of a smaller subset of five of them, excluding SAM or PAM. Merging five or eight selection methods gave similar results. Based on the misclassification rates for the three examined microarray data sets, for any examined ensemble of classifiers, the combining of gene selection methods was not superior to single PAM or SAM selection for two of the examined data sets. Additionally, the procedure of heterogeneous combining of five base classifiers—k-nearest neighbors, SVM linear and SVM radial with parameter c=1, shrunken centroids regularized classifier (SCRDA) and nearest mean classifier—proved to significantly outperform resampling classifiers such as bagging decision trees. Heterogeneously combined classifiers also outperformed double bagging for some ranges of gene numbers and data sets, but merging is generally not superior to random forests. The preliminary step of combining gene rankings was generally not essential for the performance for either heterogeneously or homogeneously combined classifiers.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"SE-11 1","pages":"117 - 138"},"PeriodicalIF":0.0,"publicationDate":"2019-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84637905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary The main estimation and hypothesis testing procedures are presented for experiments conducted in nested block designs of a certain type. It is shown that, under appropriate randomization, these experiments have the convenient orthogonal block structure. Due to this property, the analysis of experimental data can be performed in a comparatively simple way. Certain simplifying procedures are indicated. The main advantage of the presented methodology concerns the analysis of variance and related hypothesis testing procedures. Under the adopted approach one can perform these analytical methods directly, not by combining the results from analyses based on stratum submodels. The application of the presented theory is illustrated by three examples of real experiments in relevant nested block designs. The present paper is the second in the planned series concerning the analysis of experiments with orthogonal block structure.
{"title":"On a new approach to the analysis of variance for experiments with orthogonal block structure.","authors":"T. Caliński, I. Siatkowski","doi":"10.2478/bile-2018-0011","DOIUrl":"https://doi.org/10.2478/bile-2018-0011","url":null,"abstract":"Summary The main estimation and hypothesis testing procedures are presented for experiments conducted in nested block designs of a certain type. It is shown that, under appropriate randomization, these experiments have the convenient orthogonal block structure. Due to this property, the analysis of experimental data can be performed in a comparatively simple way. Certain simplifying procedures are indicated. The main advantage of the presented methodology concerns the analysis of variance and related hypothesis testing procedures. Under the adopted approach one can perform these analytical methods directly, not by combining the results from analyses based on stratum submodels. The application of the presented theory is illustrated by three examples of real experiments in relevant nested block designs. The present paper is the second in the planned series concerning the analysis of experiments with orthogonal block structure.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"50 1","pages":"147 - 178"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84538137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anderson Cristiano Neisse, Jhessica L. Kirch, Kuang Hongyu
Summary The presence of genotype-environment interaction (GEI) influences production making the selection of cultivars in a complex process. The two most used methods to analyze GEI and evaluate genotypes are AMMI and GGE Biplot, being used for the analysis of multi environment trials data (MET). Despite their different approaches, both models complement each other in order to strengthen decision making. However, both models are based on biplots, consequently, biplot-based interpretation doesn’t scale well beyond two-dimensional plots, which happens whenever the first two components don’t capture enough variation. This paper proposes an approach to such cases based on cluster analysis combined with the concept of medoids. It also applies AMMI and GGE Biplot to the adjusted data in order to compare both models. The data is provided by the International Maize and Wheat Improvement Center (CIMMYT) and comes from the 14th Semi-Arid Wheat Yield Trial (SAWYT), an experiment concerning 50 genotypes of spring bread wheat (Triticum aestivum) germplasm adapted to low rainfall. It was performed in 36 environments across 14 countries. The analysis provided 25 genotypes clusters and 6 environments clusters. Both models were equivalent for the data’s evaluation, permitting increased reliability in the selection of superior cultivars and test environments.
{"title":"AMMI and GGE Biplot for genotype × environment interaction: a medoid–based hierarchical cluster analysis approach for high–dimensional data","authors":"Anderson Cristiano Neisse, Jhessica L. Kirch, Kuang Hongyu","doi":"10.2478/bile-2018-0008","DOIUrl":"https://doi.org/10.2478/bile-2018-0008","url":null,"abstract":"Summary The presence of genotype-environment interaction (GEI) influences production making the selection of cultivars in a complex process. The two most used methods to analyze GEI and evaluate genotypes are AMMI and GGE Biplot, being used for the analysis of multi environment trials data (MET). Despite their different approaches, both models complement each other in order to strengthen decision making. However, both models are based on biplots, consequently, biplot-based interpretation doesn’t scale well beyond two-dimensional plots, which happens whenever the first two components don’t capture enough variation. This paper proposes an approach to such cases based on cluster analysis combined with the concept of medoids. It also applies AMMI and GGE Biplot to the adjusted data in order to compare both models. The data is provided by the International Maize and Wheat Improvement Center (CIMMYT) and comes from the 14th Semi-Arid Wheat Yield Trial (SAWYT), an experiment concerning 50 genotypes of spring bread wheat (Triticum aestivum) germplasm adapted to low rainfall. It was performed in 36 environments across 14 countries. The analysis provided 25 genotypes clusters and 6 environments clusters. Both models were equivalent for the data’s evaluation, permitting increased reliability in the selection of superior cultivars and test environments.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"407 1","pages":"121 - 97"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84870841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary Genotype-by-environment interaction (GEI) is frequently encountered in multi-environment trials, and represents differential responses of genotypes across environments. With the development of molecular markers and mapping techniques, researchers can go one step further and analyse the whole genome to detect specific locations of genes which influence a quantitative trait such as yield. Such a location is called a quantitative trait locus (QTL), and when these QTLs have different expression across environments we talk about QTL-by-environment interaction (QEI), which is the basis of GEI. Good understanding of these interactions enables researchers to select better genotypes across different environmental conditions, and consequently to improve crops in developed and developing countries. In this paper we present an overview of statistical methods and models commonly used to detect and to understand GEI and QEI, ranging from the simple joint regression model to complex eco-physiological genotype-to-phenotype simulation models.
{"title":"An overview of statistical methods to detect and understand genotype-by-environment interaction and QTL-by-environment interaction","authors":"P. Rodrigues","doi":"10.2478/bile-2018-0009","DOIUrl":"https://doi.org/10.2478/bile-2018-0009","url":null,"abstract":"Summary Genotype-by-environment interaction (GEI) is frequently encountered in multi-environment trials, and represents differential responses of genotypes across environments. With the development of molecular markers and mapping techniques, researchers can go one step further and analyse the whole genome to detect specific locations of genes which influence a quantitative trait such as yield. Such a location is called a quantitative trait locus (QTL), and when these QTLs have different expression across environments we talk about QTL-by-environment interaction (QEI), which is the basis of GEI. Good understanding of these interactions enables researchers to select better genotypes across different environmental conditions, and consequently to improve crops in developed and developing countries. In this paper we present an overview of statistical methods and models commonly used to detect and to understand GEI and QEI, ranging from the simple joint regression model to complex eco-physiological genotype-to-phenotype simulation models.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"6 1","pages":"123 - 138"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81439434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary This paper presents an application of Hellwig’s method for selecting concomitant variables under a growth curve model, where the values of the concomitant variables change over time and are the same for all experimental units. The authors present a simple adaptation of the growth curve model to the multiple regression model for which Hellwig’s method applies. The theoretical considerations are applied to the selection of significant concomitant variables for raspberry fruiting.
{"title":"Adapting Hellwig’s method for selecting concomitant variables under a certain growth curve model","authors":"M. Wesołowska-Janczarek, M. Różańska-Boczula","doi":"10.2478/bile-2018-0010","DOIUrl":"https://doi.org/10.2478/bile-2018-0010","url":null,"abstract":"Summary This paper presents an application of Hellwig’s method for selecting concomitant variables under a growth curve model, where the values of the concomitant variables change over time and are the same for all experimental units. The authors present a simple adaptation of the growth curve model to the multiple regression model for which Hellwig’s method applies. The theoretical considerations are applied to the selection of significant concomitant variables for raspberry fruiting.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"53 1","pages":"139 - 146"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73818054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary Linear regression with temporally delayed covariates (distributed-lag linear regression) is a standard approach to lag exposure assessment, but it is limited to a single biomarker of interest and cannot provide insights on the relationships holding among the pathogen exposures, thus precluding the assessment of causal effects in a general context. In this paper, to overcome these limitations, distributed-lag linear regression is applied to Markovian structural causal models. Dynamic causal effects are defined as a function of regression coefficients at different time lags. The proposed methodology is illustrated using a simple lag exposure assessment problem.
{"title":"Linear Markovian models for lag exposure assessment","authors":"Alessandro Magrini","doi":"10.2478/bile-2018-0012","DOIUrl":"https://doi.org/10.2478/bile-2018-0012","url":null,"abstract":"Summary Linear regression with temporally delayed covariates (distributed-lag linear regression) is a standard approach to lag exposure assessment, but it is limited to a single biomarker of interest and cannot provide insights on the relationships holding among the pathogen exposures, thus precluding the assessment of causal effects in a general context. In this paper, to overcome these limitations, distributed-lag linear regression is applied to Markovian structural causal models. Dynamic causal effects are defined as a function of regression coefficients at different time lags. The proposed methodology is illustrated using a simple lag exposure assessment problem.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"39 1","pages":"179 - 195"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88781316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary The consequences of the growing demand for water include a significant deterioration in its quality and a drastic decline in biodiversity, which is a serious threat to the hydrological and biocenotic balance of freshwater ecosystems. A good indicator of aquatic environment quality is macrophytes. Studies on macrophytes are one of the primary elements in the ecological status assessment of surface waters, in accordance with the guidelines of the Water Framework Directive. In Poland, research on the ecological status of rivers with regard to macrophytes has been carried out since 2008, using the Macrophyte Index for Rivers (MIR), which takes into account the number and coverage of macrophyte taxa. An analysis of numbers of species that need to be indicated at a site for valid assessment of the ecosystem was conducted on the basis of studies on macrophytes from 2008–2013, at 60 sites in small lowland rivers with a sandy substrate, of which 20 sites were selected on the most diverse watercourses: the least clean (quality class V), moderate (quality class III), and the cleanest (quality class I). The results of the botanical studies served to assess the completeness of the samples (the number of species recorded at a site) used to evaluate the ecological status of a river. The proposed analyses enabled estimation of the approximate number of species required to determine the MIR for rivers in each quality class.
{"title":"Determining the change point for the error in the Macrophyte Index for Rivers","authors":"Anna Budka","doi":"10.2478/bile-2018-0015","DOIUrl":"https://doi.org/10.2478/bile-2018-0015","url":null,"abstract":"Summary The consequences of the growing demand for water include a significant deterioration in its quality and a drastic decline in biodiversity, which is a serious threat to the hydrological and biocenotic balance of freshwater ecosystems. A good indicator of aquatic environment quality is macrophytes. Studies on macrophytes are one of the primary elements in the ecological status assessment of surface waters, in accordance with the guidelines of the Water Framework Directive. In Poland, research on the ecological status of rivers with regard to macrophytes has been carried out since 2008, using the Macrophyte Index for Rivers (MIR), which takes into account the number and coverage of macrophyte taxa. An analysis of numbers of species that need to be indicated at a site for valid assessment of the ecosystem was conducted on the basis of studies on macrophytes from 2008–2013, at 60 sites in small lowland rivers with a sandy substrate, of which 20 sites were selected on the most diverse watercourses: the least clean (quality class V), moderate (quality class III), and the cleanest (quality class I). The results of the botanical studies served to assess the completeness of the samples (the number of species recorded at a site) used to evaluate the ecological status of a river. The proposed analyses enabled estimation of the approximate number of species required to determine the MIR for rivers in each quality class.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"52 1","pages":"215 - 232"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78352059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Iwańska, A. Oleksy, M. Dacko, B. Skowera, T. Oleksiak, E. Wójcik-Gront
Summary Wheat is one of the modern world’s staple food sources. Its production requires good environmental conditions, which are not always available. However, agricultural practices may mitigate the effects of unfavorable weather or poor-quality soils. The influence of environmental and crop management variables on yield can be evaluated only based on representative long-term data collected on farms through well-prepared surveys.The authors of this work analyzed variation in winter wheat yield among 3868 fields in western and eastern Poland for 12 years, as dependent on both soil/weather and crop management factors, using the classification and regression tree (CART) method. The most important crop management deficiencies which may cause low wheat yields are insufficient use of fungicides, phosphorus deficiency, non-optimal date of sowing, poor quality of seeds, failure to apply herbicides, lack of crop rotation, and use of cultivars of unknown origin not suitable for the region. Environmental variables of great importance for the obtaining of high yields include large farm size (10 ha or larger) and good-quality soils with stable pH. This study makes it possible to propose strategies supporting more effective winter wheat production based on the identification of characteristics that are crucial for wheat cultivation in a given region.
{"title":"Use of classification and regression trees (CART) for analyzing determinants of winter wheat yield variation among fields in Poland","authors":"M. Iwańska, A. Oleksy, M. Dacko, B. Skowera, T. Oleksiak, E. Wójcik-Gront","doi":"10.2478/bile-2018-0013","DOIUrl":"https://doi.org/10.2478/bile-2018-0013","url":null,"abstract":"Summary Wheat is one of the modern world’s staple food sources. Its production requires good environmental conditions, which are not always available. However, agricultural practices may mitigate the effects of unfavorable weather or poor-quality soils. The influence of environmental and crop management variables on yield can be evaluated only based on representative long-term data collected on farms through well-prepared surveys.The authors of this work analyzed variation in winter wheat yield among 3868 fields in western and eastern Poland for 12 years, as dependent on both soil/weather and crop management factors, using the classification and regression tree (CART) method. The most important crop management deficiencies which may cause low wheat yields are insufficient use of fungicides, phosphorus deficiency, non-optimal date of sowing, poor quality of seeds, failure to apply herbicides, lack of crop rotation, and use of cultivars of unknown origin not suitable for the region. Environmental variables of great importance for the obtaining of high yields include large farm size (10 ha or larger) and good-quality soils with stable pH. This study makes it possible to propose strategies supporting more effective winter wheat production based on the identification of characteristics that are crucial for wheat cultivation in a given region.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"1 1","pages":"197 - 214"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83083453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary Data arranged in a two-way contingency table can be obtained as a result of many experiments in the life sciences. In some cases the categorized trait is in fact conditioned by an unobservable continuous variable, called liability. It may be interesting to know the relationship between the Pearson correlation coefficient of these two continuous variables and the entropy function measuring the corresponding relation for categorized data. After many simulation trials, a linear regression was estimated between the Pearson correlation coefficient and the normalized mutual information (both on a logarithmic scale). It was observed that the regression coefficients obtained do not depend either on the number of observations classified on a categorical scale or on the continuous random distribution used for the latent variable, but they are influenced by the number of columns in the contingency table. In this paper a known measure of dependency for such data, based on the entropy concept, is applied.
{"title":"Entropy as a measure of dependency for categorized data","authors":"E. Skotarczak, A. Dobek, K. Moliński","doi":"10.2478/bile-2018-0014","DOIUrl":"https://doi.org/10.2478/bile-2018-0014","url":null,"abstract":"Summary Data arranged in a two-way contingency table can be obtained as a result of many experiments in the life sciences. In some cases the categorized trait is in fact conditioned by an unobservable continuous variable, called liability. It may be interesting to know the relationship between the Pearson correlation coefficient of these two continuous variables and the entropy function measuring the corresponding relation for categorized data. After many simulation trials, a linear regression was estimated between the Pearson correlation coefficient and the normalized mutual information (both on a logarithmic scale). It was observed that the regression coefficients obtained do not depend either on the number of observations classified on a categorical scale or on the continuous random distribution used for the latent variable, but they are influenced by the number of columns in the contingency table. In this paper a known measure of dependency for such data, based on the entropy concept, is applied.","PeriodicalId":8933,"journal":{"name":"Biometrical Letters","volume":"28 1","pages":"233 - 243"},"PeriodicalIF":0.0,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82328573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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