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Testing Differences between Populations with Eigenvectors

100%
Stelmach J., Katowice U. o. E.
Acta Universitatis Lodziensis. Folia Oeconomica
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2013
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vol. 285
EN
Testing differences between multivariate populations is one of a crucial problems in statistical investigations. The most known – MANOVA tests being parametric ones need to fulfill the assumptions about the conformity with multivariate normal distribution. Very often these assumptions are practically unrealistic or the verification, especially for small number of observations is hard. This paper presents an approach, based on permutation tests (no needs of verification mentioned assumptions), where proposed test statistics base on the properties of eigenvectors. The investigations were carried out for simulated and real multivariate datasets, where the permutation tests were compared with variable-based and MANOVA test statistics.
PL
Testowanie różnic pomiędzy populacjami wielowymiarowymi jest jednym z kluczowych problemów w badaniach statystycznych. Najbardziej znane – testy MANOVA, jako parametryczne wymagają spełnienia założenia o zgodności z rozkładem normalnym wielowymiarowym. Bardzo często założenia te są praktycznie nierealne lub ich weryfikacja, szczególnie dla małej ilości obserwacji jest trudna. Artykuł ten przedstawia podejście, oparte o testy permutacyjne (co zwalnia z weryfikacji powyższych założeń), gdzie proponowane statystyki testowe oparte są o własności wektorów własnych. Badania zostały przeprowadzone dla symulowanych i rzeczywistych zestawów danych, gdzie testy permutacyjne zostały porównane z testami opartymi na analizie zmiennych i statystykach testowych w MANOVA.
2
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On Estimation of a Quantity of Base Models with Parametric z and Permutation Tests

100%
Stelmach J., Katowice U. o. E.
Acta Universitatis Lodziensis. Folia Oeconomica
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2013
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vol. 286
EN
One of the crucial problems in multiple-model approach of the regression is estimation of optimal number of base models. If the quantity is too low – it increases the prediction error whereas too high number of models increases time and complication of calculations. Unfortunately, the estimation of the quantity of base models based on the analysis of prediction error can lead to its overestimation. This paper proposes a formal approach where the predictions obtained with the models aggregated from different number of base models are compared. In this approach both: parametric and permutation tests were applied with the empirical data from petroleum industry.
PL
Jednym z kluczowych problemów w wielomodelowym podejściu do zagadnienia regresji jest estymacja optymalnej ilości modeli bazowych. Jeśli ich ilość jest zbyt mała – rośnie błąd predykcji, zbyt duża ilość powiększa czas i komplikację obliczeń. Niestety estymacja tej ilości na podstawie analizy błędu predykcji może prowadzić do jej przeszacowania. W artykule proponuje się formalne podejście, w którym porównywane są wyniki prognoz otrzymanych z modeli zagregowanych z różnej liczby modeli bazowych. W tym przypadku wykorzystane zostały zarówno testy parametryczne jak i testy permutacyjne, a jako dane testowe: dane empiryczne wykorzystywane w przemyśle rafineryjnym.
3
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Comparison of the beef prices in selected countries of the European Union

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Jaworski S.
Metody Ilościowe w Badaniach Ekonomicznych
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2012
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vol. 13
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issue 2
31-39
EN
Functional data analysis is used to examine beef price differences in selected countries of the European Union from 2006 to 2011. The prices are modeled as functional observations. The analysis is conducted in three steps relating to three kinds of functional data analysis. First the observations are smoothed with roughness penalty. Then functional principal analysis is applied. Finally functional analysis of variance is used to reveal significant difference between two given groups of countries.
4

On the Use of Permutation Tests in the Significance Testing of Response Surface Function Parameters

100%
Złotoś M.
Argumenta Oeconomica Cracoviensia
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2020
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issue 1(22)
21-29
EN
Objective: The methods of experimental design were first used in agricultural experiments performed by R. A. Fisher. The development of experimental design methods took place along with their effective use in production companies. The most frequently used designs of experiments are the factorial designs. One of the stages in the factorial design of experiments is the estimation of the response surface function formula which describes the influence of factors on the response variable values. The aim of this article is to propose a method to indicate the factors which have a significant influence on the response variable. Research Design & Methods: In this case, in the classical approach, the t-test of the significance of particular parameters of the response surface function is used. The t-test requires fulfilment of the assumptions about the distribution and independence of the model errors. If the assumptions are not fulfilled, or the sample size is not sufficient, the use of the t-test is unjustified. An alternative approach to verify the significance of response surface parameters is a permutation test. Permutation tests use simulation methods and do not entail the fulfilment of strict assumptions relating to the distribution of errors and the sample size of experimental data. Findings: The paper deals with the use of a permutation test that allows us to assess the significance of response surface function parameters when the quantity of experimental data is small. These results were obtained using parametric tests and permutation tests. Implications/Recommendations: Based on the performed calculations, it was found that it is possible to use permutation tests to analyse the response surface function, especially when the assumptions about the residuals of the model are not fulfilled or the number of considered experimental trials is small. Contribution: A proper analysis of the response surface function is an important stage in the design of experiments. In the case of a small quantity of experimental data, assessment of the significance of the model and the parameters of the response surface function using parametric tests may lead to incorrect conclusions. Therefore, the use of permutation tests was indicated as an alternative approach in the analysis of the response surface function.
5
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ON GENERATING MULTIVARIATE SAMPLES WITH ARCHIMEDEAN COPULAS

100%
Stelmach J.
Acta Universitatis Lodziensis. Folia Oeconomica
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2014
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vol. 3
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issue 302
EN
Archimedean copulas are one of the most known classes of copulas. They allow modeling the dependencies between variables with small number of parameters. This paper presents a method designated to generate multivariate samples of the same distribution like primary sample with Archimedean copulas. Such generator may be used in Monte Carlo investigations to create multivariate samples. Apart from theoretical considerations there are presented the examples of application of the method. All the calculations were carried out with R 2.15.0 packages.
6
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ON THE METHOD OF COMPARING STRUCTURES BASED ON THE DATA IN THE CONTINGENCY TABLES

100%
Polko D.
Acta Universitatis Lodziensis. Folia Oeconomica
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2014
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vol. 3
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issue 302
EN
Comparison of populations is one of the most important problems in statistics. The most common comparisons apply to two populations, but comparisons of k populations, where k > 2 are also carried out. Parametric methods allow to compare the means, variances or proportions. The non-parametric methods allow to compare the distributions of two or more populations. The problem of comparison structures based on data in contingency tables is analyzed in the paper. The permutation tests were applied in the multivariate nominal data structure comparison.
7
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Using Permutation Tests in Multiple Correlation Investigation

88%
Stelmach J., Uniwersytet Ekonomiczny w Katowicach; Wydział Zarządzania; Katedra Statystyki
Acta Universitatis Lodziensis. Folia Oeconomica
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2012
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vol. 269
EN
An indication of correlation between dependent variable and predictors is a crucial point in building statistical regression model. The test of Pearson correlation coefficient – with relatively good power – needs to fulfill the assumption about normal distribution. In other cases only non-parametric tests can be used. This article presents a possibility and advantages of permutation tests with the discussion about proposed test statistics. The power of proposed tests was estimated on the basis of Monte Carlo experiments. The investigations were carried out for real data – a sample of refinery process parameters, where the indication of changes in correlation, even for sample with small size is very important. It creates an opportunity to react to changes and update statistical models quickly and keep acceptable quality of prediction
8

A Comparative Study of the Power of Parametric and Permutation Tests for a Multidimensional Two-sample Location Problem

88%
Polko-Zając D.
Argumenta Oeconomica Cracoviensia
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2020
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issue 2(23)
69-79
EN
Objective: A comparison of multidimensional populations is a very interesting and common statistical problem. It most often involves verifying a hypothesis about the equality of mean vectors in two populations. The classical test for verification of this hypothesis is the Hotelling’s T2 test. Another solution is to use simulation and randomization methods to test the significance of differences between the studied populations. Permutation tests are to enable statistical inference in situations where it is not possible to use classical parametric tests. These tests are supposed to provide comparable power to parametric tests with a simultaneous reduction of assumptions, e.g. regarding the sample size taken or the distribution of the tested variable in the population. The purpose of this study is a comparative analysis of the parametric test, the (usual) permutation test, and the nonparametric permutation procedure using two-stage ASL determination. Research Design & Methods: The study considered the analysis of multivariate data. The paper presents theoretical considerations and refers to the Monte Carlo simulation. Findings: The article presents a permutational, complex procedure for assessing the overall ASL (achieved significance level) value. The applied nonparametric statistical inference procedure uses combining functions. A simulation study was carried out to determine the size and power of the test under normality. A Monte Carlo simulation made it possible to compare the empirical power of this test with that of Hotelling’s T2 test. The most powerful test was the permutation test based on a two-stage ASL determination method using the Fisher combining function. Implications/Recommendations: The advantage of the proposed method is that it can be used even when samples are taken from any type of continuous distributions in a population. Contribution: The proposed test can be used in the analysis of multidimensional economic phenomena.
9
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On permutation location–scale tests

88%
Polko-Zając D.
Statistics in Transition new series
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2019
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vol. 20
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issue 4
153-166
EN
Statisticians are constantly looking for methods of statistical inference that would be both effective and would require meeting as few assumptions as possible. Permutation tests seem to fit here, as using them makes it possible to perform statistical inference in situations where classical parametric tests do not work. Permutation tests appear to be comparably powerful to parametric tests, but require meeting fewer assumptions, e.g. regarding the size of the sample or the from of distribution of the tested variable in a population. The presented tests make it possible to verify the overall hypothesis about the identity of both location and scale parameters in the studied populations. In literature, the Lepage test and the Cucconi test are most often referred to in this context. The paper considers various forms of test statistics, and presents a simulation study carried out to determine the size and power of the tests under normality. As the study demonstrated, the advantage of the proposed method is that it can be applied to small-size samples. A nonparametric, complex procedure was used to assess the overall ASL (achieved significance level) value by applying the permutation principle. For comparative purposes, the results for the permutation Lepage test and the permutation Cucconi test are also presented.
10

O pewnej modyfikacji testu adaptacyjnego dla równości wartości oczekiwanych

63%
Kończak G.
Zeszyty Naukowe Uniwersytetu Ekonomicznego w Krakowie
|
2015
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issue 4(940)
5-17
PL
W opracowaniu przedstawiono propozycję modyfikacji testu adaptacyjnego dla porównania wartości oczekiwanych w dwóch populacjach. W zaproponowanym rozwiązaniu nie dokonuje się wyboru postaci statystyki testowej, lecz na podstawie danych pochodzących z wylosowanej próbki modyfikowane są wagi występujące w statystyce testowej. Własności rozważanego testu i testów klasycznych zostały porównane z wykorzystaniem symulacji komputerowych. Test może być wykorzystany w procedurach kontroli jakości. Nie wymaga on spełnienia ostrych założeń dotyczących postaci rozkładu zmiennej diagnostycznej i z tego powodu może być wykorzystywany do wykrywania rozregulowania procesu w przypadkach, gdy nie jest znana postać rozkładu zmiennej.
EN
The paper presents a proposal of a modification of the L. Hao and D. Houser adaptive test for comparing the locations of two distributions. The modification is based on the linear combination of three test statistics. In the Hao and Houser test, due to the values of the robust asymmetry and shape characteristics, the test statistic is chosen. A method of continuous modification of the test statistic is presented. The properties of the proposed procedure are analyzed in a Monte Carlo study. The proposal could be used in quality control monitoring processes.
11
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On Testing Significance of the Multivariate Rank Correlation Coefficient

51%
Kończak G.
Acta Universitatis Lodziensis. Folia Oeconomica
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2018
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vol. 3
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issue 335
21-34
PL
Współczynnik korelacji rang Spearmana pozwala na badanie siły zależności między dwiema zmiennymi, dla których dokonano pomiaru na skali porządkowej. W literaturze są prezentowane rozszerzenia tego współczynnika na przypadek wielowymiarowy. W tych konstrukcjach wykorzystywane są zwykle funkcje łączące (kopule). W artykule przedstawiono propozycję testowania istotności zależności wielowymiarowej dla danych mierzonych na skali rangowej. Przedstawiony test dla istotności wielowymiarowego współczynnika korelacji rang wykorzystuje metodę permutacyjną. Własności proponowanego testu scharakteryzowano z wykorzystaniem symulacji komputerowych.  
EN
The Spearman’s rho is a measure of the strength of the association between two variables. There are some extensions of this coefficient for the multivariate case. Measures of the multivariate association which are the generalisation of the bivariate Spearman’s rho are considered in the literature. These measures are based on copula functions. This article presents a proposal of the testing for the multivariate Spearman’s rank correlation coefficient. The proposed test is based on the permutation method. The test statistic used in the permutation test is based on the empirical copula function. The properties of the proposed method have been described using computer simulations.  
12
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On the Use of Permutation Tests in the Analysis of the Factorial Design of Experiment Results

51%
Złotoś M.
Acta Universitatis Lodziensis. Folia Oeconomica
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2019
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vol. 4
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issue 343
123-136
PL
Metody planowania eksperymentów są wykorzystywane w statystycznej kontroli jakości procesu produkcyjnego. Właściwe planowanie eksperymentów przed realizacją procesu produkcyjnego prowadzi do poprawy jego rezultatów technologicznych, co w efekcie powoduje poprawę rezultatów ekonomicznych procesu. W ostatnich latach na znaczeniu zyskały metody repróbkowania, wykorzystujące symulacje komputerowe. Jedną z nich są testy permutacyjne służące do weryfikacji hipotez statystycznych. W porównaniu do testów parametrycznych nie wymagają one spełnienia restrykcyjnych założeń i mogą być stosowane do niewielkiej liczby obserwacji. Przedmiotem artykułu jest wskazanie możliwości wykorzystania testów permutacyjnych w analizie wyników eksperymentu. Rozważania przeprowadzone zostały dla danych dotyczących rezultatów ustalonego procesu produkcyjnego.
EN
An experimental design is one of the tools which are used in statistical quality control. The proper implementation of experimental design results in the improvement of technological outcomes of a manufacturing process, which in turn leads to the enhancement of economic results. Permutation tests, among other things, form a group of resampling methods which are used to verify statistical hypotheses. These tests, unlike parametric ones, do not entail the fulfilment of strict criteria and may be used for a small number of observations. The presented article deals with the use of permutation tests in the design of experiments. The proposed method will be presented with reference to selected empirical data.
13
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On the Monitoring Complex Multivariate Processes

51%
Rajda-Tasior A., Kończak G.
Acta Universitatis Lodziensis. Folia Oeconomica
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2016
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vol. 3
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issue 322
PL
W artykule przedstawiono propozycję metody monitorowania złożonych wielowymiarowych procesów produkcyjnych. Rozważany problem dotyczy monitorowania jakości produkcji przy stosowaniu oceny alternatywnej jednocześnie względem wielu charakterystyk, gdy produkcja wykonywana jest na wielu różnych stanowiskach. Do opisu stanu jakości w czasie t wykorzystywana jest macierz, w której elementami są liczby wadliwych sztuk wykonanych na danym stanowiskuwedług ocenianych wielu charakterystyk.Proponowana metoda odwołuje się do testu permutacyjnego. Sygnał o nieprawidłowym przebiegu produkcji jest uzyskiwany na podstawie porównania macierzy z bieżącego okresu dla monitorowanego procesu oraz macierzy danych uzyskanej z procesu ustabilizowanego. Ze względu na dużą liczbę charakterystyk rejestrowanych na skali porządkowej konstrukcja statystyki testowej została oparta o funkcję odległości macierzy. Własności proponowanej metody zostały poddane analizie z wykorzystaniem symulacji komputerowych. Przeprowadzono również porównania wyników uzyskanych z zastosowaniem proponowanej metody i karty kontrolnej c.
EN
This article presents a proposal of the method of monitoring complex multidimensional processes. The problem relates to monitoring the quality of production with some attribute variables when the production is performed by some operators. To describe the quality status we used the matrix in which elements are the numbers of defective units.The proposed method uses permutation tests. The "out-of-order" signal is obtained by comparing the matrix in period t to the matrix from stable process. The test statistic used in permutation test is based on a function of distance between matrices. The properties of the proposed method have been described using computer simulation.
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