This study investigates how outliers of different magnitude impact the robustness of the twosample t-test. A simulation study approach is used to analyze the behavior of type I error rates when outliers are added to generated data. Outliers may distort parameter estimates such as the mean and variance and cause misleading test results. Previous research has shown that Welch’s ttest performs better than the traditional Student’s t-test when group variances are unequal. Therefore these two alternative statistics are compared in terms of type I error rates when outliers are added to the samples. The results show that control of type I error rates can be maintained in the presence of a single outlier. Depending on the magnitude of the outlier and the sample size, there are scenarios where the t-test is robust. However, the sensitivity of the t-test is illustrated by deteriorating type I error rates when more than one outlier are included. The comparison between Welch’s t-test and Student’s t-test shows that the former is marginally more robust against outlier influence.

Η μέθοδος αναγνώρισης μέσω του τρόπου πληκτρολόγησης αποτελεί μία μέθοδο αναγνώρισης βιομετρικών χαρακτηριστικών με στόχο να ελαχιστοποιηθεί ο κίνδυνος κλοπής των προσωπικών κωδικών των πελατών ενός συστήματος. Το παρόν βιομετρικό σύστημα βασίζεται στο σενάριο ότι ο ρυθμός με τον οποίο ένα πρόσωπο πληκτρολογεί είναι ξεχωριστός. Το βιομετρικό σύστημα έχει δύο λειτουργίες, την εγγραφή των πελατών στο σύστημα και τη σύγκριση. Για την εγγραφή απαραίτητη είναι η εξαγωγή των προτύπων των πελατών τα οποία αποθηκεύονται στη βάση δεδομένων του συστήματος ενώ για στη σύγκριση το πρότυπο του χρήστη συγκρίνεται με το πρότυπο του πελάτη που ισχυρίζεται ότι είναι. Στη παρούσα εργασία η εξαγωγή τον προτύπων πραγματοποιείται μέσω μία σειράς αλγοριθμικών διαδικασιών. Αρχικά η μονοδιάστατη χαρακτηριστική χρονοσειρά του χρήστη μετατρέπεται μέσω της μεθόδου Method of Delays σε ένα πολυδιάστατο διάνυσμα που λειτουργεί ως χαρακτηριστικό της ακολουθίας. Στη συνέχεια χρησιμοποιούμε δύο διαφορετικές μεθόδους για να υπολογίσουμε τις ανομοιότητες μεταξύ των πολυδιάστατων διανυσμάτων που προέκυψαν. Οι δύο αυτές μέθοδοι είναι οι Wald-Wolfowitz test και Mutual Nearest Point Distance. Οι τιμές αυτές τοποθετούνται σε έναν πίνακα κάθε στοιχείο του οποίου αναπαριστά την ανομοιότητα μεταξύ δύο χρονοσειρών. Ο πίνακας αυτός μπορεί είτε να αποτελέσει το σύνολο των προτύπων των χρηστών είτε να χρησιμοποιηθεί ως είσοδο στη μέθοδο Multidimensional Scaling που χρησιμοποιείται για μετατροπή του πίνακα ανομοιοτήτων σε διανύσματα και εξαγωγή νέων προτύπων. Τέλος, προτείνουμε ως επέκταση της εργασίας την εκπαίδευση του βιομετρικού συστήματος με χρήση των τεχνικών Support Vector Machines. Για τη λειτουργία της σύγκρισης εξάγουμε πάλι το πρότυπο του χρήστη με την ίδια διαδικασία και το συγκρίνουμε με μία τιμή κατωφλίου. Τέλος, ο έλεγχος της αξιοπιστίας του συστήματος πραγματοποιείται μέσω της χρήσης τριών δεικτών απόδοσης, Equal Error Rate, False Rejection Rate και False Acceptance Rate. / The identification method via keystroke is a method of identifying biometric features in order to minimize the risk of theft of personal codes of customers of a system. The present biometric system based on the scenario that the rate at which a person presses the keyboard buttons is special. The biometric system has two functions, the enrollment of customers in the system and their test. For enrollment, it is necessary to export standards of customers’ information stored in the system database and for the test the standard of the user is compared with the standard of the user that is intended to be the customer. In the present thesis the export of the standards is taken place via a series of algorithmic procedures. Initially,the one dimensional characteristic time series of user is converted, by the technique Method of Delays, in a multidimensional vector that acts as a feature of the sequence. Then, two different methods are used to compute the dissimilarities between multidimensional vectors obtained. These two methods are the Wald-Wolfowitz test and the Mutual Nearest Point Distance. These values are placed in an array, each element of which represents the dissimilarity between two time series. This table can be either the standards of users or can be entry in the Multidimensional Scaling method used to convert the table disparities in vectors and then produce new standards of users. Finally, we propose as extension of our thesis, the training of biometric system with using the techniques of Support Vector Machine. For the test, again the pattern of the user is extracted with the same procedure and is compared to a threshold. Finally, the reliability of the system is carried out through the use of three performance indicators, Equal Error Rate, False Rejection Rate and False Acceptance Rate.

Copy-number variation (CNV) has been implicated in many complex diseases. It is of great interest to detect and locate such regions through genetic association testings. However, the association testings are complicated by the fact that CNVs usually span multiple markers and thus such markers are correlated to each other. To overcome the difficulty, it is desirable to pool information across the markers. In this thesis, we propose a kernel-based method for aggregation of marker-level tests, in which first we obtain a bunch of p-values through association tests for every marker and then the association test involving CNV is based on the statistic of p-values combinations. In addition, we explore several aspects of its implementation. Since p-values among markers are correlated, it is complicated to obtain the null distribution of test statistics for kernel-base aggregation of marker-level tests. To solve the problem, we develop two proper methods that are both demonstrated to preserve the family-wise error rate of the test procedure. They are permutation based and correlation base approaches. Many implementation aspects of kernel-based method are compared through the empirical power studies in a number of simulations constructed from real data involving a pharmacogenomic study of gemcitabine. In addition, more performance comparisons are shown between permutation-based and correlation-based approach. We also apply those two approaches to the real data. The main contribution of the dissertation is the development of marker-level association testing, a comparable and powerful approach to detect phenotype-associated CNVs. Furthermore, the approach is extended to high dimension setting with high efficiency.

Competition between species is believed to lead to patterns of either competitive exclusion or limiting similarity within ecological communities; however, to date the amount of support for either as an outcome has been relatively weak. The two classes of null model commonly used to assess co-occurrence and limiting similarity have both been well studied for statistical performance; however, the methods used to evaluate their performance, particularly in terms of type II statistical errors, may have resulted in the underreporting of both patterns in the communities tested. The overall purpose of this study was to evaluate the efficacy of the negative co-occurrence and limiting similarity null models to detect patterns believed to result from competition between species and to develop an improved method for detecting said patterns. The null models were tested using synthetic but biologically realistic presence-absence matrices for both type I and type II error rate estimations. The effectiveness of the null models was evaluated with respect to community dimension (number of species × number of plots), and amount of pattern within the community. A novel method of subsetting species was developed to assess communities for patterns of co-occurrence and limiting similarity and four methods were assessed for their ability to isolate the species contributing signal to the pattern. Both classes of null model provided acceptable type I and type II error rates when matrices of more than 5 species and more than 5 plots were tested. When patterns of negative co-occurrence or limiting similarity were add to all species both null models were able to detect significant pattern (β > 0.95); however, when pattern was added to only a proportion of species the ability of the null models to detect pattern deteriorated rapidly with proportions of 80% or less. The use of species subsetting was able to detect significant pattern of both co-occurrence and limiting similarity when fewer than 80% of species were contributing signal but was dependent on the metric used for the limiting similarity null model. The ability of frequent pattern mining to isolate the species contributing signal shows promise; however, a more thorough evaluation is required in order to confirm or deny its utility.

With code division multiple access (CDMA) systems being the prominent multiple access scheme for the air interface for 3G cellular systems, most standardisation bodies have based their terrestrial cellular systems on DS-CDMA (W-CDMA, UMTS, cdma2000). With 4G systems fast approaching, bringing with them improved services and quality of service standards, there is growing interest in further investigating and developing more efficient multiple access techniques such as multicarrier CDMA (MC-CDMA) systems. MC-CDMA combines multicarrier modulation (MCM), namely OFDM, with CDMA profiting from the benefits of both multiplexing techniques; as such, MC-CDMA is emerging as a possible candidate for the air interface multiple access scheme for 4G cellular systems. Multiple access interference (MAI) is a limiting factor of CDMA systems in terms of system capacity as orthogonally designed spreading sequences lose their orthogonality in the presence of timing misalignments amongst mobile subscribers in a cell; such is the case over the uplink channel. Ensuring orthogonal code properties minimises the MAI over synchronous environments, however, it is when the users are allowed to transmit asynchronously, as is the case over the uplink channel, that MAI inflicts significant performance degradation. In CDMA systems, all subscribers are active on the same frequency band simultaneously and signal separation is facilitated upon reception via the properties of the assigned spreading codes. Under asynchronous conditions the code properties alone do not provide the necessary separation and an additive MAI term remains in the detection process. In addition to the separation abilities of the spreading codes, a further method of deciphering the desired subscriber signal from the interfering subscriber signals is sought. In this thesis we propose a statistical model for both the probability density function (pdf) of the total MAI power and the corresponding bit-error rate (BER) observed during asynchronous CDMA transmission. The modelling offers the full statistic the MAI power and resulting BER, not just the first and second order statistics. In addition to statistically quantifying the MAI power, the thesis also proposes a technique for the successful reduction of MAI caused by asynchronous transmission. This interference reduction technique is derived from an ambiguity domain analysis of the asynchronous CDMA detection problem and its application to both the DS-CDMA and MC-CDMA multiplexing techniques is presented and the results show significant MAI reduction, and thus an improved the BER. A methodology for the approximation of the total MAI power pdf and the resulting BER pdf is proposed for the asynchronous DS-CDMA and MC-CDMA techniques. This methodology is derived for the use of Walsh-Hadamard (WH) and Gold spreading sequences, however, it is applicable to any given set of deterministic spreading sequences. The total MAI power pdfs of both systems are statistically modelled as being Nakagamim distributed and the corresponding BER modelling is derived from the Nakagami-m formulation offering the full statistic of both the incurred MAI power and the achievable BER. The proposed pdf acquisition methodology and statistical models can be used as analysis tools to assess the relative performances of the DS-CDMA and MC-CDMA techniques for a variety of communications environments. Here the asynchronous uplink channel is considered in the absence of fading and the results show a clear distinction between the BER performances of the MC-CDMA and DS-CDMA systems, for which the MC-CDMA system offers a superior performance for the purely asynchronous channel considered. The results suggest a higher resistance to MAI in the MC-CDMA technique in comparison to the DS-CDMA system for the considered transmission scenario. Following ambiguity function analysis of the asynchronous CDMA detection problem, the concept of dual-frequency switching is introduced to the existing DS-CDMA and MC-CDMA techniques giving rise to the proposed dual-frequency DS-CDMA (DF/DSCDMA) and dual-frequency MC-CDMA (DF/MC-CDMA) schemes. Periodically switching the carrier frequency between dual frequency bands at consecutive symbol boundaries facilitates partial CDMA signal separation upon asynchronous reception. Such switching of the carrier frequency induces a separation in frequency between offset interference signals and the reference signal; this is equivalent to shifting the energy concentration of the interference signals away form the ambiguity domain origin (representing the decision variable of the matched filter). Further MAI reduction is demonstrated through careful design of the dual carrier frequencies. The newly proposed DF systems clearly outperform the standard DS-CDMA and MC-CDMA systems when adopting equivalent spreading factors. The DF/DS-CDMA technique in particular achieves the most MAI reduction and in doing so, surpasses all other considered techniques to offer the best BER performance for the purely asynchronous channel considered. In terms of bandwidth usage, the DF/DS-CDMA band width is 1.5 times that of the DF/MC-CDMA system and from the BER results presented, one may argue that DF/MC-CDMA offers the better BER given the bandwidth usage. The multicarrier systems presented, MC-CDMA and DF/MC-CDMA, offer attractive BER performances for the bandwidth used and it is concluded that MC-CDMA is a genuine candidate for the uplink air interface multiple access scheme for future mobile cellular technologies.

This study uses a simulation to gather data regarding a payment disruption. The simulation is part of a project called CCRAAAFFFTING, which examines what happens to a society when a payment disruption occurs. The purpose of this study is to develop a measure for resilience in the simulation and use machine learning to analyse the attributes in the simulation to see how they affect the resilience in the society. The resilience is defined as “the ability to bounce back to a previous state”, and the resilience measure is developed according to this definition. Two resilience measurements are defined, one which relates the simulated value to the best-case and worst-case scenarios, and the other which takes the pace of change in values into consideration. These two measurements are then combined to one measure of the total resilience. The three machine learning algorithms compared are Neural Network, Support Vector Machine and Random Forest, and the performance measure of these are the error rate. The results show that Random Forest performs significantly better than the other two algorithms, and that the most important attributes in the simulation are those concerning the customers’ ability to make purchases in the simulation. The developed resilience measure proves to respond logically to how the situation unfolded, and some suggestions to further improve the measurement is provided for future research. / I denna studie används en simulering för att samla in data. Simuleringen är en del i ett projekt som kallas för CCRAAAFFFTING, vars syfte är att undersöka vad som händer i ett samhälle om en störning i betalsystemet inträffar. Syftet med denna studie är att utveckla ett mått för resiliens i simuleringen, samt att använda machine learning för att analysera attributen i simuleringen för att se hur de påverkar resiliensen i samhället. Resiliensen definieras enligt ”förmågan att snabbt gå tillbaka till ett tidigare stadie”, och resiliensmåttet utvecklas i enlighet med denna definition. Två resiliensmått definieras, där det ena måttet relaterar det simulerade värdet till de värsta och bästa scenarierna, och det andra måttet tar i beaktning hur snabbt värdena förändrades. Dessa två mått kombineras sedan till ett mått för den totala resiliensen. De tre machine learning-algoritmerna som jämförs är Neural Network, Support Vector Machine och Random Forest, och måttet för hur de presterar är felfrekvens. Resultaten visar att Random Forest presterar märkbart bättre än de andra två algoritmerna, och att de viktigaste attributen i simuleringen är de som berör kunders möjlighet att genomföra köp i simuleringen. Det utvecklade resiliensmåttet svarar på ett logiskt sätt enligt hur situationen utvecklar sig, och några förslag för att vidare utveckla måttet ges för vidare forskning.

Este trabalho concentra-se no aumento da robustez à taxa de erros em redes de sensores sem fio industriais (RSSFI). Para alcançar tal objetivo, é preciso, de algum modo, contornar os problemas intrínsecos da comunicação sem fio, que são, o ruído ambiental, interferências e desvanecimento por multipercurso. O multipercurso pode ser considerado um dos principais fatores que tornam as comunicações nas redes de sensores sem fio (RSSF) um verdadeiro desafio quando comparadas com outros meios, como a transmissão em fibra óptica, cabo ou mesmo transmissões de rádio ponto-a-ponto. Nesse sentido, propõe-se a diversidade de antenas como uma solução para minimizar os efeitos do multicaminho, com o objetivo de melhorar a confiabilidade do enlace de rádio para permitir o emprego de RSSF densas. O potencial da diversidade de antenas em RSSF não está totalmente explorado em aplicações industriais. Esta dissertação apresenta ainda, a técnica “Combinação de seleção” para RSSFI através de um algoritmo que seleciona a porta do receptor que apresenta o melhor indicador de qualidade de enlace e realiza a comutação das antenas nos módulos de rádio. Os resultados são analisados para dois tipos de enlaces, isto é, com e sem diversidade de antenas na recepção, e discute-se opções para melhorar o PER (“Packet Error Rate”) com as atuais técnicas de diversidade. / This work concentrates on the increase of reliability and robustness in Industrial Wireless Sensor Networks (IWSNs), decreasing the Packet Error Rate (PER). To achieve this objective, is need to somehow circumvent and reduce the underlying problems of wireless communication, which are: environmental noise, interference, and multipath fading. The multipath can be seen as the main factor which becomes the communications in the Wireless Sensor Networks (WSNs) a real challenge when compared to other types of means such as a transmission in fiber, cable or even point-to-point radio transmissions. In this sense, it is proposed antenna diversity as solution to reduce these effects, with the aim to improve the reliability of the radio link to allow the use of dense WSNs. The potential of antenna diversity in WSN is not fully exploited in industrial applications. In this sense, this dissertation presents the "Combination of selection" technique for IWSNs through an internal algorithm that selects the receiver port that has the highest LQI ("Link Quality Indicator") and performs an antenna switching in the radio modules. Also, options to improve PER using diversity techniques are discussed.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / The likelihood ratio test for independence between two groups of variables allows us to identify whether there is a dependency relationship between two groups of variables, ie, if the covariance between the two groups are zero. This test assumes normality multivariate data, which limits its application, in many studies of agronomic area, times when you need use, for example, the canonical correlation analysis. The objective of this study is to evaluate the type I error and power of the likelihood ratio test (LRT) for independence between two groups of variables in different scenarios, consisting of combinations of: sample sizes 16, 40 number of combinations of two variables groups, and nine degrees of correlation between variables in matrices (for power); multivariate normal distribution under normal and contaminated, as well as compare the different scenarios, two formulas for calculating the test statistic. Thus, were evaluated the effect of 640 and 5760 scenarios on rates of type I error and power, respectively, in each one of probability distributions and formulas. The assessment of performance of LRT was performed through computational simulation by Monte Carlo method, using 2000 simulations in each of the scenarios generated. In multivariate normal situation: when the number of variables is large (p = 24), the LRT for independence between two groups of variables, controls the type I error rates and has high power in sample sizes greater than 100 and 500, with use of formulas for small and large samples, respectively; and, for sample sizes small (n = 25, 30 and 50), the test presents good performance, provided that, the number of variables does not exceed to 12; and, the formula chosen, whether for small samples. Under contaminated multivariate normal distribution, the LRT for independence between two groups of variables have high values of power, but is not robust, because it has high rates of type I error in any scenario evaluated. / O teste de razão de verossimilhança para a independência entre dois grupos de variáveis permite identificar se há ou não relação de dependência entre dois grupos de variáveis, ou seja, se as covariâncias entre os dois grupos são nulas. Esse teste pressupõe normalidade multivariada dos dados, o que limita sua aplicação, em muitos estudos da área agronômica, em que se necessita utilizar, por exemplo, a análise de correlação canônica. O objetivo deste trabalho é avaliar o erro tipo I e o poder do teste de razão de verossimilhança (TRV) para independência entre dois grupos de variáveis em diversos cenários, constituídos pelas combinações de: 16 tamanhos de amostra; 40 combinações de número de variáveis dos dois grupos; e, nove graus de correlação entre as variáveis (para o poder); sob distribuição normal multivariada e distribuição normal multivariada contaminada, bem como, comparar, nos diferentes cenários, duas fórmulas para cálculo da estatística do teste. Dessa forma, foram avaliados o efeito de 640 e 5760 cenários sobre as taxas de erro tipo I e poder, respectivamente, em cada uma das distribuições de probabilidade e fórmulas. A avaliação do desempenho do TRV foi realizada por meio de simulação computacional pelo método Monte Carlo, utilizando-se 2000 simulações em cada um dos cenários gerados. Em situação de normalidade multivariada: quando o número de variáveis é grande (p= 24), o TRV para a independência entre dois grupos de variáveis, controla as taxas de erro tipo I e apresentou poder elevado, em tamanhos de amostra superiores a 100 e 500, com uso das fórmulas para pequenas e grandes amostras, respectivamente; e, para tamanhos amostrais pequenos (n= 25, 30 e 50), o teste apresenta bom desempenho, desde que, o número de variáveis não exceda a 12; e, a fórmula escolhida, seja para pequenas amostras. Sob distribuição normal multivariada contaminada, o TRV para a independência entre dois grupos de variáveis possui elevados valores de poder, mas não é robusto, pois apresenta elevadas taxas de erro tipo I, em qualquer cenário avaliado.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2017-03-08T12:29:03Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) JOAO VICTOR DE CARVALHO EVANGELISTA_DISSERTACAO_VERSAO_FINAL_2016.pdf: 4051425 bytes, checksum: c53a5039b8aa3054c77f2ee82a10849f (MD5) / Made available in DSpace on 2017-03-08T12:29:03Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) JOAO VICTOR DE CARVALHO EVANGELISTA_DISSERTACAO_VERSAO_FINAL_2016.pdf: 4051425 bytes, checksum: c53a5039b8aa3054c77f2ee82a10849f (MD5) Previous issue date: 2016-08-16 / Message authentication, which ensures that a received message comes from its acclaimed sender, is of fundamental importance for secure communication systems. We consider in this work a physical layer authentication system employing tag signals embedded in the message to provide a robust authentication method. This work diverges from previous work in the area when it comes to the tag generation method. While the previous works use methods based on cryptographic hash functions or on the channel side information our system employs unidimensional chaotic maps to generate these tags. Due to the loss of information about the initial condition of chaotic maps, we show that they are strong candidates for the tag generation process. We prove that chaotic tags provide a positive lower bound on the unconditional security of the system. Additionally, we calculate the probability of success for three possible attacks to the authentication system: impersonation, substitution and replay.Finally, we analyze how the system parameters affect these probabilities and some performance metrics (bit error rate, outage probability, probability of false negative) and explore the tradeoff between security and performance in order to provide guidelines to design the system. / A autenticação de mensagem, o que garante que uma mensagem recebida vem de seu aclamado remetente, é de fundamental importância para sistemas de comunicação seguros. Neste contexto, considera-se neste trabalho um sistema de autenticação em camada física empregando tags embutidos nas mensagens proporcionando um robusto método de autenticação. Este trabalho diverge de trabalhos anteriores na área no que se refere ao método de geração de tags. Enquanto os trabalhos anteriores utilizam métodos baseados em funções criptográficas de hash e na informação do estado do canal, nosso sistema emprega mapas caóticos unidimensionais para gerar os tags. Devido ao fato de que a informação sobre a condição inicial se perde ao longo de uma órbita caótica mostraremos que elas são fortes candidatas para o processo de geração de tags. Provamos que tags caóticos garantem um limitante inferior positivo na segurança incondicional do sistema. Adicionalmente, nós calculamos a probabilidade de sucesso de três tipos de ataque: de personificação, de substituição e de repetição. Para finalizar, analisamos como os parâmetros do sistema afetam essas probabilidades e algumas métricas de performance (taxa de erro por bit, probabilidade de interrupção e probabilidade de falso negativo) e os compromissos entre segurança e performance para prover um guia de projeto do sistema.

O desbalanceamento de dados em experimentos está muitas vezes presente em diversas pesquisas nas mais variadas áreas do conhecimento. Embora existam muitas maneiras de análise de tais dados, além de diversos recursos computacionais já implementados em diversos softwares estatísticos, ainda perdura dúvidas entre os pesquisadores a respeito da opção de análise mais eficiente. A literatura fornece ao pesquisador direção na escolha da metodologia de análise a obter maior eficácia nos resultados de sua pesquisa, mas o número elevado de opções pode tornar a escolha difícil. Em se tratando de testes estatísticos, algumas das opções para se trabalhar com dados desbalanceados são os testes t e Wald-F, mas ainda resta ao pesquisador decidir entre as várias opções disponíveis nos pacotes, pois nem sempre as opções padrões são as mais indicadas. No presente trabalho foram realizadas simulações com diferentes cenários experimentais, utilizando-se o delineamento casualizado em blocos com um fator de tratamento em uma situação e o esquema de tratamentos em parcelas subdividas em outra, sendo comparados quatro métodos de cálculo do número aproximado de graus de liberdade (Containment, Residual, Satterthwaite e Kenward-Roger). Verificou-se que o método de Kenward-Roger controla de maneira mais eficiente a taxa de erro tipo I e não é inferior aos outros métodos com respeito ao poder do teste Wald-F. / The data imbalance in experiments is often present in several researches in various fields of knowledge. While there are many ways to analyze these data in addition to various computer resources already implemented in many statistical software, doubt still lingers among researchers about the most efficient analysis option. The literature provides the researcher direction in choosing the analysis methodology to get better in your search results, but the large number of options can make the difficult choice. When it comes to statistical tests, some of the options for working with unbalanced data are the tests t and Wald-F, but there is still the researcher to decide between the various options available in the packages because the defaults are not always the most suitable. This experiment was carried out simulations with different experimental scenarios, using the randomized block design with one factor in a situation treatment and treatment regimen subdivided parcels in another, and compared four methods of calculating the approximate number of degrees of freedom (Containment, Residual, Satterthwaite and Kenward-Roger). It has been found that the method of Kenward-Roger controls more efficiently the type I error rate and is not inferior to other methods with respect to the power of the test Wald-F.