Classic parametric statistical tests, like the analysis of variance (ANOVA), are powerful tools used for comparing population means. These tests produce accurate results provided the data satisfies underlying assumptions such as homoscedasticity and balancedness, otherwise biased results are obtained. However, these assumptions are rarely satisfied in real-life. Alternative procedures must be explored. This thesis aims at investigating the impact of heteroscedasticity and unbalancedness on effect sizes in two-way fixed-effects ANOVA models. A real-life dataset, from which three different samples were simulated was used to investigate the changes in effect sizes under the influence of unequal variances and unbalancedness. The parametric bootstrap approach was proposed in case of unequal variances and non-normality. The results obtained indicated that heteroscedasticity significantly inflates effect sizes while unbalancedness has non-significant impact on effect sizes in two-way ANOVA models. However, the impact worsens when the data is both unbalanced and heteroscedastic. / Statistics / M. Sc. (Statistics)

<p>This thesis investigates the statistical properties of preliminary test estimators of linear models with normally distributed errors. Specifically, we derive exact expressions for the mean, variance and quadratic risk (i.e. the Mean Square Error) of estimators whose form are determined by the outcome of a statistical test. In the process, some new results on the moments of truncated linear or quadratic forms in normal vectors are established.</p><p>In the first paper (Paper I), we consider the estimation of the vector of regression coefficients under a model selection procedure where it is assumed that the analyst chooses between two nested linear models by some of the standard model selection criteria. This is shown to be equivalent to estimation under a preliminary test of some linear restrictions on the vector of regression coefficients. The main contribution of Paper I compared to earlier research is the generality of the form of the test statistic; we only assume it to be a quadratic form in the (translated) observation vector. Paper II paper deals with the estimation of the regression coefficients under a preliminary test for homoscedasticity of the error variances. In Paper III, we investigate the statistical properties of estimators, truncated at zero, of variance components in linear models with random effects. Paper IV establishes some new results on the moments of truncated linear and/or quadratic forms in normally distributed vectors. These results are used in Papers I-III. In Paper V we study some algebraic properties of matrices that occur in the comparison of two nested models. Specifically we derive an expression for the inertia (the number of positive, negative and zero eigenvalues) of this type of matrices.</p>

This thesis investigates the statistical properties of preliminary test estimators of linear models with normally distributed errors. Specifically, we derive exact expressions for the mean, variance and quadratic risk (i.e. the Mean Square Error) of estimators whose form are determined by the outcome of a statistical test. In the process, some new results on the moments of truncated linear or quadratic forms in normal vectors are established. In the first paper (Paper I), we consider the estimation of the vector of regression coefficients under a model selection procedure where it is assumed that the analyst chooses between two nested linear models by some of the standard model selection criteria. This is shown to be equivalent to estimation under a preliminary test of some linear restrictions on the vector of regression coefficients. The main contribution of Paper I compared to earlier research is the generality of the form of the test statistic; we only assume it to be a quadratic form in the (translated) observation vector. Paper II paper deals with the estimation of the regression coefficients under a preliminary test for homoscedasticity of the error variances. In Paper III, we investigate the statistical properties of estimators, truncated at zero, of variance components in linear models with random effects. Paper IV establishes some new results on the moments of truncated linear and/or quadratic forms in normally distributed vectors. These results are used in Papers I-III. In Paper V we study some algebraic properties of matrices that occur in the comparison of two nested models. Specifically we derive an expression for the inertia (the number of positive, negative and zero eigenvalues) of this type of matrices.

Coefficient of Pearson r is used to compare scientific tests. In seeds technology it is used to compare results of procedures that measure vigour. When the correspondly similar results are not found in very similar conditions, Person s correlation faces criticism mainly due to two factors. The first one comes from statistics for whose usage of Person s correlation there are prescriptions that are not always observed, when they are not understood as assumption. Variables naturally associated are required with bivariated normal distribution, pairing; homoscedasticity, rectilinear dispersion; detection of outliers. Added to them, there are practical observations in what refers the correlation to be valid only in a restrict range of the data series, the necessity to create value ranges to consider this correlation as good or bad , the need of the graphical analysis, the use and interpretation of the significance, among others. The second cause of odd results would be the existence of several biological factors, which are sometimes support for the reserarcher conclusions. With the objective of identifying applicability of correlations and the causes for odd results of r, there have been compared data existent in the Seeds Analysis Laboratory of ICIAG of the Universidade Federal de Uberlândia-MG, as well as tests of germination of acelerated aging in optimal conditions of repetibility done in laboratory, and tests of field seedling emergency, as well as other simulated variables. The results showed odd results. The normal scattergram between X and Y is enough clear to elucidate only correlated variables of large samples. Although, if the covariance is not as obvious the dispersion Y=f(X) is not enough to show simultaneous increasing or decreasing between variables. With an alternative methodology of plotting the variables related to another auxiliar variable Z of the same n elements of X and Y, we could study the variable behavior in an individual way. It was possible to create graphic criteria to assess non-valid correlations, such as similarity of variables comparable to homoscedastity; influence of outliers on small or big n; grouping of outliers in a dissident range , influence of treatments effect. In the analysed cases, we concluded that, comparing seeds vigour with only laboratory results, as well as its relation with the field results and among simulated data, the results inconsistency of correlations are prevalent as they do not follow the literature prescriptions, among others. The magnitude of the distortions due to statistical causes did not leave space for measuring effects of the variation of the biological seeds conditions, temporal alterations related to management or the edafoclimatic ones. Keywords: 1. Failure in correlations 2. Correlation reliability / Coeficiente de Pearson r é usado para comparar experimentos científicos. Em tecnologias de sementes serve para comparar resultados de procedimentos que medem vigor. Quando se prognosticam resultados de correlações baseados em condições similares e eles não acontecem, a correlação de Pearson enfrenta críticas, atribuídas principalmente a duas causas. Primeiramente pela estatística, para cuja utilização da correlação de Pearson existem prescrições nem sempre observadas, talvez por não serem entendidas como pressuposições. Exigem-se variáveis métricas naturalmente associadas, com distribuição normal bivariada, pareamento, homoscedasticidade, nuvem de dispersão retilínea; detectção de outliers. Somam-se observações práticas quanto à validade restrita a um trecho da série de dados, da criação de faixas de valores para considerá-la de baixa a alta , da necessidade da análise gráfica, da interpretação de significância, entre outras. A segunda causa seria justamente a existência de variação biológica devido a fatores diversos externos e interno às sementes, servindo às vezes de sustentáculo para conclusões de interesse do pesquisador. No objetivo de identificar aplicabilidade das correlações e as causas de resultados estranhos, foram comparados dados existentes no Laboratório de Análises de Sementes do ICIAG da Universidade Federal de Uberlândia-MG, testes germinação de envelhecimento acelerado em condições ideais de repetibilidade em laboratório, e teste de emergência de plântulas em campo, e outras variáveis simuladas, havendo incidência de resultados estranhos. A representação gráfica normal da dispersão entre X e Y mostra satisfatoriamente o correlacionamento de variáveis naturalmente associadas com n grande. Entretanto, se a covariância não é tão óbvia, a disperção Y=f(X) não é suficiente para mostrar crescimento ou decréscimo simultâneo entre as variáveis. Usando metodologia alternativa de plotagem das variáveis em relação a uma variável auxiliar Z, de mesmos n elementos que X e Y, pôde-se estudar individualmente o comportamento das variáveis. O método gráfico permitiu taxar correlações em válidas ou não pela similaridade das variáveis, comparável à homoscedasticidade; verificar outliers em n pequeno ou grande; agrupamento de outliers em trecho dissidente e mostrar efeito de tratamentos. Nos casos analisados, concluiu-se que, comparando vigor de sementes com resultados só de laboratório, tão bem como no seu relacionamento com os de campo; e entre dados simulados, as inconsistências de resultados de correlações são preponderantes por não seguirem as prescrições da literatura, entre outras. A magnitude das distorções por causas estatísticas não deixou espaço para mensurar efeitos da variação de condições biológicas de sementes, alterações temporais relativas a manuseio ou edafoclimáticas. / Mestre em Agronomia