The standard methodology when building statistical models has been to use one of several algorithms to systematically search the model space for a good model. If the number of variables is small then all possible models or best subset procedures may be used, but for data sets with a large number of variables, a stepwise procedure is usually implemented. The stepwise procedure of model selection was designed for its computational efficiency and is not guaranteed to find the best model with respect to any optimality criteria. While the model selected may not be the best possible of those in the model space, commonly it is almost as good as the best model. Many times there will be several models that exist that may be competitors of the best model in terms of the selection criterion, but classical model building dictates that a single model be chosen to the exclusion of all others. An alternative to this is Bayesian model averaging (BMA), which uses the information from all models based on how well each is supported by the data. Using BMA allows a variance component due to the uncertainty of the model selection process to be estimated. The variance of any statistic of interest is conditional on the model selected so if there is model uncertainty then variance estimates should reflect this. BMA methodology can also be used for variable assessment since the probability that a given variable is active is readily obtained from the individual model posterior probabilities. The multivariate methods considered in this research are principal components analysis (PCA), canonical variate analysis (CVA), and canonical correlation analysis (CCA). Each method is viewed as a particular multivariate extension of univariate multiple regression. The marginal likelihood of a univariate multiple regression model has been approximated using the Bayes information criteria (BIC), hence the marginal likelihood for these multivariate extensions also makes use of this approximation. One of the main criticisms of multivariate techniques in general is that they are difficult to interpret. To aid interpretation, BMA methodology is used to assess the contribution of each variable to the methods investigated. A second issue that is addressed is displaying of results of an analysis graphically. The goal here is to effectively convey the germane elements of an analysis when BMA is used in order to obtain a clearer picture of what conclusions should be drawn. Finally, the model uncertainty variance component can be estimated using BMA. The variance due to model uncertainty is ignored when the standard model building tenets are used giving overly optimistic variance estimates. Even though the model attained via standard techniques may be adequate, in general, it would be difficult to argue that the chosen model is in fact the correct model. It seems more appropriate to incorporate the information from all plausible models that are well supported by the data to make decisions and to use variance estimates that account for the uncertainty in the model estimation as well as model selection. / Ph. D.

Thesis (MComm)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The main topics of study in this thesis are the Principal Component Analysis (PCA) and Canonical Variate Analysis (CVA) biplots, with the primary focus falling on the quality measures associated with these biplots. A detailed study of different routes along which PCA and CVA can be derived precedes the study of the PCA biplot and CVA biplot respectively. Different perspectives on PCA and CVA highlight different aspects of the theory that underlie PCA and CVA biplots respectively and so contribute to a more solid understanding of these biplots and their interpretation. PCA is studied via the routes followed by Pearson (1901) and Hotelling (1933). CVA is studied from the perspectives of Linear Discriminant Analysis, Canonical Correlation Analysis as well as a two-step approach introduced in Gower et al. (2011). The close relationship between CVA and Multivariate Analysis of Variance (MANOVA) also receives some attention. An explanation of the construction of the PCA biplot is provided subsequent to the study of PCA. Thereafter follows an in depth investigation of quality measures of the PCA biplot as well as the relationships between these quality measures. Specific attention is given to the effect of standardisation on the PCA biplot and its quality measures. Following the study of CVA is an explanation of the construction of the weighted CVA biplot as well as two different unweighted CVA biplots based on the two-step approach to CVA. Specific attention is given to the effect of accounting for group sizes in the construction of the CVA biplot on the representation of the group structure underlying a data set. It was found that larger groups tend to be better separated from other groups in the weighted CVA biplot than in the corresponding unweighted CVA biplots. Similarly it was found that smaller groups tend to be separated to a greater extent from other groups in the unweighted CVA biplots than in the corresponding weighted CVA biplot. A detailed investigation of previously defined quality measures of the CVA biplot follows the study of the CVA biplot. It was found that the accuracy with which the group centroids of larger groups are approximated in the weighted CVA biplot is usually higher than that in the corresponding unweighted CVA biplots. Three new quality measures that assess that accuracy of the Pythagorean distances in the CVA biplot are also defined. These quality measures assess the accuracy of the Pythagorean distances between the group centroids, the Pythagorean distances between the individual samples and the Pythagorean distances between the individual samples and group centroids in the CVA biplot respectively. / AFRIKAANSE OPSOMMING: Die hoofonderwerpe van studie in hierdie tesis is die Hoofkomponent Analise (HKA) bistipping asook die Kanoniese Veranderlike Analise (KVA) bistipping met die primêre fokus op die kwaliteitsmaatstawwe wat daarmee geassosieer word. ’n Gedetailleerde studie van verskillende roetes waarlangs HKA en KVA afgelei kan word, gaan die studie van die HKA en KVA bistippings respektiewelik vooraf. Verskillende perspektiewe op HKA en KVA belig verskillende aspekte van die teorie wat onderliggend is tot die HKA en KVA bistippings respektiewelik en dra sodoende by tot ’n meer breedvoerige begrip van hierdie bistippings en hulle interpretasies. HKA word bestudeer volgens die roetes wat gevolg is deur Pearson (1901) en Hotelling (1933). KVA word bestudeer vanuit die perspektiewe van Linieêre Diskriminantanalise, Kanoniese Korrelasie-analise sowel as ’n twee-stap-benadering soos voorgestel in Gower et al. (2011). Die noue verwantskap tussen KVA en Meerveranderlike Analise van Variansie (MANOVA) kry ook aandag. ’n Verduideliking van die konstruksie van die HKA bistipping word voorsien na afloop van die studie van HKA. Daarna volg ’n indiepte-ondersoek van die HKA bistipping kwaliteitsmaatstawwe sowel as die onderlinge verhoudings tussen hierdie kwaliteitsmaatstawe. Spesifieke aandag word gegee aan die effek van die standaardisasie op die HKA bistipping en sy kwaliteitsmaatstawe. Opvolgend op die studie van KVA is ’n verduideliking van die konstruksie van die geweegde KVA bistipping sowel as twee veskillende ongeweegde KVA bistippings gebaseer op die twee-stap-benadering tot KVA. Spesifieke aandag word gegee aan die effek wat die inagneming van die groepsgroottes in die konstruksie van die KVA bistipping op die voorstelling van die groepstruktuur onderliggend aan ’n datastel het. Daar is gevind dat groter groepe beter geskei is van ander groepe in die geweegde KVA bistipping as in die oorstemmende ongeweegde KVA bistipping. Soortgelyk daaraan is gevind dat kleiner groepe tot ’n groter mate geskei is van ander groepe in die ongeweegde KVA bistipping as in die oorstemmende geweegde KVA bistipping. ’n Gedetailleerde ondersoek van voorheen gedefinieerde kwaliteitsmaatstawe van die KVA bistipping volg op die studie van die KVA bistipping. Daar is gevind dat die akkuraatheid waarmee die groepsgemiddeldes van groter groepe benader word in die geweegde KVA bistipping, gewoonlik hoër is as in die ooreenstemmende ongeweegde KVA bistippings. Drie nuwe kwaliteitsmaatstawe wat die akkuraatheid van die Pythagoras-afstande in die KVA bistipping meet, word gedefinieer. Hierdie kwaliteitsmaatstawe beskryf onderskeidelik die akkuraatheid van die voorstelling van die Pythagoras-afstande tussen die groepsgemiddeldes, die Pythagoras-afstande tussen die individuele observasies en die Pythagoras-afstande tussen die individuele observasies en groepsgemiddeldes in die KVA bistipping.

A demanda crescente de alimentos gera problemas ambientais, principalmente devido à remoção das coberturas vegetais nativas para a expansão da agricultura no Brasil. Essas alterações no uso da terra acarretam mudanças na dinâmica da matéria orgânica do solo. O tipo de vegetação também tem influência sobre a biota do solo e, por isso, a sua alteração ocasiona mudanças na biomassa microbiana, como revelam estudos envolvendo desmatamento. Os microorganismos representam a maior diversidade biológica e fisiológica do solo, além de serem responsáveis por mais de 95% dos processos de decomposição e ciclagem de nutrientes. Por este motivo, a demanda por indicadores rápidos e confiáveis para detectar alterações na qualidade do solo, principalmente por aqueles com base nos microrganismos, vem crescendo continuamente. O objetivo geral foi verificar se há diferenciação dos padrões da atividade e da diversidade funcional da biomassa microbiana dos solos sob as diferentes vegetações naturais, pastagens em uso e sistemas agrícolas recém implantados. A área de abrangência desta pesquisa corresponde aos estados de Rondônia (RO) e Mato Grosso (MT), que foi dividida em 11 ecorregiões relativamente homogêneas em termos de solos, vegetação nativa, geologia, clima e relevo. Foi realizada a análise da variável canônica de alguns atributos físicos, químicos e microbiológicos de cada ecorregião e uso da terra, buscando evidenciar padrões e as variáveis que os diferenciam. As áreas nativas apresentaram padrões distintos na dinâmica dos atributos microbiológicos relacionadas principalmente à quantidade de serapilheira em cada bioma estudado. Em relação às áreas antropizadas, observaram-se resultados similares entre pastagens e áreas nativas, significativamente diferentes dos resultados obtidos em áreas agrícolas, que devido à diferença de manejo e diversidade de culturas analisadas apresentou uma grande variabilidade no seu resultado final. Os resultados obtidos reforçam a recomendação da utilização dos atributos microbiológicos como indicadores de mudança de uso da terra, aliados a fatores químicos e físicos. A respiração induzida por substratos (ou diversidade catabólica) mostrou-se eficiente na separação dos usos da terra e ecorregiões. Estes resultados suportam a idéia de que cada forma de uso da terra induz a presença de grupos específicos de microrganismos, independente de tipo de solo, clima ou outras influências externas. / The increasing demand for food creates environmental problems, mainly due to the removal of native vegetation cover for agriculture expansion in Brazil. These changes in land use leads to changes in the soil organic matter dynamics. The vegetation type also affects the soil biota and, therefore, its change causes modifications in the microbial biomass, as shown by studies involving deforestation. Microorganisms represent the most biological and physiological diverse in soil, as well as being responsible for more than 95% of the decomposition and nutrient cycling. For this reason, the demand for fast and reliable indicators to detect changes in soil quality, especially for those based on micro-organisms has been growing continuously. The main objective in this research was to check the differentiation of patterns in activity and functional diversity of soil microbial biomass under natural vegetation, pastures and agricultural systems in use. The area covered by this study corresponds to the states of Rondônia (RO) and Mato Grosso (MT), which was divided into 11 ecoregions relatively homogeneous in terms of soil, vegetation, geology, climate and landscape. A canonical variate analysis was used in some physical, chemical and microbiological factors in each ecoregion and land use, looking for patterns and variables that can differentiate them. The native areas showed distinct patterns in the dynamics of microbiological attributes mainly related to the amount of litter in each biome studied. For the disturbed areas, there were similar results between pastures and native areas, significantly different from the results obtained in agricultural areas, which due to differences in management and kind of cultures analyzed showed a great variability in its final result. The results support the recommendation for use of microbiological attributes as indicators of land use change, combined with chemical and physical factors. The substrate-induced respiration (or catabolic diversity) was efficient in the separation of land uses and ecoregions. These findings support the idea that specific land uses have specific groups of microbial biomass, regardless soil type, climate or other external influences.

A demanda crescente de alimentos gera problemas ambientais, principalmente devido à remoção das coberturas vegetais nativas para a expansão da agricultura no Brasil. Essas alterações no uso da terra acarretam mudanças na dinâmica da matéria orgânica do solo. O tipo de vegetação também tem influência sobre a biota do solo e, por isso, a sua alteração ocasiona mudanças na biomassa microbiana, como revelam estudos envolvendo desmatamento. Os microorganismos representam a maior diversidade biológica e fisiológica do solo, além de serem responsáveis por mais de 95% dos processos de decomposição e ciclagem de nutrientes. Por este motivo, a demanda por indicadores rápidos e confiáveis para detectar alterações na qualidade do solo, principalmente por aqueles com base nos microrganismos, vem crescendo continuamente. O objetivo geral foi verificar se há diferenciação dos padrões da atividade e da diversidade funcional da biomassa microbiana dos solos sob as diferentes vegetações naturais, pastagens em uso e sistemas agrícolas recém implantados. A área de abrangência desta pesquisa corresponde aos estados de Rondônia (RO) e Mato Grosso (MT), que foi dividida em 11 ecorregiões relativamente homogêneas em termos de solos, vegetação nativa, geologia, clima e relevo. Foi realizada a análise da variável canônica de alguns atributos físicos, químicos e microbiológicos de cada ecorregião e uso da terra, buscando evidenciar padrões e as variáveis que os diferenciam. As áreas nativas apresentaram padrões distintos na dinâmica dos atributos microbiológicos relacionadas principalmente à quantidade de serapilheira em cada bioma estudado. Em relação às áreas antropizadas, observaram-se resultados similares entre pastagens e áreas nativas, significativamente diferentes dos resultados obtidos em áreas agrícolas, que devido à diferença de manejo e diversidade de culturas analisadas apresentou uma grande variabilidade no seu resultado final. Os resultados obtidos reforçam a recomendação da utilização dos atributos microbiológicos como indicadores de mudança de uso da terra, aliados a fatores químicos e físicos. A respiração induzida por substratos (ou diversidade catabólica) mostrou-se eficiente na separação dos usos da terra e ecorregiões. Estes resultados suportam a idéia de que cada forma de uso da terra induz a presença de grupos específicos de microrganismos, independente de tipo de solo, clima ou outras influências externas. / The increasing demand for food creates environmental problems, mainly due to the removal of native vegetation cover for agriculture expansion in Brazil. These changes in land use leads to changes in the soil organic matter dynamics. The vegetation type also affects the soil biota and, therefore, its change causes modifications in the microbial biomass, as shown by studies involving deforestation. Microorganisms represent the most biological and physiological diverse in soil, as well as being responsible for more than 95% of the decomposition and nutrient cycling. For this reason, the demand for fast and reliable indicators to detect changes in soil quality, especially for those based on micro-organisms has been growing continuously. The main objective in this research was to check the differentiation of patterns in activity and functional diversity of soil microbial biomass under natural vegetation, pastures and agricultural systems in use. The area covered by this study corresponds to the states of Rondônia (RO) and Mato Grosso (MT), which was divided into 11 ecoregions relatively homogeneous in terms of soil, vegetation, geology, climate and landscape. A canonical variate analysis was used in some physical, chemical and microbiological factors in each ecoregion and land use, looking for patterns and variables that can differentiate them. The native areas showed distinct patterns in the dynamics of microbiological attributes mainly related to the amount of litter in each biome studied. For the disturbed areas, there were similar results between pastures and native areas, significantly different from the results obtained in agricultural areas, which due to differences in management and kind of cultures analyzed showed a great variability in its final result. The results support the recommendation for use of microbiological attributes as indicators of land use change, combined with chemical and physical factors. The substrate-induced respiration (or catabolic diversity) was efficient in the separation of land uses and ecoregions. These findings support the idea that specific land uses have specific groups of microbial biomass, regardless soil type, climate or other external influences.

With both a heterogeneous population and a large number of unidentified persons in South Africa, an accurate method to estimate ancestry is needed. The purpose of this study was to evaluate variation in nasal aperture shape in black, white and coloured South Africans, using linear measures and geometric morphometrics (GM), the latter which includes both procrustes analysis (GPA) and elliptical fourier analysis (EFA). To test statistical significance among groups, discriminant function analysis (DFA) and principal component analysis (PCA) was used. A total of 310 (164 male, 145 female) crania of black, white and coloured South Africans were used. Thirteen standard landmarks, namely, glabella, nasion, nasale superior, dacryon, nasale inferius, alare, most inferior nasal border and subspinale, were digitised with a MicroScribe G2™ (Immersion: San Jose, CA). Five linear measures, nasion-dacryon angle (NDA), nasal breadth (NLB), nasal height (NLH), inter-orbital breadth (DKB) and nasion-dacryon subtense (NDS), were calculated. For EFA, photographs were taken in a frontal plane of skulls that had been positioned in the Frankfort horizontal plane on a craniophore. All classification accuracies for all groups were better than chance. Using linear measures and GPA, black South Africans classified 55-71% correctly, coloured classified 53-61% correctly and whites classified 85-95% correctly. Black and coloured South Africans demonstrated bell-shaped nasal apertures with nasal spines superior to the inferior nasal border. White South Africans had pear-shaped nasal apertures with a nasal spine inferior of the inferior nasal border. Using EFA black South Africans classified 62% correctly. While coloured South Africans only classified 39% correctly, which demonstrates high within group variability. Due to their unique historical development, large variation (heterogeneity) within the coloured group was expected. White South Africans had the highest correct classification accuracy of 85%. For all methods, misclassification rarely occurred between white and non-white (black and coloured) groups and most difficulties arose in distinguishing non-white groups from each other. High rates of misclassification was also noted between sex designations within a group, which suggests less or an absence of sexual dimorphism for these variables The distinct separation of white South Africans may reflect the mid-to late 20th century political and social separation of white and non-white groups in South Africa. Nasal aperture shape, alone, is less useful for separating groups such that all groups have relatively intermediate nasal aperture shapes; however the pinched nasal bone structure of white South Africans clearly separates them from the other groups. When using nasal bone and aperture landmarks, linear measures are as accurate as the modern geometric techniques in distinguishing groups. All methods are feasible to use in the estimation of ancestry on modern South Africans, with craniometry a sensible solution as the data can be rapidly collected, accurately analysed and compared to current reference samples. Copyright / Dissertation (MSc)--University of Pretoria, 2012. / Anatomy / unrestricted