On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.

Early biomarkers to predict grade of encephalopathy following hypoxic ischaemic injury

The standard early markers for identifying and grading HIE severity, are not sufficient to ensure all children who would benefit from treatment are identified in a timely fashion. The aim of this thesis was to explore potential early biomarkers of HIE. Methods: To achieve this a cohort of infants with perinatal depression was prospectively recruited. All infants had cord blood samples drawn and biobanked, and were assessed with standardised neurological examination, and early continuous multi-channel EEG. Cord samples from a control cohort of healthy infants were used for comparison. Biomarkers studied included; multiple inflammatory proteins using multiplex assay; the metabolomics profile using LC/MS; and the miRNA profile using microarray. Results: Eighty five infants with perinatal depression were recruited. Analysis of inflammatory proteins consisted of exploratory analysis of 37 analytes conducted in a sub-population, followed by validation of all significantly altered analytes in the remaining population. IL-6 and IL-6 differed significantly in infants with a moderate/severely abnormal vs. a normal-mildly abnormal EEG in both cohorts (Exploratory: p=0.016, p=0.005: Validation: p=0.024, p=0.039; respectively). Metabolomic analysis demonstrated a perturbation in 29 metabolites. A Cross- validated Partial Least Square Discriminant Analysis model was developed, which accurately predicted HIE with an AUC of 0.92 (95% CI: 0.84-0.97). Analysis of the miRNA profile found 70 miRNA significantly altered between moderate/severely encephalopathic infants and controls. miRNA target prediction databases identified potential targets for the altered miRNA in pathways involved in cellular metabolism, cell cycle and apoptosis, cell signaling, and the inflammatory cascade. Conclusion: This thesis has demonstrated that the recruitment of a large cohortof asphyxiated infants, with cord blood carefully biobanked, and detailed early neurophysiological and clinical assessment recorded, is feasible. Additionally the results described, provide potential alternate and novel blood based biomarkers for the identification and assessment of HIE.

Statistical methods for polyhedral shape classification with incomplete data - application to cryo-electron tomographic images

A certain type of bacterial inclusion, known as a bacterial microcompartment, was recently identified and imaged through cryo-electron tomography. A reconstructed 3D object from single-axis limited angle tilt-series cryo-electron tomography contains missing regions and this problem is known as the missing wedge problem. Due to missing regions on the reconstructed images, analyzing their 3D structures is a challenging problem. The existing methods overcome this problem by aligning and averaging several similar shaped objects. These schemes work well if the objects are symmetric and several objects with almost similar shapes and sizes are available. Since the bacterial inclusions studied here are not symmetric, are deformed, and show a wide range of shapes and sizes, the existing approaches are not appropriate. This research develops new statistical methods for analyzing geometric properties, such as volume, symmetry, aspect ratio, polyhedral structures etc., of these bacterial inclusions in presence of missing data. These methods work with deformed and non-symmetric varied shaped objects and do not necessitate multiple objects for handling the missing wedge problem. The developed methods and contributions include: (a) an improved method for manual image segmentation, (b) a new approach to 'complete' the segmented and reconstructed incomplete 3D images, (c) a polyhedral structural distance model to predict the polyhedral shapes of these microstructures, (d) a new shape descriptor for polyhedral shapes, named as polyhedron profile statistic, and (e) the Bayes classifier, linear discriminant analysis and support vector machine based classifiers for supervised incomplete polyhedral shape classification. Finally, the predicted 3D shapes for these bacterial microstructures belong to the Johnson solids family, and these shapes along with their other geometric properties are important for better understanding of their chemical and biological characteristics.