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.

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.

Physical function, psychosocial adaptation, and hardiness post stroke

Introduction: Stroke is a chronic condition that significantly impacts on morbidity and mortality (Balanda et al. 2010). Globally, the complexity of stroke is well documented and more recently, in Ireland, as part of the National Survey of Stroke Survivors (Horgan et al. 2014). There are a number of factors that are known to influence adaptation post stroke. However, there is a lack of research to explain the variability in how survivors adapt post stroke. Hardiness is a broad personality trait that leads to better outcome. This study investigated the influence of hardiness and physical function on psychosocial adaptation post stroke. Methods: A quantitative cross-sectional, correlational, exploratory study was conducted between April and November 2013. The sample consisted of stroke survivors (n=100) who were recruited from three hospital outpatient departments and completed a questionnaire package. Results: The mean age of participants was 76 years (range 70-80), over half (56%) of the participants achieved the maximum score of 20 on the Barthel Index indicating independence in activities of daily living. The median number of days since stroke onset was 91 days (range 74-128). The total mean score and standard deviation for hardiness was 1.89 (0.4) as measured by the Dispositional Resilience Scale, indicating medium hardiness (possible range 0-3). Psychosocial adaptation was measured using the Psychosocial Adjustment to Illness Scale, the total weighted mean and standard deviation was 0.54 (0.3) indicating a satisfactory level of psychosocial adaptation (possible range 0-3). A hierarchical multiple linear regression was performed which contained 6 independent variables (hardiness, living arrangement, and length of hospital stay, number of days since stroke onset, physical function and self-rated recovery). Findings demonstrated that physical function (p<0.001) and hardiness (p=0.008) were significantly related to psychosocial adaptation. Altogether, 65% of the variation in psychosocial adaptation can be explained by the combined effect of the independent variables. Physical functioning had the highest unique contribution (11%) to explain the variance in psychosocial adaptation while self-rated recovery, hardiness, and living arrangements contributed 3% each. Conclusion: This research provides important information regarding factors that influence psychosocial adaptation post stroke at 3 months. Physical function significantly contributed to psychosocial adaptation post stroke. The personality trait of hardiness provides insight into how behaviour influenced adaptation post stroke. While hardiness also had a strong relationship with psychosocial adaptation, further research is necessary to fully comprehend this process.