The field of medicine is regularly faced with the challenge of utilizing information that is complicated or difficult to characterize. Physicians often must use their best judgment in reaching decisions or recommendations for treatment in the clinical setting. The goal of this thesis is to use innovative statistical tools in tackling three specific challenges of this nature from current healthcare applications. The first aim focuses on developing a novel approach to meta-analysis when combining binary data from multiple studies of paired design, particularly in cases of high heterogeneity between studies. The challenge is in properly accounting for heterogeneity when dealing with a low or moderate number of studies, and with a rarely occurring outcome. The proposed approach uses a Rasch model for translating data from multiple paired studies into a unified structure that allows for properly handling variability associated with both pair effects and study effects. Analysis is then performed using a Bayesian hierarchical structure, which accounts for heterogeneity in a direct way within the variances of the separate generating distributions for each model parameter. This approach is applied to the debated topic within the dental community of the comparative effectiveness of materials used for pit-and-fissure sealants. The second and third aims of this research both have applications in early detection of breast cancer. The interpretation of a mammogram is often difficult since signs of early disease are often minuscule, and the appearance of even normal tissue can be highly variable and complex. Physicians often have to consider many important pieces of the whole picture when trying to assess next steps. The final two aims focus on improving the interpretation of findings in mammograms to aid in early cancer detection. When dealing with high frequency and irregular data, as is seen in most medical images, the behaviors of these complex structures are often difficult or impossible to quantify by standard modeling techniques. But a commonly occurring phenomenon in high-frequency data is that of regular scaling. The second aim in this thesis is to develop and evaluate a wavelet-based scaling estimator that reduces the information in a mammogram down to an informative and low-dimensional quantification of the innate scaling behavior, optimized for use in classifying the tissue as cancerous or non-cancerous. The specific demands for this estimator are that it be robust with respect to distributional assumptions on the data, and with respect to outlier levels in the frequency domain representation of the data. The final aim in this research focuses on enhancing the visualization of microcalcifications that are too small to capture well on screening mammograms. Using scale-mixing discrete wavelet transform methods, the existing detail information contained in a very small and course image will be used to impute scaled details at finer levels. These "informed" finer details will then be used to produce an image of much higher resolution than the original, improving the visualization of the object. The goal is to also produce a confidence area for the true location of the shape's borders, allowing for more accurate feature assessment. Through the more accurate assessment of these very small shapes, physicians may be more confident in deciding next steps.