The first chapter proposes multifractal analysis to measure inhomogeneity of regularity of 1H-NMR spectrum using wavelet-based multifractal tools. The geometric summaries of multifractal spectrum are informative summaries, and as such employed to discriminate 1H-NMR spectra associated with different treatments. The methodology is applied to evaluate the effect of sulfur amino acids. The second part of this thesis provides essential materials for understanding engineering background of a nano-particle fabrication process. The third chapter introduces a constrained random effect model. Since there are certain combinations of process variables resulting to unproductive process outcomes, a logistic model is used to characterize such a process behavior. For the cases with productive outcomes a normal regression serves the second part of the model. Additionally, random-effects are included in both logistics and normal regression models to describe the potential spatial correlation among data. This chapter researches a way to approximate the likelihood function and to find estimates for maximizing the approximated likelihood. The last chapter presents a method to decide the sample size under multi-layer system. The multi-layer is a series of layers, which become smaller and smaller. Our focus is to decide the sample size in each layer. The sample size decision has several objectives, and the most important purpose is the sample size should be enough to give a right direction to the next layer. Specifically, the bottom layer, which is the smallest neighborhood around the optimum, should meet the tolerance requirement. Performing the hypothesis test of whether the next layer includes the optimum gives the required sample size.