In the lifetime process in some systems, most data cannot belong to one single population. In fact, it can represent several subpopulations. In such a case, the known distribution cannot be used to model data. Instead, a mixture of distribution is used to modulate the data and classify them into several subgroups. The mixture of Rayleigh distribution is best to be used with the lifetime process. This paper aims to infer model parameters by the expectation-maximization (EM) algorithm through the maximum likelihood function. The technique is applied to simulated data by following several scenarios. The accuracy of estimation has been examined by the average mean square error (AMSE) and the average classification success rate (ACSR). The results showed that the method performed well in all simulation scenarios with respect to different sample sizes.

Multilocus haplotype analysis of candidate variants with genome wide association studies (GWAS) data may provide evidence of association with disease, even when the individual loci themselves do not. Unfortunately, when a large number of candidate variants are investigated, identifying risk haplotypes can be very difficult. To meet the challenge, a number of approaches have been put forward in recent years. However, most of them are not directly linked to the disease-penetrances of haplotypes and thus may not be efficient. To fill this gap, we propose a mixture model-based approach for detecting risk haplotypes. Under the mixture model, haplotypes are clustered directly according to their estimated disease penetrances. A theoretical justification of the above model is provided. Furthermore, we introduce a hypothesis test for haplotype inheritance patterns which underpin this model. The performance of the proposed approach is evaluated by simulations and real data analysis. The simulation results show that the proposed approach outperforms an existing multiple testing method in terms of average specificity and sensitivity. We apply the proposed approach to analyzing two datasets on coronary artery disease and hypertension in the Wellcome Trust Case Control Consortium, identifying many more disease associated haplotype blocks than does the existing method.

Breast cancer has got much attention in the recent years as it is a one of the complex diseases that can threaten people lives. It can be determined from the levels of secreted proteins in the blood. In this project, we developed a method of finding a threshold to classify the probability of being affected by it in a population based on the levels of the related proteins in relatively small case-control samples. We applied our method to simulated and real data. The results showed that the method we used was accurate in estimating the probability of being diseased in both simulation and real data. Moreover, we were able to calculate the sensitivity and specificity under the null hypothesis of our research question of being diseased or not.

Methods of estimating statistical distribution have attracted many researchers when it comes to fitting a specific distribution to data. However, when the data belong to more than one component, a popular distribution cannot be fitted to such data. To tackle this issue, mixture models are fitted by choosing the correct number of components that represent the data. This can be obvious in lifetime processes that are involved in a wide range of engineering applications as well as biological systems. In this paper, we introduce an application of estimating a finite mixture of Inverse Rayleigh distribution by the use of the Bayesian framework when considering the model as Markov chain Monte Carlo (MCMC). We employed the Gibbs sampler and Metropolis – Hastings algorithms. The proposed techniques are applied to simulated data following several scenarios. The accuracy of estimation has been examined by the average mean square error (AMSE) and the average classification success rate (ACSR). The results showed that the method was well performed in all simulation scenarios with respect to different sample sizes.

The region-based association analysis has been proposed to capture the collective behavior of sets of variants by testing the association of each set instead of individual variants with the disease. Such an analysis typically involves a list of unphased multiple-locus genotypes with potentially sparse frequencies in cases and controls. To tackle the problem of the sparse distribution, a two-stage approach was proposed in literature: In the first stage, haplotypes are computationally inferred from genotypes, followed by a haplotype co-classification. In the second stage, the association analysis is performed on the inferred haplotype groups. If a haplotype is unevenly distributed between the case and control samples, this haplotype is labeled as a risk haplotype. Unfortunately, the in-silico reconstruction of haplotypes might produce a proportion of false haplotypes which hamper the detection of rare but true haplotypes. Here, to address the issue, we propose an alternative approach: In Stage 1, we cluster genotypes instead of inferred haplotypes and estimate the risk genotypes based on a finite mixture model. In Stage 2, we infer risk haplotypes from risk genotypes inferred from the previous stage. To estimate the finite mixture model, we propose an EM algorithm with a novel data partition-based initialization.The performance of the proposed procedure is assessed by simulation studies and a real data analysis. Compared to the existing multiple Z-test procedure, we find that the power of genomewide association studies can be increased by using the proposed procedure.