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A Bayesian Framework for the Analysis of Microarray Expression Data: Regularized tTest and Statistical Inferences of Gene Changes
 Bioinformatics
, 2001
"... Motivation: DNA microarrays are now capable of providing genomewide patterns of gene expression across many different conditions. The first level of analysis of these patterns requires determining whether observed differences in expression are significant or not. Current methods are unsatisfactory ..."
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Cited by 491 (6 self)
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Motivation: DNA microarrays are now capable of providing genomewide patterns of gene expression across many different conditions. The first level of analysis of these patterns requires determining whether observed differences in expression are significant or not. Current methods are unsatisfactory due to the lack of a systematic framework that can accommodate noise, variability, and low replication often typical of microarray data. Results: We develop a Bayesian probabilistic framework for microarray data analysis. At the simplest level, we model logexpression values by independent normal distributions, parameterized by corresponding means and variances with hierarchical prior distributions. We derive point estimates for both parameters and hyperparameters, and regularized expressions for the variance of each gene by combining the empirical variance with a local background variance associated with neighboring genes. An additional hyperparameter, inversely related to the number of empirical observations, determines the strength of the background variance. Simulations show that these point estimates, combined with a ttest, provide a systematic inference approach that compares favorably with simple ttest or fold methods, and partly compensate for the lack of replication. Availability: The approach is implemented in a software called CyberT accessible through a Web interface at www.genomics.uci.edu/software.html. The code is available as Open Source and is written in the freely available statistical language R. and Department of Biological Chemistry, College of Medicine, University of California, Irvine. To whom all correspondence should be addressed. Contact: pfbaldi@ics.uci.edu, tdlong@uci.edu. 1
Discriminating between the Weibull and LogNormal distributions
 Naval Research Logistics
, 2004
"... LogNormal andWeibull distributions are the most popular distributions for modeling skewed data. In this paper, we consider the ratio of the maximized likelihood in choosing between the two distributions. The asymptotic distribution of the logarithm of the maximized likelihood ratio has been obtain ..."
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Cited by 7 (2 self)
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LogNormal andWeibull distributions are the most popular distributions for modeling skewed data. In this paper, we consider the ratio of the maximized likelihood in choosing between the two distributions. The asymptotic distribution of the logarithm of the maximized likelihood ratio has been obtained. It is observed that the asymptotic distribution is independent of the unknown parameters. The asymptotic distribution has been used to determine the minimum sample size required to discriminate between two families of distributions for a user speci¯ed probability of correct selection. We perform some numerical experiments to observe how the asymptotic methods work for di®erent sample sizes. It is observed that the asymptotic results work quite well even for small samples also. Two real data sets have been analyzed.
Discriminating between the LogNormal and gamma distributions
 Journal of the Applied Statistical Sciences
, 2005
"... For a given data set the problem of selecting either lognormal or gamma distribution with unknown shape and scale parameters is discussed. It is well known that both these distributions can be used quite e®ectively for analyzing skewed nonnegative data sets. In this paper, we use the ratio of the ..."
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Cited by 5 (2 self)
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For a given data set the problem of selecting either lognormal or gamma distribution with unknown shape and scale parameters is discussed. It is well known that both these distributions can be used quite e®ectively for analyzing skewed nonnegative data sets. In this paper, we use the ratio of the maximized likelihoods in choosing between the lognormal and gamma distributions. We obtain asymptotic distributions of the ratio of the maximized likelihoods and use them to determine the minimum sample size required to discriminate between these two distributions for user speci¯ed probability of correct selection and tolerance limit. Key Words and Phrases: Asymptotic distribution; KolmogorovSmirnov distances; probability of correct selection; tolerance level. 1
Discriminating Between The Normal and The Laplace Distributions
"... Both normal and Laplace distributions can be used to analyze symmetric data. In this paper we consider the logarithm of the ratio of the maximized likelihoods to discriminate between the two distribution functions. We obtain the asymptotic distributions of the test statistics and it is observed that ..."
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Cited by 1 (0 self)
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Both normal and Laplace distributions can be used to analyze symmetric data. In this paper we consider the logarithm of the ratio of the maximized likelihoods to discriminate between the two distribution functions. We obtain the asymptotic distributions of the test statistics and it is observed that they are independent of the unknown parameters. If the underlying distribution is normal the asymptotic distribution works quite well even when the sample size is small. But if the underlying distribution is Laplace the asymptotic distribution does not work well for the small sample sizes. For the later case we propose a bias corrected asymptotic distribution and it works quite well even for small sample sizes. Based on the asymptotic distributions, minimum sample size needed to discriminate between the two distributing functions is obtained for a given probability of correct selection. Monte Carlo simulations are performed to examine how the asymptotic results work for small sizes and two data sets are analyzed for illustrative purposes.
Discriminating Among the LogNormal, Weibull and Generalized Exponential Distributions
"... In this paper we consider the model selection / discrimination among the three important lifetime distributions. All these three distributions have been used quite e®ectively to analyze lifetime data in the reliability analysis. We study the probability of correct selection using the maximized likel ..."
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In this paper we consider the model selection / discrimination among the three important lifetime distributions. All these three distributions have been used quite e®ectively to analyze lifetime data in the reliability analysis. We study the probability of correct selection using the maximized likelihood method, as it has been used in the literature. We further compute the asymptotic probability of correct selection and compare the theoretical and simulation results for di®erent sample sizes and for di®erent model parameters. The results have been extended for TypeI censored data also. The theoretical and simulation results match quite well. Two real data sets have been analyzed for illustrative purposes. We also suggest a method to determine the minimum sample size required to discriminate among the three distributions for a given probability of correct selection and a user speci¯ed protection level.