The null distribution of the likelihood ratio test for a mixture of two normals after a restricted box-cox transformation

Ning, Yuming; Finch, Stephen J.

doi:10.1080/03610910008813621

Cited by 5 publications

(3 citation statements)

References 16 publications

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“…Note that the 2 distribution is approximate. The P value provided by this test has been shown to be liberal (12), and an improved approximation by a 2 distribution with 2.5 df has been suggested (13). When the two normal components in the mixture distribution have unequal variances, simulation results have indicated that the limiting distribution is bounded by 2 distributions with 4 and 6 df (14,15).…”

Section: Methodsmentioning

confidence: 93%

Bimodality of 2-h Plasma Glucose Distributions in Whites

et al. 2005

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OBJECTIVE -Several studies have shown a bimodal curve in the distribution of glucose in populations with a high prevalence of type 2 diabetes, but bimodality has not been reported among whites of Northern European ancestry. It is not clear whether this difference reflects the lower prevalence of diabetes, obscuring a second mode, or implies a more fundamental difference between whites and nonwhites. We investigate this issue by studying glucose distributions in older white patients. RESEARCH DESIGN AND METHODS-A study of diabetes was conducted among older community-dwelling white residents of a suburban Southern California community between 1984 and 1987. Two-hour plasma glucose data were collected from 2,326 older white men and women aged 23-92. To investigate bimodality of glucose distributions, we fit unimodal and bimodal normal models to 2-h plasma glucose concentrations transformed by the Box-Cox family of transformations.RESULTS -We found that the bimodal normal mixture model fit the data significantly better than the unimodal skewed distribution model for both sexes and all age-groups except those Ն80 years. The cut points separating the two modes were generally within the 11.1-to 13.6-mmol/l range.CONCLUSIONS -The bimodality of glucose distributions among whites, combined with previous findings, indicates that this phenomenon may be universal. A smaller second mode in our study compared with other studies suggests that whites have diabetes susceptibility but may require more obesity to demonstrate it. With increasing obesity in the U.S., the predicted epidemic of diabetes may affect all ethnic groups including whites. Diabetes Care 28:1451-1456, 2005A bimodal distribution of plasma glucose concentrations in a population was first described in a 1971 study on Pima Indians, who have the highest prevalence of type 2 diabetes in the world (1). A bimodal curve showing two distinct distributions of plasma glucose concentrations, instead of a single skewed distribution, is compatible with a genetic condition (diabetes) manifest by hyperglycemia and a second population without diabetes (without hyperglycemia). Subsequently, bimodality was reported in other populations with a high prevalence of diabetes, including Nauruans from Micronesia (2), Samoans (3), Asian Indians who had migrated to South Africa (4), and Mexican Americans who were ϳ50% white (5). The diversity of populations with glucose bimodality suggests that this may be a universal phenomenon, readily detected in populations with a high prevalence of diabetes. To our knowledge, no studies have reported a bimodal glucose distribution among whites of Northern European ancestry, suggesting either genetic/environmental differences or a prevalence of diabetes too low to detect a second mode (6,7).We hypothesized that statistically significant bimodality might be detectable in older white patients because the prevalence of diabetes increases to nearly 20% in old age (8) and because most genetically susceptible people might be expected to develop diabetes if ...

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Section: Methodsmentioning

confidence: 93%

Bimodality of 2-h Plasma Glucose Distributions in Whites

et al. 2005

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“…Second, we considered a wider interval [0, 1.5] than the one recommended by Maclean et al [ 2 ] for selecting the optimal power parameter. Third, as suggested by Ning et al [ 11 ], we used 6.9 as the 0.05 critical value for LRT of "a single component distribution" vs. "a mixture distribution of two components."…”

Section: Methodsmentioning

confidence: 99%

Mixture modeling of microarray gene expression data

et al. 2007

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About 28% of genes appear to have an expression pattern that follows a mixture distribution. We use first- and second-order partial correlation coefficients to identify trios and quartets of non-sex-linked genes that are highly associated and that are also mixtures. We identified 18 trio and 35 quartet mixtures and evaluated their mixture distribution concordance. Concordance was defined as the proportion of observations that simultaneously fall in the component with the higher mean or simultaneously in the component with the lower mean based on their Bayesian posterior probabilities. These trios and quartets have a concordance rate greater than 80%. There are 33 genes involved in these trios and quartets. A factor analysis with varimax rotation identifies three gene groups based on their factor loadings. One group of 18 genes has a concordance rate of 56.7%, another group of 8 genes has a concordance rate of 60.8%, and a third group of 7 genes has a concordance rate of 69.6%. Each of these rates is highly significant, suggesting that there may be strong biological underpinnings for the mixture mechanisms of these genes. Bayesian factor screening confirms this hypothesis by identifying six single-nucleotide polymorphisms that are significantly associated with the expression phenotypes of the five most concordant genes in the first group.

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“…When the two normal components in the mixture distribution have equal variance, i.e., σ 1 = σ 2 , the p-value provided by the traditional LRT has been shown to be liberal (Thode et al, 1988) and an improved approximation by a chi-square distribution with 2.5 degrees of freedom has been suggested (Ning and Finch, 2000). When the two normal components in the mixture distribution have unequal variances, i.e., σ 1 = σ 2 , simulation results have indicted that the limiting distribution is bounded by chi-square distributions with 4 and 6 degrees of freedom (McLachlan, 1987;Gutierrez et al, 1995).…”

Section: Introductionmentioning

confidence: 99%

Bimodality of Plasma Glucose Distributions in Whites: A Bootstrap Approach to Testing Mixture Models

Yang¹,

Fan²,

May³

2021

Journal of Data Science

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The null distribution of the likelihood ratio test (LRT) of a onecomponent normal model versus two-component normal mixture model is unknown. In this paper, we take a bootstrap approach to the likelihood ratio test for testing bimodality of plasma glucose concentrations from Rancho Bernardo Diabetes Study. The small p-values from this approach support the hypothesis that a bimodal normal mixture model fits the data significantly better than a unimodal normal model. The size and power of the bootstrap based LRT are evaluated through simulations. The results suggest that a sample size of close to 500 would be necessary in order to attain a power of 90% for detecting the unbalanced mixtures with means and variances similar to those in the Rancho Bernardo data. Besides sample size, the power also depends on the two means and variances of the two components in the data.

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The null distribution of the likelihood ratio test for a mixture of two normals after a restricted box-cox transformation

Cited by 5 publications

References 16 publications

Bimodality of 2-h Plasma Glucose Distributions in Whites

Bimodality of 2-h Plasma Glucose Distributions in Whites

Mixture modeling of microarray gene expression data

Bimodality of Plasma Glucose Distributions in Whites: A Bootstrap Approach to Testing Mixture Models

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