Tong Tong Wu

Coordinate descent algorithms for lasso penalized regression

Imposition of a lasso penalty shrinks parameter estimates toward zero and performs continuous model selection. Lasso penalized regression is capable of handling linear regression problems where the number of predictors far exceeds the number of cases. This paper tests two exceptionally fast algorithms for estimating regression coefficients with a lasso penalty. The previously known l 2 algorithm is based on cyclic coordinate descent. Our new l 1 algorithm is based on greedy coordinate descent and Edgeworths algorithm for ordinary l 1 regression. Each algorithm relies on a tuning constant that can be chosen by cross-validation. In some regression problems it is natural to group parameters and penalize parameters group by group rather than separately. If the group penalty is proportional to the Euclidean norm of the parameters of the group, then it is possible to majorize the norm and reduce parameter estimation to l 2 regression with a lasso penalty. Thus. the existing algorithm can be extended to novel settings. Each of the algorithms discussed is tested via either simulated or real data or both. The Appendix proves that a greedy form of the l 2 algorithm converges to the minimum value of the objective function.

Genome-wide association analysis by lasso penalized logistic regression

MOTIVATION In ordinary regression, imposition of a lasso penalty makes continuous model selection straightforward. Lasso penalized regression is particularly advantageous when the number of predictors far exceeds the number of observations. METHOD The present article evaluates the performance of lasso penalized logistic regression in case-control disease gene mapping with a large number of SNPs (single nucleotide polymorphisms) predictors. The strength of the lasso penalty can be tuned to select a predetermined number of the most relevant SNPs and other predictors. For a given value of the tuning constant, the penalized likelihood is quickly maximized by cyclic coordinate ascent. Once the most potent marginal predictors are identified, their two-way and higher order interactions can also be examined by lasso penalized logistic regression. RESULTS This strategy is tested on both simulated and real data. Our findings on coeliac disease replicate the previous SNP results and shed light on possible interactions among the SNPs. AVAILABILITY The software discussed is available in Mendel 9.0 at the UCLA Human Genetics web site. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

Repeated Measures Longitudinal Analyses of HIV Virologic Response as a Function of Percent Adherence, Dose Timing, Genotypic Sensitivity, and Other Factors

Background: Adherence to antiretroviral medications is critical to achieving HIV viral suppression. Studies have been limited to cross-sectional analyses using measures that reflect only the percentage of prescribed doses taken (percent adherence), however. The contribution of dose timing and other factors to achieving virologic suppression has received less scrutiny. Methods: In a longitudinal study, we collected detailed adherence information using multiple tools along with demographic, clinical, social-behavioral, and virologic measures. Subjects were followed for 48 weeks. Percent adherence, dose-timing, genotypic sensitivity, and virologic outcomes were collected every 4 weeks. Repeated measures mixed effects models (RMMEMs) were used to model the relation between virologic outcomes and adherence as well as genotypic sensitivity and others. Results: Of the 141 subjects, mean percent adherence was 73% with a downward trend. Viral load (VL) dropped significantly (P = 0.01) over time. RMMEMs revealed that higher genotypic sensitivity, higher percent adherence, lower baseline VL, longer inclusion in the study, earlier HIV stage, and smaller dose-timing error were significantly associated with lower VL. In multivariate modeling, a 0.50 increase in the genotypic sensitivity score, a 10% increase in adherence, and a decrease of 3 hours of dose-timing error were associated with a decrease in log10 HIV RNA at 48 weeks of 0.69, 0.54, and 0.48, respectively (P < 0.05 for each). Conclusions: Long-term viral suppression requires consistent and high percent adherence accompanied by optimal interdose intervals. Efforts to improve viral outcomes should address not only missed doses but excessive variation in dose timing and prevention of adherence decline over time. Preventing the development and transmission of resistant variants is also critically important.

The MM Alternative to EM

The EM algorithm is a special case of a more general algorithm called the MM algorithm. Specific MM algorithms often have nothing to do with missing data. The first M step of an MM algorithm creates a surrogate function that is optimized in the second M step. In minimization, MM stands for majorize--minimize; in maximization, it stands for minorize--maximize. This two-step process always drives the objective function in the right direction. Construction of MM algorithms relies on recognizing and manipulating inequalities rather than calculating conditional expectations. This survey walks the reader through the construction of several specific MM algorithms. The potential of the MM algorithm in solving high-dimensional optimization and estimation problems is its most attractive feature. Our applications to random graph models, discriminant analysis and image restoration showcase this ability.

Fine mapping of regulatory loci for mammalian gene expression using radiation hybrids.

We mapped regulatory loci for nearly all protein-coding genes in mammals using comparative genomic hybridization and expression array measurements from a panel of mouse–hamster radiation hybrid cell lines. The large number of breaks in the mouse chromosomes and the dense genotyping of the panel allowed extremely sharp mapping of loci. As the regulatory loci result from extra gene dosage, we call them copy number expression quantitative trait loci, or ceQTLs. The −2log10P support interval for the ceQTLs was <150 kb, containing an average of <2–3 genes. We identified 29,769 trans ceQTLs with −log10P > 4, including 13 hotspots each regulating >100 genes in trans. Further, this work identifies 2,761 trans ceQTLs harboring no known genes, and provides evidence for a mode of gene expression autoregulation specific to the X chromosome.

Physical Activity Trajectories and Multilevel Factors Among Adolescent Girls

PURPOSE Although the decline of physical activity in adolescent girls is well-documented, there are girls whose physical activity does not follow this pattern. This study examined the relationships between physical activity trajectories and personal, psychosocial, and environmental factors among adolescent girls. METHODS Participants were from the University of Maryland field site of the Trial of Activity for Adolescent Girls. Of 730 girls measured in 8th grade, 589 were remeasured in 11th grade. Moderate-to-vigorous physical activity was assessed by accelerometers; participants were categorized as active maintainers (n = 31), inactive maintainers (n = 410), adopters (n = 64), or relapsers (n = 56). Height and weight were measured, personal and psychosocial information was collected from surveys, and distance from home to school and parks was assessed from Geographical Information Systems. Multivariable logistic regression was used for data analysis. RESULTS Variables at individual, social, and environmental levels predicted active maintainers and inactive maintainers, while only individual-level variables predicted adoption. None predicted relapse. Higher (favorable) scores for physical self-concept, perceived body fat, friend and family physical activity support, frequency of physical activity with friends, and shorter distance from home to a park predicted active maintainers. Overweight/obese status, earlier age at menses, and lower scores for physical self-concept, perceived body fat, friend physical activity support, and frequency of physical activity with friends, and farther distance from home to school predicted inactive maintainers. High physical self-concept and not being overweight/obese predicted adopters. CONCLUSIONS Multilevel factors appear to predict behavior maintenance rather than actual change.

Nitric oxide is negatively correlated to pain during acute inflammation.

BackgroundThe role that nitric oxide (NO) plays in modulating pain in the periphery is unclear. We show here, the results of two independent clinical studies (microdialysis and gene expression studies) and a pilot dose finding study (glyceryl trinitrate study), to study the role of NO in the early phase of acute inflammatory pain following oral surgery. The effect of ketorolac on NO production and nitric oxide synthase (NOS) gene expression was also studied.ResultsMicrodialysis samples showed significantly higher levels of NO at the first 100 min compared to the last 80 minutes in the placebo treated group. In the ketorolac group, on the other hand, NO levels gradually decreased over the first 60 min but were similar to placebo over the later 100-180 min, with no significant change in NO level over time. The levels of NO were negatively correlated to pain intensity scores. Local infusion of the NO donor glyceryl trinitrate at the site of surgery, showed a small analgesic effect that did not reach statistical significance in the sample size used. While the gene expression of iNOS and eNOS were not up-regulated, 3 hours after surgery, nNOS was downregulated in both treatment groups and eNOS gene expression was significantly lower in the ketorolac group compared to the placebo group. Further, there was a positive correlation between the change in gene expression of nNOS and eNOS in the placebo goup but not in the ketorolac group.ConclusionWe suggest that at this early stage of inflammatory pain in man, NO is analgesic in the periphery. Further, ketorolac down-regulates eNOS gene expression.

A transcriptome analysis by lasso penalized Cox regression for pancreatic cancer survival.

Pancreatic cancer is the fourth leading cause of cancer deaths in the United States with five-year survival rates less than 5% due to rare detection in early stages. Identification of genes that are directly correlated to pancreatic cancer survival is crucial for pancreatic cancer diagnostics and treatment. However, no existing GWAS or transcriptome studies are available for addressing this problem. We apply lasso penalized Cox regression to a transcriptome study to identify genes that are directly related to pancreatic cancer survival. This method is capable of handling the right censoring effect of survival times and the ultrahigh dimensionality of genetic data. A cyclic coordinate descent algorithm is employed to rapidly select the most relevant genes and eliminate the irrelevant ones. Twelve genes have been identified and verified to be directly correlated to pancreatic cancer survival time and can be used for the prediction of future patients survival.

An MM Algorithm for Multicategory Vertex Discriminant Analysis

This article introduces a new method of supervised learning based on linear discrimination among the vertices of a regular simplex in Euclidean space. Each vertex represents a different category. Discrimination is phrased as a regression problem involving ϵ-insensitive residuals and a quadratic penalty on the coefficients of the linear predictors. The objective function can by minimized by a primal MM (majorization–minimization) algorithm that (a) relies on quadratic majorization and iteratively re-weighted least squares, (b) is simpler to program than algorithms that pass to the dual of the original optimization problem, and (c) can be accelerated by step doubling. Limited comparisons on real and simulated data suggest that the MM algorithm is competitive in statistical accuracy and computational speed with the best currently available algorithms for discriminant analysis.

MULTICATEGORY VERTEX DISCRIMINANT ANALYSIS FOR HIGH-DIMENSIONAL DATA

In response to the challenges of data mining, discriminant analysis continues to evolve as a vital branch of statistics. Our recently introduced method of vertex discriminant analysis (VDA) is ideally suited to handle multiple categories and an excess of predictors over training cases. The current paper explores an elaboration of VDA that conducts classification and variable selection simultaneously. Adding lasso (l-norm) and Euclidean penalties to the VDA loss function eliminates unnecessary predictors. Lasso penalties apply to each predictor coefficient separately; Euclidean penalties group the collective coefficients of a single predictor. With these penalties in place, cyclic coordinate descent accelerates estimation of all coefficients. Our tests on simulated and benchmark real data demonstrate the virtues of penalized VDA in model building and prediction in high-dimensional settings.