Kenneth Lange

Fast model-based estimation of ancestry in unrelated individuals

Population stratification has long been recognized as a confounding factor in genetic association studies. Estimated ancestries, derived from multi-locus genotype data, can be used to perform a statistical correction for population stratification. One popular technique for estimation of ancestry is the model-based approach embodied by the widely applied program structure. Another approach, implemented in the program EIGENSTRAT, relies on Principal Component Analysis rather than model-based estimation and does not directly deliver admixture fractions. EIGENSTRAT has gained in popularity in part owing to its remarkable speed in comparison to structure. We present a new algorithm and a program, ADMIXTURE, for model-based estimation of ancestry in unrelated individuals. ADMIXTURE adopts the likelihood model embedded in structure. However, ADMIXTURE runs considerably faster, solving problems in minutes that take structure hours. In many of our experiments, we have found that ADMIXTURE is almost as fast as EIGENSTRAT. The runtime improvements of ADMIXTURE rely on a fast block relaxation scheme using sequential quadratic programming for block updates, coupled with a novel quasi-Newton acceleration of convergence. Our algorithm also runs faster and with greater accuracy than the implementation of an Expectation-Maximization (EM) algorithm incorporated in the program FRAPPE. Our simulations show that ADMIXTUREs maximum likelihood estimates of the underlying admixture coefficients and ancestral allele frequencies are as accurate as structures Bayesian estimates. On real-world data sets, ADMIXTUREs estimates are directly comparable to those from structure and EIGENSTRAT. Taken together, our results show that ADMIXTUREs computational speed opens up the possibility of using a much larger set of markers in model-based ancestry estimation and that its estimates are suitable for use in correcting for population stratification in association studies.

Robust statistical modeling using the t distribution

Abstract The t distribution provides a useful extension of the normal for statistical modeling of data sets involving errors with longer-than-normal tails. An analytical strategy based on maximum likelihood for a general model with multivariate t errors is suggested and applied to a variety of problems, including linear and nonlinear regression, robust estimation of the mean and covariance matrix with missing data, unbalanced multivariate repeated-measures data, multivariate modeling of pedigree data, and multivariate nonlinear regression. The degrees of freedom parameter of the t distribution provides a convenient dimension for achieving robust statistical inference, with moderate increases in computational complexity for many models. Estimation of precision from asymptotic theory and the bootstrap is discussed, and graphical methods for checking the appropriateness of the t distribution are presented.

A Tutorial on MM Algorithms

Most problems in frequentist statistics involve optimization of a function such as a likelihood or a sum of squares. EM algorithms are among the most effective algorithms for maximum likelihood estimation because they consistently drive the likelihood uphill by maximizing a simple surrogate function for the log-likelihood. Iterative optimization of a surrogate function as exemplified by an EM algorithm does not necessarily require missing data. Indeed, every EM algorithm is a special case of the more general class of MM optimization algorithms, which typically exploit convexity rather than missing data in majorizing or minorizing an objective function. In our opinion, MM algorithms deserve to be part of the standard toolkit of professional statisticians. This article explains the principle behind MM algorithms, suggests some methods for constructing them, and discusses some of their attractive features. We include numerous examples throughout the article to illustrate the concepts described. In addition to surveying previous work on MM algorithms, this article introduces some new material on constrained optimization and standard error estimation.

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.

Optimization Transfer Using Surrogate Objective Functions

Abstract The well-known EM algorithm is an optimization transfer algorithm that depends on the notion of incomplete or missing data. By invoking convexity arguments, one can construct a variety of other optimization transfer algorithms that do not involve missing data. These algorithms all rely on a majorizing or minorizing function that serves as a surrogate for the objective function. Optimizing the surrogate function drives the objective function in the correct direction. This article illustrates this general principle by a number of specific examples drawn from the statistical literature. Because optimization transfer algorithms often exhibit the slow convergence of EM algorithms, two methods of accelerating optimization transfer are discussed and evaluated in the context of specific problems.

Prioritizing GWAS results: A review of statistical methods and recommendations for their application.

Genome-wide association studies (GWAS) have rapidly become a standard method for disease gene discovery. A substantial number of recent GWAS indicate that for most disorders, only a few common variants are implicated and the associated SNPs explain only a small fraction of the genetic risk. This review is written from the viewpoint that findings from the GWAS provide preliminary genetic information that is available for additional analysis by statistical procedures that accumulate evidence, and that these secondary analyses are very likely to provide valuable information that will help prioritize the strongest constellations of results. We review and discuss three analytic methods to combine preliminary GWAS statistics to identify genes, alleles, and pathways for deeper investigations. Meta-analysis seeks to pool information from multiple GWAS to increase the chances of finding true positives among the false positives and provides a way to combine associations across GWAS, even when the original data are unavailable. Testing for epistasis within a single GWAS study can identify the stronger results that are revealed when genes interact. Pathway analysis of GWAS results is used to prioritize genes and pathways within a biological context. Following a GWAS, association results can be assigned to pathways and tested in aggregate with computational tools and pathway databases. Reviews of published methods with recommendations for their application are provided within the framework for each approach.

Use of population isolates for mapping complex traits

Geneticists have repeatedly turned to population isolates for mapping and cloning Mendelian disease genes. Population isolates possess many advantages in this regard. Foremost among these is the tendency for affected individuals to share ancestral haplotypes derived from a handful of founders. These haplotype signatures have guided scientists in the fine mapping of scores of rare disease genes. The past successes with Mendelian disorders using population isolates have prompted unprecedented interest among medical researchers in both the public and private sectors. Despite the obvious genetic and environmental complications, geneticists have targeted several population isolates for mapping genes for complex diseases.

Extensions to pedigree analysis III. Variance components by the scoring method

The classic variance components for simple polygenic traits - additive, dominance, and environmental variance - have traditionally been estimated from sample covariances between first-degree relatives. If data is gathered on pedigrees, this statistical procedure wastes information. Recently Elston & Stewart suggested an alternative likelihood procedure that uses all the information in a set of pedigrees. A refinement of their method based on the scoring technique gives rapidly converging maximum likelihood estimates of the variance components and of the male and female means. Tests of statistical hypotheses about the various parameters can then be made by the likelihood ratio method. Furthermore, using classical regression analysis, the estimated parameter values allow prediction of unknown trait values from known trait values within a pedigree. These methods should apply to traits like total finger ridge count and to quantitative measurements associated with disease traits. Since the model postulates independent environmental effects and no assortative mating, its utility in human behaviour genetics seems limited.

Convergence of EM image reconstruction algorithms with Gibbs smoothing

P.J. Green has defined an OSL (one-step late) algorithm that retains the E-step of the EM algorithm (for image reconstruction in emission tomography) but provides an approximate solution to the M-step. Further modifications of the OSL algorithm guarantee convergence to the unique maximum of the log posterior function. Convergence is proved under a specific set of sufficient conditions. Several of these conditions concern the potential function of the Gibbs prior, and a number of candidate potential functions are identified. Generalization of the OSL algorithm to transmission tomography is also considered.