MULTIPLE COMPARISONS OF LOG-LIKELIHOODS AND COMBINING NONNESTED MODELS WITH APPLICATIONS TO PHYLOGENETIC TREE SELECTION

We consider multiple comparisons of log-likelihood's to take account of the multiplicity of testings in selection of nonnested models. A resampling version of the Gupta procedure for the selection problem is used to obtain a set of good models, which are not significantly worse than the maximum likelihood model; i.e., a confidence set of models. Our method is to test which model is better than the other, while the object of the classical testing methods is to find the correct model. Thus the null hypotheses behind these two approaches are very different. Our method and the other commonly used approaches, such as the approximate Bayesian posterior, the bootstrap selection probability, and the LR test against the full model, are applied to the selection of molecular phylogenetic tree of mammal species. Tree selection is a version of the model-based clustering, which is an example of nonnested model selection. It is shown that the structure of the tree selection problem is equivalent to that of the variable selection problem of the multiple regression with some constraints on the combinations of the variables. It turns out that the LR test rejects all the possible trees because of the misspecification of the models, whereas our method gives a reasonable confidence set. For a better understanding of the uncertainty in the selection, we combine the maximum likelihood estimates (MLE's) of the trees to obtain the full model that includes the trees as the submodels by using a linear approximation of the parametric models. The MLE of the phylogeny is then represented as a network of species rather than a tree. A geometrical interpretation of the problem is also discussed.     

CONDITIONING VS. STANDARDIZATION FOR CONTRASTS ON ERRORS ATTRACTED TO STABLE LAWS

In multiple regression and other settings one encounters the problem of estimating sampling distributions for contrast operations applied to i.i.d. errors. Permutation bootstrap applied to least squares residuals has been proven to consistently estimate conditionalsampling distributions of contrasts, conditional upon order statistics of errors, even for long-tailed error distributions. How does this compare with the unconditional sampling distribution of the contrast when standardizing by the sample s.d. of the errors (or the residuals)? For errors belonging to the domain of attraction of a normal we present a limit theorem proving that these distributions are far closer to one another than they are to the limiting standard normal distribution. For errors attracted to α-stable laws with α ≤ 2 we construct random variables possessing these conditional and unconditional sampling distributions and develop a Poisson representation for their a.s. limit correlation ρ_α. We prove that ρ₂= 1, ρ_α→ 1 for α → 0 + or 2 ?, and ρ_α< 1 a.s. for α < 2.     

Detecting Abrupt Leaks in Blended Underground Storage Tanks

We suggest and compare two multiple change-point algorithms (segmentation and sequential) for accurate detection of the onset of abrupt leaks in blended underground storage tanks. We apply these algorithms to two simulated scenarios to demonstrate the advantages of the sequential algorithm, and then we apply the sequential algorithm to the Cary blended site data. In addition, we obtain a confidence set for the locations of the change points conditional on the number of change points by inverting the related hypothesis test.     

Fitting a Semi-Parametric Mixture Model for Competing Risks in Survival Data

A model for survival analysis is studied that is relevant for samples which are subject to multiple types of failure. In comparison with a more standard approach, through the appropriate use of hazard functions and transition probabilities, the model allows for a more accurate study of cause-specific failure with regard to both the timing and type of failure. A semiparametric specification of a mixture model is employed that is able to adjust for concomitant variables and allows for the assessment of their effects on the probabilities of eventual causes of failure through a generalized logistic model, and their effects on the corresponding conditional hazard functions by employing the Cox proportional hazards model. A carefully formulated estimation procedure is presented that uses an EM algorithm based on a profile likelihood construction. The methods discussed, which could also be used for reliability analysis, are applied to a prostate cancer data set.     

Estimation of Slope for Linear Regression Model with Uncertain Prior Information and Student-t Error

This article considers estimation of the slope parameter of the linear regression model with Student-t errors in the presence of uncertain prior information on the value of the unknown slope. Incorporating uncertain non sample prior information with the sample data the unrestricted, restricted, preliminary test, and shrinkage estimators are defined. The performances of the estimators are compared based on the criteria of unbiasedness and mean squared errors. Both analytical and graphical methods are explored. Although none of the estimators is uniformly superior to the others, if the non sample information is close to its true value, the shrinkage estimator over performs the rest of the estimators.     

Likelihood Function and Canonical Correlation Analysis of the Peña–Box Model

The Peña–Box model is a type of dynamic factor model whose factors try to capture the time-effect movements of a multiple time series. The Peña–Box model can be expressed as a vector autoregressive (VAR) model with constraints. This article derives the maximum likelihood estimates and the likelihood ratio test of the VAR model for Gaussian processes. Then a test statistic constructed by canonical correlation coefficients is presented and adjusted for conditional heteroscedasticity. Simulations confirm the validity of adjustments for conditional heteroscedasticity, and show that the proposed statistics perform better than the statistics used in the existing literature.     

Multiple Interval Mapping for Quantitative Trait Loci via EM Algorithm in the Presence of Crossover Interference

The rapid advance in molecular biology has made feasible systematic studies of mapping quantitative trait loci (QTL) in experiment organisms. The method of multiple interval mapping provides an appropriate way for mapping QTL using genetic makers. However, crossover interference has not been considered sufficiently in the current QTL mapping in which no crossover interference is assumed, and the length of maker interval is always kept fixed. In this article, we consider the issue of statistical inference in multiple interval mapping for QTL when crossover interference is present. The marker interval can be chosen appropriately in our method without keeping the maker interval lengths fixed in advance, and the asymptotic variance–covariance matrix of the MLEs is also derived. Two simulations are performed to evaluate the proposed method and show the impact of crossover interference to QTL mapping.     

Rank Tests for Two-Sample Problems Based on Multiple Type-II Censored Data

In this article, we study the effect of censoring on the asymptotic efficiency of the two-sample rank tests based on multiple Type-II censored data. Since the scores generating functions associated with these test statistics have a finite number of jump discontinuities, we use a slightly modified version of a theorem of Dupac and Hajek (1969 Dupac , V. , Hajek , J. ( 1969 ). Asymptotic normality of simple linear rank statistics under alternatives II . Ann. Math. Statist. 40 : 1992 – 2017 .[Crossref] , [Google Scholar]) to obtain their asymptotic distributions under fixed alternatives. This modified version, which leads to a simpler centering constant, is proved by Dupac (1970 Dupac , V. ( 1970 ). A contribution to the asymptotic normality of simple linear rank statistics . In: Puri , M. L. , ed. Nonparametric Techniques in Statistical Inference . Cambridge : Cambridge University Press . [Google Scholar]) in the light of results of Hoeffding (1968 Hoeffding , W. ( 1968 ). On the Centering of Simple Linear Rank Statistics. Instit. Statist. Mimeo Series No. 585, University of North Carolina . [Google Scholar]), an earlier version of Hoeffding (1973 Hoeffding , W. ( 1973 ). On the centering of simple linear rank statistics . Ann. Statist. 1 : 54 – 66 .[Crossref], [Web of Science ®] , [Google Scholar]). Hence, we obtain the Pitman ARE's of these rank tests relative to the corresponding tests based on the complete samples. The ARE's are computed for some well known rank tests for two-sample location and scale problems, when the combined ordered samples from different underlying distributions are censored using triple and lower order Type-II censoring schemes. The effect of all these censoring schemes on the ARE's of the different tests is examined numerically. It is found that there is a gain in efficiency due to censoring in many of the cases considered here. This suggests that in such cases it is possible to improve the efficiency of rank tests by discarding suitable portions of the data.     

Spatial Health Factors with Selection Among Multiple Causes: Lung Cancer in U.S. Counties

Composite morbidity indices summarize geographic inequalities in disease, and are used to distribute resources. A spatial latent variable approach is developed for such an index, focusing on lung cancer in 3,141 U.S. counties. The model incorporates multiple indicators (cancer deaths and incidence), but also allows for population risk variables (area socio-economic, environmental, and smoking indicators) that affect lung cancer, and for missingness among indicators or risk variables. Selection of significant causes is illustrated, including nonadaptive and adaptive selection. To reflect geographic clustering in lung cancer, the latent morbidity index is spatially correlated, although the level of correlation is data determined.