Cristina Rueda

Estimation of Parameters Subject to Order Restrictions on a Circle With Application to Estimation of Phase Angles of Cell Cycle Genes

Motivated by a problem encountered in the analysis of cell cycle gene expression data, this article deals with the estimation of parameters subject to order restrictions on a unit circle. A normal eukaryotic cell cycle has four major phases during cell division, and a cell cycle gene has its peak expression (phase angle) during the phase that may correspond to its biological function. Because the phases are ordered along a circle, the phase angles of cell cycle genes are ordered unknown parameters on a unit circle. The problem of interest is to estimate the phase angles using the information regarding the order among them. We address this problem by developing a circular version of the well-known isotonic regression for Euclidean data. Because of the underlying geometry, the standard pool adjacent violator algorithm (PAVA) cannot be used for deriving the circular isotonic regression estimator (CIRE). However, PAVA can be modified to obtain a computationally efficient algorithm for deriving the CIRE. We illustrate the CIRE by estimating the phase angles of some of well-known cell cycle genes using the unrestricted estimators obtained in the literature.

Identification of a core set of signature cell cycle genes whose relative order of time to peak expression is conserved across species

A cell division cycle is a well-coordinated process in eukaryotes with cell cycle genes exhibiting a periodic expression over time. There is considerable interest among cell biologists to determine genes that are periodic in multiple organisms and whether such genes are also evolutionarily conserved in their relative order of time to peak expression. Interestingly, periodicity is not well-conserved evolutionarily. A conservative estimate of a number of periodic genes common to fission yeast (Schizosaccharomyces pombe) and budding yeast (Saccharomyces cerevisiae) (‘core set FB’) is 35, while those common to fission yeast and humans (Homo sapiens) (‘core set FH’) is 24. Using a novel statistical methodology, we discover that the relative order of peak expression is conserved in ∼80% of FB genes and in ∼40% of FH genes. We also discover that the order is evolutionarily conserved in six genes which are potentially the core set of signature cell cycle genes. These include ace2 (a transcription factor) and polo-kinase plo1, which are well-known hubs of early M-phase clusters, cdc18 a key component of pre-replication complexes, mik1 which is critical for the establishment and maintenance of DNA damage check point, and histones hhf1 and hta2.

The Loss of Efficiency Estimating Linear Functions under Restrictions

This article is motivated by the problem of estimating contrast in a one-way ANOVA model with restrictions in the parameter vector. We prove that when the restrictions are given by a tree order or a simple order the MLE of some contrast has greater MSE than the unrestricted estimator. A similar behaviour of the MLE is exhibited in a general restricted setting given by a multivariate normal distribution with mean vector constrained to belong to a circular cone. The approach we use focuses on the central direction of the cones. These directions appear to have the greater MSE when the dimension of the restricted cone is big enough.

Incorporating Additional Information to Normal Linear Discriminant Rules

The most useful and broadly known rule in the classical two-group linear normal discriminant analysis is Andersons rule. In this article we propose some alternative procedures that prove useful when prior constraints on the mean vectors are known. These rules are based on new estimators of the difference of means. We prove under mild conditions that the new rules perform better when the common covariance matrix is known. Simulated experiments show that the misclassification errors are lower for the restricted rules defined here in the general case of an unknown covariance matrix. The prior constraints on the mean vector restrict the parameter space to a cone. A family of estimators indexed by a parameter γ, with 0 ≤ γ ≤ 1, is defined using an iterative procedure in such a way that the estimator with a higher value for γ takes values closer to the center of the cone with a greater probability. When γ = 0, the restricted maximum likelihood estimator is given, although the most interesting rule from a theoretical and practical standpoint is obtained when the estimator chosen is given by γ = 1. The usefulness of the proposed rules with real data is demonstrated by their application to two medical examples, the first dealing with heart attack patients and the second dealing with a diabetes dataset. In the former case, restrictions among surviving and nonsurviving patients are used; in the latter, the restrictions arise from differences between the healthy and diabetic populations.

Reduction of risk using restricted estimators

We consider the problem of estimating a linear function, c′θ, of the mean of a random vector, X -Nk(θ, D), where D is diagonal and known and θ is subject to p linear restrictions. Let X* be the maximum likelihood estimatoi of θ. We establish that for p=2 and any cϵ RkE( c′X* - c′θ )2≥ E( c′X* - c′θ)2 . We also provide conditions which guarantee that c′X* universally dominates c′X.

Identification of two clusters within schizophrenia with different structural, functional and clinical characteristics.

BACKGROUND Several biologically distinct subgroups may coexist within schizophrenia, which may hamper the necessary replicability to translate research findings into clinical practice. METHODS Cortical thickness, curvature and area values and subcortical volumes of 203 subjects (121 schizophrenia patients, out of which 64 were first episodes), 60 healthy controls and 22 bipolar patients were used to identify clusters using principal components and canonical discriminant analyses. Regional glucose metabolism using positron emission tomography, P300 event related potential, baseline clinical data and percentage of improvement with treatment were used to validate possible clusters based on MRI data. RESULTS All the controls, the bipolar patients and most of the schizophrenia patients were grouped in a cluster (cluster A). A group of 24 schizophrenia patients (12 first episodes), characterized by large intrinsic curvature values, was identified (cluster B). These patients, but not those in cluster A, showed reduced thalamic and cingulate glucose metabolism in comparison to controls, as well as a worsening of negative symptoms at follow-up. Patients in cluster A showed a significant putaminal metabolic increase, which was not observed for those in cluster B. P300 amplitude was reduced in patients of both clusters, in comparison to controls. CONCLUSIONS Results of this study support the existence of a biologically distinct group within the schizophrenia syndrome, characterized by increased cortical curvature values, reduced thalamic and cingulate metabolism, lack of the expected increased putaminal metabolism with antipsychotics and persistent negative symptoms.

Degrees of freedom and model selection in semiparametric additive monotone regression

The degrees of freedom of semiparametric additive monotone models are derived using results about projections onto sums of order cones. Two important related questions are also studied, namely, the definition of estimators for the parameter of the error term and the formulation of specific Akaike Information Criteria statistics. Several alternatives are proposed to solve both problems and simulation experiments are conducted to compare the behavior of the different candidates. A new selection criterion is proposed that combines the ability to guess the model but also the efficiency to estimate the variance parameter. Finally, the criterion is used to select the model in a regression problem from a well known data set.

Testing non-oblique hypotheses

This paper considers a likelihood ratio test for testing hypotheses defined by non-oblique closed convex cones, satisfying the so called iteration projection property, in a set of k normal means. We obtain the critical values of the test using the Chi-Bar-Squared distribution. The obtuse cones are introduced as a particular class of cones which are non-oblique with every one of their faces. Examples with the simple tree order cone and the total order cone are given to illustrate the results.

Classification of samples into two or more ordered populations with application to a cancer trial

In many applications, especially in cancer treatment and diagnosis, investigators are interested in classifying patients into various diagnosis groups on the basis of molecular data such as gene expression or proteomic data. Often, some of the diagnosis groups are known to be related to higher or lower values of some of the predictors. The standard methods of classifying patients into various groups do not take into account the underlying order. This could potentially result in high misclassification rates, especially when the number of groups is larger than two. In this article, we develop classification procedures that exploit the underlying order among the mean values of the predictor variables and the diagnostic groups by using ideas from order-restricted inference. We generalize the existing methodology on discrimination under restrictions and provide empirical evidence to demonstrate that the proposed methodology improves over the existing unrestricted methodology. The proposed methodology is applied to a bladder cancer data set where the researchers are interested in classifying patients into various groups.

Small area semiparametric additive monotone models

In this paper, semiparametric monotone mixed models are introduced, exploring, in particular, the problems of estimation and bootstrapping. The models are defined in a small area setting, using the assumption that some of the auxiliary variables have a monotone relationship with the response, and with the incorporation of linear terms to model other auxiliaries as the dummy variables. An estimator for the variance of the random effects is proposed and two bootstrap approaches, specially designed for monotone regression, are given to estimate the mean squared error for the area means. A simulation experiment is carried out to compare the performance of the new model-based estimators against the Fay–Herriot approach and to confirm the good performance of the bootstrap. The semiparametric model-based area estimators are also compared with the parametric-based estimators using data on a survey of lakes, where the questions of the prediction of missing data and model selection are nicely solved using simple proposals.