Using biplots to interpret gene expression patterns in plants

Sum: Plant biologists in fields of ecology, evolution, genetics and breeding frequently use multivariate methods. This paper illustrates Principal Component Analysis (PCA) and Gabriel's biplot as applied to microarray expression data from plant pathology experiments. Availability: An example program in the publicly distributed statistical language R is available from the web site (www.tpp.uq.edu.au) and by e-mail from the contact. Contact: scott.chapman@csiro.au.

Maximum Likelihood Estimation of Mixture Densities for Binned and Truncated Multivariate Data

Binning and truncation of data are common in data analysis and machine learning. This paper addresses the problem of fitting mixture densities to multivariate binned and truncated data. The EM approach proposed by McLachlan and Jones (Biometrics, 44: 2, 571-578, 1988) for the univariate case is generalized to multivariate measurements. The multivariate solution requires the evaluation of multidimensional integrals over each bin at each iteration of the EM procedure. Naive implementation of the procedure can lead to computationally inefficient results. To reduce the computational cost a number of straightforward numerical techniques are proposed. Results on simulated data indicate that the proposed methods can achieve significant computational gains with no loss in the accuracy of the final parameter estimates. Furthermore, experimental results suggest that with a sufficient number of bins and data points it is possible to estimate the true underlying density almost as well as if the data were not binned. The paper concludes with a brief description of an application of this approach to diagnosis of iron deficiency anemia, in the context of binned and truncated bivariate measurements of volume and hemoglobin concentration from an individual's red blood cells.

Multivariate monitoring of a biological wastewater treatment process: a case study at Melbourne Water's Western Treatment Plant

Biological wastewater treatment is a complex, multivariate process, in which a number of physical and biological processes occur simultaneously. In this study, principal component analysis (PCA) and parallel factor analysis (PARAFAC) were used to profile and characterise Lagoon 115E, a multistage biological lagoon treatment system at Melbourne Water's Western Treatment Plant (WTP) in Melbourne, Australia. In this study, the objective was to increase our understanding of the multivariate processes taking place in the lagoon. The data used in the study span a 7-year period during which samples were collected as often as weekly from the ponds of Lagoon 115E and subjected to analysis. The resulting database, involving 19 chemical and physical variables, was studied using the multivariate data analysis methods PCA and PARAFAC. With these methods, alterations in the state of the wastewater due to intrinsic and extrinsic factors could be discerned. The methods were effective in illustrating and visually representing the complex purification stages and cyclic changes occurring along the lagoon system. The two methods proved complementary, with each having its own beneficial features. (C) 2003 Elsevier B.V. All rights reserved.

Studying phenotypic evolution using multivariate quantitative genetics

Quantitative genetics provides a powerful framework for studying phenotypic evolution and the evolution of adaptive genetic variation. Central to the approach is G, the matrix of additive genetic variances and covariances. G summarizes the genetic basis of the traits and can be used to predict the phenotypic response to multivariate selection or to drift. Recent analytical and computational advances have improved both the power and the accessibility of the necessary multivariate statistics. It is now possible to study the relationships between G and other evolutionary parameters, such as those describing the mutational input, the shape and orientation of the adaptive landscape, and the phenotypic divergence among populations. At the same time, we are moving towards a greater understanding of how the genetic variation summarized by G evolves. Computer simulations of the evolution of G, innovations in matrix comparison methods, and rapid development of powerful molecular genetic tools have all opened the way for dissecting the interaction between allelic variation and evolutionary process. Here I discuss some current uses of G, problems with the application of these approaches, and identify avenues for future research.

Protein fold recognition without Boltzmann statistics or explicit physical basis

We present a fast method for finding optimal parameters for a low-resolution (threading) force field intended to distinguish correct from incorrect folds for a given protein sequence. In contrast to other methods, the parameterization uses information from >10(7) misfolded structures as well as a set of native sequence-structure pairs. In addition to testing the resulting force field's performance on the protein sequence threading problem, results are shown that characterize the number of parameters necessary for effective structure recognition.

Robust Mixture Modelling Using the t Distribution

Normal mixture models are being increasingly used to model the distributions of a wide variety of random phenomena and to cluster sets of continuous multivariate data. However, for a set of data containing a group or groups of observations with longer than normal tails or atypical observations, the use of normal components may unduly affect the fit of the mixture model. In this paper, we consider a more robust approach by modelling the data by a mixture of t distributions. The use of the ECM algorithm to fit this t mixture model is described and examples of its use are given in the context of clustering multivariate data in the presence of atypical observations in the form of background noise.

Standard errors of fitted component means of normal mixtures

In this paper use consider the problem of providing standard errors of the component means in normal mixture models fitted to univariate or multivariate data by maximum likelihood via the EM algorithm. Two methods of estimation of the standard errors are considered: the standard information-based method and the computationally-intensive bootstrap method. They are compared empirically by their application to three real data sets and by a small-scale Monte Carlo experiment.

Monte Carlo evidence on cointegration and causation

The small sample performance of Granger causality tests under different model dimensions, degree of cointegration, direction of causality, and system stability are presented. Two tests based on maximum likelihood estimation of error-correction models (LR and WALD) are compared to a Wald test based on multivariate least squares estimation of a modified VAR (MWALD). In large samples all test statistics perform well in terms of size and power. For smaller samples, the LR and WALD tests perform better than the MWALD test. Overall, the LR test outperforms the other two in terms of size and power in small samples.

Estimating and evaluating the statistics of gapped local-alignment scores

We present a novel maximum-likelihood-based algorithm for estimating the distribution of alignment scores from the scores of unrelated sequences in a database search. Using a new method for measuring the accuracy of p-values, we show that our maximum-likelihood-based algorithm is more accurate than existing regression-based and lookup table methods. We explore a more sophisticated way of modeling and estimating the score distributions (using a two-component mixture model and expectation maximization), but conclude that this does not improve significantly over simply ignoring scores with small E-values during estimation. Finally, we measure the classification accuracy of p-values estimated in different ways and observe that inaccurate p-values can, somewhat paradoxically, lead to higher classification accuracy. We explain this paradox and argue that statistical accuracy, not classification accuracy, should be the primary criterion in comparisons of similarity search methods that return p-values that adjust for target sequence length.

Bridging the gap between different statistical approaches: An integrated framework for modelling

This paper proposes a template for modelling complex datasets that integrates traditional statistical modelling approaches with more recent advances in statistics and modelling through an exploratory framework. Our approach builds on the well-known and long standing traditional idea of 'good practice in statistics' by establishing a comprehensive framework for modelling that focuses on exploration, prediction, interpretation and reliability assessment, a relatively new idea that allows individual assessment of predictions. The integrated framework we present comprises two stages. The first involves the use of exploratory methods to help visually understand the data and identify a parsimonious set of explanatory variables. The second encompasses a two step modelling process, where the use of non-parametric methods such as decision trees and generalized additive models are promoted to identify important variables and their modelling relationship with the response before a final predictive model is considered. We focus on fitting the predictive model using parametric, non-parametric and Bayesian approaches. This paper is motivated by a medical problem where interest focuses on developing a risk stratification system for morbidity of 1,710 cardiac patients given a suite of demographic, clinical and preoperative variables. Although the methods we use are applied specifically to this case study, these methods can be applied across any field, irrespective of the type of response.

Genetic assignment methods for the direct, real-time estimation of migration rate: a simulation-based exploration of accuracy and power

Genetic assignment methods use genotype likelihoods to draw inference about where individuals were or were not born, potentially allowing direct, real-time estimates of dispersal. We used simulated data sets to test the power and accuracy of Monte Carlo resampling methods in generating statistical thresholds for identifying F-0 immigrants in populations with ongoing gene flow, and hence for providing direct, real-time estimates of migration rates. The identification of accurate critical values required that resampling methods preserved the linkage disequilibrium deriving from recent generations of immigrants and reflected the sampling variance present in the data set being analysed. A novel Monte Carlo resampling method taking into account these aspects was proposed and its efficiency was evaluated. Power and error were relatively insensitive to the frequency assumed for missing alleles. Power to identify F-0 immigrants was improved by using large sample size (up to about 50 individuals) and by sampling all populations from which migrants may have originated. A combination of plotting genotype likelihoods and calculating mean genotype likelihood ratios (D-LR) appeared to be an effective way to predict whether F-0 immigrants could be identified for a particular pair of populations using a given set of markers.