Statistical significance analysis of longitudinal gene expression data

MOTIVATION: Time-course microarray experiments are designed to study biological processes in a temporal fashion. Longitudinal gene expression data arise when biological samples taken from the same subject at different time points are used to measure the gene expression levels. It has been observed that the gene expression patterns of samples of a given tumor measured at different time points are likely to be much more similar to each other than are the expression patterns of tumor samples of the same type taken from different subjects. In statistics, this phenomenon is called the within-subject correlation of repeated measurements on the same subject, and the resulting data are called longitudinal data. It is well known in other applications that valid statistical analyses have to appropriately take account of the possible within-subject correlation in longitudinal data. RESULTS: We apply estimating equation techniques to construct a robust statistic, which is a variant of the robust Wald statistic and accounts for the potential within-subject correlation of longitudinal gene expression data, to detect genes with temporal changes in expression. We associate significance levels to the proposed statistic by either incorporating the idea of the significance analysis of microarrays method or using the mixture model method to identify significant genes. The utility of the statistic is demonstrated by applying it to an important study of osteoblast lineage-specific differentiation. Using simulated data, we also show pitfalls in drawing statistical inference when the within-subject correlation in longitudinal gene expression data is ignored.     

Statistical methods suitable for the analysis of plant tissue culture data

Statistical analyses are an essential part of biological research. Statistical methods are available to biological researchers that range from very simple to extremely complex. Therefore, caution should be used when selecting a statistical method. When possible it is best to avoid complicated statistical procedures that are difficult to interpret and may hinder the researcher's ability to make treatment comparisons. Instead a method should be chosen that compliments a logical and practical treatment design. Statistics should be used as a tool to compare treatments of interest and should not dictate the treatments. Experimental designs should take into account the eventual analysis, otherwise one could conceive of a design that could not be analyzed or, when analyzed, would not answer the desired questions. Therefore, time should be spent before conducting an experiment to plan an experimental design and analysis that best compliments the treatment scheme and questions to be answered. The purpose of this paper is to present examples of experimental designs, means separation procedures, data transformations and presentation methods suitable for plant cell and tissue culture data.Abbreviations ANOVA analysis of variance - BA benzyladenine - CV coefficient of variation - DF degrees of freedom - IAA indole-3-acetic acid - IBA indole-3-butyric acid - LOF lack-of-fit - MSE mean square error - P-ITB phenyl indole-3-thiolobutyrate - S standard deviation - SE standard error of the mean - TDZ thidiazuron     

Bivariate line-fitting methods for allometry

Fitting a line to a bivariate dataset can be a deceptively complex problem, and there has been much debate on this issue in the literature. In this review, we describe for the practitioner the essential features of line-fitting methods for estimating the relationship between two variables: what methods are commonly used, which method should be used when, and how to make inferences from these lines to answer common research questions. A particularly important point for line-fitting in allometry is that usually, two sources of error are present (which we call measurement and equation error), and these have quite different implications for choice of line-fitting method. As a consequence, the approach in this review and the methods presented have subtle but important differences from previous reviews in the biology literature. Linear regression, major axis and standardised major axis are alternative methods that can be appropriate when there is no measurement error. When there is measurement error, this often needs to be estimated and used to adjust the variance terms in formulae for line-fitting. We also review line-fitting methods for phylogenetic analyses. Methods of inference are described for the line-fitting techniques discussed in this paper. The types of inference considered here are testing if the slope or elevation equals a given value, constructing confidence intervals for the slope or elevation, comparing several slopes or elevations, and testing for shift along the axis amongst several groups. In some cases several methods have been proposed in the literature. These are discussed and compared. In other cases there is little or no previous guidance available in the literature. Simulations were conducted to check whether the methods of inference proposed have the intended coverage probability or Type I error. We identified the methods of inference that perform well and recommend the techniques that should be adopted in future work.     

Methods for analysis of unbalanced,longitudinal, growth data

We describe an approach to analysis of growth that does not depend on assumptions about the underlying functional growth pattern and that allows for multiple observations arising from individual-specific, irregularly spaced data. We produce estimated growth curves for predefined subject groups by using LOWESS, a nonparametric smoothing algorithm. We describe how statistical significance of curve features may be evaluated by using the “jackknife,” a sample re-use method; this technique can be used to assess differences between subject groups. We then obtain residuals at each data point by reference to the estimated curve. Consistency of residuals is evaluated as a characteristic of individual subjects, and in the presence of individual consistency, relative size-for-age is then scored by the average residual for each individual. This allows study of relationships between relative size and other individual characteristics such as birth order, dominance rank, or age of maturation. Finally, we indicate flexibility of these methods and alternatives, propose uses related to other questions about growth, and suggest potential applications to variables other than body size. Appendices demonstrate application of the LOWESS and jackknife algorithms to the problem of testing sex differences in growth. © 1992 Wiley-Liss, Inc.     

Quantitative methods for genome-scale analysis of in situ hybridization and correlation with microarray data

With the emergence of genome-wide colorimetric in situ hybridization (ISH) data sets such as the Allen Brain Atlas, it is important to understand the relationship between this gene expression modality and those derived from more quantitative based technologies. This study introduces a novel method for standardized relative quantification of colorimetric ISH signal that enables a large-scale cross-platform expression level comparison of ISH with two publicly available microarray brain data sources.     

Statistical methods in regression and calibration analysis of chromosome aberration data

The method of iteratively reweighted least squares for the regression analysis of Poisson distributed chromosome aberration data is reviewed in the context of other fit procedures used in the cytogenetic literature. As an application of the resulting regression curves methods for calculating confidence intervals on dose from aberration yield are described and compared, and, for the linear quadratic model lambda = beta 0 + beta 1 chi + beta 2 chi 2 a confidence interval for the ratio beta 1/ beta 2 is given. Emphasis is placed on the rationale, interpretation and the limitations of various methods from a statistical point of view.     

Statistical methods for detecting natural selection from genomic data

In the study of molecular and phenotypic evolution, understanding the relative importance of random genetic drift and positive selection as the mechanisms for driving divergences between populations and maintaining polymorphisms within populations has been a central issue. A variety of statistical methods has been developed for detecting natural selection operating at the amino acid and nucleotide sequence levels. These methods may be largely classified into those aimed at detecting recurrent and/or recent/ongoing natural selection by utilizing the divergence and/or polymorphism data. Using these methods, pervasive positive selection has been identified for protein-coding and non-coding sequences in the genomic analysis of some organisms. However, many of these methods have been criticized by using computer simulation and real data analysis to produce excessive false-positives and to be sensitive to various disturbing factors. Importantly, some of these methods have been invalidated experimentally. These facts indicate that many of the statistical methods for detecting natural selection are unreliable. In addition, the signals that have been believed as the evidence for fixations of advantageous mutations due to positive selection may also be interpreted as the evidence for fixations of deleterious mutations due to random genetic drift. The genomic diversity data are rapidly accumulating in various organisms, and detection of natural selection may play a critical role for clarifying the relative role of random genetic drift and positive selection in molecular and phenotypic evolution. It is therefore important to develop reliable statistical methods that are unbiased as well as robust against various disturbing factors, for inferring natural selection.     

Joint regression analysis for discrete longitudinal data

Summary We introduce an approximation to the Gaussian copula likelihood of Song, Li, and Yuan (2009, Biometrics 65, 60–68) used to estimate regression parameters from correlated discrete or mixed bivariate or trivariate outcomes. Our approximation allows estimation of parameters from response vectors of length much larger than three, and is asymptotically equivalent to the Gaussian copula likelihood. We estimate regression parameters from the toenail infection data of De Backer et al. (1996, British Journal of Dermatology 134, 16–17), which consist of binary response vectors of length seven or less from 294 subjects. Although maximizing the Gaussian copula likelihood yields estimators that are asymptotically more efficient than generalized estimating equation (GEE) estimators, our simulation study illustrates that for finite samples, GEE estimators can actually be as much as 20% more efficient.     

Evaluation of some methods for hybrid analysis, exemplified by hybridization in Argyrantheimim (Asteraceae)

A grouping of transformations is proposed: 1) "Element transformations", aimed at changing relations between elements within a single character vector; and 2) "vector transformations", aimed at changing relations between different character vectors. Logarithmic element transformation seemed suitable for revealing variation in size characters.
Principal coordinate analysis (PCO) was appropriate for determination of dimensionality and structural extremes (parentage). Due to polynomial distortions, however, variation in extreme populations was underestimated and variation in intermediate populations exaggerated.
A "character index", the mean of a specimen's ranged characters, is suggested to replace Anderson's hybrid index. Knowledge of parentage and parental maxima, but not of variation in pure parental populations, is required. The character index combined with modified Gay triangles was found suitable for revealing the structure of the material, which showed mainly one-dimensional variation. The material analysed comprised Argyranthemum broussonetü, A. frutescens , a hybrid swarm and experimental F₁ hybrids between these species; and A. sundingü , which was found to be a stabilized hybrid derivative, probably evolved by hybrid speciation with external barriers.     

Statistical analysis of radioimmunoassay data

The statistical processing of radioimmunoassay data is discussed, with special emphasis on fitting the standard curve, screening the data for aberrant readings and combining separate estimations from a single sample.     

Functional data analysis for longitudinal data with informative observation times

In functional data analysis for longitudinal data, the observation process is typically assumed to be noninformative, which is often violated in real applications. Thus, methods that fail to account for the dependence between observation times and longitudinal outcomes may result in biased estimation. For longitudinal data with informative observation times, we find that under a general class of shared random effect models, a commonly used functional data method may lead to inconsistent model estimation while another functional data method results in consistent and even rate-optimal estimation. Indeed, we show that the mean function can be estimated appropriately via penalized splines and that the covariance function can be estimated appropriately via penalized tensor-product splines, both with specific choices of parameters. For the proposed method, theoretical results are provided, and simulation studies and a real data analysis are conducted to demonstrate its performance.     

Statistical analysis of in situ hybridization data: derivation and use of the zmax test.

There are many situations in which grain distributions resulting from in situ hybridization of radioactively labeled probes to unique genes should be subjected to a statistical analysis. However, the problems posed by analysis of in situ hybridization data are not straightforward, and no completely satisfying method is currently available. We have developed a procedure in which the major and any number of minor site(s) of hybridization may be specifically located and the significance of each tested. This zmax procedure first tests the overall distribution for departure from randomness and then identifies significantly overlabeled whole chromosomes (or chromosome arms or other large segments), a process that may be repeated to pinpoint significantly overlabeled regions within these chromosomes. We describe in detail the derivation of the zmax statistic, present tables of significant zmax levels, and show with examples how zmax is used in tests of significance of in situ hybridization data.     

Multivariate methods for clustered binary data with multiple subclasses, with application to binary longitudinal data.

Clustered binary data occur frequently in biostatistical work. Several approaches have been proposed for the analysis of clustered binary data. In Rosner (1984, Biometrics 40, 1025-1035), a polychotomous logistic regression model was proposed that is a generalization of the beta-binomial distribution and allows for unit- and subunit-specific covariates, while controlling for clustering effects. One assumption of this model is that all pairs of subunits within a cluster are equally correlated. This is appropriate for ophthalmologic work where clusters are generally of size 2, but may be inappropriate for larger cluster sizes. A beta-binomial mixture model is introduced to allow for multiple subclasses within a cluster and to estimate odds ratios relating outcomes for pairs of subunits within a subclass as well as in different subclasses. To include covariates, an extension of the polychotomous logistic regression model is proposed, which allows one to estimate effects of unit-, class-, and subunit-specific covariates, while controlling for clustering using the beta-binomial mixture model. This model is applied to the analysis of respiratory symptom data in children collected over a 14-year period in East Boston, Massachusetts, in relation to maternal and child smoking, where the unit is the child and symptom history is divided into early-adolescent and late-adolescent symptom experience.