A model-based analysis of microarray experimental error and normalisation

A statistical model is proposed for the analysis of errors in microarray experiments and is employed in the analysis and development of a combined normalisation regime. Through analysis of the model and two-dye microarray data sets, this study found the following. The systematic error introduced by microarray experiments mainly involves spot intensity-dependent, feature-specific and spot position-dependent contributions. It is difficult to remove all these errors effectively without a suitable combined normalisation operation. Adaptive normalisation using a suitable regression technique is more effective in removing spot intensity-related dye bias than self-normalisation, while regional normalisation (block normalisation) is an effective way to correct spot position-dependent errors. However, dye-flip replicates are necessary to remove feature-specific errors, and also allow the analyst to identify the experimentally introduced dye bias contained in non-self-self data sets. In this case, the bias present in the data sets may include both experimentally introduced dye bias and the biological difference between two samples. Self-normalisation is capable of removing dye bias without identifying the nature of that bias. The performance of adaptive normalisation, on the other hand, depends on its ability to correctly identify the dye bias. If adaptive normalisation is combined with an effective dye bias identification method then there is no systematic difference between the outcomes of the two methods.     

A comparison on effects of normalisations in the detection of differentially expressed genes

Background  

Various normalisation techniques have been developed in the context of microarray analysis to try to correct expression measurements for experimental bias and random fluctuations. Major techniques include: total intensity normalisation; intensity dependent normalisation; and variance stabilising normalisation. The aim of this paper is to discuss the impact of normalisation techniques for two-channel array technology on the process of identification of differentially expressed genes.     

Variance stabilization and normalization for one-color microarray data using a data-driven multiscale approach

MOTIVATION: Many standard statistical techniques are effective on data that are normally distributed with constant variance. Microarray data typically violate these assumptions since they come from non-Gaussian distributions with a non-trivial mean-variance relationship. Several methods have been proposed that transform microarray data to stabilize variance and draw its distribution towards the Gaussian. Some methods, such as log or generalized log, rely on an underlying model for the data. Others, such as the spread-versus-level plot, do not. We propose an alternative data-driven multiscale approach, called the Data-Driven Haar-Fisz for microarrays (DDHFm) with replicates. DDHFm has the advantage of being 'distribution-free' in the sense that no parametric model for the underlying microarray data is required to be specified or estimated; hence, DDHFm can be applied very generally, not just to microarray data. RESULTS: DDHFm achieves very good variance stabilization of microarray data with replicates and produces transformed intensities that are approximately normally distributed. Simulation studies show that it performs better than other existing methods. Application of DDHFm to real one-color cDNA data validates these results. AVAILABILITY: The R package of the Data-Driven Haar-Fisz transform (DDHFm) for microarrays is available in Bioconductor and CRAN.     

Variance stabilization applied to microarray data calibration and to the quantification of differential expression

We introduce a statistical model for microarray gene expression data that comprises data calibration, the quantification of differential expression, and the quantification of measurement error. In particular, we derive a transformation h for intensity measurements, and a difference statistic Deltah whose variance is approximately constant along the whole intensity range. This forms a basis for statistical inference from microarray data, and provides a rational data pre-processing strategy for multivariate analyses. For the transformation h, the parametric form h(x)=arsinh(a+bx) is derived from a model of the variance-versus-mean dependence for microarray intensity data, using the method of variance stabilizing transformations. For large intensities, h coincides with the logarithmic transformation, and Deltah with the log-ratio. The parameters of h together with those of the calibration between experiments are estimated with a robust variant of maximum-likelihood estimation. We demonstrate our approach on data sets from different experimental platforms, including two-colour cDNA arrays and a series of Affymetrix oligonucleotide arrays.     

Weighted analysis of microarray gene expression using maximum-likelihood

MOTIVATION: The numerical values of gene expression measured using microarrays are usually presented to the biological end-user as summary statistics of spot pixel data, such as the spot mean, median and mode. Much of the subsequent data analysis reported in the literature, however, uses only one of these spot statistics. This results in sub-optimal estimates of gene expression levels and a need for improvement in quantitative spot variation surveillance. RESULTS: This paper develops a maximum-likelihood method for estimating gene expression using spot mean, variance and pixel number values available from typical microarray scanners. It employs a hierarchical model of variation between and within microarray spots. The hierarchical maximum-likelihood estimate (MLE) is shown to be a more efficient estimator of the mean than the 'conventional' estimate using solely the spot mean values (i.e. without spot variance data). Furthermore, under the assumptions of our model, the spot mean and spot variance are shown to be sufficient statistics that do not require the use of all pixel data.The hierarchical MLE method is applied to data from both Monte Carlo (MC) simulations and a two-channel dye-swapped spotted microarray experiment. The MC simulations show that the hierarchical MLE method leads to improved detection of differential gene expression particularly when 'outlier' spots are present on the arrays. Compared with the conventional method, the MLE method applied to data from the microarray experiment leads to an increase in the number of differentially expressed genes detected for low cut-off P-values of interest.     

A targeted maximum likelihood estimator of a causal effect on a bounded continuous outcome

Targeted maximum likelihood estimation of a parameter of a data generating distribution, known to be an element of a semi-parametric model, involves constructing a parametric model through an initial density estimator with parameter ? representing an amount of fluctuation of the initial density estimator, where the score of this fluctuation model at ? = 0 equals the efficient influence curve/canonical gradient. The latter constraint can be satisfied by many parametric fluctuation models since it represents only a local constraint of its behavior at zero fluctuation. However, it is very important that the fluctuations stay within the semi-parametric model for the observed data distribution, even if the parameter can be defined on fluctuations that fall outside the assumed observed data model. In particular, in the context of sparse data, by which we mean situations where the Fisher information is low, a violation of this property can heavily affect the performance of the estimator. This paper presents a fluctuation approach that guarantees the fluctuated density estimator remains inside the bounds of the data model. We demonstrate this in the context of estimation of a causal effect of a binary treatment on a continuous outcome that is bounded. It results in a targeted maximum likelihood estimator that inherently respects known bounds, and consequently is more robust in sparse data situations than the targeted MLE using a naive fluctuation model. When an estimation procedure incorporates weights, observations having large weights relative to the rest heavily influence the point estimate and inflate the variance. Truncating these weights is a common approach to reducing the variance, but it can also introduce bias into the estimate. We present an alternative targeted maximum likelihood estimation (TMLE) approach that dampens the effect of these heavily weighted observations. As a substitution estimator, TMLE respects the global constraints of the observed data model. For example, when outcomes are binary, a fluctuation of an initial density estimate on the logit scale constrains predicted probabilities to be between 0 and 1. This inherent enforcement of bounds has been extended to continuous outcomes. Simulation study results indicate that this approach is on a par with, and many times superior to, fluctuating on the linear scale, and in particular is more robust when there is sparsity in the data.     

Differential and trajectory methods for time course gene expression data

MOTIVATION: The issue of high dimensionality in microarray data has been, and remains, a hot topic in statistical and computational analysis. Efficient gene filtering and differentiation approaches can reduce the dimensions of data, help to remove redundant genes and noises, and highlight the most relevant genes that are major players in the development of certain diseases or the effect of drug treatment. The purpose of this study is to investigate the efficiency of parametric (including Bayesian and non-Bayesian, linear and non-linear), non-parametric and semi-parametric gene filtering methods through the application of time course microarray data from multiple sclerosis patients being treated with interferon-beta-1a. The analysis of variance with bootstrapping (parametric), class dispersion (semi-parametric) and Pareto (non-parametric) with permutation methods are presented and compared for filtering and finding differentially expressed genes. The Bayesian linear correlated model, the Bayesian non-linear model the and non-Bayesian mixed effects model with bootstrap were also developed to characterize the differential expression patterns. Furthermore, trajectory-clustering approaches were developed in order to investigate the dynamic patterns and inter-dependency of drug treatment effects on gene expression. RESULTS: Results show that the presented methods performed significant differently but all were adequate in capturing a small number of the potentially relevant genes to the disease. The parametric method, such as the mixed model and two Bayesian approaches proved to be more conservative. This may because these methods are based on overall variation in expression across all time points. The semi-parametric (class dispersion) and non-parametric (Pareto) methods were appropriate in capturing variation in expression from time point to time point, thereby making them more suitable for investigating significant monotonic changes and trajectories of changes in gene expressions in time course microarray data. Also, the non-linear Bayesian model proved to be less conservative than linear Bayesian correlated growth models to filter out the redundant genes, although the linear model showed better fit than non-linear model (smaller DIC). We also report the trajectories of significant genes-since we have been able to isolate trajectories of genes whose regulations appear to be inter-dependent.     

Variable Selection for Semiparametric Mixed Models in Longitudinal Studies

Summary .  We propose a double-penalized likelihood approach for simultaneous model selection and estimation in semiparametric mixed models for longitudinal data. Two types of penalties are jointly imposed on the ordinary log-likelihood: the roughness penalty on the nonparametric baseline function and a nonconcave shrinkage penalty on linear coefficients to achieve model sparsity. Compared to existing estimation equation based approaches, our procedure provides valid inference for data with missing at random, and will be more efficient if the specified model is correct. Another advantage of the new procedure is its easy computation for both regression components and variance parameters. We show that the double-penalized problem can be conveniently reformulated into a linear mixed model framework, so that existing software can be directly used to implement our method. For the purpose of model inference, we derive both frequentist and Bayesian variance estimation for estimated parametric and nonparametric components. Simulation is used to evaluate and compare the performance of our method to the existing ones. We then apply the new method to a real data set from a lactation study.     

Clinically validated benchmarking of normalisation techniques for two-colour oligonucleotide spotted microarray slides

Acquisition of microarray data is prone to systematic errors. A correction, called normalisation, must be applied to the data before further analysis is performed. With many normalisation techniques published and in use, the best way of executing this correction remains an open question. In this study, a variety of single-slide normalisation techniques, and different parameter settings for these techniques, were compared over many replicated microarray experiments. Different normalisation techniques were assessed through the distribution of the standard deviation of replicates from one biological sample across different slides. It is shown that local normalisation outperformed global normalisation, and intensity-based 'LOWESS' outperformed trimmed mean and median normalisation techniques. Overall, the top performing normalisation technique was a print-tip-based LOWESS with zero robust iterations. Lastly, we validated this evaluation methodology by examining the ability to predict oestrogen receptor-positive and -negative breast cancer samples with data that had been normalised using different techniques.     

Replication, Variation and Normalisation in Microarray Experiments

INTRODUCTION: Microarray experiments often have complex designs that include sample pooling, biological and technical replication, sample pairing and dye-swapping. This article demonstrates how statistical modelling can illuminate issues in the design and analysis of microarray experiments, and this information can then be used to plan effective studies. METHODS: A very detailed statistical model for microarray data is introduced, to show the possible sources of variation that are present in even the simplest microarray experiments. Based on this model, the efficacy of common experimental designs, normalisation methodologies and analyses is determined. RESULTS: When the cost of the arrays is high compared with the cost of samples, sample pooling and spot replication are shown to be efficient variance reduction methods, whereas technical replication of whole arrays is demonstrated to be very inefficient. Dye-swap designs can use biological replicates rather than technical replicates to improve efficiency and simplify analysis. When the cost of samples is high and technical variation is a major portion of the error, technical replication can be cost effective. Normalisation by centreing on a small number of spots may reduce array effects, but can introduce considerable variation in the results. Centreing using the bulk of spots on the array is less variable. Similarly, normalisation methods based on regression methods can introduce variability. Except for normalisation methods based on spiking controls, all normalisation requires that most genes do not differentially express. Methods based on spatial location and/or intensity also require that the nondifferentially expressing genes are at random with respect to location and intensity. Spotting designs should be carefully done so that spot replicates are widely spaced on the array, and genes with similar expression patterns are not clustered. DISCUSSION: The tools for statistical design of experiments can be applied to microarray experiments to improve both efficiency and validity of the studies. Given the high cost of microarray experiments, the benefits of statistical input prior to running the experiment cannot be over-emphasised.     

Detecting differential gene expression with a semiparametric hierarchical mixture method

Mixture modeling provides an effective approach to the differential expression problem in microarray data analysis. Methods based on fully parametric mixture models are available, but lack of fit in some examples indicates that more flexible models may be beneficial. Existing, more flexible, mixture models work at the level of one-dimensional gene-specific summary statistics, and so when there are relatively few measurements per gene these methods may not provide sensitive detectors of differential expression. We propose a hierarchical mixture model to provide methodology that is both sensitive in detecting differential expression and sufficiently flexible to account for the complex variability of normalized microarray data. EM-based algorithms are used to fit both parametric and semiparametric versions of the model. We restrict attention to the two-sample comparison problem; an experiment involving Affymetrix microarrays and yeast translation provides the motivating case study. Gene-specific posterior probabilities of differential expression form the basis of statistical inference; they define short gene lists and false discovery rates. Compared to several competing methodologies, the proposed methodology exhibits good operating characteristics in a simulation study, on the analysis of spike-in data, and in a cross-validation calculation.     

Cross-platform microarray data normalisation for regulatory network inference

Background

Inferring Gene Regulatory Networks (GRNs) from time course microarray data suffers from the dimensionality problem created by the short length of available time series compared to the large number of genes in the network. To overcome this, data integration from diverse sources is mandatory. Microarray data from different sources and platforms are publicly available, but integration is not straightforward, due to platform and experimental differences.

Methods

We analyse here different normalisation approaches for microarray data integration, in the context of reverse engineering of GRN quantitative models. We introduce two preprocessing approaches based on existing normalisation techniques and provide a comprehensive comparison of normalised datasets.

Conclusions

Results identify a method based on a combination of Loess normalisation and iterative K-means as best for time series normalisation for this problem.     

Modeling gene expression measurement error: a quasi-likelihood approach

Background

Using suitable error models for gene expression measurements is essential in the statistical analysis of microarray data. However, the true probabilistic model underlying gene expression intensity readings is generally not known. Instead, in currently used approaches some simple parametric model is assumed (usually a transformed normal distribution) or the empirical distribution is estimated. However, both these strategies may not be optimal for gene expression data, as the non-parametric approach ignores known structural information whereas the fully parametric models run the risk of misspecification. A further related problem is the choice of a suitable scale for the model (e.g. observed vs. log-scale).

Results

Here a simple semi-parametric model for gene expression measurement error is presented. In this approach inference is based an approximate likelihood function (the extended quasi-likelihood). Only partial knowledge about the unknown true distribution is required to construct this function. In case of gene expression this information is available in the form of the postulated (e.g. quadratic) variance structure of the data.As the quasi-likelihood behaves (almost) like a proper likelihood, it allows for the estimation of calibration and variance parameters, and it is also straightforward to obtain corresponding approximate confidence intervals. Unlike most other frameworks, it also allows analysis on any preferred scale, i.e. both on the original linear scale as well as on a transformed scale. It can also be employed in regression approaches to model systematic (e.g. array or dye) effects.

Conclusions

The quasi-likelihood framework provides a simple and versatile approach to analyze gene expression data that does not make any strong distributional assumptions about the underlying error model. For several simulated as well as real data sets it provides a better fit to the data than competing models. In an example it also improved the power of tests to identify differential expression.

     

Rosetta error model for gene expression analysis

MOTIVATION: In microarray gene expression studies, the number of replicated microarrays is usually small because of cost and sample availability, resulting in unreliable variance estimation and thus unreliable statistical hypothesis tests. The unreliable variance estimation is further complicated by the fact that the technology-specific variance is intrinsically intensity-dependent. RESULTS: The Rosetta error model captures the variance-intensity relationship for various types of microarray technologies, such as single-color arrays and two-color arrays. This error model conservatively estimates intensity error and uses this value to stabilize the variance estimation. We present two commonly used error models: the intensity error-model for single-color microarrays and the ratio error model for two-color microarrays or ratios built from two single-color arrays. We present examples to demonstrate the strength of our error models in improving statistical power of microarray data analysis, particularly, in increasing expression detection sensitivity and specificity when the number of replicates is limited.     

High-resolution spatial normalization for microarrays containing embedded technical replicates

MOTIVATION: Microarray data are susceptible to a wide-range of artifacts, many of which occur on physical scales comparable to the spatial dimensions of the array. These artifacts introduce biases that are spatially correlated. The ability of current methodologies to detect and correct such biases is limited. RESULTS: We introduce a new approach for analyzing spatial artifacts, termed 'conditional residual analysis for microarrays' (CRAM). CRAM requires a microarray design that contains technical replicates of representative features and a limited number of negative controls, but is free of the assumptions that constrain existing analytical procedures. The key idea is to extract residuals from sets of matched replicates to generate residual images. The residual images reveal spatial artifacts with single-feature resolution. Surprisingly, spatial artifacts were found to coexist independently as additive and multiplicative errors. Efficient procedures for bias estimation were devised to correct the spatial artifacts on both intensity scales. In a survey of 484 published single-channel datasets, variance fell 4- to 12-fold in 5% of the datasets after bias correction. Thus, inclusion of technical replicates in a microarray design affords benefits far beyond what one might expect with a conventional 'n = 5' averaging, and should be considered when designing any microarray for which randomization is feasible. AVAILABILITY: CRAM is implemented as version 2 of the hoptag software package for R, which is included in the Supplementary information.     

Modeling spatiotemporal abundance of mobile wildlife in highly variable environments using boosted GAMLSS hurdle models

Modeling organism distributions from survey data involves numerous statistical challenges, including accounting for zero‐inflation, overdispersion, and selection and incorporation of environmental covariates. In environments with high spatial and temporal variability, addressing these challenges often requires numerous assumptions regarding organism distributions and their relationships to biophysical features. These assumptions may limit the resolution or accuracy of predictions resulting from survey‐based distribution models. We propose an iterative modeling approach that incorporates a negative binomial hurdle, followed by modeling of the relationship of organism distribution and abundance to environmental covariates using generalized additive models (GAM) and generalized additive models for location, scale, and shape (GAMLSS). Our approach accounts for key features of survey data by separating binary (presence‐absence) from count (abundance) data, separately modeling the mean and dispersion of count data, and incorporating selection of appropriate covariates and response functions from a suite of potential covariates while avoiding overfitting. We apply our modeling approach to surveys of sea duck abundance and distribution in Nantucket Sound (Massachusetts, USA), which has been proposed as a location for offshore wind energy development. Our model results highlight the importance of spatiotemporal variation in this system, as well as identifying key habitat features including distance to shore, sediment grain size, and seafloor topographic variation. Our work provides a powerful, flexible, and highly repeatable modeling framework with minimal assumptions that can be broadly applied to the modeling of survey data with high spatiotemporal variability. Applying GAMLSS models to the count portion of survey data allows us to incorporate potential overdispersion, which can dramatically affect model results in highly dynamic systems. Our approach is particularly relevant to systems in which little a priori knowledge is available regarding relationships between organism distributions and biophysical features, since it incorporates simultaneous selection of covariates and their functional relationships with organism responses.     

The use of wavelets for spatial pattern analysis in ecology

Abstract. We investigate the characteristics of the wavelet transform as an approach to analyzing spatial pattern. Compared to the familiar methods of paired quadrat or blocked quadrat variance calculations, the wavelet method seems to offer several advantages. First, when wavelet variance is plotted as a function of scale, the peak variance height is determined by pattern intensity and does not increase with scale and, depending on the wavelet chosen, the position of the variance peak matches the scale exactly. Second, the method produces only faint resonance peaks, if any, and third, by using several different wavelet forms, different characteristics of the pattern can be investigated. Fourth, the method is able to portray very clearly trends in the data, when the pattern is non-stationary. Lastly, the wavelet position variance can be used to identify patches and gaps in data with random error. We demonstrate these characteristics using artificial data and data from previously published studies for comparison. We show that two versions of the familiar blocked quadrat variance technique are forms of wavelet analysis.     

Statistical tests for differential expression in cDNA microarray experiments

Extracting biological information from microarray data requires appropriate statistical methods. The simplest statistical method for detecting differential expression is the t test, which can be used to compare two conditions when there is replication of samples. With more than two conditions, analysis of variance (ANOVA) can be used, and the mixed ANOVA model is a general and powerful approach for microarray experiments with multiple factors and/or several sources of variation.