Random-effects (RE) models are commonly applied to account for heterogeneity in effect sizes in gene expression meta-analysis. The degree of heterogeneity may differ due to inconsistencies in sample quality. High heterogeneity can arise in meta-analyses containing poor quality samples. We applied sample-quality weights to adjust the study heterogeneity in the DerSimonian and Laird (DSL) and two-step DSL (DSLR2) RE models and the Bayesian random-effects (BRE) models with unweighted and weighted data, Gibbs and Metropolis-Hasting (MH) sampling algorithms, weighted common effect, and weighted between-study variance. We evaluated the performance of the models through simulations and illustrated application of the methods using Alzheimer’s gene expression datasets. Sample quality adjusting within study variance ( w P 6 ) models provided an appropriate reduction of differentially expressed (DE) genes compared to other weighted functions in classical RE models. The BRE model with a uniform(0,1) prior was appropriate for detecting DE genes as compared to the models with other prior distributions. The precision of DE gene detection in the heterogeneous data was increased with the DSLR2 w P 6 weighted model compared to the DSL w P 6 weighted model. Among the BRE weighted models, the w P 6 weighted- and unweighted-data models and both Gibbs- and MH-based models performed similarly. The w P 6 weighted common-effect model performed similarly to the unweighted model in the homogeneous data, but performed worse in the heterogeneous data. The w P 6 weighted data were appropriate for detecting DE genes with high precision, while the w P 6 weighted between-study variance models were appropriate for detecting DE genes with high overall accuracy. Without the weight, when the number of genes in microarray increased, the DSLR2 performed stably, while the overall accuracy of the BRE model was reduced. When applying the weighted models in the Alzheimer’s gene expression data, the number of DE genes decreased in all metadata sets with the DSLR2 w P 6 weighted and the w P 6 weighted between study variance models. Four hundred and forty-six DE genes identified by the w P 6 weighted between study variance model could be potentially down-regulated genes that may contribute to good classification of Alzheimer’s samples. The application of sample quality weights can increase precision and accuracy of the classical RE and BRE models; however, the performance of the models varied depending on data features, levels of sample quality, and adjustment of parameter estimates.

BACKGROUND: Networks are ubiquitous in modern cell biology and physiology. A large literature exists for inferring/proposing biological pathways/networks using statistical or machine learning algorithms. Despite these advances a formal testing procedure for analyzing network-level observations is in need of further development. Comparing the behaviour of a pharmacologically altered pathway to its canonical form is an example of a salient one-sample comparison. Locating which pathways differentiate disease from no-disease phenotype may be recast as a two-sample network inference problem. RESULTS: We outline an inferential method for performing one- and two-sample hypothesis tests where the sampling unit is a network and the hypotheses are stated via network model(s). We propose a dissimilarity measure that incorporates nearby neighbour information to contrast one or more networks in a statistical test. We demonstrate and explore the utility of our approach with both simulated and microarray data; random graphs and weighted (partial) correlation networks are used to form network models. Using both a well-known diabetes dataset and an ovarian cancer dataset, the methods outlined here could better elucidate co-regulation changes for one or more pathways between two clinically relevant phenotypes. CONCLUSIONS: Formal hypothesis tests for gene- or protein-based networks are a logical progression from existing gene-based and gene-set tests for differential expression. Commensurate with the growing appreciation and development of systems biology, the dissimilarity-based testing methods presented here may allow us to improve our understanding of pathways and other complex regulatory systems. The benefit of our method was illustrated under select scenarios.

We thank Professors Nagarajan and Upreti for their interest in our paper, Mukhopadhyay and Chatterjee (2007). There, we propose using Granger causality-based pathway detection in an acyclic, homoscedastic framework for microarray time-series expressions; which are generally short-duration time series involving very large number of genes. Professors Nagarajan and Upreti point out that in the presence of heteroscedasticity, and a cycle like ‘gene x regulates the expression of gene y and simultaneously gene y regulates the expression of gene x’, Granger causality tests may not be informative. Here, we adopt the term ‘heteroscedasticity’ (‘homoscedasticity’) to mean the unconditional variance of the white noise, represented as a bivariate vector in the Euclidean co-ordinate system, is different (same) in different co-ordinate directions. Thus, in essence, if the assumptions about the acyclic and homoscedastic nature of the time series are violated, tests for causality detection may fail. This is an important point, since when a contemporaneous cyclic relationship is present, the notion of causality makes little sense. In the context of economics, Eichler (2007) present a treatment of contemporaneous correlation as well as Granger causality. Extreme heteroscedasticity may be indicative of improper normalization of gene expressions. At the end of their letter, Dr Nagarajan and Dr Upreti mention the normalization step. Proper normalization should remove wide discrepancy in noise variance, hence nowadays microarray datasets are typically available in de facto normalized version. The data used in Mukhopadhyay and Chatterjee (2007) is also normalized. However, difference in technical variance, as indicated by Professors Nagarajan and Upreti, may still be present. And that will violate the assumption of our method (as well as many other statistical comparison methods relying on common unknown variance). Professor Nagarajan, in review, kindly suggested references for two-gene systems whose time-profile may not fit into to a homoscedastic, cause-effect framework. Thus, a full vector autoregression structure may be needed to capture their mutual dependence at various lags (including lag zero). It can be guessed that multi-gene systems exist whose temporal co-dependency nature is extremely complex. Although current knowledge about gene regulatory networks is limited, some biology experts we consulted believe that cyclical patterns may be found in large multi-gene networks as a part of a feedback procedure, if they are studied over long enough time spans. A proper approach to elicit such patterns would be to conduct multivariate, possibly non-stationary, time-series analysis with all the genes over a long time horizon. This is not feasible currently, since present state-of-the-art microarray time series experiments are of short duration and typically involve very large number of genes. Hence, restricting the network to acyclic ones is, in our opinion, a small price to pay to produce informative analysis. Future microarray experiments over longer duration, along with discoveries of biological and chemical properties relating to gene and protein interactions, will no doubt lead to better understanding of gene networks. We would like to point out in Model 1 (Equation 2), α12, α21, and need to be known constants for the mathematical displays (4)–(7) to hold. As they stand, displays (4)–(7) are missing the O (n−1) terms with each estimated parameters if some (or all) of are estimated from data, where n is the length of the time series data. Also, the equation for s1 does not account for the fact that as univariate time series, both xt and yt are AR(2) (autoregressive of order 2) process and not AR(1). Similar comments hold for Model 2 (Equation 11). The difficulty of modeling microarray time-series can be appreciated from the fact that in the human cell cycle data considered in Mukhopadhyay and Chatterjee (2007), n was 12 in one experiment, while the time-series itself was 802 dimensional.

MOTIVATION: Interaction among time series can be explored in many ways. All the approach has the usual problem of low power and high dimensional model. Here we attempted to build a causality network among a set of time series. The causality has been established by Granger causality, and then constructing the pathway has been implemented by finding the Minimal Spanning Tree within each connected component of the inferred network. False discovery rate measurement has been used to identify the most significant causalities. RESULTS: Simulation shows good convergence and accuracy of the algorithm. Robustness of the procedure has been demonstrated by applying the algorithm in a non-stationary time series setup. Application of the algorithm in a real dataset identified many causalities, with some overlap with previously known ones. Assembled network of the genes reveals features of the network that are common wisdom about naturally occurring networks.