Using the ratio of means as the effect size measure in combining results of microarray experiments
© Hu et al; licensee BioMed Central Ltd. 2009
Received: 26 June 2009
Accepted: 05 November 2009
Published: 05 November 2009
Development of efficient analytic methodologies for combining microarray results is a major challenge in gene expression analysis. The widely used effect size models are thought to provide an efficient modeling framework for this purpose, where the measures of association for each study and each gene are combined, weighted by the standard errors. A significant disadvantage of this strategy is that the quality of different data sets may be highly variable, but this information is usually neglected during the integration. Moreover, it is widely known that the estimated standard deviations are probably unstable in the commonly used effect size measures (such as standardized mean difference) when sample sizes in each group are small.
We propose a re-parameterization of the traditional mean difference based effect measure by using the log ratio of means as an effect size measure for each gene in each study. The estimated effect sizes for all studies were then combined under two modeling frameworks: the quality-unweighted random effects models and the quality-weighted random effects models. We defined the quality measure as a function of the detection p-value, which indicates whether a transcript is reliably detected or not on the Affymetrix gene chip. The new effect size measure is evaluated and compared under the quality-weighted and quality-unweighted data integration frameworks using simulated data sets, and also in several data sets of prostate cancer patients and controls. We focus on identifying differentially expressed biomarkers for prediction of cancer outcomes.
Our results show that the proposed effect size measure (log ratio of means) has better power to identify differentially expressed genes, and that the detected genes have better performance in predicting cancer outcomes than the commonly used effect size measure, the standardized mean difference (SMD), under both quality-weighted and quality-unweighted data integration frameworks. The new effect size measure and the quality-weighted microarray data integration framework provide efficient ways to combine microarray results.
Microarray technology has been widely used in identifying differentially expressed genes [1, 2] and in building predictors for disease outcome diagnosis [3–7]. Although individual microarray studies can be highly informative for this purpose (e.g. van 'tVeer et al., ), it is difficult to make a direct comparison among the results obtained by different groups addressing similar biological problems, since laboratory protocols, microarray platforms and analysis techniques used in each study may not be identical [8, 9]. Moreover, most individual studies have relatively small sample sizes, and hence prediction models trained on individual studies by using cross-validation procedures are prone to over-fitting, leading to prediction accuracies that are over-estimated and lack generalizability .
Recent studies show that systematic integration of gene expression data from different sources can increase statistical power to detect differentially expressed genes while allowing for an assessment of heterogeneity [11–18], and may lead to more robust, reproducible and accurate predictions . Therefore, our ability to develop powerful statistical methods for efficiently integrating related genomic experiments is critical to the success of the massive investment made on genomic studies. Broadly speaking, the strategies to integrate microarray studies can be divided into three categories:
The first category is a combined analysis of all the data. Each data set is first preprocessed to clean and align the signals, and then these preprocessed datasets are put together so that the integrated data set can be treated as though it comes from a single study. In this way, the effective sample size is greatly increased. Several transformation methods have been proposed to process gene expression measures from different studies [9, 14, 17, 20]. For example, Jiang et al.  transformed the normalized data sets to have similar distributions and then put the data sets together. Wang et al.  standardized gene expression levels based on the means and standard deviations of expression measurements from the arrays of healthy prostate samples. These methods are simple and in many cases, if the transformation is carefully made, the performance of disease outcome prediction can be improved . Nevertheless, there are no consensus or clear guidelines on the best way to perform the necessary data transformations.
The second strategy is to combine analysis results obtained from each study. The basic idea is to combine evidence of differential expression using a summary statistic, such as the p-value, across multiple gene profiling studies and then to adjust for multiple testing. For example, Rhodes et al. [11, 12] combined results from four prostate cancer microarray datasets analyzed on different platforms. Differential expression between the prostate tumor group and the normal group was first assessed independently for each gene in each dataset using the statistical confidence measure, the p-value. Then the study-specific p-values were combined, using the result that -2 log(p-value) has a chi-squared distribution under the null hypothesis of no differential expression. The analysis revealed that stronger significance was obtained from the combined analysis than from the individual studies. Combining p-values is useful in obtaining more precise estimates of significance, but this method does not indicate the direction of significance (e.g., up-or down-regulation) . Instead of integrating p-values directly, some studies explored combining ranks of statistics from different studies [18, 22]. For example, DeConde et al.  proposed a rank-aggregation method to combine final microarray results from five prostate cancer studies. The method summarizes majority preferences between pairs of genes across ranked list from different studies. They found this method more reliably identifies differentially expressed genes across studies.
The third strategy involves taking inter-study variability into account when estimating the overall effect for each gene across studies, and then basing conclusions on the distribution of these overall measures. For example, Choi et al.  focused on integrating effect size estimates in individual studies into an overall estimate of the average effect size. The effect size is normally used to measure the magnitude of treatment effect in a given study. Inter-study variability was included in the model with an associated prior distribution. This type of model, also termed hierarchical Bayesian random effects, has been used broadly in non-microarray contexts (e.g., DuMouchel and Harris ; Smith et al., ). Using the same microarray datasets as those used by Rhodes et al. , they demonstrated that their method can lead to the discovery of small but consistent expression changes with increased sensitivity and reliability among the datasets. The hierarchical Bayesian random effects meta-analysis model has several favorable features: it provides an overall effect size, and it accounts for inter-study variability, which may improve accuracy of results.
The widely used effect size measure in this type of models is the standardized mean difference [25, 26]. It has been well-known in microarray data analysis that the estimated standard deviation is probably unstable when sample size in each group is small. Therefore, many efforts have been made to overcome the shortcoming by estimating a penalty parameter for smoothing the estimates using information from all genes rather than relying solely on the estimates from an individual gene [1, 27].
However, recent studies show that differentially expressed genes may be best identified using fold-change measures rather than t-like statistics . Fold change is a commonly used measure in small laboratory experiments of gene expression; it is considered to be a natural measure for gene expression changes . In high-throughput microarray analysis, properties of fold change statistics have received little attention. Therefore, more investigation on reparameterization of effect size measures is needed.
Most data integration papers in microarray analysis have not used measures of quality to refine their analyses [9, 11–15, 17, 20, 22]. Nevertheless, in classical meta-analysis, quality measures have often been used when combining results across studies. It has been argued that studies of a higher quality will give more accurate estimates of the true parameter of interest, and therefore studies of high quality should receive a higher weight in the analysis summarizing across studies . In gene expression microarrays, many genes may be "off" or not detectable in a particular adult tissue, and in addition, some genes may be poorly measured due to probes that are not sufficiently sensitive or specific. Therefore, the signal strength and clarity will vary across the genes, suggesting that a quality measurement could highlight strong clear signals [31, 32]. Although it is still an open question how to best measure the quality of a gene expression measurement, and how best to use such a quality measure, different strategies can be considered for incorporating quality weights into meta-analysis of microarray studies. For example, we can define a quality threshold and only include genes that are above this threshold in the meta-analysis. However, the choice of threshold will be arbitrary. In a recent study, we proposed a quality measure based on the detection p-values estimated from Affymetrix microarray raw data [16, 31]. Using an effect-size model, we demonstrated that the incorporation of quality weights into the study-specific test statistics, within a meta-analysis of two Affymetrix microarray studies, produced more biological meaningful results than the unweighted analysis .
In this paper, we reparameterize the effect size measure for each gene in each study as the log ratio of the mean expressions of the two groups being compared. Following the method proposed by Hu et al. , we then place the new effect size measure into a quality-weighted modeling framework. We evaluate and compare the effect size measures (new and old) under the quality-weighted and quality-unweighted data integration frameworks using simulated data sets and real data sets with focus on identifying differentially expressed biomarkers and their performance on cancer outcome prediction.
Quality score measure for Affymetrix microarray data
For Affymetrix expression data, we previously developed a quality measure based on the detection p-values  that reflects whether the transcript is reliably expressed above the background in at least one experimental group in each study [16, 31] (see Additional file 1). The sensitivity parameter, v, that alters the tolerance of the quality weight to the detection p-value significance levels, was set to 0.05.
Using log ratio of means as effect size measure
For a study with n(n = n t +n c ) samples, an approximately unbiased estimate of yg 1is given by .
where and are the variances of the treatment and control groups, respectively.
Integrative analysis of effect sizes in a quality-adjusted modeling framework
We evaluated the statistical significance of gene g by calculating the p-value corresponding to the z statistic; then we estimated the false discovery rates (FDR) for each significance level, to take into account the number of tests performed . A detailed description of the integrative analysis of effect sizes can be found in the see Additional file 1.
We refer the approaches of estimating z gm using either the log ratio of means (m = 2) or the standardized mean difference (m = 1) as WROM and WSMD, respectively, in the quality-adjusted modeling framework, and as UWROM and UWSMD, respectively, in the quality-unadjusted modeling framework, where q ig = 1.
Model probe-level gene expression profile in a single study
where Y jgk and W jgk are PM and MM intensities for the probe j in probeset g on array k respectively. O denotes optical noise, independently drawing from and . represent non-specific binding (NSB) noise for PM (XX = PM) and MM (XX = MM), respectively. We set μ MM = μ PM = 4.6 and assumed that and follow a bivariate normal distribution with mean 0, variance 1, and correlation 0.88. We then generated identically and independently distributed random variates e ~ N(0,0.08), so that and similarly . are quantity proportional to RNA expression for PM (XX = PM) and MM (XX = MM), respectively, and the coefficient 0 < Φ < 1 accounts for the fact that for some probe-pairs the MM detects signal; When probe j of gene g is attached by picking up stray signal, Φ jg is generated as Φ jg ~Beta(0.5,5), otherwise, Φ gj = 0. Since S follows a power law, we set its base to 2. Therefore, if we denote γ g as the baseline log expression level for probeset g, we can select log2(γ g ) expression levels from 0 to 12, which can be generated from γ g ~12* Beta(1,3)+1. δ g is the expected differential expression of gene g in covariate X. α jgk is the signal detecting ability of probe j in gene g on array k, which is assumed to follow a normal distribution with mean zero and signal detection variance . We generated multiplicative errors and independently from N(0, ).
Generate simulated data sets for multiple studies
Parameters used in simulation of probe-level gene expression profile
Number of genes
Proportion of expressed genes
Proportion of differentially expressed genes
25 arrays in groups t and c, respectively
50 arrays in groups t and c, respectively
Number of probes in each probeset
We used summarized receiver operating characteristic (SROC) curves to compare performance, where the test sensitivities and specificities (true positive and true negative proportions) for a range of p-value cutoffs were averaged over 500 simulated datasets in each study. The SROC curve's overall behavior can be measured by the area under the curve (AUC) .
Affymetrix Microarray data
Main characteristics of the Affymetrix microarray data sets
Number of Normal Samples
Number of Prostate Cancer Samples
(Singh et al., 2002)
(Welsh et al., 2001)
(LaTulippe et al., 2002)
(Stuart et al., 2004)
Analysis of simulated data sets
Area under the curves of the four meta-analysis models (s = 0.05)
δ g = 2.0
δ g = 1.5
δ g = 1.0
Analysis of prostate cancer Affymetrix microarray data sets
Comparing gene ranks among different meta-analytic procedures
Rank of known and validated prostate cancer markers
Welsh et al. (2001)
Welsh et al. (2001)
Welsh et al. (2001)
Welsh et al. (2001)
Tricoli et al. 2004
Tricoli et al. 2004
Tricoli et al. 2004
Tricoli et al. 2004
Tricoli et al. 2004
It should be noted that some of the known tumor genes identified by our new methods have much better ranks than the conventional methods. For example, the ranks of tumor genes FASN and TACSTD1 are 15 and 6 by WROM and 13 and 6 by UWROM while the ranks of these genes are 72 and 289 by WSMD and 231 and 413 by UWSMD.
Comparing prediction performance of top-ranked meta-signatures among meta-analytic procedures
Many microarray experiments include only a few replications, therefore, it is critical to improve the effect size estimation in meta-analytic procedure. With small sample sizes, the traditional SMD estimates are prone to unpredictable changes, since gene-specific variability can easily be underestimated resulting in large statistics values. In this study, we re-parameterized the traditional SMD-based effect size measure by using a log ratio of means as an effect size measure for each gene in each study. Our results show the new effect size measure has better performance than the traditional one.
Traditional wisdom for statistical analysis recommends that highly skewed data should be transformed prior to analysis. It is therefore unexpected, perhaps, that the ROM measure (where log transforms are taken after calculating means) gives better prediction accuracy than the SMD measure (where log transformation is done prior to calculating means). Since the signals from Affymetrix are expected to be a mixture of background or non-specific binding and true signal, and only the true signal is expected to follow a power law, using the log transformation up front may be introducing variability, in particular for genes with low levels of expression. Furthermore, for genes whose expression levels change dramatically between experimental groups, the apriori log transformation may be inappropriate in the group with low expression levels.
We noticed that the ranks of some of the known tumor genes (e.g. five candidate markers discussed by Tricoli et al.  are relatively low in all four data integration methods (WROM, WSMD, ROM and SMD). There are several possible reasons for this. For example, since the patients used in these studies were collected in different places, there may be clinical heterogeneity, which may result in very different expression profiles of the same gene in different studies. It is also possible that the lower ranks of these tumor genes result from the relatively small sample sizes. Integration of more microarray data sets may lead to the discovery of more robust prostate cancer biomarkers.
Our results show that different predictors, including various combinations of differentially expressed genes can lead to similar prediction accuracy. This can make it challenging to select optimal biomarker sets for clinical use. Our recent study  showed that many of the differentially expressed genes which have similar classification results are involved in the same or similar biological pathways. In other words, the genes with the best discriminative power likely correspond to a limited set of biological functions or pathways. Hence, the selection of biomarkers for prediction may need to be based on a combination of statistical results and knowledge of pathways.
It is widely known that data from various sources might contain different informativity for a given biological task (such as differential analysis of gene expression levels between case and control). Some data sources might, for example, be more informative than others. A statistically sound data integration framework should, therefore, take these into account. One approach towards this goal is to develop suitable quality measures for different data types and these measures are then integrated into the statistical models. We used a simple quality measure associated with both log-ratio of means based and standardized mean difference based effect sizes. Our analysis showed this measure works well in the real and simulated data sets.
In summary, we combined estimated ROM-based effect sizes for all studies under two data integration frameworks: the quality-unweighted random effects models and the quality-weighted random effects models . Comparing with the SMD-based effect size measure, our real examples and simulation studies showed that the proposed methods have better power to identify differential expressed genes and the detected genes have better accuracies in predicting cancer outcomes. In conclusion, the new effect size measure and the quality-weighted microarray data integration framework provide efficient way to combine microarray results.
List of abbreviations
ratio of mean
log ratio of mean used as the effect size measure in weighted meta-analysis Framework
log ratio of mean used as the effect size measure in unweighted meta-analysis framework
standardized mean difference
standardized mean difference used as the effect size measure in weighted meta-analysis framework
standardized mean difference used as the effect size measure in unweighted meta-analysis framework
maximum likelihood estimation
robust multi-array average
summarized receiver operating characteristic
area under the curve
false discovery rate
support vector machines
diagonal linear discriminant analysis.
This work was partially supported by grants from the Canadian Institutes of Health Research (CIHR) (grant number 84392), the Natural Sciences and Engineering Research, Council of Canada (NSERC), the Mathematics of Information Technology and Complex Systems (MITACS), and Genome Canada through the Ontario Genomics Institute. We would like to thank three anonymous reviewers for their helpful comments and suggestions.
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