 Proceedings
 Open Access
Integrated analysis of the heterogeneous microarray data
 Sung Gon Yi^{1} and
 Taesung Park^{2}Email author
https://doi.org/10.1186/1471210512S5S3
© Yi and Park; licensee BioMed Central Ltd. 2011
 Published: 27 July 2011
Abstract
Background
As the magnitude of the experiment increases, it is common to combine various types of microarrays such as paired and nonpaired microarrays from different laboratories or hospitals. Thus, it is important to analyze microarray data together to derive a combined conclusion after accounting for heterogeneity among data sets. One of the main objectives of the microarray experiment is to identify differentially expressed genes among the different experimental groups. We propose the linear mixed effect model for the integrated analysis of the heterogeneous microarray data sets.
Results
The proposed linear mixed effect model was illustrated using the data from 133 microarrays collected at three different hospitals. Though simulation studies, we compared the proposed linear mixed effect model approach with the metaanalysis and the ANOVA model approaches. The linear mixed effect model approach was shown to provide higher powers than the other approaches.
Conclusions
The linear mixed effect model has advantages of allowing for various types of covariance structures over ANOVA model. Further, it can handle easily the correlated microarray data such as paired microarray data and repeated microarray data from the same subject.
Keywords
 Microarray Data
 Covariance Structure
 ANOVA Model
 Linear Mixed Effect Model
 Statistical Test Procedure
Background
Microarray technology has important applications in pharmaceutical and clinical research. For example, microarrays can be used to identify tumorrelated genes and targets for therapeutic drugs. In microarray experiments, the identification of differentially expressed genes (DEG) is an important issue. Statistical test procedures have served as useful tools for identifying the DEGs which can be candidate genes for a specific disease or can be used for the further analysis such as clustering analysis and gene regulatory network construction.
As the cost of producing microarrays has become lower costs and the importance of replication in microarray experiments has been demonstrated by many researchers [1], replicated microarrays are commonly used in microarray experiments. In order to handle replicated microarrays, many statistical test procedures have been developed, such as tstatistics, to identify DEGs between two groups [2]. The analysis of variance (ANOVA) model approach was proposed to identify DEGs among multiple groups [3]. In addition, many statistical models have been proposed to identify the DEGs on replicated microarrays [4–11].
When the magnitude of a microarray experiment increases, it is common to use the same type of microarrays from different laboratories or hospitals. Thus, it is important to analyze microarray data together to derive a combined conclusion after accounting for the differences. Recently, statistical approaches based on metaanalysis have been proposed in order to combine independent and heterogeneous microarray studies [12–15]. In these approaches, microarrays were classified into several independent groups and integration methods to analyze microarray data sets from different laboratories were proposed. The key idea of metaanalysis is to combine summary statistics from each study in which significant levels (pvalues) and effect sizes are commonly used as summary statistics. Metaanalysis requires data be homogeneous within the data set. When there are microarrayspecific covariates such as gender and smoking status, metaanalysis can be less effective.
Shen et al. (2004) introduced the probability of expression (POE) and proposed a method to estimate the POE using MCMC [16]. The POE is the scalefree measure transformed from raw gene expression defined by the difference between probabilities of over and underexpressed gene expression. Using the POE, the gene expressions of heterogenous microarray experiments can be uniquely scaled from 1 and 1 and combined easily. Choi et al. (2007) proposed EM algorithm to estimate the POE instead of MCMC, which can reduce the estimation time of the POE [17]. Standardized POE can combine multiple microarray data sets, however, the POE method can be more efficient when the microarrayspecific covariates are applied.
Park et al. [18] proposed a twostage ANOVA model approach for the integrated analysis, which uses the ANOVA model with controlling variables for additional variability of heterogeneous microarray studies. The usual ANOVA model was extended to account for an additional variability resulting from many confounding variables. When variability among data sets is relatively small, the ANOVA model is effective. Otherwise, the ANOVA model is not recommended. Further, when the microarrays are correlated, the ANOVA model cannot handle such correlation appropriately, because it requires the independence of samples. Therefore, correlated microarray data can violate the assumption of the ANOVA model and thus the extended model to allow for various types of covariance structure of errors is needed.
In this paper, we propose the linear mixed effect (LMe) model for the integrated analysis of the heterogeneous microarray data sets. The LMe model contains various random effects which effectively account for the heterogeneous variability in the data from many different sources. Further, the LMe model has advantages of allowing for various types of covariance structures over metaanalysis and ANOVA model approaches. Thus, it can handle easily the correlated microarray data such as paired and nonpaired microarray data. The proposed method is illustrated using the liver cancer microarray data sets obtained from three different hospitals [14].
Materials and methods
Four independent microarray data sets were generated from three hospitals using two different chips [15]. The first chip, C_{1}, contains 10,336 human cDNA probes that were verified by single pass sequencing. The second chip, C_{2}, contains 10,368 human cDNA probes. Two chips shared the common 9,984 cDNA probes. The chips were cDNA chips with twocolors, where the way of labeling samples and controls is described in Choi et al. (2004). A further detailed description of the chips has been uploaded to the Gene Expression Omnibus (GEO) site (http://www.ncbi.nlm.nih.gov/geo/) with GEO accession number GPL2911.
The chip type (1 and 2), labeling scheme, hospital and number of samples are shown in this table. Here, the data were normalized by locally weighted scatterplot smoothing (LOWESS; Cleveland, 1979). For LOWESS normalization, the value of the span parameter was 0.75 and the tricubic function was used as a weight function. For robustness analysis, Tukey’s biweight function was used [18]. Hepatocellular carcinoma (HCC) and adjacent control (normal) samples were obtained with informed consent from patients at three hospitals. All the HCC samples were hepatitis B virus (HBV) positive. Sample preparation, microarray hybridizations, and fluorescence signal acquisitions were carried out independently at each institution according to similar but not identical experimental protocols and laboratory conditions.
Descriptive information for the liver cancer microarray data
Data set ID  Hospital  Chip type  Number of paired samples  Number of nonpaired samples  Total number of samples  

tumor  control  tumor  control  
D1  A  C _{1}  15  15  1  1  32 
D2  B  C _{1}  23  23  0  0  46 
D3  C  C _{1}  4  4  25  1  34 
D4  C  C _{2}  8  8  4  1  21 
The LMe models
Suppose there are H multiple data sets denoted by h = 1, …, H. There are n_{ h } patients for the h th data set. In our study, H = 4 and treatment groups consist of two levels denoted by k = T, C, where one (k = T) is the tumor tissue group and the other (k = C) is the control tissue group. For the paired observations, k has two values T and C. For the nonpaired observation, k has only one value of T or C. Assume there are N common probes on each chip for all data sets. We denote genes by l (= 1,…, N). The linear mixed effects (LMe) model consists of both fixed effects and random effects. The LMe model for the l th gene is given by
Y_{ hil } = X_{ hil }β_{ l } + Z_{ hil }b_{ hil } + ε_{ hil },
h = 1, …, H, i = 1, …, n_{ h }, l = 1, …, N, (1)
where Y_{ hil } is a response vector for the i th subject (patient) of the h th data set, β_{ l } is the fixed effect parameter vector, b_{ hil } is the random effect parameter vector, and ε_{ hil } is the error vector. Random effects and errors are assumed to be independent and normally distributed:
b_{ hil } ~ N(0, Φ_{ l }), ε_{ hil } ~ N(0, I σ^{2}). (2)
The variance of random effects Φ_{ h } can have several forms. When the offdiagonal terms are zero, then the random effects are uncorrelated. Otherwise, they are correlated. By allowing different forms of Φ_{ h }, we can model variability among samples efficiently. When there are no random effects, say Z_{ hil } = 0, the LMe models become equivalent to the ANOVA models.
where l = 1,…, 9984, h = 1,…, 4, β_{ Tl } represents the treatment effect of differences between tumor tissue and control tissue, β_{ Cl } represents the effect of differences between two chips, and two parameters, β_{ H }_{1}l and β_{ H }_{2}l, represent the effect of differences among hospitals.
Types of covariance structure
The most general form of covariance matrix in the LMe models assumes the covariance matrix of gene expressions within each data set is unstructured and differs among data sets. However, this covariance matrix requires many parameters to be estimated, which could result in a possible loss of power. Therefore, we need to consider simplified forms of the covariance matrices of b_{ hil }. We consider four types of covariance forms for the integrated analysis of microarray data. For simplicity, we start with the case when the data consist of all paired observations.
Paired microarrays
 1.
 2.
 3.
 4.
Type 2 assumes the covariance matrix of gene expressions within each data set is unstructured like Type 1 but it is the same over the data sets, which is a simplified form of Type 1. Type 3 assumes each covariance matrix within the data set is compound symmetric and differs over the data sets. Type 4 is simplified version of Type 3 assuming the same covariance matrix over the data sets.
For all types of covariance structure, the variance of Y_{ hil } is given by
Y_{ hil } = Var(b_{hil}) + I σ^{2}.
Nonpaired microarrays
Tests
LMe model parameters can be estimated via maximum likelihood estimation. The DEGs can be identified by testing whether β_{ Tl } = 0 or not. LMe models also suffer from the multiple testing problem. We apply the FDR adjustment method proposed by Benjamini et al.[19].
Results
Analysis of the liver cancer microarray data
We applied the integrated analysis using LMe models, twostage ANOVA model, and metaanalysis to liver cancer data. The LMe model is given in Equation 3. We fit this LMe model by assuming that b_{ hil } has the covariance structure of Types 1 to 4. These four models are denoted by M1, M2, M3, and M4, respectively. The last LMe model M5 assumes no random effects and is expected to provide similar results to the twostage ANOVA model.
Genes that are identified as differentially expressed when FDR is controlled 1%, 5%, 10%, and 20%, respectively
FDR  Meta analysis  Twostage ANOVA  LMe  

M1  M2  M3  M4  M5  
1%  57  46  119  184  205  124  37 
5%  197  145  214  543  589  375  114 
10 %  303  203  339  879  978  740  181 
20 %  478  336  585  1500  1761  1323  342 
Common genes detected by metaanalysis, twostage ANOVA model, and LMe M3 model when FDR is controlled by 1% (9 known genes)
Unigene ID  Description 

Hs.82084  Integrin beta 3 binding protein (beta3endonexin) (ITGB3BP), mRNA 
Hs.514  Cyclin H (CCNH), mRNA 
Hs.167529  Cytochrome P450, subfamily IIC (mephenytoin 4hydroxylase), polypeptide 9 (CYP2C9), mRNA 
Hs.117367  Solute carrier family 22 (organic cation transporter), member 1 (SLC22A1), mRNA 
Hs.54900  Serologically defined colon cancer antigen 1 (SDCCAG1), mRNA 
Hs.80756  Betainehomocysteine methyltransferase (BHMT), mRNA 
Hs.8765  RNA helicaserelated protein (RNAHP), mRNA 
Hs.755990  Haptoglobin (HP), mRNA 
Hs.35101  Prolinerich Gla (Gcarboxyglutamic acid) polypeptide 2 (PRRG2), mRNA 
The number of genes identified only by M3 was 183 in the Figure 1 Some genes have been found to be related with liver disorders (BChE, C6, C9, CAP2, CDKN2A, CtBP, Cul4A, Gab1, Id1, NTRK1, PSG1, and PSMG). HChE was shown to exhibit highly elevated aryl acylamidase activity (AAA). The absolute levels of AAA were increased as BChE activity decreased while deviating from normal samples and such deviation was directly proportional to the severity of the liver disorder [20].
C6 is a component of the complement system, which plays an important role as a humoral effector system during inflammation and infection, and consists of more than 25 components, including regulatory proteins. C6 was shown to lateacting complement proteins that participate in the assembly of the membrane attack complex, which causes cell lysis by the formation of pores in the cell membrane of certain microorganisms. [21]. C9 was related to the medication of tumor PDT by photosensitizer Photofrin using mouse Lewis lung carcinoma (LLC) model [22]. Cyclaseassociated protein 2 (CAP2) was listed as an upregulated gene in early hepatocellular carcinoma (HCC) [23]. CDKN2A was reported to be differentially regulated by methylation between normal tissue and HCC. Low levels of methylation in normal tissue and adjacent tissue but high levels in HCC [24]. Cterminal binding protein (CtBP) was reported to relate with INK4A/ARF tumor suppressor gene. The INK4A/ARF tumor suppressor locus is frequently inactivated in HCC. Inhibition of cell invasion by p19Arf was dependent on its Cterminal binding protein (CtBP) [25]. The Cul4A gene is amplified in human breast and liver cancers, and lossoffunction of Cul4 results in the accumulation of the replication licensing factor CDT1 in Caenorhabditis elegans embryos and ultraviolet (UV)irradiated human cells [26].
Gab1 was reported to be related with hepatic insulin action. Deletion of Gab1 in the liver leads to enhanced glucose tolerance and improved hepatic insulin action. It was also shown that association of Gab1 adaptor protein and Shp2 tyrosine phosphatase is a critical event at the early phase of liver regeneration [27, 28]. Id1 was identified as TGFβ/ALK1/Smad1 target gene in HSCs and represents a critical mediator of transdifferentiation that might be involved in hepatic fibrogenesis. Transforming growth factor (TGF)β is critically involved in the activation of hepatic stellate cells (HSCs) that occurs during the process of liver damage, for example, by alcohol, hepatotoxic viruses, or aflatoxins [29, 30]. NTRK1 was reported to be a favorable neuroblastoma (NB) genes. NB is a common pediatric solid tumor that exhibits a striking clinical bipolarity: favorable and unfavorable. Highlevel expression of NTRK1 predicts favorable NB outcome and inhibits growth of unfavorable NB cells [31]. PSG1 was reported to an upregulated gene in a fetal liver [32]. PSMG was reported to significantly elevated expression in HCC [33].
Simulation study
In order to evaluate the proposed methods, we simulated the two sets of microarray data and then performed the integrated analysis by using the proposed LMe method as well as other methods. For simplicity, we assume the logtransformed ratio of two intensities are normally distributed. To mimic the liver cancer microarray data, we assume that a pair of microarrays are obtained from the same patient. The first microarray data set consists of 60 microarrays from 30 patients and the second data set consists of another 60 microarrays from 30 patients. Suppose that two microarrays from the same patient are from different groups, say from tumor and control tissues. The main objective of the analysis is to identify the DEGs between two groups.
where β_{ Dl } represents a fixed effect of the difference between two data sets and β_{ Tl } represents a fixed effect for difference of expression levels between tumor and control tissues. The values of β_{ Tl }s are 1.5 for l = 1, ⋯, 3, and 1.5 for l = 4, ⋯, 6, respectively, and zero for l = 7, …, 30. The values of β_{ Dl } are randomly determined by generating random variables from the standard normal distribution. Errors are also generated from the normal distribution with mean 0 and variance σ^{2} = 0.5^{2}.
Setting for random effects b_{ hik }_{ l }
Type  random effects of subject  covariance of random effects 

1 


2 


3 


For the simulated data sets, we perform the analyses using the metaanalysis, the twostage ANOVA model and five LMe models. We fit this LMe model by assuming that b_{ il } has the covariance structure of Types 1 to 4. These four models are denoted by M1, M2, M3, and M4, respectively. The last LMe model M5 is the one assuming no random effects, which is expected to provide similar results to the twostage ANOVA model.
Power and FDR of methods under simulated data of Types 1, 2, and 3 covariance structures when FDR was controlled by 0.05
Type  ρ  Meta analysis  Twostage ANOVA  LMe  

M1  M2  M3  M4  M5  
1  0  Power  0.2087  0.1983  0.2770  0.2073  0.2610  0.2273  0.2240 
FDR  0.0740  0.0600  0.1150  0.0730  0.0863  0.0746  0.0807  
0.2  Power  0.1863  0.1683  0.3493  0.2570  0.2847  0.2607  0.1903  
FDR  0.0345  0.0307  0.0958  0.0666  0.0707  0.0668  0.0371  
0.4  Power  0.1543  0.1403  0.4783  0.3950  0.4007  0.3953  0.1580  
FDR  0.0170  0.0186  0.0718  0.0558  0.0573  0.0557  0.0104  
2  0  Power  0.1453  0.1423  0.3650  0.3067  0.3093  0.3073  0.1570 
FDR  0.0224  0.0251  0.0920  0.0564  0.0569  0.0563  0.0248  
0.2  Power  0.1290  0.1347  0.3373  0.2867  0.2867  0.2867  0.1450  
FDR  0.0203  0.0098  0.0858  0.0591  0.591  0.0591  0.0247  
0.4  Power  0.1490  0.1497  0.3700  0.3150  0.3170  0.3150  0.1503  
FDR  0.0283  0.0323  0.1091  0.0680  0.0676  0.0680  0.0363  
3  Power  0.1517  0.0010  1.000  1.0000  1.0000  1.0000  0.0043  
FDR  0.0000  0.0000  0.0712  0.0455  0.0455  0.0455  0.0000 
When ρ was zero, powers and FDRs showed very consistent results for all methods, although the variances of tumor tissue and control tissue are assumed to be different. This means all methods perform similarly when the correlation between tumor and control tissues does not exist.
Table 5 summarizes the simulation results for Type 1 covariance matrix. In general, metaanalysis, twostage ANOVA model analysis, and M5 provided similar results in powers and FDRs. On the other hand, other LMe models provided quite different results. For example, the FDRs tend to be larger but maintain 5% level approximately except for M1. Powers of LMe models tend to be much larger than metaanalysis and twostage ANOVA model analysis. Among the five LMe models, M1 and M5 provide distinct results from the other three models M2, M3, and M4.
It is interesting to note that the performance of each method depends on the value of ρ. For metaanalysis, twostage ANOVA, and M5, the powers decrease as ρ increases. On the other hand, the powers of LMe models M1 to M4 increase. These tendencies illustrate that metaanalysis and twostage ANOVA do not handle correlations efficiently as LMe models do.
FDRs of LMe models, M2, M3, and M4 are slightly larger than 0.05. However, the FDR of M1 is much larger than 0.05, especially when ρ is close to zero. Thus, M1 is not appropriate to use when there is no correlation between tumor and control tissues.
Table 5 also summarizes the simulation result for the Type 2 covariance matrix showing similar patterns with those of Type 1 except that the results are less sensitive to ρ. In summary, metaanalysis, ANOVA model analysis, and M5 provided similar results in powers and FDRs. On the other hand, other LMe models provided quite different results. Among the five LMe models, M1 and M5 provided distinct results from the other three LMe models. The powers of LMe models M1 to M4 are larger than metaanalysis, ANOVA, and M5. Although M1 has the largest power, it also shows the largest FDR.
Finally, Table 5 also summarizes the simulation result for the Type 3 covariance matrix. Though correlation parameter ρ was not considered in this case, the correlation between tumor tissue and control tissue of same patient was assumed by the shared random parameter b_{ hil }. The results of simulated data under Type 3 are quite different from those obtained from Types 1 and 2. That is, all LMe models, M1, M2, M3, and M4 show extremely good performance. The powers are all 1 and FDRs are wellcontrolled around 0.05. LMe models work very well for this high correlation case. On the other hand, metaanalysis, ANOVA, and M5 performed worse. Among these, metaanalysis showed a slightly better performance. It is probably due to the fact that the metaanalysis allows different variances between two data sets, while others do not.
Discussion
The LMe model is much more flexible than metaanalysis. One of the main limitations of metaanalysis is that it cannot handle the samplespecific covariates appropriately. Effectsize is simply the standardized mean difference between tumor tissue and control tissue [14]. Metaanalysis requires data are homogeneous within the data set, although data may be heterogeneous across data sets. For example, when there is sex information in data, the effectsize statistic cannot account for the sex effect directly. On the other hand, LMe models can handle individual specific covariates easily. In microarray studies, many researchers want to account for the individual characteristics in the analysis by including them as controlling variables. For example, the covariates such as age, sex, tumor stage, and weight might be important controlling variables. These covariates are usually samplespecific and differ across samples.
When there are no random effects, the LMe models become equivalent to the ANOVA models. The heterogeneity among data sets is only represented by the fixed effects. When heterogeneity among data sets is small, the ANOVA model can easily handle the variability among the data sets. However, when data sets have high variability and contain the correlated data, the addition of only fixed effects may not be satisfactory. In this case, the LMe model is more appropriate to analyze data sets, because it can model the heterogeneous variance and correlation structure more appropriately. The proposed LMe model is capable of handling heterogeneous covariance structures by allowing for various random effects.
When the data set contains paired and nonpaired microarrays simultaneously, both metaanalysis and ANOVA model approaches cannot handle them appropriately. For example, the metaanalysis and the ANOVA analysis treated paired microarrays as independent microarrays. On the other hand, the proposed LMes can handle appropriately the correlation between the paired microarrays.
Finally, note that the proposed LMe model is valid when the normality assumption holds. We do not expect this assumption to hold for real microarray data. However, we expect the assumption is decreased when sufficiently large number of microarrays were combined. In future studies, we will develop permutation tests for the LMe models which do not require any distributional assumption.
Conclusion
We proposed the LMe model for the integrated analysis of microarray data to identify DEGs in the presence of many controlling variables. We analyzed the liver cancer microarray data set and simulated microarray data to evaluate the performance of the integration methods. LMe models except M1 maintained FDRs approximately. Powers of LMe models except M5 tended to be much larger than metaanalysis and twostage ANOVA model analysis. These tendencies illustrated that metaanalysis and twostage ANOVA do not handle correlations efficiently as LMe models do.
Declarations
Acknowledgements
This work was supported by the National Research Foundation (KRF2008313C00086) and the Brain Korea 21 Project of the Ministry of Education.
This article has been published as part of BMC Bioinformatics Volume 12 Supplement 5, 2011: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine 2010. The full contents of the supplement are available online at http://www.biomedcentral.com/14712105/12?issue=S5.
Authors’ Affiliations
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