# Classical and Bayesian random-effects meta-analysis models with sample quality weights in gene expression studies

**Part of the following topical collections:**

## Abstract

### Background

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.

### Results

Sample quality adjusting within study variance (*w*_{P6}) 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*_{P6} weighted model compared to the DSL*w*_{P6} weighted model. Among the BRE weighted models, the *w*_{P6}weighted- and unweighted-data models and both Gibbs- and MH-based models performed similarly. The *w*_{P6} weighted common-effect model performed similarly to the unweighted model in the homogeneous data, but performed worse in the heterogeneous data. The *w*_{P6}weighted data were appropriate for detecting DE genes with high precision, while the *w*_{P6}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*_{P6}weighted and the *w*_{P6}weighted between study variance models. Four hundred and forty-six DE genes identified by the *w*_{P6}weighted between study variance model could be potentially down-regulated genes that may contribute to good classification of Alzheimer’s samples.

### Conclusions

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.

## Keywords

Random-effects model Bayesian random-effects model Meta-analysis Study heterogeneity Gene expression Sample quality weights Alzheimer’s disease## Abbreviations

- AUC
Area under receiver operating characteristic curve

- BH
Benjamini and Hochberg

- BRE
Bayesian random-effects model

- DE
Differentially expressed

- DSL
DerSimonian and Laird

- DSLR2
Two-step DerSimonian and Laird estimate

- FDR
False discovery rate

- FE
Fixed-effects

- GAPDH
Glyceraldehyde-3-phosphate dehydrogenase

- MH
Metropolis Hasting

- MSSE
Minimum sum of squared error

- QC
Quality control

- RE
Random-effects

- SD
Standard deviation

## Background

Although modern sequencing technologies such as ribonucleic acid sequencing and next-generation sequencing have been developed, microarrays have been a widely used high-throughput technology in gathering large amounts of genomic data [1, 2]. Due to small sample sizes in single microarray studies, microarray studies are combined with meta-analytic techniques to increase statistical power and generalizability of the results [1, 3].

Common meta-analysis techniques applied in gene expression studies included combining of *p*-values, rank values, and effect sizes. Examples of the p-value based methods include Fisher’s method, Stouffer’s method, minimum *p*-value method, maximum p-value method, and adaptively weighted Fisher’s method. The rank-based methods include *rt*h ordered *p*-value method, naïve sum of ranks, naïve product of ranks, rank product, and rank sum methods. The effect-size based methods include fixed-effects (FE) and random-effects (RE) models.

Appropriateness of the meta-analysis techniques in gene expression data depends on types of hypothesis testing: HSA, HSB, or HSC as described in [4, 5, 6]. Maximum p-value and naïve sum of rank methods were appropriate for HSA hypothesis that detected DE genes across all studies. The *rt*h ordered *p*-value method and two-step DerSimonian and Laird estimated RE models were appropriate for HSB hypothesis that detected DE genes in one or more studies. DerSimonian and Laird (DSL) and empirical Bayes estimated RE models, including our two-step estimated RE model using DSL and random coefficient of determination (*R*^{2}) method were appropriate for HSC hypothesis that detected DE genes in a majority of combined studies [4, 5, 6].

Some of these methods may be limited in their application. The p-value based methods are limited in reporting summary effects and addressing study heterogeneity [3, 7, 8, 9]. The rank-based methods are robust towards outliers and applied without assuming a known distribution [8, 10]; however, their results are dependent on the influence of other genes included in microarrays [1]. The FE model assumes that total variation is derived from a true effect size and a measurement error [3]; however, the effect may vary across studies in real-world applications. Concurrently, although the RE model can address study-specific effects and accounts for both within and between study variation, the between study variation or the heterogeneity in effect sizes is unknown. Many frequentist-based methods have been developed to estimate the between study variation. More details can be found in [6, 9, 11, 12].

The RE models are commonly applied in gene expression meta-analysis. Classical RE models assume studies are independently and identically sampled from a population of studies. However, an infinite population of studies may not exist and studies may be designed based on results of previous studies, thus potentially violating an independence assumption. Bayesian random-effects (BRE) models have been used to allow for uncertainty of parameters. The uncertainty is expressed through a prior distribution and a summary of evidence provided by the data is expressed by the likelihood of the models. Multiplying the prior distribution and the likelihood function results in a posterior distribution of the parameters [13, 14].

Sample quality has substantial influence on results of gene expression studies [15, 16]. The degree of heterogeneity may differ due to inconsistencies in sample quality. Low heterogeneity can be found in meta-analyses containing good quality samples, while high heterogeneity arises in meta-analyses containing poor quality samples. In our recent study, we evaluated the relationships between DE and heterogeneous genes in meta-analyses of Alzheimer’s gene expression data. We detected some overlapped DE and heterogeneous genes in meta-analyses containing borderline quality samples, while no heterogeneous genes were detected in meta-analyses containing good quality samples [6]. Obviously, data obtained from borderline (poor) quality samples can increase study heterogeneity and reduce the efficiency of meta-analyses in detecting DE genes [17, 18].

In this study, we implemented a meta-analytic approach that includes sample-quality weights to take study heterogeneity into account in RE and BRE models. The gene expression data therefore would consist of up-weighted good quality samples and down-weighted borderline quality samples. Therefore in the Methods section we first review quality assessments of microarray samples, sample-quality weights, RE models, BRE models, weighted RE models, and weighted BRE models. We then describe our simulation studies and application data. Our results are then presented followed by discussion and conclusions.

## Methods

This section describes quality assessments of microarray samples, sample-quality weights, RE models, BRE models, weighted RE models, and weighted BRE models.

### Microarray quality assessments

Affymetrix GeneChips and Illumina BeadArrays have been widely used single channel microarrays. Quality assessments in Affymetrix arrays include the 3′:5′ ratios of two-control genes: beta-actin, and glyceraldehyde-3-phosphate dehydrogenase (GAPDH); the percent of number of genes called present; the array-specific scale factor; and the average background [15, 19]. A 3′:5′ ratio close to 1 indicates a good quality sample while a ratio > 3 suggests a poor quality sample, resulting from problems of RNA extraction, cDNA synthesis reaction, or conversion to cRNA [15, 20]. Additionally, the percent present calls should be consistent among all arrays hybridized and generally should range from 30 to 60% [21]. The scale factor is used to assess overall expression levels with an acceptable value within 3-fold of one another. The proportion of up- and down-regulated genes should be consistent at the average signal intensity so that the expression among arrays can be comparable. The average background should also be consistent across all arrays [15]. For Illumina BeadArrays, quality assessments include the average and standard deviation of intensities, the detection rate, and the distance of specific probe intensities to the overall mean intensities of all samples [22, 23, 24].

### Random-effects models

In this section, we provided a brief summary of the random-effects models implemented in this study. The hypothesis settings for detecting DE genes in meta-analysis of gene expression data are described in the supplemental material.

#### DerSimonian-Laird model (DSL)

*y*

_{ig}) can be obtained for each gene

*g*as described in Hedges et.al. (1985) and Choi et.al. (2003) as:

*i*th study,

*i = 1,…,k*; s

_{ig}and

*n*

_{ig}are an estimate of the pooled standard deviation across groups and the total sample size in the

*i*th study; and

*y*

_{ig}is obtained as the correction for sample size bias. The estimated variance of

*y*

_{ig}is \( {\sigma}_{ig}^2=\left({n}_{ig(a)}^{-1}+{n}_{ig\left(\mathrm{c}\right)}^{-1}\right)+{y}_{ig}^2{\left(2\left({n}_{ig(a)}+{n}_{ig(c)}\right)\right)}^{-1} \). The model of effect-size combination is based on a two-level hierarchical model:

*y*

_{ig}is the effect for gene

*g*in

*i*th study,

*i = 1,…,k*;

*θ*

_{ig}is the true difference in mean expression; \( {\sigma}_{ig}^2 \)is the within-study variability representing sampling errors conditional on the

*i*th study;

*β*

_{g }is the common effects or average measure of differential expression across datasets for each gene or the parameter of interest;

*δ*

_{ig}is the random effect; and \( {\tau}_g^2 \) is the between-study variability. The RE model is defined when there is between-study variation [11, 25]. The estimator for \( {\tau}_g^2 \)is typically obtained using DerSimonian-Laird (DSL) estimator [26, 27] as

where \( {Q}_g={\sum}_{i=1}^k{w}_{ig}{\left({y}_{ig}-{\widehat{\beta}}_g\right)}^2,\kern0.5em {w}_{ig}={\sigma}_{ig}^{-2},\kern0.5em {\widehat{\beta}}_g=\frac{\sum_{i=1}^k{w}_{ig}{y}_{ig}}{\sum_{i=1}^k{w}_{ig}},\kern0.5em {S}_{rg}={\sum}_{i=1}^k{w}_{ig}^r \), and *r* = {1, 2}. For each gene, we estimated \( {\widehat{\beta}}_g\left({\widehat{\tau}}_{DSL(g)}^2\right) \) with \( {w}_{ig}={\left({\sigma}_{ig}^2+{\widehat{\tau}}_{DSL(g)}^2\right)}^{-1} \) using a generalized least squares method to obtain statistics *z*_{DSL(g)}. More details can be found in [11, 25].

#### Two-step estimate model (DSLR2)

*τ*

^{2}(

**Y**

_{g}−

**β**

_{g}), so its bias does not influence the unbiasedness of the treatment and random effects [6, 12]. The \( {\widehat{\tau}}_{DSLR2(g)}^2 \)on the zero-to-one scale provides a lower minimum sum of squared error (MSSE) than the \( {\widehat{\tau}}_{DSL(g)}^2 \) estimate. The \( {R}_{DSL(g)}^2 \) measuring the strength of study heterogeneity can also be used to compare variation of genes in different meta-analyses to decide which studies should be included in the meta-analysis [28]. The estimates of treatment effects, its variance, z-statistics, and random effects are obtained as

When compared to the DSL method, the DSLR2 method had a relatively better sensitivity and accuracy in detecting DE genes under HSC hypothesis testing and a higher precision when the proportion of truly DE genes in the metadata was higher [6]. The DSLR2 method performed well with a low computational cost and almost all significantly DE genes identified were genes among the significantly DE genes identified using the DSL method. However, similar to the DSL method, the performance of the DSLR2 method can be reduced when sample sizes in single studies are restricted (e.g., < 60 in both arms) and the normality assumption of the meta-analysis outcome does not hold [6].

*z*

_{(g)}= (

*z*

_{(1)}≤ ⋯ ≤

*z*

_{(G)}) and \( {z}_{(g)}^r=\left({z}_{(1)}^r\le \cdots \le {z}_{(G)}^r\right) \) as

*α*is the significance threshold of the single test,

*g*is an index of genes

*1,…,G*, and

*r*is an index of permutation

*1,…,R.*

#### Bayesian random-effects model (BRE)

*g*is given by

where \( {\mathbf{y}}_g=\left({y}_{1g},\dots, {y}_{kg}\right),{\boldsymbol{\upsigma}}_g^2=\left({\sigma}_{1g}^2,\dots, {\sigma}_{kg}^2\right) \), and **θ**_{g} = (*θ*_{1g}, …, *θ*_{kg}) for gene *g* in the *i*th study; *i = 1,…,k.* The *π*(*β*_{g}) and \( \pi \left({\tau}_g^2\right) \) are non-informative priors given as *β*_{g} ∼ *N*(0, 1000), and*τ*_{g}∼uniform (a,b) and gamma (α,β).

The choice of prior distributions for scale parameters can affect analysis results, particularly in small samples. With scale parameters, the distributional form and the location of the prior distributions are decided [31]. Uniform distributions are appropriate non-informative priors for \( {\tau}_g^2 \) [13]. We conducted simulations to select appropriate priors for \( {\tau}_g^2 \), allowing the maximum (b) of the uniform distribution to be b∈{0.005, 0.001, 0.05, 0.01, 0.5, 0.1, 1, 5, 10} and b~Gamma(1,2). The potential choices of the appropriate priors were selected based on parameters obtained from an Alzheimer’s gene expression data [6] in order to further apply the results.

### Sample-quality weights

*y*

_{ig}. A high variance therefore gives low weights in meta-analysis [32, 33]. In this study, the weights corresponding to the QC indicators fall into two categories: standardized ratio weights and zero-to-one weights (Table 1).

List of sample quality weights

Standardized ratio weights ( | Zero-to-one weights ( | |
---|---|---|

\( {w}_{S1}={\left({\sigma}_g^2+{s}_{ij}{\widehat{\tau}}_g^2\right)}^{-1} \) \( {w}_{S2}={\left({s}_{ij}{\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)}^{-1} \) \( {w}_{S3}={\left({s}_{ij}\left({\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)\right)}^{-1} \) \( {w}_{S4}={2}^{-\left({\sigma}_{ig}^2+{s}_{ij}{\widehat{\tau}}_g^2\right)} \) \( {w}_{S5}={2}^{-\left({s}_{ij}{\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)} \) \( {w}_{S6}={2}^{-\left({s}_{ij}\left({\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)\right)} \) | \( {w}_{P1}\in \left\{{2}^{-{s}_{ij}},0.01{\overset{\sim }{p}}_{ij}\right\} \) \( {w}_{P2}={\left({\sigma}_{ig}^2+\left(1-{w}_{P1}\right){\widehat{\tau}}_g^2\right)}^{-1} \) \( {w}_{P3}={\left(\left(1-{w}_{P1}\right){\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)}^{-1} \) \( {w}_{P4}={\left(\left(1-{w}_{P1}\right)\left({\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)\right)}^{-1} \) \( {w}_{P5}={\left({\sigma}_{ig}^2+{\widehat{\tau}}_g^{2\left({w}_{P1}\right)}\right)}^{-1} \) \( {w}_{P6}={\left({\sigma}_{ig}^{2\left({w}_{P1}\right)}+{\hat{\tau}}_g^2\right)}^{-1} \) \( {w}_{P7}={\left({\left({\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)}^{\left({w}_{P1}\right)}\right)}^{-1} \) | \( {w}_{P8}={2}^{-\left({\sigma}_{ig}^2+\left(1-{w}_{P1}\right){\widehat{\tau}}_g^2\right)} \) \( {w}_{P9}={2}^{-\left(\left(1-{w}_{P1}\right){\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)} \) \( {w}_{P10}={2}^{-\left(\left(1-{w}_{P1}\right)\left({\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right)\right)} \) \( {w}_{P11}={2}^{-\left({\sigma}_{ig}^2+{\widehat{\tau}}_g^{2\left({w}_{P1}\right)}\right)} \) \( {W}_{P12}={2}^{-\left({\sigma}_{ig}^{2\left({w}_{P1}\right)}+{\widehat{\tau}}_g^2\right)} \) \( {w}_{P13}={2}^{-\left(\left({\sigma}_{ig}^2+{\widehat{\tau}}_g^2\right){w}_{P1}\right)} \) |

#### Standardized ratio weights (*w* _{S,ij})

*R*

_{ij}is a quality indicator, i.e. 3′:5’ GAPDH ratio of the

*j*th sample in the

*i*th study,

*SD*(

*R*

_{i})is the standard deviation of the quality indicator in the

*i*th study,

*w*

_{S1 − S3}∈ (0, ∞), and

*w*

_{S4 − S8}∈ (0, 1).

*f*(.) is a function of sample-quality weights with the within and between study variances as shown in Table 1. A low value of the

*S*

_{ij}indicates good quality samples, providing high values of standardized ratio weights (

*w*

_{S,ij}) to give more weight on the expression data.

#### Zero-to-one weights (*w* _{P,ij})

*S*

_{ij}is the percent of present calls and the standardized quality indicators of the

*j*th sample in the

*i*th study, respectively,

*w*

_{P1 − P7}∈ (0, ∞), and

*w*

_{P8 − P13}∈ (0, 1). A high value of the

*P*

_{ij}weights indicate good quality samples, providing high values of zero-to-one weights (

*w*

_{P,ij}) to give more weight on the expression data.

The weights are primarily selected based on availability of quality indicators, such as 3′:5’ GAPDH ratio in Affymetrix arrays or detection rate in Affymetrix arrays and Illumina BeadArrays. Both the 3′:5’ GAPDH ratio and detection rate can be converted to the zero-to-one weights via *w*_{P1}.

### Weighted random-effects models

An appropriate weight was chosen based on the precision and accuracy of the DSL weighted and DSLR2 weighted models in detecting DE genes via simulations and were used to weight the expression data and to adjust the common effect and the between-study variance in the BRE model.

#### Weighted DSL and DSLR2 models

_{2}normalized intensity data were weighted with an appropriate weight obtained from the DSL and DSLR2 weighted models. The weighted mean \( \left({\overline{x}}_{ig(a)}\right) \) and weighted sample variance \( \left({\mathrm{s}}_{ig(a)}^2\right) \) of the normalized intensity data in each group were calculated:

*x*_{ijg(a)} is the log_{2} normalized intensity data for gene *g* of the *j*th sample in the case (a) group and in the *i*th study, *n*_{ig(a)} is the sample size of case (a) group for gene *g* in the *i*th study, and *w*_{ijg(a)} is the sample-quality weight of the *j*th sample in the case (a) group in the *i*th study for the gene *g*. The same calculations were applied for the weighted mean \( \left({\overline{x}}_{ig(c)}\right) \) and the weighted sample variance \( \left({\mathrm{s}}_{ig(c)}^2\right) \) in the control (c) group. The unbiased standardized mean difference of the expression between groups were re-calculated and re-combined using the DSL and DSLR2 models (Eq.1 and Eq.2).

#### Weighted common effect model

*i*th study for gene

*g*\( \left({\overline{w}}_{ig}={\sum}_{j=1}^{n_{ig(a)}+{n}_{ig\left(\mathrm{c}\right)}}{w}_{ijg}/\left({n}_{ig(a)}+{n}_{ig\left(\mathrm{c}\right)}\right)\right) \). The BRE weighted common effect model for gene

*g*is given by

#### Weighted between-study variance model

*i*th study for gene

*g*\( \left({\overline{w}}_{ig}={\sum}_{j=1}^{n_{ig(a)}+{n}_{ig\left(\mathrm{c}\right)}}{w}_{ijg}/\left({n}_{ig(a)}+{n}_{ig\left(\mathrm{c}\right)}\right)\right) \). The BRE weighted between-study variance model for gene

*g*is given by

Example WinBUGS code appears in the supplemental material.

Bayesian random-effects (BRE) models by data features, sampling algorithms, and weighted inference models

BRE Models | |||||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | |

Data features | |||||||

Unweighted normalized intensity data | ✓ | ✓ | ✓ | ||||

Weighted normalized intensity data | ✓ | ✓ | ✓ | ✓ | |||

Sampling algorithms | |||||||

Gibbs sampling | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |

Metropolis-Hasting sampling | ✓ | ||||||

Weighted inference models | |||||||

Unweighted model | ✓ | ✓ | ✓ | ||||

Weighted common effect | ✓ | ✓ | |||||

Weighted between-study variance | ✓ | ✓ |

The DE genes were defined as those with FDR less than 5%. Unsupervised hierarchical clustering using Ward’s method and one minus Pearson’s correlation coefficient for measures of similarities were used to graphically present the DE genes in the individual analysis of Alzheimer’s gene expression data using a heatmap.

### Simulation setting

- 1.
Five studies each with 2000 genes were generated (800 clustering and 1200 non-clustering genes). The clustering genes with the same correlation pattern within their clusters were equally allocated into 40 clusters.

- 2.
Gene expression levels among clustering and non-clustering genes were assumed to follow a multivariate normal distribution \( {\left({X}_{gc{1}^{\prime }},\dots, {X}_{gc{40}^{\prime }}\right)}^T\sim MVN\left(0,{\Sigma}_{ck}\right), \) 1 ≤

*k*≤ 5, 1 ≤*c*≤ 40, \( {\sum}_{ck^{\prime }}\sim {W}^{-1}\left(\psi, 60\right), \) and*ψ*= 0.5*I*_{20 × 20}+ 0.5*J*_{20 × 20}, and a standard normal distribution, respectively. - 3.
Truly DE genes were generated with uniform(0.5,3), accounted for 10% of the total genes, and equally classified into 5 groups (

*t*_{g}= 1, …, 5). On average each group included 200 true genes. As the RE models appropriated under HSC, 120 genes in more than 50% of the combined studies were defined as the truly DE genes. - 4.
Truly heterogeneous genes constituted 15% of the total genes, implied by the random effects with uniform(0.5,3), and proportionally allocated into truly DE and not truly DE gene groups. The heterogeneous gene was defined by a significant random effect, where the gene expression was not identical across studies.

- 5.
Sample-quality weights were assumed to follow beta distributions(

*α*= 10,*β*= 1) for the zero-to-one weights and normal distributions*N*(0, 0.6) for the standardized ratio weights.

The N, G, K, and H denote the number of samples, the number of genes, the number of studies, the number of studies containing heterogeneous genes, respectively, all of which varied in different simulations. Because the simulation results under the same algorithms on 2000 and 10,000 genes were similar [6] and implementing Bayesian models requires intensive computations, we conducted the simulations on 2000 genes. Eight simulated metadata sets: two sets for the weighted and unweighted methods in the homogeneous data (H0), and each two of six sets for the weighted and unweighted methods in the heterogeneous data (H1, H2, and H3) were generated. A thousand simulations each with 1000 permutations of group labels were implemented for all DSL and DSLR models, and without permutation for the BRE models with different uniform(0,b) priors; b∈{0.005, 0.001, 0.05, 0.01, 0.5, 0.1, 1, 5, 10, and 100}, and b~Gamma(1,2) prior.

### Evaluations of methods in simulations

Because RE models were suitable under HSC hypothesis: detecting DE genes in a majority of combined studies [5, 6], the models were anticipated to detect DE genes in more than 50% of combined studies, *r = 3* for meta-analysis of five studies. We evaluated the number of detected DE genes, minimum sum squared error (MSSE), precision, accuracy, and area under receiver operating characteristic curve (AUC). Precision was calculated as the proportion of truly DE genes correctly identified as significant over the total number of genes declared significant. Accuracy was calculated as the proportion of genes correctly identified as being truly DE genes or not truly DE genes over the total of evaluated genes. The accuracy of the tests was also determined using AUC, where AUC ∈ (0.5, 0.7], AUC ∈ (0.7, 0.9] and AUC ∈ (0.9, 1.0] represent low, moderate, and high accuracy, respectively [35, 36]. All statistical methods and simulations were implemented using programs and modified programs from *limma, metafor, GeneMeta, MAMA, Rjags, R2jags, Coda* in the R programming environment.

Four publicly available Alzheimer’s disease (AD) gene expression datasets of post-mortem hippocampus brain samples were applied: GSE1297 [37], GSE5281 [38], GSE29378 [39], and GSE48350 [40]. After data preprocessing, quantile normalization, and data aggregating [20, 41, 42, 43, 44], our meta-analysis was performed on 12,037 target genes in 131 subjects (68 AD cases and 63 controls). We examined the strength of study heterogeneity by considering five ways of metadata sets as previously described in [6] and defined in the caption of Figs. 5 and 6. The metadata A, B, D, E may contain heterogeneous data due to a relatively high *R*^{2}, while the metadata C had a relatively low *R*^{2}or contained homogenous data. The 3′:5’ GAPDH ratio was used as a quality indicator in this analysis. The 3′:5’ GAPDH ratio was converted to the zero-to-one weight, *w*_{P6}, via *w*_{P1}.

## Results

Performance of random-effects models applied in simulated data

Model | Prior | No. DE Genes | MSSE | Precision | Accuracy | AUC | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

H0 | H1 | H2 | H3 | H0 | H1 | H2 | H3 | H0 | H1 | H2 | H3 | H0 | H1 | H2 | H3 | H0 | H1 | H2 | H3 | ||

DSL | – | 65 | 74 | 92 | 124 | 2.9 | 2.9 | 2.9 | 2.9 | 0.95 | 0.95 | 0.91 | 0.79 | 0.97 | 0.97 | 0.98 | 0.98 | 0.76 | 0.79 | 0.84 | 0.90 |

DSLR2 | – | 69 | 104 | 139 | 198 | 1.7 | 1.7 | 1.7 | 1.7 | 0.95 | 0.91 | 0.79 | 0.59 | 0.97 | 0.98 | 0.98 | 0.96 | 0.77 | 0.89 | 0.95 | 0.97 |

BRE | U(0,0.001) | 126 | 157 | 254 | 305 | 18.1 | 25.8 | 33.0 | 39.9 | 0.82 | 0.70 | 0.45 | 0.39 | 0.98 | 0.97 | 0.93 | 0.91 | 0.93 | 0.94 | 0.94 | 0.94 |

BRE | U(0,0.01) | 218 | 324 | 404 | 436 | 10.5 | 16.0 | 20.0 | 22.3 | 0.55 | 0.37 | 0.30 | 0.28 | 0.95 | 0.90 | 0.86 | 0.84 | 0.97 | 0.95 | 0.92 | 0.92 |

BRE | U(0,0.1) | 181 | 269 | 354 | 391 | 9.4 | 14.3 | 17.8 | 19.8 | 0.66 | 0.45 | 0.34 | 0.31 | 0.97 | 0.93 | 0.88 | 0.86 | 0.98 | 0.96 | 0.94 | 0.93 |

BRE | U(0,1) | 80 | 108 | 141 | 203 | 1.7 | 2.2 | 2.4 | 2.6 | 1.00 | 0.94 | 0.80 | 0.58 | 0.98 | 0.99 | 0.98 | 0.96 | 0.84 | 0.92 | 0.96 | 0.97 |

BRE | U(0,10) | 11 | 9 | 9 | 12 | 1.0 | 1.1 | 1.1 | 1.1 | 1.00 | 1.00 | 1.00 | 0.96 | 0.95 | 0.94 | 0.94 | 0.95 | 0.54 | 0.54 | 0.54 | 0.55 |

BRE | U(0,100) | 10 | 8 | 8 | 11 | 1.0 | 1.0 | 1.0 | 1.0 | 1.00 | 1.00 | 1.00 | 0.96 | 0.94 | 0.94 | 0.94 | 0.94 | 0.54 | 0.53 | 0.53 | 0.54 |

BRE | U(0,0.005) | 329 | 447 | 520 | 546 | 10.6 | 16.1 | 20.1 | 22.4 | 0.37 | 0.27 | 0.23 | 0.22 | 0.90 | 0.84 | 0.80 | 0.79 | 0.94 | 0.91 | 0.89 | 0.89 |

BRE | U(0,0.05) | 184 | 275 | 359 | 395 | 10.3 | 15.7 | 19.6 | 21.8 | 0.65 | 0.44 | 0.33 | 0.30 | 0.97 | 0.92 | 0.88 | 0.86 | 0.98 | 0.96 | 0.94 | 0.93 |

BRE | U(0,0.5) | 137 | 167 | 253 | 330 | 3.0 | 4.4 | 5.3 | 5.7 | 0.86 | 0.71 | 0.47 | 0.36 | 0.99 | 0.98 | 0.93 | 0.89 | 0.98 | 0.98 | 0.96 | 0.94 |

BRE | U(0,5) | 13 | 11 | 12 | 17 | 1.1 | 1.1 | 1.1 | 1.1 | 1.00 | 1.00 | 1.00 | 0.97 | 0.95 | 0.95 | 0.95 | 0.95 | 0.55 | 0.54 | 0.55 | 0.57 |

BRE | G(1,2) | 41 | 53 | 69 | 94 | 1.7 | 2.0 | 2.1 | 2.1 | 1.00 | 1.00 | 0.97 | 0.89 | 0.96 | 0.97 | 0.97 | 0.98 | 0.67 | 0.72 | 0.78 | 0.84 |

Therefore, the DSLR2 and BRE models with a uniform(0,1) prior were appropriate for detecting DE genes in terms of an appropriate number of DE genes, a lower MSSE, a higher precision, and a higher AUC, particularly in the heterogeneous data. The BRE model with a uniform(0,1) prior particularly performed better than the DSLR2 model in the homogeneous data but performed similarly in the heterogeneous data.

### Weighted DSL and DSLR2 models

*w*

_{P6}function was most appropriate for detecting DE genes in the DSL and DSLR2 models. The QC indicators adjusted the within study variance in the weighted function as:

*S*

_{ij}denoted standardized quality indicators of the

*j*th sample in the

*i*th study. Fig. 2 presents the precision of the DSLR2 model with and without the

*w*

_{P6}function under different hypotheses in the homogeneous and heterogeneous data. The precision was increased with the DSLR2 weighted model in the heterogeneous data. The

*w*

_{P6}model provided an appropriate reduction of detected DE genes and MSSEs and higher precision as compared to the other weighted functions (Additional file 1: Tables S1 and S2). Similar results were found under different levels of sample quality (results not shown). The DSLR2

*w*

_{P6}weighted model had a lower MSSE and detected more DE genes than the DSL

*w*

_{P6}weighted model in the heterogeneous data.

### Weighted Bayesian random-effects models

*w*

_{P6}and DSLR2

*w*

_{P6}models, and BRE weighted models. A uniform(0,1) prior for between study variance was applied in all BRE models. The BRE weighted Models 1, 3, 4, 6, and 7 in Table 4 detected more DE genes with a higher AUC than the DSL

*w*

_{P6}and DSLR2

*w*

_{P6}models. The

*w*

_{P6}weighted-data models performed similarly to the unweighted-data models (Models 2 vs. 5 and 3 vs. 6). The

*w*

_{P6}weighted common-effect model performed similarly to the unweighted model in the homogeneous data, but performed worse in the heterogeneous data (Models 1 vs. 2). Additionally, the Gibbs- and MH-based models performed similarly on the

*w*

_{P6}weighted-data model. The numbers of detected DE genes were reduced close to the number of truly DE genes and the precisions were increased while maintaining a high accuracy as compared to the performance in the unweighted-data Gibbs-based model (Models 4 and 7 vs. 1). For homogeneous and heterogeneous data, the Gibbs- and MH-based models with the

*w*

_{P6}weighted-data performed similarly and were most appropriate for detecting DE genes with high precision (Models 4 and 7). The

*w*

_{P6}weighted between-study variance models were most appropriate for detecting DE genes with high overall accuracy (Models 3 and 6).

Performance of weighted random-effects models applied in simulated data

Model | No. DE Genes | MSSE | Precision | Accuracy | AUC | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

H0 | H1 | H2 | H3 | H0 | H1 | H2 | H3 | H0 | H1 | H2 | H3 | H0 | H1 | H2 | H3 | H0 | H1 | H2 | H3 | |

DSL | 62 | 62 | 64 | 65 | 2.9 | 3.0 | 3.0 | 3.0 | 0.95 | 0.96 | 0.96 | 0.96 | 0.97 | 0.97 | 0.97 | 0.97 | 0.75 | 0.75 | 0.75 | 0.76 |

DSLR2 | 66 | 72 | 78 | 85 | 1.6 | 1.6 | 1.6 | 1.6 | 0.96 | 0.95 | 0.94 | 0.92 | 0.97 | 0.97 | 0.97 | 0.98 | 0.76 | 0.78 | 0.80 | 0.82 |

BRE with a uniform(0,1) prior | ||||||||||||||||||||

Model 1: Unweighted data, Gibbs | 81 | 109 | 140 | 204 | 1.7 | 2.1 | 2.4 | 2.6 | 1.00 | 0.94 | 0.81 | 0.58 | 0.98 | 0.99 | 0.98 | 0.96 | 0.84 | 0.92 | 0.96 | 0.97 |

Model 2: Unweighted data, Gibbs, \( \beta {\overline{w}}_{P6} \) | 81 | 66 | 51 | 39 | 6.0 | 6.2 | 6.5 | 6.9 | 1.00 | 1.00 | 0.97 | 0.93 | 0.98 | 0.97 | 0.96 | 0.96 | 0.84 | 0.77 | 0.71 | 0.65 |

Model 3: Unweighted data, Gibbs, \( {\tau}^2{\overline{w}}_{P6} \) | 161 | 157 | 151 | 142 | 0.8 | 1.5 | 2.1 | 2.7 | 0.74 | 0.76 | 0.77 | 0.79 | 0.98 | 0.98 | 0.98 | 0.98 | 0.99 | 0.99 | 0.97 | 0.96 |

Model 4: Weighted data, Gibbs | 81 | 87 | 92 | 100 | 1.8 | 2.2 | 2.7 | 3.1 | 1.00 | 0.99 | 0.97 | 0.93 | 0.98 | 0.98 | 0.98 | 0.98 | 0.84 | 0.86 | 0.87 | 0.89 |

Model 5: Weighted data, Gibbs, \( \beta {\overline{w}}_{P6} \) | 81 | 65 | 51 | 39 | 6.3 | 6.5 | 6. 9 | 7.3 | 1.00 | 1.00 | 0.97 | 0.93 | 0.98 | 0.97 | 0.96 | 0.96 | 0.84 | 0.77 | 0.70 | 0.65 |

Model 6: Weighted data, Gibbs, \( {\tau}^2{\overline{w}}_{P6} \) | 162 | 157 | 151 | 142 | 1.6 | 2.6 | 3.6 | 4.5 | 0.74 | 0.76 | 0.77 | 0.79 | 0.98 | 0.98 | 0.98 | 0.98 | 0.99 | 0.99 | 0.97 | 0.96 |

Model 7: Weighted data, MH | 81 | 87 | 93 | 102 | 2.2 | 2.7 | 3.1 | 3.5 | 1.00 | 0.98 | 0.97 | 0.92 | 0.98 | 0.98 | 0.98 | 0.98 | 0.84 | 0.86 | 0.87 | 0.89 |

### Additional simulation results

Simulations with varying sample size, number of genes, and different levels of sample quality were conducted and some results were presented in the supplemental material. It is noteworthy that the BRE models identified less genes for sample sizes < 60. The DE gene detection and the MSSE were stable for sample sizes > 60. Specifically, the BRE with a U(0,1) had consistently high precisions and was able to maintain overall accuracies for all sample sizes > 60 (Additional file 1: Table S3). As anticipated, these findings were similar to the findings in the classical RE models [6]. When the number of genes in the analyses increased, the classical RE models performed stably, while the overall accuracy in the BRE model with a uniform(0,1) prior was reduced (Additional file 1: Table S4). For different levels of sample quality, the weights with higher sample quality detected more DE genes and had higher overall accuracy than the weights with lower sample quality (Additional file 1: Table S5).

### Application in Alzheimer’s gene expression data

*R*

^{2}, while the metadata C had a relatively low

*R*

^{2}or contained homogenous data. Figure 3 presents distribution of unbiased standardized mean differences of gene expression in the GSE5281 dataset, different from the other datasets. Figure 4 presents the percent of present calls and the 3′:5’ GAPDH ratio of the heterogeneous dataset.

*w*

_{P6}weighted model, the number of DE genes decreased in all metadata sets. Almost all the DE genes identified by the weighted model were genes among the significant DE genes identified by the unweighted DSL and DSLR2 models. The DE genes identified using the weighted model in the metadata C concurrently detected approximately 13% of the unweighted DSL and DSLR2 models (266/2116 genes and 213/1696 genes), respectively (Fig. 5). Likewise, the number of DE genes decreases with the

*w*

_{P6}weighted between study variance (Models 3 and 6). Those DE genes were genes among the significant DE genes identified by the unweighted model (Model 1). Sixty and 446 DE genes were detected across the three weighted BRE models in the metadata C and D, respectively (Fig. 6). Among the unweighted or weighted classical RE and BRE models, 446 genes could potentially be down-regulated genes that may contribute to good classification of Alzheimer’s samples. Additional file 1: Figure S2 presents potential down-regulations of those genes in Alzheimer’s samples in each microarray dataset. Of note, no genes were detected using the weighted common-effect models (Models 2 and 5) and the weighted-data model (Models 4 and 7).

The lists of 213 and 446 DE genes can be found in Additional file 1: Tables S6 and S7, respectively, where 98 were the same genes. The identified DE genes participate in significant pathways such as cytoskeleton organization, actin filament bundle organization, synaptic transmission, regulation of biological quality, neutral lipid biosynthetic process, acylglycerol biosynthetic process, intermediate filament-based process, negative regulation of neuron projection development, cell-cell signaling, glutamate decarboxylation to succinate, stress fiber assembly, single-organism behavior, single-organism behavior, response to ethanol, cellular component assembly, neuron projection development, learning and long-term memory.

## Discussion

This study presents the performance of the classical RE and BRE models in meta-analysis of gene expression studies. We found 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 BRE model with a uniform(0,1) prior performed better than the DSLR2 model in the homogeneous data, but performed similarly in the heterogeneous data in terms of an appropriate number of detected DE genes, lower MSSE, higher precision, and higher AUC.

This is the first study to reveal an application of sample-quality weights to adjust the study heterogeneity in the classical RE and BRE models in microarray gene expression studies. The DSL and DSLR2 weighted models were implemented for the classical RE models. The unweighted and weighted data, Gibbs and MH sampling algorithms, weighted common effect, and weighted between-study variance were applied for the BRE models. We evaluated the performance of the models through simulation studies and through application to Alzheimer’s gene expression datasets.

With simulation results, the sample quality indicators adjusting the within study variance (*w*_{P6}) in the classical RE models provided an appropriate reduction of detected DE genes and MSSEs, and higher precision as compared to the other weighted functions. The precision in detecting DE genes was increased with the DSLR2 *w*_{P6} weighted model in the heterogeneous data. The DSLR2 *w*_{P6} weighted model had a lower MSSE and detected more DE genes than the DSL *w*_{P6} weighted model in the heterogeneous data. Among the BRE weighted models, the *w*_{P6}weighted- and unweighted-data models and both Gibbs- and MH-based models performed similarly. The *w*_{P6} weighted common-effect model performed similarly to the unweighted model in the homogeneous data, but performed worse in the heterogeneous data. The *w*_{P6}weighted-data were appropriate for detecting DE genes with high precision, while the *w*_{P6}weighted between-study variance models were appropriate for detecting DE genes with high overall accuracy.

The sample quality has substantial influence on results of gene expression studies [15]. Because variation of sample quality limited meta-analysis techniques to properly detect DE genes [45, 46] and the classical RE and BRE models allow flexibility in calculating *y*_{ig}and its variance \( {\sigma}_{ig}^2 \) as well as study-specific adjustments [47], we developed approaches to up-weight good quality samples and down-weight borderline quality samples in the models. This compromised approach utilizes sample-quality information in the meta-analysis of microarray studies in detecting DE genes. The results in this study would benefit microarray gene expression studies because a large amount of microarray data are available in public repositories and unfortunately the data quality are often overlooked. However, the performance of the proposed models depends on not only degree of sample quality but also the number of studies, the number of genes, and sample sizes in the individual studies. The methods for controlling FDR under multiple testing would be another important aspect influencing gene expression results. Further intensive investigation of the topics would be the subject of future research.

The BRE models have the ability to allow for uncertainty of the parameter estimates in the model. Because the classical RE models tended to estimate \( {\tau}_g^2 \) as being zero, the variance of \( {\widehat{\beta}}_g \)were underestimated. The BRE models, in contrast, used the marginal posterior distribution of \( {\tau}_g^2 \) for \( {\widehat{\beta}}_g \)estimation, which do not depend on the point estimate of \( {\tau}_g^2 \). The BRE models can in turn increase the fitness of the models [48]. To illustrate, the precision was increased in the BRE*w*_{P6}weighted-data models and the accuracy was increased in the BRE*w*_{P6}weighted between-study variance models as compared to the classical RE weighted models. The BRE weighted models could be strengthened further in future research with informative priors using prior knowledge and historical information.

In real-world applications, BRE modeling in gene expression meta-analysis may be computationally intensive. To illustrate, a Gibbs-based model requires approximately 6 h per 10,000 gene set under supercomputers. A MH-based model requires twice longer than a Gibbs-based model. The computational time for a BRE model is highly dependent on not only types of the model, but also computer capacity. Computation time can indeed be another concern for model selection.

## Conclusions

This study applies sample-quality weights to adjust the study heterogeneity in the random-effects meta-analysis models. This meta-analytic approach can increase precision and accuracy of the classical and Bayesian random-effects models in gene expression meta-analysis. However, the performance of the weighted models varied depending on data feature, levels of sample quality, and adjustment of parameter estimates.

## Notes

### Acknowledgements

We would like to thank anonymous reviewers for their suggestions and insightful comments.

### Funding

Research reported in this work was supported in part by the National Library Of Medicine of the National Institutes of Health under Award Number R01LM011169. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

### Availability of data and materials

Additional file 1 Supporting online material. PDF document with WinBUGS code, supplementary tables (Additional file 1: Table S1 – S7) and figure (Additional file 1: Figure S1 and S2). The program for implementing the weighted models can be modified from existing R packages and are available upon request.

### Authors’ contributions

US conceived the study, conducted the simulations, interpreted the results, wrote and edited the manuscript. KA and NM advised the study, interpreted the results, reviewed and edited the manuscript. All authors read and approved the final manuscript.

### Ethics approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary material

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