In biological research, high-throughput assays, such as microarrays, are widely used to effectively select potential targets by studying a large number of candidates in a single experiment. However a high-throughput assay is often subject to substantial variability. Reproducibility of high-throughput assays, such as the level of agreement across replicate samples, test sites or data analytical platforms, is a concerned topic in scientific applications, and has been discussed in [1] for microarray and [2] for ChIP-seq technology. Therefore quantitative analysis for the reproducibility of high-throughput assays is an important exercise for evaluating the reliability and robustness of scientific discoveries across studies.

Reproducibility is nonstandard and unsettled across the sciences. Goodman et al. [3] provides a survey on the papers with the word reproducibility included in titles, abstracts and keywords, and concludes that the interpretation of reproducibility varies among different papers. Goodman et al. [3] further allies the word reproducibility in the papers and classifies them into three terms: methods reproducibility, results reproducibility and inferential reproducibility. In [3], methods reproducibility refers to the provision of enough detail about study procedures and data so the same procedures could, in theory or in actuality, be exactly repeated, such as [1] and [2]; results reproducibility refers to obtaining the same results from the conduct of an independent study whose procedures are as closely matched to the original experiment as possible, such as [4] and [5]; Inferential reproducibility refers to the drawing of qualitatively similar conclusions from either an independent replication of a study or a reanalysis of the original study, such as [1] and [2].

In this paper, our reproducibility analysis aims to identify reproducible targets with consistent and significant signals across replicate studies, which belongs to the category of inferential reproducibility as defined in [3]. Our reproducibility analysis is different from meta-analysis, such as [6] and [7]. Meta-analysis combines the data from multiple studies to gain extra power for identifying targets with signals. The identified targets may not necessarily be significant across all studies.

A few methods have been developed for our reproducibility analysis. Hong et al. [8] proposed a permutation based method through estimating the empirical distribution of the rank product. Benjamini & Heller [9] developed a framework for testing partial conjunction hypothesis that the discovery is true in at least u studies out of total n studies. Most recently, [10] proposed a copula mixture model for estimating the irreproducible discovery rate across studies.

However all existing methods potentially have limitations. The permutation based method [8] can be computationally expensive when dealing with a large number of candidates. Benjamini & Heller method [9] aims at identifying candidates with reproduced signals in a few but not all the studies, which is a related but generally weaker goal than ours. The special case of Benjamini & Heller method testing whether signals are reproduced in all studies is identical to using the largest p-value. The copula mixture [10] method builds the copula mixture using the rank transformation of the original data, which might be less powerful than modeling the original data with a proper probabilistic model as in our proposed method. A major drawback of both Benjamini & Heller method [9] and the copula mixture [10] method is that they both use the significant score of signals, such as p-value, without taking into account the directionality of signals, hence is prune to selecting candidates with significant scores but different directions across studies. For example, in the context of two replicate microarray studies with a treatment and a control group, consider genes with significant p-values in both experiments, but are up-regulated in one study and down-regulated in the other. Although those genes have inconsistent signals across studies, both methods will likely classify them as reproducible based on p-values alone. In contrast, our proposed method models the test statistics directly and is expected to correctly classify those genes as irreproducible most of the time.

In this paper, we propose a Bayesian hierarchical model and show the test statistics from replicate studies can be approximated by a mixture of multivariate Gaussian distributions. The proposed Gaussian mixture model classifies the signals into three components: one irreproducible component and two reproducible components for consistent up-regulated and down-regulated signals respectively. The posterior probability of belonging to the reproducible components is used as a measure for reproducibility.

where μ _gi is the expected group mean difference for gene g in the i-th study, and \(\delta _{S_{i}}=\tilde {\sigma }_{i}^{-1}(1/n_{i1}+1/n_{i2})^{-1/2}\) with \(\tilde {\sigma }_{i}\) being the common standard deviation for {x _{g i j1}}, j=1,2,…,n _i1 and {x _{g i j2}}, j=1,2,…,n _i2. When the sample size is small, the same procedure as in [11] can be applied to construct z-tests based on two-sample t-tests. For simplicity we assume the within-group between-sample standard deviation is the same for all the genes. The general case can be derived in a similar fashion but a bit more tedious.

where μ _g is the “true" group mean difference for gene g across all studies and \(\sigma _{g}^{2}\) models the between-study variability due to various experiment conditions.

where π _i ≥0, i=0,1,2, with π ₀+π ₁+π ₂=1, \(\mu _{G_{1}}>0\) and \(\mu _{G_{2}}<0\). The distribution has three components: the null case where there is no differentially expressed gene, the “up-regulated” case where the treatment stimulates the gene expression, and the “down-regulated” case where the treatment suppresses the gene expression. Generally for microarray studies π ₀≃1. Similar mixture models have been considered in [11–16]. Particularly we choose to model the cluster of up-regulated (or down-regulated) genes with a Gaussian distribution for the computational convenience, same as in [12]. Alternative choices include the semiparametric mixture model in [11, 14], mixture of Gaussian distributions in [13, 15] and mixture of t-distributions in [16].

where \(\mathcal {N}(\mathcal {\mu }_{l},\Sigma _{l})\) (l=0,1,2) is the biviariate normal distribution with mean vector \(\mathcal {\mu }_{l}\) and covariance matrix Σ _l . Let I ₂ and J ₂ be the identity matrix and the square matrix of ones respectively, both with order 2. This mixture model classify the candidates into three components: \(\mathcal {N}(\mathcal {\mu }_{0},\Sigma _{0})\) is the irreproducible component with zero-mean \(\mathcal {\mu }_{0}=(0,0)^{T}\) and covariance structure \(\Sigma _{0}=\left (\sigma _{g}^{2}+1\right)I_{2}\); \(\mathcal {N}(\mathcal {\mu }_{1},\Sigma _{1})\) and \(\mathcal {N}(\mathcal {\mu }_{2},\Sigma _{2})\) are two reproducible components with \(\mathcal {\mu }_{1}=(\delta _{S_{1}}\mu _{G_{1}}, \delta _{S_{2}}\mu _{G_{1}})>0\) and \(\Sigma _{1}=\left (\sigma _{g}^{2}+1\right)I_{2} + \sigma _{G_{1}}^{2}J_{2}\) representing the up-regulated genes, and \(\mathcal {\mu }_{2}=(\delta _{S_{1}}\mu _{G_{2}}, \delta _{S_{2}}\mu _{G_{2}})<0\) and \(\Sigma _{2}=\left (\sigma _{g}^{2}+1\right)I_{2} + \sigma _{G_{2}}^{2}J_{2}\) representing the down-regulated genes, where the inequalities are meant to be interpreted component-wise.

Note with increased sample sizes or decreased within-group between-sample variability, the mean \(\mathcal {\mu }_{1}\) and \(\mathcal {\mu }_{2}\) of the reproducible components move further away from the origin, making the three components more separable. Also note the test statistics from replicate studies have zero correlations in the irreproducible components; in the reproducible components, the correlations become larger when the between-study variability becomes smaller; for all components, the variance is smaller with less between-study variability, resulting in more separable components.

where ϕ(·|·) is the density function of bivariate normal distribution. According to [10], the posterior probability of being in the irreproducible/null component p _i0 can be introduced as the individual significant score, namely local false discovery rate. When p _g0 is less than a significant level α, gene g is classified as reproducible.

In our algorithm, we start with some initials value for the parameters \(\mathcal {\theta }^{0}\), then iterate between two steps: (1) Evaluate the current posterior probabilities p _gl using the current parameters; (2) Maximize the likelihood estimator given current posterior probabilities. The details of the EM procedures are provided in Appendix. Multiple random initial vaues are used to avoid being trapped at the local maximum.

From this model, the mean expression level of gene g for group 1 of study s is modeled as μ _{g s1}=μ+α _g +β _i +(α β) _gi , where μ is the overall mean; α _g is the main effect of gene g; β _i is the main effect of study i; (α β) _gi is the gene-study interaction. We set μ=0, \(\alpha _{g} \sim \mathcal {N}(0,1)\), β _i =0.1, and \((\alpha \beta)_{gi} \sim \mathcal {N}(0,0.5^{2})\). For non-differentially expressed genes, the mean expression level for both groups are the same, i.e., μ _{g s1}=μ _{g s2}. For differentially expressed genes, (8) models the difference between the two comparison groups as μ _{g i2}−μ _{g i1}=δ+γ _g +(γ β) _gi , where δ is the fixed effect of group difference; γ _g is the effect of gene on the group difference; (γ β) _gi is the gene-study interaction of the group difference. We set δ=0, generate γ _g from \(\mathcal {N}(2, 0.5^{2})\) or \(\mathcal {N}(-2, 0.5^{2})\) to mimic two possible directions of signals, \((\gamma \beta)_{gi}\sim \mathcal {N}(0,0.5^{2})\). ε _gijk is the random error term, and following the distribution \(\mathcal {N}(0,0.5^{2})\).

This paper proposes a new method for identifying consistent and significant signals across replicate high-throughput experiments. Existing methods ignore the directionality of signals, and can incorrectly identify signals with opposite directions as reproducible ones. Our proposed method considers both the significant scores and directions of signals by modeling the test statistics directly, leading to improved performance in selecting reproducible candidates. When the proposed method is applied to a real data example for identifying reproducible genes in studies of idiopathic pulmonary fibrosis samples, it is shown to have better performance in detecting significant and reproducible genes compared to other methods. Simulations also demonstrate that our method compares favorably to the existing methods.