A Bayesian mixture model for metaanalysis of microarray studies

Abstract

The increased availability of microarray data has been calling for statistical methods to integrate findings across studies. A common goal of microarray analysis is to determine differentially expressed genes between two conditions, such as treatment vs control. A recent Bayesian metaanalysis model used a prior distribution for the mean log-expression ratios that was a mixture of two normal distributions. This model centered the prior distribution of differential expression at zero, and separated genes into two groups only: expressed and nonexpressed. Here, we introduce a Bayesian three-component truncated normal mixture prior model that more flexibly assigns prior distributions to the differentially expressed genes and produces three groups of genes: up and downregulated, and nonexpressed. We found in simulations of two and five studies that the three-component model outperformed the two-component model using three comparison measures. When analyzing biological data of Bacillus subtilis, we found that the three-component model discovered more genes and omitted fewer genes for the same levels of posterior probability of differential expression than the two-component model, and discovered more genes for fixed thresholds of Bayesian false discovery. We assumed that the data sets were produced from the same microarray platform and were prescaled.

Appendices

Appendix A

Two-study simulation data

We produced synthetic data for two studies, with format similar to the B. subtilis mutant and induction biological data (see Appendix A: Biological data), and 3,000 genes for each study. We simulated three separate data sets, with simulated percent of differentially expressed genes p _s  = 5, 10, 25%, with equivalent overexpressed and underexpressed genes. Study 1 contained five replicate slides within three replicate experiments, and Study 2 contained four replicate slides within three replicate experiments. We simulated data from Model (1), with model parameters specified to resemble the biological data. We set $\eta ^{2}_{{jg0}} = 0.015$ and c _j  = 80, j = 1, 2; the interstudy variability was assumed to be normally distributed with mean 0.3, variance 0.1 for the differentially expressed genes, and mean 0.03, variance 0.004 for the nonexpressed genes, similar to the biological data. For Study 1, we assigned variance across slides to 0.074 and across experiments to 0.026; for Study 2, we assigned slide variation to 0.023 and experiment variation to 0.022. Each slide was standardized to have mean zero and unit standard deviation (see also Shen et al. 2004; Conlon et al. 2007). Although gene expression is expected to be correlated among genes, this has been shown to be difficult to model in simulations. We thus simulate gene expression independently for each gene, similar to other authors (e.g. Lönnstedt and Speed 2002; Gottardo et al. 2003; Bhowmick et al. 2006).

Simulation data for five studies

For the five study simulations, we used Study 1 and Study 2 data from the previous section, and simulated three additional studies, each with 3,000 genes. We again simulated three separate data sets, with percent differentially expressed genes p _s  = 5, 10, 25%, and equivalent overexpressed and underexpressed genes. For Studies 3, 4, and 5, we used the data format similar to Study 1, with five replicate slides within three repeated experiments. We again simulated data from Model (1), with model parameters specified to be similar to the biological data. We set $\eta ^{2}_{{jg0}} = 0.015$ and c _j  = 80, j = 1,...,5; the interstudy variability was assigned a normal distribution with mean 0.3, variance 0.1 for the differentially expressed genes, and mean 0.03, variance 0.004 for the nonexpressed genes, similar to the biological data. For the slide and experiment variance parameters for Studies 3, 4, and 5, we assigned values that were either within the range of values for Studies 1 and 2, or somewhat outside the range. For Study 3, we assigned slide variance of 0.05 and experiment variance 0.02; for Study 4, slide variance 0.04 and experiment variance 0.022; for Study 5, slide variance 0.06 and experiment variance 0.03. Each slide was standardized to have zero mean and unit standard deviation.

Biological data

B. subtilis is a bacterium that has the ability to form spores, which allow the organism to endure extreme environmental conditions. Two reciprocal B. subtilis microarray studies were implemented to identify sporulation genes controlled by a key transcription factor σ^E (sigma E). The studies had complementary designs; the first was a deletion of σ^E, and the second was an induction of σ^E, described in the following (see also Eichenberger et al. 2003, Conlon et al. 2004).

Mutant study

In the mutant study, the sample with a knockout of the gene for σ^E (the mutant sample) was compared to the wild-type sample, using cDNA microarrays. The wild-type/mutant gene expression ratios identified genes in the σ^E regulon. In total, five microarray slides were created from three separate identical experiments; the first experiment had three replicate slides and the second and third experiments each created one slide. The number of genes spotted on each array was somewhat larger than the genome size of 4,106 of B. subtilis due to replicate spotting of particular genes on the slides. The missing data rate due to low quality spots ranged from 18.6 to 64.5% for the five arrays; in total, 3,713 genes contained measurable expression levels for at least one array. We used log-ratios of expression values (Dudoit et al. 2002), normalized slides by a rank-invariant method (Schadt et al. 2001; Tseng et al. 2001), and standardized each array to have mean zero and unit standard deviation (Shen et al. 2004; Conlon et al. 2007).

Induction study

For the induction study, σ^E was overexpressed in response to a specific inducer; thus inducer-treated cells were compared to wild-type cells. Induction/wild-type ratios identified genes under the control of σ^E. In total, four microarray slides were produced from three separate identical experiments. The first two experiments each created one array and the third experiment created two replicate arrays. The percent of missing data due to low quality ranged from 52.6 to 67.0% for the four arrays; 2,552 genes contained measurable expression for at least one array. We again analyzed the post-normalized log-expression ratios, and standardized each slide to have mean zero and unit standard deviation.

Appendix B

Markov chain Monte Carlo implementation

We use a Markov chain Monte Carlo (MCMC) algorithm to simulate the posterior distributions of the parameters. MCMC procedures produce samples from the density p(ω) for the parameter ω [note that p(ω) may be known only to a proportionality constant] by generating a Markov chain on the state space of ω that has p as its stationary distribution (further details in Liu 2001).

Joint posterior distributions

For Model (1), the joint posterior distribution of the data and parameters is:

$$\begin{array}{*{20}c} {{p{\left( {y_{{jgse}} ,\xi _{{jge}} ,\phi ^{2}_{{jg}} ,\sigma ^{2}_{{jg}} ,\theta _{{jg}} ,I_{g} ,p,\eta ^{2}_{{jg0}} ,c_{j} } \right)} = }} \\ {{\left[ {{\prod\limits_{j = 1}^J {{\prod\limits_{g = 1}^G {{\prod\limits_{e = 1}^E {{\left\{ {{\prod\limits_{s = 1}^{S_{e} } {p{\left( {y_{{jgse}} \left| {\xi _{{jge}} ,\phi ^{2}_{{jg}} } \right.} \right)}p{\left( {\xi _{{jge}} \left| {\theta _{{jg,}} \sigma ^{2}_{{jg}} } \right.} \right)}} }} \right\}} \times } }} }} }} \right.}} \\ {{\left. {p{\left( {\theta _{{jg}} ,I_{g} \left| {p,\Omega _{j} } \right.} \right)}p{\left( {\phi ^{2}_{{jg}} } \right)}p{\left( {\sigma ^{2}_{{jg}} } \right)}p{\left( p \right)}p{\left( {\Omega _{j} } \right)}} \right]}} \\ \end{array} $$

where $\Omega _{j} = {\left\{ {\eta ^{2}_{{jg0}} ,c_{j} } \right\}}$, j = study, g = gene, e = experiment, s = slide.

For Model (2), the joint posterior distribution of the data and parameters is:

$$\begin{array}{*{20}c} {{p{\left( {y_{{jgse}} ,\xi _{{jge}} ,\phi ^{2}_{{jg}} ,\sigma ^{2}_{{jg}} ,\theta _{{jg}} ,I_{g} ,p,\Omega _{j} } \right)} = }} \\ {{\left[ {{\prod\limits_{j = 1}^J {{\prod\limits_{g = 1}^G {{\prod\limits_{e = 1}^E {{\left\{ {{\prod\limits_{s = 1}^{S_{e} } {p{\left( {y_{{jgse}} \left| {\xi _{{jge}} ,\phi ^{2}_{{jg}} } \right.} \right)}p{\left( {\xi _{{jge}} \left| {\theta _{{jg,}} \sigma ^{2}_{{jg}} } \right.} \right)}} }} \right\}} \times } }} }} }} \right.}} \\ {{\left. {p{\left( {\theta _{{jg}} ,I_{g} \left| {p,\Omega _{{jg}} } \right.} \right)}p{\left( {\phi ^{2}_{{jg}} } \right)}p{\left( {\sigma ^{2}_{{jg}} } \right)}p{\left( p \right)}p{\left( {\Omega _{{jg}} } \right)}} \right]}} \\ \end{array} $$

where p = {p ₀,p ₁,p ₂}, $\Omega _{{{\mathbf{jg}}}} = {\left\{ {\mu _{{jg1}} ,\delta ^{2}_{{jg1}} ,\tau ^{2}_{{jg1}} ,\mu _{{jg2}} ,\delta ^{2}_{{jg2}} ,\tau ^{2}_{{jg2}} } \right\}}$, j = study, g = gene, e = experiment, s = slide.

Prior distributions for parameters common to Models (1) and (2)

We assigned as uninformative prior distributions as possible to the parameters of Models (1) and (2) that still resulted in convergence of the models. The prior distributions of the slide effect and experiment effect variance parameters, $\phi ^{2}_{{jg}}$ and $\sigma ^{2}_{{jg}}$, respectively, required some information from the data. We assigned the following inverse chi-squared prior distributions for these parameters (see also Tseng et al. 2001; Lönnstedt and Speed 2002; Gottardo et al. 2003; Conlon et al. 2006, 2007):

$$\begin{array}{*{20}c} {\phi ^{2}_{{jg}} \sim {\text{ }}{k\widetilde{\phi }^{2}_{j} } \mathord{\left/ {\vphantom {{k\widetilde{\phi }^{2}_{j} } {\chi ^{2}_{k} }}} \right. \kern-\nulldelimiterspace} {\chi ^{2}_{k} }} \\ {\sigma ^{2}_{{jg}} \sim {\text{ }}{h\widetilde{\sigma }^{2}_{j} } \mathord{\left/ {\vphantom {{h\widetilde{\sigma }^{2}_{j} } {\chi ^{2}_{h} }}} \right. \kern-\nulldelimiterspace} {\chi ^{2}_{h} }} \\ \end{array} $$

Here, $\widetilde{\phi }^{2}_{j}$ and $\widetilde{\sigma }^{2}_{j}$ are the scale parameters for the inverse chi-squared distributions determined from the data. $\widetilde{\phi }^{2}_{j}$ is derived as follows:

$$ \widetilde{\phi }^{2}_{j} {\text{ }} = {\text{ }}\frac{1} {{G{\left( {{\sum {S_{e} } } - 1} \right)}}}{\sum\limits_{g = 1}^G {{\sum\limits_{e = 1}^E {{\sum\limits_{s = 1}^{S_{e} } {{\left( {y_{{jgse}} - y_{{jg \cdot e}} } \right)}^{2} } }} },} } $$

where y _jg.e is the mean log-ratio of expression over the slides within experiment:

$$ y_{{jg \cdot e}} = \frac{1} {{S_{e} }}{\sum\limits_{s = 1}^{S_{e} } {y_{{jgse}} } }. $$

The scale parameter for $\sigma ^{2}_{{jg}}$ is obtained similarly, as follows:

$$ \widetilde{\sigma }^{2}_{j} = \frac{1} {{G{\left( {E - 1} \right)}}}{\sum\limits_{g = 1}^G {{\sum\limits_{e = 1}^E {{\left( {y_{{jg.e}} - y_{{jg..}} } \right)}^{2} } }} }, $$

where y _jg.. is the mean log-ratio of expression over both slides and experiments. We specified three degrees of freedom for each study for both $\phi ^{2}_{{jg}}$ and $\sigma ^{2}_{{jg}}$, i.e. h = k = 3, which corresponds to a distribution that is as uninformative as possible (for details, see Conlon et al. 2007).

Prior distributions for parameters specific to Model (1) only

For the parameter p, we assigned a noninformative Uniform(0,1) distribution. The remaining parameters were assigned the following prior distributions, which were as uninformative as possible while still allowing the models to converge.

$$\begin{array}{*{20}c} {\eta ^{2}_{{jg0}} {\text{ }} \sim {\text{ }}{as^{2}_{1} } \mathord{\left/ {\vphantom {{as^{2}_{1} } {\chi ^{2}_{a} }}} \right. \kern-\nulldelimiterspace} {\chi ^{2}_{a} }} \\ {c_{j} {\text{ }} \sim {\text{ }}{bs^{2}_{2} } \mathord{\left/ {\vphantom {{bs^{2}_{2} } {\chi ^{2}_{b} }}} \right. \kern-\nulldelimiterspace} {\chi ^{2}_{b} }} \\ \end{array} $$

The degrees of freedom and scale parameters a, $s^{2}_{1}$, respectively, were assigned so that the prior mean of $\eta ^{2}_{{jg0}}$ was 1 with variance 0.1, and b, $s^{2}_{2}$ were assigned so that the prior mean of c _j was 100 with variance 10,000 (further details in Conlon et al. 2007).

Prior distributions for parameters specific to Model (2) only

We assigned the following prior distributions to the parameters for Model (2).

$$\begin{array}{*{20}c} {\mu _{{jg1}} \sim {\text{ }}N{\left( {m_{1} ,{\text{ }}\delta ^{2}_{{jg1}} } \right)}{\left[ {{\text{truncated}}{\left( {{\text{0,}}\infty } \right)}} \right]}} \\ {\mu _{{jg2}} \sim {\text{ }}N{\left( {m_{2} ,{\text{ }}\delta ^{2}_{{jg2}} } \right)}{\left[ {{\text{truncated}}{\left( { - \infty {\text{,0}}} \right)}} \right]}} \\ {\delta ^{2}_{{jgi}} \sim {\text{ }}IG{\left( {u_{i} ,v_{i} } \right)},{\text{ }}i = 1,2} \\ {\tau ^{2}_{{jgi}} \sim {\text{ }}IG{\left( {u_{i} ,v_{i} } \right)},{\text{ }}i = 1,2} \\ \end{array} $$

Here, the hyperparameters m ₁ and m ₂ were set equal to the trimmed means of the highest and lowest p percent of data, respectively, based on the average of the individual study analyses (see also Shen et al. 2004; Conlon et al. 2007). The hyperparameters u ₁ and u ₂ were set equal to 3, and v ₁ and v ₂ were set equal to 2 × (trimmed variance) for the highest and lowest p percent of data, respectively, based on the average of the individual study analyses (see also Shen et al. 2004; Conlon et al. 2007). Based on these specifications, the prior means of $\delta ^{2}_{{jgi}}$ and $\tau ^{2}_{{jgi}}$, i = 1, 2 were equal to the variance of the top genes, with small variance, similar to Jung et al. (2006). Note that we used some information from the data but assigned as uninformative priors as possible. For p, we assigned an uninformative Dirichlet(α ₀,α ₁,α ₂) prior with α ₀ = α ₁ = α ₂ = 1.

Full conditional posterior distributions

Each parameter was sampled from the full conditional posterior distributions by an MCMC algorithm using the WinBUGS software (Spiegelhalter et al. 2003; Lunn 2003). For the two-study simulation and biological data sets, convergence was achieved after approximately 1,000 iterations for both models. For the five study simulation data sets, Model (1) converged after approximately 5,000 iterations; however, Model (2) required approximately 30,000 iterations until convergence. We used 5,000 iterations after convergence for all posterior inference, which was more than sufficient (similar to Tseng et al. 2001; Conlon et al. 2004, 2006, 2007). Further MCMC algorithm details can be found in Conlon et al. (2006, 2007).