A probabilistic undirected network model is defined as an undirected graph with an associated probability measure 34. An undirected graph G is specified as a pair G = V , E ,with V a set of vertices (i.e. gene expression products, single nucleotide polymorphisms, or downstream phenotypes), and ℰ a set of unordered pairs of vertices, specifying undirected edges between vertices 34. We focus on a type of Gaussian graphical model (GGM) defined with respect to a conditional multivariate normal distribution, where we model the distribution of expression traits, conditional on the genotypic states (see Methods for parametric details of a conditional GGM). This particular type of model only captures linear interactions between variables (we do not consider nonlinear or epistatic interactions in this paper). The goal of any network discovery algorithm is to determine the relevant set of edges for each vertex in the graph (i.e. the neighborhood of other vertices connected to each vertex), based on the observed genetic polymorphism, expression, and downstream phenotype data. Most practical genomics applications have far more variables (i.e. vertices) than samples, and therefore require a carefully constrained solution to the neighborhood selection problem, to prevent over-fitting and high false discovery rates 312. We therefore focus our attention on the regularized solutions to this problem 12131422, where the identified neighborhood size can be restricted based on the choice of penalization, and where the neighborhood selection problem is solved through penalized regression of each phenotype on all other phenotypes and genotypes 121322.

In our algorithm, we treat the neighborhood selection problem for any given phenotype y as a regularized regression problem. More specifically, this neighborhood selection problem involves identifying a subset of expression products or genotypes with non-zero regression coefficients in a multiple regression equation y_i = μ + ∑_j x_ij β_j + e_i, where a penalty is defined over the regression coefficients (β₁,...,β_{p + m}) for the i^th sample with j = 1,...,p + m possible expression phenotypes and genotypes. We use a mixture spike-and-slab prior as our penalty, β j ~ p β = 0 I β = 0 + p β ≠ 0 N 0 , σ β 2 , with the spike (p_{β = 0}I[β = 0]) being related to an l₀ type penalty which drives the sparse feature selection and the slab p β ≠ 0 N 0 , σ β 2 being related to a ridge or l 2 2 type penalty which effectively smooths the identified sparse model. The combination of the two different types of penalization is similar in principle to the elastic net penalty 35, which incorporates a combination of l₁ and l22 penalties. There is both theoretical 27 and empirical evidence 29 that the spike-and-slab prior is more effective than other penalties such as the lasso 1213 at generating sparse solutions with low false discovery rates, for ultra-high-dimensional problems when p ≫ n, where p is the number of variables, and n is the sample size.

We employ a fully Bayesian framework to handle the spike-and-slab prior, where the mixture proportions are estimated directly from the data. The algorithm finds an appropriate level of sparsity supported by the data via the probabilistic bound on model size without relying on cross-validation type approaches or information theoretic model selection criterion. Specifically, this is done by constraining the model dimension such that the number of selected features (s) for any given problem are on the order s n = O n . This is done by truncating the distribution of p_β≠0 such that p β ≠ 0 ≤ n / m + p - 1 for p gene expression or downstream phenotypes and m genotypes. Given mild regularity conditions it has been shown in the context of linear regression that consistency can be established for both s n = o n 36 and under further mild assumptions s(n) = o(n/log(n)) 37. Note that the latter bound is a much weaker constraint on the model size as a function of the number of observations than the former bound. In addition, Zhang et al. 25 show that the s n = On choice of model size will asymptotically lead to minimum prediction error at a rate O n - 1 / 2 . This justifies our choice of the strength of the penalization to be sufficiently conservative in terms of ensuring few irrelevant features enter the model when p ≫ n, especially for data with at least hundreds of observations, because of the optimal rate. This bound is also consistent with the results from the simulations within this paper, as well as results from previous applications of this bound 242529. The Bayesian framework also allows the algorithm to take advantage of the multiple modality of the posterior with Bayesian model averaging, a particularly valuable approach when any well-supported sparse solution is expected to capture only a portion of the true network connections.

Another feature of our algorithm is that we also simultaneously correct for systematic error, or other large scale confounding factors among samples, based upon the expression data, by including the top twenty eigenvectors from a principal component analysis as unpenalized fixed effects in our model selection procedure. Therefore, the previous multiple regression equation becomes y_i = μ + ∑_j x_ij β_j + ∑_k t_ik α_k + e_i, with t₁,...,t₂₀ being the top 20 across sample eigenvectors obtained from a standard principal component analysis of the joint gene expression data. The motivation for this is analogous to the use of eigenvectors from principal component analysis to correct for confounding population structure in genetic association analyses 2130, which aims to remove any potentially confounding effects from the inference of the neighborhood of any given phenotype.

As in Logsdon et al. 29, we use a variational Bayes approximate inference approach to solve the high-dimensional feature selection problem. For the feature selection problem, the variational Bayes approximation is a good tradeoff between speed, since it is much faster than alternative exact inference approaches, and quality of the identified solutions, where empirical evidence shows that it performs well for underlying sparse models 29. The variational Bayes approximation consists of minimizing the Kullback-Leibler divergence between an approximate factorized posterior distribution q β 1 β 1 ⋯ q β m β m q p β ≠ 0 p β ≠ 0 q σ e σ e q σ β σ β and the full posterior distribution p(β₁,...,β_m, p_{β ≠ 0}, σ_e, σ_β), using iterative expectation-type steps as in an Expectation-Maximization algorithm 3839. The relevant statistic for the j^th expression product or genotype produced by the algorithm for the problem of feature selection is the posterior probability of inclusion in the model denoted a p ^ j from thereon. This p^j parameter comes from the approximate posterior inference of the mixture parameters in the spike-and-slab prior. A detailed description of this statistic in terms of the other model parameters is given in the Additional file 1, Equations 2, 4, and 17. In our approach we perform a two-step reconstruction of the joint genotype, expression, and downstream phenotype network, where we first perform neighborhood selection for each downstream phenotype individually on all expression traits and genotypes. Then, in the second step, we perform neighborhood selection for each expression trait on all other expression traits and genotypes. The procedure is split into two steps because of the primary interest in the neighborhoods of the downstream phenotypes, followed by interest in the expression Quantitative Trait Loci (eQTL) networks associated with the neighborhoods of the downstream phenotypes. To resolve discrepancies in neighborhoods identified in two directions of regression, we average the p^j scores across both directions of regression at a cutoff of p ^ j > 0 . 99 . This approach is supported by the significant improvement in results obtained from simulations (see Figures 1, 2). We then combine the neighborhoods of the first and second step of the algorithm through a simple union operation.

We performed a simulation study to compare our approach to other comparable methods that use the lasso with mechanisms for bounding the number of type I errors, as well as to lasso methods using a standard cross-validation approach, shrinkage estimation, partial least squares estimation, and ridge estimation methods. For the bounded type I error lasso methodologies, this included the randomized lasso with stability selection 40 and the regular lasso with the penalization chosen to bound the number of type I errors as in Meinshausen and Bühlmann 12. We simulated twenty networks with a random underlying topology, p = 1000 variables, n = 300 observations, and on average 1.47 edges per variable (further details of the simulation are presented in the Methods). In Figure 1 we show the precision-recall curves for two different strategies for defining the posterior probability of edge inclusion for the variational Bayes methodology: vb^α where the posterior probabilities are averaged in both directions of regression, and vb^β, where the posterior probabilities are not averaged in both directions of regression. We also show four different strategies for defining the empirical selection probabilities defined by the randomized lasso with stability selection: ℓ1a where the penalization parameter is chosen as in Meinshausen and Bühlmann 40 to bound the number of false positives to be less than one, and the empirical selection probabilities are averaged in both directions, ℓ1β where the penalty parameter is chosen similarly, but the empirical selection probabilities are not averaged, ℓ1γ where the penalization parameter is chosen as in Meinshausen and Bühlmann 40 to bound the number of false positives to be less than 1000, and finally ℓ1δ, where the penalty parameter is chosen similarly, but the empirical selection probabilities are not averaged. All curves are generated as a function of the threshold for declaring significance based on the associated probability statistics. We see that the variational Bayes approach significantly outperforms the randomized lasso with stability selection in terms of both power and type I error control for the averaged and non-averaged (vb^α, vb^β) posterior probability statistics across most thresholds for declaring significance.

In Figure 2, we illustrate the performance in terms of the average number of false positives observed in the entire network per replicate network (the left panel), and the overall power (the right panel) for specific thresholds of the variational Bayes and lasso approaches. Specifically, we investigate the variational Bayes approach for the conservative posterior probability thresholds of p^j > 0.99, not averaged (vb^a), and averaged (vb^b), as well as for the more liberal p^j > 0.5, not averaged (vb^c), and averaged (vb^d) as described above. We also show the randomized lasso with stability selection for the more conservative strategy where the number of false positives is bounded to be less than 1, not averaged (ℓ1a), and averaged (ℓ1b), as well as for the more liberal strategy of the number of false positives bounded to be less than 1000, not averaged (ℓ1c), and averaged (ℓ1d). Finally, we show the method of choosing the penalization of the regular lasso to bound the number of false positives to be less than 1 12 (ℓ1e), and less than 1000 (ℓ1f). Across all of these results, we see that the more conservative variational Bayes approaches (vb^a and vb^b) outperforms the conservative lasso approaches (ℓ1a, ℓ1b, and ℓ1e), as well as most of the more liberal lasso approaches (except with vb^b and ℓ1c), while at the same time recovering far fewer false positives. At the more liberal threshold of p^j > 0.5, the performance of the variational Bayes algorithm is further improved, especially for the averaged solution, vb^d, which has a comparable number of false positives to any of the liberal lasso solutions, but has much greater power. The lasso methods had the most comparable performance to our algorithm based on additional simulated data considering five competing methods: lasso, adaptive lasso, shrinkage estimation 14, partial least squares estimator 41, and a ridge estimator 22 (Figures 1, 2, Additional file 1: Figure S4, Additional file 1: Figure S5, Additional file 1: figure S6, Additional file 1: figure S7) (see the Additional file 1 for a detailed description of additional simulations and network reconstruction methods). Therefore we only compared the lasso approaches to our algorithm when analyzing the experimental data.

For the data analysis, we analyzed the F₂ progeny of a cross between the C57BL/6 J (B6) and C3H/HeJ (C3H) strains on an apolipoprotein E null (ApoE -/-) background (BXH.ApoE^-/-), as presented in Ghazalpour et al. and Wang et al. 931. We focused on the gene expression data that was collected in the liver of the mice where expression was assayed on 23,574 custom probes 9. In addition, there were 22 downstream phenotypes that were assayed, including weight, cholesterol, glucose, free fatty acid, among other metabolic phenotypes, as well as 1,347 genetic markers 9. A total of 298 individuals were retained after filtering down to those for which both expression and genetic markers were collected. Previous authors have shown the antagonistic sex effects within this data 31, i.e. the effect of a risk locus is opposite between males and females. To address the sex specific effects, as well as other possibly confounding factors, we included both the sex and the 20 first eigenvectors from a principal component analysis computed across samples for expression phenotypes as unpenalized fixed effects in our linear model for all methods that we compared.

We first ran our variational algorithm on each of the 22 obesity related downstream phenotypes individually, where we performed sparse feature selection on all gene products and genetic markers. Our variational algorithm produced a sparse set of expression and genetic markers for each downstream phenotype, with the phenotypes with more than seven expression or genotype features identified shown in Additional file 1: Table S1. We ran the randomized lasso with stability selection to bound the number of false positives to be less than 1 40, and the regular lasso with the choice of penalty to bound the false positives to be less than 1 12, and found that no expression traits or genotypes were identified as having non-zero effects by either approach. We also ran the lasso and adaptive lasso with ten-fold cross-validation for the same set of downstream phenotypes, as shown in Additional file 1: Table S1. The number of identified genetic interactions of each downstream phenotype was on average much larger for the lasso and the adaptive lasso with ten-fold cross-validation. The variational algorithm identifies additional features, with only 55% overlap with the lasso, and 46% overlap with the adaptive lasso for the seven phenotypes shown in Additional file 1: Table S1.

To assess the statistical confidence of our initial mouse obesity analysis, we determined the confidence intervals for each of the downstream phenotype network connections recovered with each feature selection method, in an independent, non-penalized linear multiple regression model. Both the lasso and the adaptive lasso contained many features that were not statistically significant at the P < 0.05 significance level, indicating that the use of cross-validation as a method to control the sparsity of the model for the lasso or adaptive lasso allows an unacceptable number of false positives to be included in the model. We depict the network model and confidence intervals for the network connections identified by the lasso and adaptive lasso for weight in Additional file 1: Figure S1 and Additional file 1: Figure S2. For all phenotypes, the features identified as having a downstream network connection by the variational algorithm were all significant (the models and confidence intervals are depicted in Figures 3 and 4). We also recapitulated a similar result in additional simulations where we demonstrate that at the p^j > 0.99 cutoff, the variational Bayes method returns fewer false positives and tighter confidence intervals on all predicted network connections as opposed to the lasso and adaptive lasso (Additional file 1: Figure S3). Given the increased performance in terms of learning a statistically robust model and the appropriate sparsity of that model, we proceeded with only the variational Bayes algorithm for the expanded network analysis.

In the second step of our analysis, we used our variational Bayes algorithm to generate an undirected network among genotypes, expressed genes, and downstream phenotypes, by solving the neighborhood selection problem for each gene expression product individually, against all other genes and genetic markers. We resolved the neighborhoods of the networks very conservatively, by averaging the p^j scores in both directions of regression for the expression phenotypes, and only declaring an interaction between genes present in the model if the averaged p^j scores were greater than 0.99. To determine the most relevant aspects of this sparse network with respect to weight and other related phenotypes, we combined the neighborhoods produced for each of the downstream phenotypes from the first phase of the analysis and the second phase expression undirected network, to depict the local sub-networks associated with each downstream phenotype, for weight, total cholesterol, high density lipoprotein (HDL) cholesterol, unesterified cholesterol (UC), free fatty acids (FFA), glucose levels, and low density lipoprotein + very low density lipoprotein (LDL + VLDL) levels (Figure 5). Table 1 summarizes identified genes which have been previously implicated in obesity, or related diseases and pathologies (Additional file 1: Table S2 is a version of this table with references available in the Additional file 1).

The network recovered by our algorithm is enriched for interactions that have been previously associated with these phenotypes: a total of 18 out of 118 recovered. While this may appear modest, it still suggests that this list of 118 variables is enriched for good candidate genes for follow up studies. From the first step of the analysis we find eleven genes (Gch1, Zfp69, Dlgap1, Gna14, Yy1, Gabarapl1, Folr2, Fdft1, Cnr2, Slc24a3, and Ccl19), as well as a single nucleotide polymorphism (rs3664823) that are directly linked to the downstream metabolic phenotypes and have independent evidence of being associated with obesity or obesity related pathologies (along with 52 novel genetic variables). We further identify six genes that feed into the genes that directly interact with the metabolic phenotypes (Ier2, H11r, Wisp1, Crhr1, Qpctl, Vcam-1), as well as an additional 54 novel interactions. These other novel interactions included Dcamkl1, Ercc1, and Cyp7a1, implicating a possible connection with weight, intestinal stem cell lineage 42, and DNA damage repair 43, along with a connection between the levels of free fatty acids and bile acid production 44.

Similarly to Yin and Li 55, we assume that the expression data for the i^th sample (y_i) conditioned on a set of genotypes and unpenalized covariates (x_i) is distributed normally y i | x i i i d ~ N Γ x i , Θ y y - 1 , with means determined by possibly sparse linear functions of genotypes and unpenalized covariates (Γx_i) as well as a sparse precision matrix (Θ_yy) (i.e. the inverse covariance matrix). Yet, in contrast to Yin and Li 55, we define an alternative parameterization of the mean effects Γ, such that Γ = Θ y x T Θ y y - 1 . This lets us consider not only the conditional independencies among phenotypes correcting for the effect of genotypes and covariates as in Yin and Li 55, but this also allows us to identify a set of genotypic effects Θ y x T that directly takes into account the conditional independence structure among the expression phenotypes. The log-likelihood defined by this model is as follows 55:

log Y | X , Θ ∝ log det Θ y y - Tr S Θ ,

where:

Θ = Θ y y Θ y x Θ y x T Θ x x ,

and

S = 1 n Y T Y Y T X X T Y X T X ,

being the sample covariance matrix, X and Y mean-centered, X being an n × m matrix of genotypes and fixed effects, and Y being an n × p matrix of expression or downstream phenotypes. The elements θ_ij of the matrix Θ represent the pairwise Markov dependencies of the random variables Y 34. Intuitively, the set of non-zero θ_ij parameters for a given random variable y_i, defines the set of other phenotypes once conditioned on, make y_i probabilistically independent from the rest of the variables in the model (also known as the neighborhood of y_i). In this model everything is conditional on the state of the entire set of genotypes and fixed effects. The non-zero structure of the Θ_yy sub-matrix specifies a conditional Markov random field among the expression or downstream phenotypes. Accordingly, the element θ y y i j for i ≠ j of the Θ_yy matrix is zero iff

p y i , y j | Y - ( i , j ) , X = p y i | Y - ( i , j ) , X p y j | Y - ( i , j ) , X ,

i.e. the probability distribution satisfies the local Markov property 34 with respect to an undirected graph G = V , E , with Y_{-(i, j)} indicating the set of other phenotypes, excluding the single variables y_i and y_j. Since this is a Markov random field conditioned on X assuming an underlying linear model, the non-zero structure of the Θ_xy sub-matrix does not imply a factorization over an underlying probability density, but the element θ x y i j is zero iff

p y j | X - i , Y - j , x i = p y j | X - i , Y - j ,

i.e. the conditional distribution of y_j is the same, when conditioning on X_-i and Y_-j, whether one conditions on x_i or not. Finally, since this is a conditional Markov random field, the rank of the matrix Θ is p and Θ x x = Θ x y Θ y y - 1 Θ y x .

To infer the structure of the underlying undirected graph, many authors have proposed putting different forms of element-wise penalties on the Θ matrix, such as the lasso (l₁ norm) 1256. Additionally, as other authors have noted 57, the positive-semi definite constraint on Θ imposed by the log{det} function in the log likelihood makes optimization of the full likelihood problem challenging for large scale problems, especially when the number of phenotypes and genotypes p + m greatly exceeds the sample size n. Therefore, instead of solving the full likelihood optimization problem, we follow the general strategy of Meinshausen and Bühlmann, Zhou et al., and Kraemer et al. 121322, and treat the structure learning problem as a neighborhood identification problem; i.e. we perform model selection on a set of uncoupled regression equations, where each expression phenotype is regressed on every other phenotype, and genotype. At the end of this process we resolve the neighborhoods of each gene expression product by averaging the posterior probabilities of edge inclusion in both directions of regression.

We define a given multiple regression equation as:

y i = μ + ∑ j p - 1 z i j β j y + ∑ l m x i l β l x + ∑ k l t i k α k + e i ,

where y_i is i^th sample of a given phenotype, the population mean is modeled as a fixed effect μ, z_ij is the i^th sample of the j^th phenotype, excluding the phenotype y , β j y is the effect of the j^th phenotype, x_il is the i^th sample of the l^th genotype, β l x is the effect of the l^th genotype, t_ik is the i^th sample of the k^th non-penalized effect, α_k is the effect of this k^th feature, and eⁱ is the residual error term, assumed to be normally distributed with mean zero, and variance σ e 2 . In general we include the top 20 eigenvectors from a principal component analysis of the expression phenotypes as unpenalized covariates t_ik for each penalized regression model.

While the likelihood defined in Equation 1 corresponds to a conditional GGM corresponding to the joint distribution of the gene expression phenotypes (Y) conditional on some set of genotypes and fixed effects (X), our approach focuses on solving a set of penalized regression equations for each phenotype (as in Equation 6). Our assumption (as in Meinshausen and Bühlmann 12) is that the set of variables that are selected in a particular regression model for a given phenotype (e.g. which β j y and βlx are non-zero will exactly specify which set of elements of Θ are non-zero). For example, if in the penalized regression model for the 5^th phenotype we find that β 1 y , β 3 y , and β 4 x are non-zero, then this would indicate that the corresponding elements of Θ, θ y y 15 , θ y y 35 , and θ y x 54 would be non-zero, and the corresponding conditional independence properties implied by Equation (4) and Equation (5) would be true for these variables.

Given the regression equation defined in Equation 6, we define the following hierarchical model, similar in vein to Zhang et al. and Logsdon et al. 2529:

β j ~ p β = 0 I β = 0 + p β ≠ 0 N 0 , σ β 2 ,

p β = 0 , p β ≠ 0 ~ Beta 1 , 1 ,

σ β - 2 ~ Γ 2 , 1 / 2 ,

σ e - 2 ~ Γ 2 , 1 / 2 ,

with the additional truncation restriction on the prior distribution over p_{β ≠ 0} of p β ≠ 0 ≤ n / m + p - 1 .

This mixture penalty in a Bayesian framework has attractive theoretical properties, including bounded shrinkage and indications that it may approach optimal efficiency for sparse underlying parameter spaces 58 and may still be model selection consistent when the irrepresentability condition is not met 27. One of the main advantages of this approach is that the hierarchical model can adaptively shrink the penalty to match the sparsity of the underlying parameter space, without having to resort to prediction based metrics like cross-validation which can overestimate model size or possibly heuristic model complexity measures based on information criterion such as Akaike's Information Criterion (AIC) or Bayesian Information Criterion (BIC).

Because the mixture penalty is non-convex, the posterior surface can be highly multi-modal and each mode in the posterior density can represent a different set of identified features (i.e. neighborhood). A well characterized weakness of the l₀ type penalty (i.e. best subset selection) is the instability of the identified solutions 59. One of the most important novel contributions of our algorithm is a Bayesian model averaging step 60. We perform Bayesian model averaging across the identified modes by re-weighting the posterior probability of inclusion for each feature (p^j), proportional to the estimated volume underneath each identified mode (a measure of the relative evidence of a given model) based on the lower bound, an approach similar to bagging 61. Because the algorithm is very fast, we can run it many times (up to thousands) and identify many models, along with the relative evidence of each model identified, based on the lower bound (Equation 17 in the Supporting Information), and integrate the evidence across the models. This approach is effective at integrating out model uncertainty, and generating the best estimates of which interactions are most strongly supported by the data.

The variational Bayes approximation consists of minimizing the Kullback-Leibler divergence between an approximate factorized posterior distribution q β 1 β 1 ⋯ q β m β m q p β ≠ 0 p β ≠ 0 q σ e σ e q σ β σ β and the full posterior distribution p(β₁,...,β_m, p_{β ≠ 0}, σ_e, σ_β), using iterative expectation type steps as in an Expectation-Maximization algorithm 3839. Given this optimization procedure, the variational Bayes distributional approximation for the posterior distribution of an arbitrary parameter θ (i.e. β₁,...,β_{p + m-1}, σ_e, σ_β, p_{β ≠ 0}) is given as follows:

q θ j t + 1 θ j = 1 Z θ j exp ∫ q θ - j t θ - j d θ - j log p θ | y ,

where a factorization is defined over the joint approximate posterior distribution of parameters:

q θ θ = ∏ i q θ i θ i ,

and the integral in Equation 11 at iteration t is taken with respect to every approximate distribution except q θ j t θ j . The posterior density p(θ|y) in Equation 11 is defined based on multiplying the likelihood for the model in Equation 6 with the priors in Equation 7-10. The details of this approximation for each density are presented in the Additional file 1. This factorization is required to solve for closed form iterative updates associated with the spike-and-slab prior distribution. A probability of inclusion statistic, p_j is computed after the algorithm converges, and this statistic is averaged across all models identified (i.e. modes in the posterior surface), based on the total evidence for each model (i.e. Equation 17 in the Supporting Information). This model averaged probability of inclusion statistic, p^j is used to determine whether the j^th feature is included in the model, at a given threshold.

We compared our variational spike-and-slab algorithm with two alternative methods proposed to bound the number of type I errors when using lasso penalized regression. The first method we compared was the randomized lasso with stability selection, as described by Meinshausen and Bühlmann 40. As with our method, we performed this approach by solving a penalized regression model for each phenotype in the network individually, then afterwards we compared different methods for combining the results. As in Meinshausen and Bühlmann 40, we implemented the randomized lasso with sampling weights for the variables in the regression sampled from a Unif(0.2, 1.0), then we performed the stability selection procedure by sampling n/2 observations, and running 100 replicate instances of this two-level randomized algorithm. This was run using the glmnet package in R 62 on a grid of 100 logarithmically spaced penalty parameters λ = {10^-2,...,10²} (this range of penalization was sufficient such that the approach never chose a level of penalization on the boundaries). As in Meinshausen and Bühlmann 40, we focused on bounding the number of type I errors by choosing an empirical probability of selection cutoff of 0.9, then choosing the level of penalization that returns the expected number of selected variables which satisfies the bound on the number of false positives 40. We looked at bounds of both 1 and 1000 expected false positives for the entire network, to explore both conservative and liberal choices of the cutoffs.

The alternative approach we used to bound the number of type I errors was based on the original Meinshausen and Bühlmann network algorithm 12, where they bound the number of type I errors between connectivity components in a graph. Intuitively, a connectivity component is just the set of variables for any given variable that can be reached through some path within the graph. We used their choice of penalty parameter λ α = 2 σ ^ n Φ ̃ - 1 α 2 p 2 , where α is the probability of making a type I error, σ ^ = n - 1 ∑ i y i 2 for any given phenotype y, p is the number of variables in the model, and Φ ̃ - 1 = 1 - Φ (with Φ the c.d.f. of the standard normal density function). We investigated bounds of both 1 and 1000 expected false positives for the entire network, by running the regular lasso penalized regression method with the glmnet package in R 62 and this level of penalization.

For the simulations depicted in Figures 1, 2, we simulated twenty replicate networks through the following procedure: first, a random directed graph with p = 1000 variables and adjacency matrix A was generated by sampling edges between variables with probability 1/p = 10^-3 for all p(p-1) edges. Next, each edge was weighted by a N 0 , 1 random variable. Third, the diagonal of A was set to one. Finally, we constructed the precision matrix Θ = AA^T (i.e. inverse covariance matrix), and sampled a data-set Y from the multivariate normal distribution N 0 , Θ - 1 , with 300 independent observations. Because of the moralization (i.e. edges induced between nodes that share parents 39) produced by converting the directed graph A into an undirected graph Θ, the average number of edges per node was 1.47 instead of ≈1. It is important to note that we did not include any unpenalized covariates or genotypes in this particular simulation. Additional algorithms used for comparison with analysis of further simulated data are shown in the Additional file 1.

We analyzed the F₂ progeny of a cross between the C57BL/6 J (B6) and C3H/HeJ (C3H) strains on an apolipoprotein E null (ApoE -/-) background (BXH.ApoE^-/-) 931. After further filtering the data to a set of shared samples across all variables, we were left with 298 individuals, 22 downstream phenotypes, 1,347 genetic markers, and 23,574 expression probes. We included sex as well as the 20 first eigenvectors from a principal component analysis computed across samples for expression phenotypes as fixed effects, and ran the first phase of the algorithm (i.e. feature selection on the downstream phenotypes) with 1,000 random restarts of the algorithm to get good coverage of the posterior probability surface associated with model uncertainty for each downstream phenotype. We then ran the second phase of the algorithm between just expression traits and genotypes, still incorporating the 20 eigenvectors from principal component analysis and sex as fixed effects, with 50 random restarts. All network diagrams in Figures 3, 4, 5 were generated with the network package in R 63.