Correlation-based tests of causal models
For simplicity, we assume that the genetic markers are single nucleotide polymorphisms (SNPs). For a given sample (e.g. a mouse), a bi-allelic SNP can take on one of three possible genotypes. By default, we assume an additive genetic effect and encode these genotypes as 0, 1, or 2, but alternative marker codings could also be considered. To quantify the linear relationship between a SNP marker M and a trait A, we use the correlation coefficient cor(M, A). Ordinal variables are routinely used in path analysis and structural equation modelling 3240.
To determine whether trait A mediates the effect of marker M on trait B (M → A → B) one can assess how conditioning on A affects the correlation between M and B. To quantify the linear relationship between M and B after conditioning on A, we use the partial correlation coefficient:
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If the causal model M → A → B is correct, then the partial correlation coefficient cor(M, B|A) is expected to be 0.
We use Fisher's Z transform to assess the statistical significance of a sample correlation coefficient r 23:
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where N denotes the sample size; ZFisher(r) asymptotically follows a normal distribution (Normal(μ, 1) with mean μ and variance 1. Under the null hypothesis of zero correlation, μ = 0 and ZFisher(r) follows a standard normal distribution. For brevity, we denote the Fisher transformations of the correlation coefficients cor(A, B) and cor(M, B|A) by Z(A, B) = ZFisher(cor(A, B)) and Z(M, B|A) = ZFisher(cor(M, B|A)), respectively.
If the causal graph M → A → B (Figure 1a) is correct, Z(M, B|A) follows a standard normal distribution. Thus, if the p-value corresponding to Z(M, B|A) is high (non-significant), the data fit the assumed causal graph. Using path analysis rules, the causal graph M → A → B (Figure 1a) implies the following relationships between the correlation coefficients
<p>Figure 1</p>
Approaches for genetic marker-based causal inference
Approaches for genetic marker-based causal inference. Here we contrast different approaches for causality testing based on genetic markers. (a) single marker edge orienting involving a candidate pleiotropic anchor (CPA) M. The upper half of (a) shows the starting point of network edge orienting based on a single genetic marker M which is associated with traits A and B. The undirected edge between A and B indicates a significant correlation cor(A, B) between the two traits. The causal model in the lower half of (a) implies the following relationship between the correlation coefficients cor(M, B) = cor(M, A) × cor(A, B). Further it implies that the absolute value of the correlations |cor(M, A)| and |cor(M, B)| are high whereas the partial correlation |cor(M, B|A)| (Eq. 1) is low. Figure (b) generalizes the single marker situation to the case of multiple genetic markers MA={MA(1),MA(2),...}
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cor(M, B) = cor(M, A)cor(A, B)
We refer to the marker M as a candidate common pleiotropic anchor (CPA) of A and B. If the expected values of cor(M, A) and cor(A, B) are non-zero and the causal model holds, Eq. (2) implies that the genetic marker M will be significantly correlated with both A and B. Thus, the marker M can be confirmed as a pleiotropic anchor of A and B by confirming the fit of the causal model M → A → B. We will now consider a situation where the correlation between A and B stems from a hidden confounder C, i.e. M → A ← C → B. The graph implies that A and B are correlated due to the shared confounder C. The correlation cor(M, B) is expected to be 0 since the arrows between M and B collide at A, i.e, M and B are d-separated without conditioning. In this situation Z(M, B) = ZFisher(cor(M, B)) follows a standard normal distribution. If the p-value corresponding to Z(M, B) is high (non-significant), the data fit a confounded model. In contrast, the partial correlation cor(M, B|A) is expected to be non-zero since conditioning on A 'activates' the causal flow through the collider node, i.e. it induces conditional dependence 32.
The opposite (reactive) causal graph M → A ← B also implies that the expected value of cor(M, B) is zero since the causal paths collide at A. Conditioning on A activates this collider node, and the partial correlation cor(M, B|A) is expected to be non-zero.
Similarly, one can show that the model A ← M → B implies that cor(A, B|M) is expected to be zero. Under this causal model, Z(A, B|M) asymptotically follows a standard normal distribution. In contrast, cor(M, B) is expected to be non-zero.
These considerations illustrate that one can test the predicted correlational consequences of a causal model and thus evaluate its fit.
We will now consider the situation of multiple markers (Figure 1b). Denote by
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a set of candidate common pleiotropic anchors of A and B. Analogous to Eq. (2), the causal model MA → A → B implies cor(MA(i),B)=cor(MA(i),A)cor(A,B)
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Frequently an additional set of markers MB is also available for trait B (Figure 1c). For example, when one marker is available for each trait, i.e. MA(1)→A→B←MB(1)
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4yamMaem4Ba8MaemOCaiNaeiikaGIaemyqaeKaeiilaWIaemyta00aa0baaSqaaiabdkeacbqaaiabcIcaOiabigdaXiabcMcaPaaakiabcMcaPaaa@389E@ is expected to be 0 since the causal arrows 'collide' at B 30. Geometrically speaking, the expected zero correlation between A and MB(1)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyta00aa0baaSqaaiabdkeacbqaaiabcIcaOiabigdaXiabcMcaPaaaaaa@30D4@ implies that the corresponding standardized vectors are orthogonal. Therefore, we refer to marker MB(1)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyta00aa0baaSqaaiabdkeacbqaaiabcIcaOiabigdaXiabcMcaPaaaaaa@30D4@ as an orthogonal causal anchor (OCA) with respect to the edge A → B. We will argue that the availability of orthogonal causal anchors significantly improves the recovery of the causal signal (see the simulations in Additional File 1). If the model MA(1)→A→B←MB(1)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyta00aa0baaSqaaiabdgeabbqaaiabcIcaOiabigdaXiabcMcaPaaakiabgkziUkabdgeabjabgkziUkabdkeacjabgcziSkabd2eannaaDaaaleaacqWGcbGqaeaacqGGOaakcqaIXaqmcqGGPaqkaaaaaa@3DB6@ is correct, cor(MA(1),B|A),cor(A,MB(1))
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4yamMaem4Ba8MaemOCaiNaeiikaGIaemyta00aa0baaSqaaiabdgeabbqaaiabcIcaOiabigdaXiabcMcaPaaakiabcYcaSiabdkeacjabcYha8jabdgeabjabcMcaPiabcYcaSiabdogaJjabd+gaVjabdkhaYjabcIcaOiabdgeabjabcYcaSiabd2eannaaDaaaleaacqWGcbGqaeaacqGGOaakcqaIXaqmcqGGPaqkaaGccqGGPaqkaaa@48D2@ are expected to be zero and Z(MA(1),B|A)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOwaOLaeiikaGIaemyta00aa0baaSqaaiabdgeabbqaaiabcIcaOiabigdaXiabcMcaPaaakiabcYcaSiabdkeacjabcYha8jabdgeabjabcMcaPaaa@3843@ and Z(A,MB(1))
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOwaOLaeiikaGIaemyqaeKaeiilaWIaemyta00aa0baaSqaaiabdkeacbqaaiabcIcaOiabigdaXiabcMcaPaaakiabcMcaPaaa@35B8@ asymptotically follow standard normal distributions.
<p>Additional file 1</p>
Single edge simulation study. This document describes our single edge simulation studies involving the LEO.NB.CPA score (Eq. 5) and the LEO.NB.OCA score (Eq. 6). We describe the parameters used in the single edge A ← B simulation model. A hidden confounder C affects the correlation between A and B. The effect of SNP markers on traits A and B is parameterized with the restricted heritabilities. The single edge simulation model is used i) to study the choice of thresholds for the LEO.NB scores, ii) to compare the LEO.NB.CPA with LEO.NB.OCA scores, and iii) to evaluate automatic SNP selection methods.
Click here for file
Figure 1(c) depicts a situation where two sets of genetic markers MA={MA(1),MA(2),...MA(KA)}
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Local SEM-based edge orienting scores
While SEMs can be used to study the fit of multi-trait causal models 1720 we only consider the local causal models depicted in Figure 2 since the proposed NEO method evaluates the orientation of each edge separately based on the best causal anchors available. The fit of each single marker model in Figure 2(a) can be tested using a chi-square test with 1 degree of freedom. We refer to the resulting p-value as the model p-value. In the Methods section, we review and discuss the use of model p-values for quantifying the fit of a causal model. The main point is that the higher the model p-value, the better the causal model fits the data.
<p>Figure 2</p>
Illustrating the single genetic marker versus multi-marker local SEMs used in the definition of the LEO.NB score
Illustrating the single genetic marker versus multi-marker local SEMs used in the definition of the LEO.NB score. The single genetic marker is denoted by M in (a) and the multiple genetic markers are denoted by MA(i)
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To summarize the genetic evidence in favor of a given edge orientation A → B, we propose the use of edge orienting scores. The higher the value of an edge orienting score for the orientation A → B, the stronger genetic evidence favors this causal model.
In the following, we propose local SEM-based edge orienting (LEO) scores for orientation A → B. For a single genetic marker M and traits A and B, we consider the 5 different local causal models depicted in Figure 2(a). Additional single marker models are possible. However, under the constraint that the markers are causal anchors (graphically, arrows flow only from M and not into M), then the five models pictured for nodes (M, A, B) in Figure 2(a) exhaust all possible three node models that both (1) explain A – B and (M – A or M – B) associations and (2) can be tested. The critical technical issue is having degrees of freedom (d.f.) remaining after estimating the model parameters. If the degrees of freedom are 0, the model p-values cannot be calculated. The 5 different local causal models depicted in Figure 2(a) are used to compute the following model p-values: P(M → A → B), P(A ← B ← M), P(A ← M → B), P(M → A ← B), and P(A → B ← M). While a detailed analysis should consider all model p-values, we find it useful to summarize the genetic evidence in favor of a given orientation A → B (model 1) using a single number: the Local SEM-based Edge Orienting Next Best (LEO.NB) score. The LEO.NB score is defined by dividing the model p-value for A → B by the p-value of the best fitting alternative model, i.e. the best of models 2–5 in Figure 2(a). The chi-square test p-value of the best fitting alternative model is the maximum p-value of the alternative causal models. Specifically, we define the single-marker LEO.NB score as follows:
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A positive LEO.NB(A → B) score indicates that the p-value in favor of model A → B is higher than that of any of the competing models in Figure 2(a). A negative LEO.NB score indicates that the A → B model is inferior to at least one alternative model. In our simulations, we use a threshold of 1 for LEO.NB.SingleMarker(A → B|M).
Multi-marker LEO.NB score
It is straightforward to generalize the single marker LEO.NB score (Eq. 4) to a set of genetic markers MA={MA(1),MA(2),...}
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Note that the multi-marker models used in the definition of LEO.NB.CPA correspond to the single marker models of Figure 2(b) with M replaced by MA.
If an additional genetic marker set MB={MB(1),MB(2),...MB(KB)}
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Note that model 4 in the denominator involves a hidden confounder C. The use of two independent genetic marker sets (MA and MB) alleviates the problem of model identifiability that may plague a CPA based edge orienting score.
Model equivalence is also a key consideration in choosing which models to compare. From the standpoint of model equivalence, we note that the multiple anchor models presented in Figure 2(b) include a model with a hidden (latent) variable connecting A and B, and that no such model is included in the single anchor model comparisons. Such a model was found to be indistinguishable from the models with a collider node, such as single anchor models 4 and model 5. In the single marker case, both the collider node and the hidden variable models test for independence in the marginal relationship between the anchor and the more distal trait node. Future research may lead to an understanding of what type of data allow one to consider additional alternative models for the edge score computation. It should be straightforward to adapt the proposed LEO score to additional models as long as their model p-values can be calculated. Correlated markers, which are frequently encountered in practice such as in haplotype blocks, may compromise the performance of edge orienting scores. LEO scoring allows multiple parents of a node to be correctly accounted for within each model. Moreover, the parents (causal anchors) of a model are allowed to co-vary. By contrast, the orthogonal causal anchor set is, by definition, penalized for any covariation with the pleiotropic anchors.
Thresholds for the edge orienting scores
For the single marker score LEO.NB.SingleMarker, we use a threshold of 1, which implies that the model p-value of the causal model is 101 = 10 fold higher than that of the next best model. For the LEO.NB.CPA and the LEO.NB.OCA, we use lower thresholds of 0.8 and 0.3, respectively. Using simulation studies presented in the Additional File, we found that these thresholds lead to false positive rates that are often substantially below 0.05. Similar to other statistical procedures, NEO is susceptible to the pitfalls of multiple testing that may inflate the false positive rate. Permutation procedures and data dependent schemes (e.g. based on the false discovery rate) may inform the user on how to pick a threshold for a particular application. Further, we provide R software code for carrying out both single edge and multi-edge simulation studies. Simulation studies can be used to determine the power and false positive rates in different settings (sample size, causal signal, confounders, etc).
In practice, one often observes strong dependence relationships between genetic markers. Our simulations show that correlations between genetic markers can reduce the power of edge orienting scores. Further, we mention that the NEO software implements an option for removing redundant markers that are highly correlated with each other. The removal of redundant markers may alleviate the loss of power.
Overview of network edge orienting with NEO
We now provide a detailed step-by-step description of a typical NEO analysis. An overview is also provided in Figure 3.
<p>Figure 3</p>
Overview of the network edge orienting method
Overview of the network edge orienting method. The steps of the network overview analysis are described in the text.
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Step 1: Integrate traits (gene expression traits and clinical traits) and SNPs
NEO takes trait and genetic marker data as input. Traits can include microarray gene expression data, clinical phenotypes, or other quantitative variables. Each SNP or trait is a node in the network, and the NEO software evaluates and scores the edge between traits A and B if the absolute correlation |cor(A, B)| lies above a user-specified threshold. For each edge A – B, NEO generates edge orienting scores for both possible orientations: A → B and B → A. If an erroneous edge exists between two traits, then it is meaningless to orient it. The NEO software can be used to orient any edge that the user chooses to consider. To allow the user to judge whether the existence of an edge is supported by the data, the NEO software outputs a Wald test statistic of the path coefficient, the corresponding p-value, and the correlation between the two traits. If the Wald test p-value is insignificant, orienting the edge may be meaningless.
Step 2: Genetic marker selection and assignment to traits
Edge orienting scores will only be generated for edges whose traits have been anchored to at least one genetic marker. Two basic approaches for anchoring traits to markers are implemented in the NEO software: a manual selection by the user or an automatic selection by the software itself.
Manual SNP selection
NEO provides great flexibility to the user on how to anchor traits to markers. For example, the user can manually assign SNPs to the traits (see the example in Figure 4). This flexibility entails that the user carefully studies what constitutes a significant relationship between traits and markers and between the traits in the data set. The user may wish to anchor traits to SNPs that have been implicated by prior genetic analyses. For example, results from previous quantitative trait locus studies may implicate genetic markers associated with a trait. Multiple comparison issues are just starting to be addressed in the SEM literature 394142. Edge scores cannot be computed when an overly strict multiple testing control results in no causal anchors. On the other hand, an overly lax multiple testing control may result in spurious causal anchors which may lead to erroneous edge scores. We recommend that conservative measures of genome-wide QTL significance 43 and false discovery rate be applied when selecting the initial causal anchor(s). Once a causal anchor has been established as obtaining genome-wide significance, NEO can be used to evaluate the fit of different causal models.
<p>Figure 4</p>
Manual SNP selection to study Insig1 → Dhrc7 and Insig1 → Fdft1 in mouse liver
Manual SNP selection to study Insig1 → Dhrc7 and Insig1 → Fdft1 in mouse liver. Using female liver gene expression data and SNP markers from the BxH mouse intercross, NEO retrieves known causal relationships in the cholesterol biosynthesis pathway: Insig1 → Dhrc7 and Insig1 → Fdft1. The single marker LOD score curves in (a) motivate our choice of manually selected SNPs (one SNP on chromosome 16 and another on chromosome 8). These SNP markers can also be used to screen for genes that are reactive to Insig1, see Table 2. Figures (b) and (c) show the causal models used to compute the model p-values in favor of edge orientations Insig1 → Dhcr7 and Insig1 → Fdft1, respectively. More details on the individual edges are presented in Table 1.
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Automatic SNP selection
NEO can also be used to automatically relate (anchor) traits to SNPs. The automated SNP selection methods consider each trait A in isolation from the other traits when defining a preliminary genetic marker set (denoted by M'A). Toward this end, the user can choose 1) a greedy approach based on univariate linear regression models, 2) a forward-stepwise approach based on multivariate linear regression models, or 3) both. The greedy SNP selection approach defines M'A as the set of markers with the K highest absolute correlations with A. The greedy approach is equivalent to using univariate linear regression models to relate A to each marker separately and subsequently picking the K most significant markers.
For creating multivariate linear QTL models, NEO also implements forward-stepwise marker selection. The forward-stepwise marker selection method may avoid a pitfall that plagues the greedy SNP selection: if several genetic markers are located very close to each other (and are highly correlated), the greedy SNP selection may pick all of them before considering SNPs at other loci associated with the same trait. For this reason, we recommend combining greedy and forward-stepwise SNP selection methods.
Once the preliminary sets of markers M'A and M'B are obtained, NEO evaluates the consistency of each set. We utilize a marker assignment consistency heuristic: a genetic marker can only serve as causal anchor for one trait. To fulfill this heuristic, a SNP is moved from M'A to M'B if its correlation with B is stronger than that with trait A. We denote the resulting consistent genetic marker sets by M"A and M"B. The resulting consistent genetic marker sets may be comprised of dozens of SNPs. Therefore, it can be useful to further filter the SNPs according to their joint predictive power for the trait. Toward this end, we use the Akaike Information Criterion (AIC) in conjunction with multivariate regression models to select genetic markers from within the consistent genetic marker sets 44. Specifically, to define the final genetic marker set MA for trait A, we use the AIC criterion to find a parsimonious multivariate regression model of A using predictors from within M"A. The final sets of markers MA and MB are thus comprised of consistent genetic markers that according to the AIC criterion best predict their respective traits; we use these final sets as causal anchors in computing the edge orienting scores.
The forward-stepwise approach based on multivariate linear regression models is akin to a legal courtroom where two cases are built, weighed, and judged. Broadly, the strongest genetic support (multivariate eQTL models) for the genetic influence on A and B are built independently, using AIC-based halting criteria. After consistency checks, these multivariate eQTL models are weighed by embedding them in causal models (one principal causal model in favor of edge orientation A → B and alternative causal models) and models are then compared using SEM fitting indices. When candidate CPA markers can be found for A and OCAs for B, the NEO method provides stringent consistency checks and balances against over-fitting. We consider automated SNP selection particularly useful when no prior evidence suggests causal anchors for the traits.
Step 3: Compute local edge orienting scores for aggregating the genetic evidence in favor of a causal orientation
Both LEO.NB.CPA and LEO.NB.OCA scores are computed for each edge orientation (A → B and B → A). We recommend using the LEO.NB.OCA score (Eq. 6) as the primary edge orienting score if markers affect both A and B. However, if the results of the LEO.NB.CPA score strongly disagree with those of the LEO.NB.OCA score, the latter should not be trusted. As described in the next step, all fitting indices should be considered before calling an edge causal.
Step 4: For each edge, evaluate the fit of the underlying local SEM models
Edges with high edge orienting scores may not necessarily correspond to causal relationships. Although edge orienting scores flag interesting edges, they are no substitute for carefully evaluating the fit of the underlying local SEMs. Since a LEO.NB score is defined as a ratio of two model p-values, it is advisable to check whether both p-values are small, as this would indicate poor fit of either model. If the model p-value of the confounded model A ← C → B is high, the correlation between A and B may be largely due to a hidden confounder C. NEO (using the underlying sem R package) also report a Wald test statistic for the path coefficient from A → B. If the Wald test for an edge is significant, the data support its existence. Apart from the model p-value, many other SEM model fitting indices have been defined by contrasting the observed covariance matrix Sm × m with the fitted covariance matrix Σ(θ^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqiUdeNbaKaaaaa@2D9B@) as detailed in the Methods section. The NEO software reports the standard SEM fitting indices 3245 that are implemented in the R package sem 46 including the Root Mean Square Error of Approximation (RMSEA), Comparative Fit Index (CFI), Standardized Root Mean Square Residual (SRMSR), BIC. Since a single fitting index reflects only a particular aspect of model fit, a favorable value of that index does not by itself demonstrate good model fit; it is important to assess the model fit based on multiple indices. We follow the following standard guidelines for interpreting these indices 45. Before calling an edge A → B causal, we recommend verifying that the corresponding causal model has a high model p-value (say > 0.05), a low RMSEA score (say ≤ 0.05), a low SRMSR (say ≤ 0.10), a high CFI (say ≥ 0.90), and a significant Wald test p-value (say p ≤ 0.05).
Step 5: Robustness analysis with respect to SNP selection parameters
Since the edge orienting scores for an edge A – B critically depend on the input genetic marker sets MA and MB, we also recommend carrying out a robustness analysis with respect to different marker sets. In particular, the automated SNP selection results should be carefully evaluated with regard to the threshold parameters that were used to define the marker sets. For example, when using a greedy SNP selection strategy, it is advisable to study how the LEO.NB score is affected by altering the number of most highly correlated SNPs. For a given edge and a given edge orienting score (e.g. LEO.NB.OCA), NEO implements a robustness analysis with respect to automatic marker selection (see Figures 5, 6, and 7). A robustness plot shows how the LEO.NB.OCA score (y-axis) depends on sets of automatically selected SNP markers (x-axis). When using the default SNP selection method (combined greedy and forward stepwise method), robustness step K corresponds to choosing the top K SNPs by greedy and forward selection for each trait. Since the greedy and forward SNP selection may select the same SNPs, step K typically involves fewer than 2K SNPs per trait. The edge orienting results should be relatively robust with respect to different choices of K.
<p>Figure 5</p>
Automatic SNP selection to score Insig1 → Dhrc7 and Insig1 → Fdft1 in female and male mouse livers
Automatic SNP selection to score Insig1 → Dhrc7 and Insig1 → Fdft1 in female and male mouse livers. These robustness plots show how the LEO.NB scores (y-axis) depend on sets of automatically selected SNP markers (x-axis). Here we use the default SNP selection method: combined greedy and forward stepwise method. Step K corresponds to choosing the top K greedy and top K forward selected SNPs for each trait. Since the greedy and the forward SNP selection may select the same SNPs, step K typically involves fewer than 2K SNPs per trait. Figures (a, b, top row) and (c, d) correspond to female and male BxH mice, respectively. Figures (a) and (c) report the results for edge Insig1 → Dhrc7 in female and male mouse livers, respectively. Figures (b) and (d) report the analogous results for Insig1 → Fdft1. NEO robustly retrieves the known causal relationship between these genes.
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<p>Figure 6</p>
Fsp27 is a causal driver of a biologically important co-expression module
Fsp27 is a causal driver of a biologically important co-expression module. Prior work using mouse liver expression data found the 'blue' co-expression module to be biologically important [7]. Here we used automatic SNP selection to determine whether Fsp27 is causal of the blue module gene expression profiles. The expression profiles of the blue module were summarized by their first principal component (referred to as module eigengene). The blue module eigengene MEblue can be considered as the most representative gene expression profile of the blue module. The figure shows the results of a robustness analysis regarding LEO.NB(Fsp27 → MEblue) (y-axis) with respect to different choices of genetic markers sets (x-axis). Both LEO.NB.CPA and LEO.NB.OCA scores show that the relationship is causal, i.e. the Fsp27 is upstream of the blue module expressions.
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<p>Figure 7</p>
Multi-edge simulation study involving 5 gene expression traits (E1-E5) and one clinical trait Trait
Multi-edge simulation study involving 5 gene expression traits (E1-E5) and one clinical trait Trait. The heatmap plot in (a) depicts the true causal model. Note that a red square in the i-th row and j-th column indicates that trait i causally affects trait j, e.g. E1 → E2. The rows and columns of the heatmap are ordered according to a hierarchical clustering tree, which was constructed using average linkage hierarchical clustering based on the pairwise correlations of the traits. Figure (b) depicts the corresponding heatmap of the observed network that was reconstructed using the LEO.NB.OCA score. Figure (c) shows an alternative output graph of NEO. Blue edges indicate significant correlations and a LEO.NB.OCA score is added to each edges whose LEO.NB.OCA score passes a user-supplied threshold. We find that all true causal edges are correctly retrieved at the recommended LEO.NB.OCA threshold of 0.3. Figure (d) shows the results of a robustness analysis for the LEO.NB.OCA and LEO.NB.CPA scores for the edge orientation E4 → Trait. The LEO.NB.OCA scores exceed the recommended threshold of 0.3 (red horizontal line), i.e. they retrieve the orientation correctly. Similarly, the LEO.NB.CPA scores exceed the threshold of 0.8.
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Step 6: Repeat the analysis for the next A-B trait-trait edge and apply edge score thresholds to orient the network
NEO orients each edge separately in an undirected input trait network. The results are order-independent. For each edge, NEO repeats steps 1–3 until all edges have been assigned edge orienting scores. Once each edge has been scored, the user can generate a global, directed network by choosing an edge score (e.g. LEO.NB.OCA) and a corresponding threshold (Figure 7).
NEO output and R software
The primary output of NEO is an Excel spreadsheet which reports likelihood-based edge scores (LEO.NB.CPA, LEO.NB.OCA) and other edge scores that are described in the NEO manual. For each edge, the NEO spreadsheet also contains hyperlinks that allows the user to access the log file for each edge. The log file contains a host of information regarding computation of the edge orienting scores including SEM model p-values, Wald test statistics for each path coefficient, and the SNP identifiers for the causal anchor sets MA and MB.
Although the main output of NEO are scores for every edge orientation, one can construct a global directed network by thresholding an edge orienting score. NEO uses the R software package sem 46 to compute model p-values and other fitting indices. The NEO software is documented in a series of separate tutorials that illustrate real data applications and simulation studies. These tutorials and the real data can be downloaded from our webpage.