Abstract
Background
Knockdown or overexpression of genes is widely used to identify genes that play important roles in many aspects of cellular functions and phenotypes. Because nextgeneration sequencing generates highthroughput data that allow us to detect genes, it is important to identify genes that drive functional and phenotypic changes of cells. However, conventional methods rely heavily on the assumption of normality and they often give incorrect results when the assumption is not true. To relax the Gaussian assumption in causal inference, we introduce the nonparanormal method to test conditional independence in the PCalgorithm. Then, we present the nonparanormal interventioncalculus when the directed acyclic graph (DAG) is absent (NPNIDA), which incorporates the cumulative nature of effects through a cascaded pathway via causal inference for ranking causal genes against a phenotype with the nonparanormal method for estimating DAGs.
Results
We demonstrate that causal inference with the nonparanormal method significantly improves the performance in estimating DAGs on synthetic data in comparison with the original PCalgorithm. Moreover, we show that NPNIDA outperforms the conventional methods in exploring regulators of the flowering time in Arabidopsis thaliana and regulators that control the browning of white adipocytes in mice. Our results show that performance improvement in estimating DAGs contributes to an accurate estimation of causal effects.
Conclusions
Although the simplest alternative procedure was used, our proposed method enables us to design efficient intervention experiments and can be applied to a wide range of research purposes, including drug discovery, because of its generality.
Keywords:
Nonparanormal; Gaussian assumption; Causal effect; Interventioncalculus; Directed acyclic graph; Machine learning; Causal inference; Experiment designBackground
Intervention experiments, e.g., knockdown or overexpression, are commonly conducted to identify genes that determine cell fates such as differentiation [1], induction of pluripotency [2], and direct reprogramming [3]. Those experiments are now indispensable in biological and medical research. Although intervention experiments identify a causal gene responsible for a phenotype of interest, they are timeconsuming and costintense. Therefore, it is very important to prioritize and focus on causal genes with high confidence. However, it is difficult to infer causal effects only from observational data. This task coincides with inferring causal effects that are established in Statistics. Note that in this problem setting, a causal effect is different from a regressiontype effect of association [4]. In fact, previous studies suggested that representative regression methods such as lasso and elastic net are not appropriate for our goal [46].
Recently, there has been much progress to address this problem by employing the interventioncalculus when the directed acyclic graph (DAG) is absent (IDA) [46] for the design of efficient intervention experiments. IDA combines two methods: (1) inference the unknown underlying DAG model from observational data by the PCalgorithm [7] and (2) estimating causal effects on the basis of DAG using interventioncalculus; furthermore, it provides estimated lower bounds of total causal effects from observational data. The PCalgorithm is computationally feasible and appropriate for highdimensional settings. In addition, it has the desirable property to achieve high computational efficiency as a function of sparseness of the true underlying DAG model.
In spite of these advantages, the PCalgorithm requires the Gaussianity assumption to use partial correlations to test conditional independence, and this assumption is not necessarily true in real data sets. Because the normality assumption is restrictive and conclusions inferred under this assumption could be misleading, it is desirable to relax the Gaussian assumption.
On the other hand, nonparanormal methods that use a semiparametric Gaussian copula have been proposed for estimating sparse undirected graphs and exhibit significant improvement in the performance because the normality assumption is relaxed [8,9]. The main idea of the nonparanormal method is to exploit the nonparametric correlation coefficient instead of Pearson’s correlation coefficient for estimation. Although this is the simplest alternative procedure, the nonparanormal graphical model could be a viable alternative for the Gaussian graphical model.
Consequently, we present nonparanormal IDA (NPNIDA), which uses nonparametric partial correlations to test conditional independencies in the PCalgorithm for interventioncalculus. In our method, the Gaussian assumption in the PCalgorithm is naturally relaxed by using nonparametric partial correlation. Although the nonparanormal method has been successfully applied to estimating undirected graphs in previous studies, we show that it works well for estimating DAGs in the PCalgorithm. Next, we applied our method to Arabidopsis thaliana microarray data and mouse microarray data to demonstrate that NPNIDA outperforms IDA in exploring regulators of the flowering time in A. thaliana and regulators that control the browning of white adipocytes in mice.
In summary, the three main contributions of this work are: (1) introduction of a nonparanormal method for inference of the unknown underlying DAG model from observational data in the expansive framework of the PCalgorithm, (2) combination of the nonparanormal method and the PCalgorithm significantly improves the performance in estimating DAGs on synthetic data, and (3) NPNIDA is effective in exploring regulators that control specific phenotypes of interest.
Methods
We first introduce the IDA procedure. IDA consists of (1) inference of the unknown underlying DAG model from observational data by PCalgorithm and (2) estimation of causal effects based on the DAG using interventioncalculus. Then, we introduce the nonparanormal method for PCalgorithm. Finally, we present the combination of the nonparanormal method for PCalgorithm and estimating causal effects as NPNIDA algorithm.
Inference DAGs with the PCalgorithm
Let G = (V, E) be a graph consisting of vertices V and a set of edges E ⊆ V × V. In our context, the vertices represent random variables X_{1}, …, X_{p}, and Y. The edges represent relationships between pairs of these variables. It is possible that some DAGs fulfill the Markov condition and encode the same list of conditional independencies. Two DAGs are observationally equivalent only if they have the same skeleton and same sets of vstructures, i.e., two converging arrows whose tails are connected by an arrow. In this way, DAGs can be partitioned into equivalent classes, where all members are observationally equivalent and represent the same conditional independence. In a given conditional independence set of DAGs, one can only determine a DAG up to its equivalence class. The equivalence class is called completed partially directed acyclic graph (CPDAG). It has the same skeleton as every DAG in the equivalence class and directed edges only where all DAGs in the equivalence class have the same directed edge. Arrows that point into one direction for some DAGs in the equivalence class and in the other direction for other DAGs in the equivalence class are represented by undirected edges (Figure 1). By assuming that random variables are multivariate normally distributed, conditional independencies can be inferred from a partial correlation between X_{i} and X_{j} given a set of other variables S that equals zero:
Figure 1. Example of CPDAG. CPDAG G with the DAGs G_{1}, G_{4} that are in its equivalence class.
We then used the sample version of the PCalgorithm to estimate the corresponding CPDAG. This involves multiple testing for Fisher’s Ztransformed partial correlations,
Because has a N(0, (n − S − 3)^{− 1}) distribution if ρ_{ijS} = 0, we conclude that ρ_{ijS} ≠ 0 if
where Φ is the standard normal distribution function and α is a tuning parameter, which can be interpreted as the significance level of a single partial correlation test. Choosing an appropriate value for α is difficult but, for example, can be done with the Bayesian information criterion.
First, the PCalgorithm generates a skeleton on the basis of conditional independencies. The outline of the PCalgorithm is shown in Figure 2. The complete PCalgorithm is described in detail in a precious work [7]. Note that the PCalgorithm employs partial correlation to test conditional independency.
Figure 2. PCalgorithm for generating the skeleton.
Estimating causal effects using interventioncalculus
Again, we considered p + 1 random variables X_{1}, … X_{p}, Y (also referred to as X_{1}, … X_{p}, X_{p + 1}). Any distribution that is generated from a DAG with independent error is called Markovian with respect to the DAG. Therefore, any Markovian distribution can be factorized as
To represent the effect of an intervention of a set of variables, a do operator is introduced. We denoted the distribution of Y that would occur if the treatment condition was enforced uniformly over the population via some intervention as . For a Markovian model, the distribution generated by an intervention on the set of variables X_{1}, …, X_{p + 1} is given by the following truncated factorization formula:
Where f(x_{j}pa_{j}) are the preintervention conditional distributions. Note that this formula employs the DAG structure to write the interventional distribution on the lefthand side in terms of preintervention conditional distributions on the righthand side. The distribution of Y = X_{p + 1} after an intervention can be obtained by marginalizing out x_{1}, …, x_{p} in equation (2). This can be written as follows:
where f(·) and represent preintervention distributions. We can summarize the distribution generated by the intervention by its mean
and define the causal effect of on Y by
Under the assumption that X_{1}, … X_{p}, Y are jointly Gaussian, it is easy to compute the causal effects. Gaussianity implies that is linear in and pa_{i} That is
for some values, γ_{0}, γ_{i}, and , where pa_{i} is the cardinality of the set pa_{i}. Then, we derive
which is linear in . The causal effect of X_{i} on Y when Y ∉ pa_{i} can be computed with (3) yielding γ_{i}, which is simply the regression coefficient of X_{i} in the regression of Y on X_{i} and pa_{i}. When Y ∈ pa_{i}, the causal effect becomes zero, because Y is a direct cause of X_{i}. Therefore, the causal effect of X_{i} on Y is given by the following equation:
where Y ∼ X_{i} + pa_{i} is a shorthand notation for the linear regression of Y on X_{i} and pa_{i}. Consequently, in the Gaussian case, the causal effect does not depend on the value of and can be interpreted as
Next, we will describe the IDA algorithm in detail. For simplicity, we only show the case in which the PCalgorithm gives the correct CPDAG. Details of the sample version of the IDA algorithm can be found in previously conducted studies [57]. On the basis of CPDAG, which is inferred from the PCalgorithm, we can compute the causal effect for every DAG in the equivalence class, which consists of multisets. Multisets differ from normal sets in that the multiplicity of the elements is taken into account. The IDA algorithm is given in Figure 3.
Figure 3. IDA algorithm.
For computing the causal effects θ_{ij} of X_{i} on Y in DAG G_{j}, the IDA algorithm simply computes the regression coefficient of X_{i} in a regression of Y on X_{i} and pa_{i}, which is denoted by . This procedure is performed for every DAG in the equivalence class yielding a vector of length j of possible causal effects, where j is the number of DAGs in the equivalence class. Computing the effect of every X_{1}, …, X_{p} on Y yields a matrix Θ with θ_{ij} entries. We describe the illustrative procedure of IDA in Figure 4. When we obtain a single value for the estimated causal effect, we can take the minimal absolute value of the multiset as a lower bound from Θ for the true absolute intervention effect. This procedure intends to reduce the number of false positives. From a practical point of view, because the number of false positive should be kept as low as possible, it is desirable to provide conservative results.
Figure 4. Illustrative procedure of IDA.
Nonparanormal method for NPNIDA
In the PCalgorithm, the conditional independency is inferred from a sample partial correlation between X_{i} and X_{j} given a set of other variables S. As described in the section of the PCalgorithm, a sample partial correlation coefficient can be obtained by calculating a sample Pearson’s correlation coefficient. This fact enables us to relax the Gaussianity assumption of the PCalgorithm by using the nonparanormal method. On the basis of [9], we derive the nonparanormal version of the PCalgorithm (NPNPC algorithm). Let f = (f_{1}, …, f_{p}) be a set of monotonic univariate functions and let ∑ ^{0} ∈ ℝ^{p × p} be a positive definite correlation matrix with diag(∑^{0}) = 1. We suppose that the pdimensional random variable X = (X_{1}, …, X_{p})^{T} has a nonparanormal distribution X ∼ NPN_{p}(f, ∑ ^{0}) if f(X) ≡ (f_{1}(X_{1}), …, f_{p}(X_{p}))^{T} ∼ N(0, ∑ ^{0}). For continuous distributions, the nonparanormal family is equivalent to the Gaussian copula family [8]. It has been shown that the nonparanormal family is much richer than the normal family. The conditional independence is encoded by Ω^{0} = (∑^{0})^{− 1}. Therefore, we can write the following formula given a set of other variables S:
For Gaussian copula distributions, the following important lemma connects Spearman’s correlation coefficient to the underlying Pearson’s correlation coefficient [8,9].
Lemma 1.[10] By assuming X ∼ NPN(f, ∑ ^{0}), we have .
According to the lemma, we can define the following estimator for the unknown correlation matrix ∑ ^{0}: for i ≠ j, and . Finally, if we define , we obtain the following formula that connects Spearman’s correlation coefficient and the nonparanormal partial correlation coefficient .
Instead of ρ_{ijS} in (1), we employ to test conditional independence for estimating CPDAG in the PCalgorithm. For simplicity, we denoted the method that combines the NPNPC algorithm and IDA as NPNIDA. We describe the pseudo code of the NPNIDA algorithm in Figure 5.
Figure 5. NPNIDA algorithm.
Results
Experimental settings
Two experiments were conducted for different purposes. The first purpose was to evaluate the performance of the NPNPC algorithm on synthetic data when the Gaussianity assumption is not true. According to a previous study [7], the four metrics of performance, i.e., true positive rate (TPR), false positive rate (FPR), true discovery rate (TDR), and structural hamming distance (SHD), are used. TPR, FPR, and TDR are defined as follows:
where gaps indicate the pairs of vertex that are not linked.
SHD is the number of edge insertions, deletions, and flips to transfer the estimated DAG into the true DAG [7]. A large SHD indicates a poor fit, whereas a small SHD indicates a good fit.
To simulate the case that the Gaussian assumption is not true, we generated multivariate data with dependency structure by a given DAG with nodes corresponding to random variables and added standard Cauchy distribution in partial samples using the rmvDAG function in the pcalg package. Figure 6 depicts the normal distribution and the Cauchy distribution. As shown in Figure 6, the standard Cauchy distribution is tailheavier than the standard normal distribution, and is quite different from the standard normal distribution.
Figure 6. Comparison of standard normal distribution and standard Cauchy distribution.
The second purpose was to evaluate the performance of NPNIDA when applied to predicting regulators of the flowering time as phenotype of interest using an A. thaliana microarray data set and regulators that control the browning of white adipocytes in mice using mouse microarray data. These two data sets are entirely different in terms of species and target variables. Therefore, if we obtain the consistent results thorough performance comparison of the methods, the consequence will be solid. We implemented the R language and employed the pcalg package for the PCalgorithm and IDA [11].
Performance evaluation of the NPNPC algorithm
We evaluated the performance under all combinations of settings listed in Table 1. The mixing rate indicates the percentage of samples whose error distribution was drawn from the standard Cauchy distribution. The higher the mixing rate, the less accurate is the Gaussianity assumption. To evaluate this hypothesis thoroughly, we repeated this experiment 50 times and averaged the values of TPR, FPR, TDR, and SHD. To explain the essence of these experiments, we show the representative results of the settings (α = 10^{− 4}, cp = 0.005) in Table 2. All experimental results are shown in Additional file 1. To estimate the causal effects, estimated DAGs inferred by NPNPC were employed. Because we considered the situation that the Gaussianity assumption is violated, we determined the significance level α on the basis of the average SHD when the mixing rate was 1. Average SHDs under the different probabilities of connecting one node to another node are shown in Figure 7. Because the average SHD was very small, we employed estimated DAGs when the significance level α was set to 10^{−4} to further estimate the causal effects.
Table 1. Parameter setting for performance evaluation of the NPNPC algorithm
Table 2. Performance comparison between PCalgorithm and NPNPC algorithm
Additional file 1. Experimental results of all combined parameter settings for performance comparison between the NPNPC algorithm and the PCalgorithm.
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Figure 7. Average SHD for p = 100 and 200 under different conditions of significance levels. (a) (c) cp = 0.005, (b) (d) cp = 0.01.
Performance evaluation of the NPNIDA algorithm
Exploring regulators of the flowering time in A. thaliana
We employed the microarray data set of A. thaliana and the flowering time used in a previous study [7]. The data set consisted of 21326 probes and 47 samples. We regarded the nine known regulators of the flowering time (DWF4, FLC, FRI, RPA2B, SOC1, PDHE1 BETA, ATGH9B5, LRR, and OTLD1) as truepositive genes. Because IDA requires a significant amount of computation time, we selected genes for further estimation that had higher absolute correlation coefficients against flowering time until a predefined number was reached (correlation screening). In this study, the numbers of remaining genes obtained by correlation screening were set to 500, 1000, 1500, and 2000. According to the description in the previous section, we set the significance level α to 10^{−4} for estimating DAGs. Figure 8 shows the enrichment curves under the different conditions of correlation screening. Vertical axes represent the average numbers of true positives and horizontal axes indicate the ranks of causal effects. To quantitatively compare the performances of IDA and NPNIDA, we also summarized the area under the enrichment curves (AUCs) under the different conditions of correlation screening in Table 3.
Figure 8. Enrichment curves on exploring regulators of the flowering time in Arabidopsis thaliana. (a) Correlation screening is set to 500, (b) correlation screening is set to 1000, (c) correlation screening is set to 1500, and (d) correlation screening is set to 2000.
Table 3. AUC values under the different conditions of correlation screening on exploring regulators of the flowering time in Arabidopsis thaliana
Exploring the regulators that control the browning of white adipocytes in mice
We employed the mouse microarray data set of white adipose tissue (WAT) obtained from a previous study [12]. The data set consisted of 43681 probes and 349 WAT samples. According to a previous review [13], we regarded Ucp1 as marker of brown adipose tissue (BAT). We selected genes that had higher absolute correlation coefficients against Ucp1 until a predefined number was reached. The numbers of remaining genes were set to 2000, 3000, and 4000. Table 4 shows the known regulators of the differentiation of WAT to BAT for the different conditions of correlation screening. Note that there are no true positive genes when the number of remaining genes obtained by correlation screening is below 1000.
Table 4. Known regulators of the differentiation of WAT to BAT
Because there are two probes for Tbx15, i.e., merckNM_009323_at and merckNM_01154_a_at, we regarded the probe that had the larger causal effect as Tbx15. Figure 9 shows the enrichment curves under different conditions of correlation screening. Vertical axes represent the average numbers of true positives and horizontal axes indicate the ranks of causal effects. The AUCs under the different conditions of correlation screening are summarized in Table 5.
Figure 9. Enrichment curves on exploring the regulators that control the browning of white adipocytes. (a) Correlation screening is 2000, (b) correlation screening is 3000, and (c) correlation screening is 4000.
Table 5. AUC values under the different conditions of correlation screening on exploring the regulators that control the browning of white adipocytes in mice
Discussion
From Table 2, it appears that the NPNPC algorithm consistently outperforms the PCalgorithm with regard to all performance metrics, TPR, FPR, TDR, and SHD. Furthermore, as the mixing rate increases, the performance values of the PCalgorithm decrease. This result clearly shows that the PCalgorithm does not work when the Gaussianity assumption is not true. In contrast, the NPNPC algorithm works well when mixing rate is high. In other words, the NPNPC algorithm does not require the Gaussianity assumption of data distribution. In terms of FPR, it appears that the FPR of the NPNPC algorithm is strictly suppressed under all set values. In the NPNPC algorithm, all performance metrics improved when the number of samples was 100 compared to when the number of samples was 50. From these observations, it can be concluded that the NPNPC algorithm is robust and does not depend on data distribution, unlike the PCalgorithm. This characteristic is very attractive from the practical point of view.
From Table 3 and Figure 8, NPNIDA consistently outperformed IDA on exploring regulators of the flowering time in A. thaliana. In particular, when the correlation screenings were set to 1000 and 1500, the difference in the AUCs between NPNIDA and IDA increased. When the correlation screenings were set to 500, both NPNIDA and IDA worked well. This result indicates that the known regulators were sufficiently recovered against the flowering time within genes that have high absolute correlation coefficients. Therefore, although we do not know whether the unknown regulators have high absolute correlation coefficients against the flowering time, it would be a good strategy from a practical perspective to explore novel regulators from genes that have high absolute correlation coefficients against the flowering time.
From Table 5 and Figure 9, NPNIDA consistently outperformed IDA on exploring regulators that control the browning of white adipocytes in mice. In particular, when the correlation screenings were set to 3000 and 4000, the difference in the AUCs between NPNIDA and IDA increased. These results suggest that NPNIDA successfully enriches known regulators at the top ranks when the number of available genes increases. Consequently, our two experiments clearly demonstrated that NPNIDA performs better than IDA.
To discuss whether the Gaussian assumption is true or not in the data sets used in this study, we applied the ShapiroWilk test to microarray data and a target variable of interest, which tests the null hypothesis that a sample came from a normally distributed population [14]. We show a histogram of the pvalues of the ShapiroWilk test for target phenotypes of interest (flowering time in A. thaliana and gene expression of Ucp1 in mice) and gene expression levels in Figure 10. We also show individual histograms of phenotypes of interest and expression levels of the known regulators in Figure 11; the respective pvalues of the ShapiroWilk test are summarized in Table 6. From Figure 10, it appears that the pvalues of most genes were <0.05 for both A. thaliana and mouse WAT microarray data. In other words, the normality assumption was violated in most genes. These results justify the use of NPNIDA rather than IDA, because the latter method requires a normal distribution. With regard to the A. thaliana data, it appears that the pvalues for flowering time, FLC, FRI, RPA2B, ATGH9B5, and LRR were <0.05 (Table 6); as shown in Figure 11, their distributions were skewed. With regard to the mouse WAT data, the pvalues of all genes were very small (Table 6). As shown in Figure 12, their distributions were also skewed. These results strongly suggest that NPNIDA, which does not require the Gaussian distribution, works better than IDA.
Figure 10. Histogram of pvalues of the ShapiroWilk test. (a)Arabidopsis thaliana microarray data and (b) mouse WAT microarray data (logarithmic scale).
Figure 11. Distributions of flowering time and expression levels of the nine known regulators of the flowering time. Each panel is a histogram of the flowering time and each gene’s expression.
Table 6. pvalues of the ShapiroWilk test of target variables and known regulators
Figure 12. Distributions of expression levels of the BAT marker gene Ucp1 and the five known genes.Tbx has two different probes. Each panel is a histogram of each gene’s expression.
Although the method presented here performs significantly better than IDA, one might consider the difference in the performance as too small. However, within the known regulators of flowering time in A. thaliana, four genes (PDHE1 BETA, ATGH9B5, LRR, and OTLD1) were experimentally validated using the IDAbased method [6]. Therefore, one should take into account that some of the known regulators were obtained on the basis of the Gaussian assumption. Thus, it is possible that the improvement achieved with NPNIDA is greater than the experimental results shown in this study.
In summary, the two main results of our experimental study are: (1) the NPNPC algorithm works better than the PCalgorithm in estimating DAGs on synthetic data, and (2) NPNIDA performs better than does IDA in predicting regulators of the flowering time in A. thaliana and regulators of the differentiation of WAT to BAT in mice on the basis of microarray data. From these two results, we conclude that the performance to estimate DAGs contributes to the accurate prediction of causal effects.
From a practical point of view, because regulators that control the browning of white adipocytes have not been identified only from microarray data of WAT so far, it might be worthwhile to demonstrate this possibility for the first time using our novel method.
For further performance enhancement, we consider that NPNIDA could be embedded into CStaR (causal stability ranking) [6] in the future. CStaR uses IDA with stability selection [6,15] and the performance was greatly improved compared to plain IDA. By simply replacing IDA with NPNIDA in the estimation process for causal effects, it would be easy to combine NPNIDA with CStaR.
Conclusions
We presented NPNIDA, which uses nonparametric partial correlations, to test conditional independencies in the PCalgorithm for interventioncalculus. To reveal the contribution of estimating DAGs, we evaluated the NPNPC algorithm to estimate DAGs with the nonparanormal method. The two main results of our experimental study are: (1) the NPNPC algorithm works better than the PCalgorithm in estimating DAGs on synthetic data, and (2) NPNIDA performs better than IDA does in predicting regulators of the flowering time in A. thaliana and regulators of the differentiation of WAT to BAT in mice. Considering these two results, we concluded that the performance to estimate DAGs contributes to the accurate prediction of causal effects.
Thus, the simplest alternative procedure we used for our proposed method enables us to design efficient intervention experiments and can be applied to a wide range of research purposes, including drug discovery and medicine, because of its generality.
Availability of supporting data
The microarray data sets used in this study are deposited in Additional file 2.
Additional file 2. The mouse microarray data set of microarray data set of white adipose tissue (WAT) and the microarray data set of A. thaliana and the flowering time.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
RT proposed the method, implemented, conduct experiments and wrote the manuscript. CS edited the tables. SF supervised the research project. All authors read and approved the final manuscript.
Acknowledgements
We are grateful to Tatsumi Yamazaki, the president of Forerunner Pharma Research, for his support and encouragement throughout the study. We also thank Takumi Nagata and Kenji Yoshida for their valuable discussions.
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