Differential expression analysis identifies differences in gene expression levels between groups Y (Costa-Silva et al., 2017). A gene Xi may be differentially expressed due to multiple reasons. For example, Xi may cause Y, or a confounder C may explain the relation between Xi and Y such that Xi←C→Y. In this paper, we take expression analysis a step further by pinpointing causal relations from expression levels regardless of the variable type of Y (discrete or continuous). We in particular seek to discover patient-specific root causal genes from bulk RNA-seq data, which we carefully define below.

We represent a biological system in bulk RNA-seq as a causal graph G – such as in Figure 2a – where p vertices X~ represent true gene expression levels in a bulk sample and Y denotes the patient symptoms or diagnosis. The set X~ contains thousands of genes in practice. Directed edges between the vertices in G refer to direct causal relations. We assume that gene expression causes patient symptoms but not vice versa so that no edge from Y is directed towards X~. The set Pa(X~i) refers to the parents of X~i∈X~, or those variables with an edge directed into X~i. For example, Pa(X~2)={X~1,X~3} in Figure 2a. A root vertex corresponds to a vertex with no parents.

We can associate G with the structural equation X~i=fi(Pa(X~i),Ei) for each X~i∈X~ that links each vertex to its parents and error term Ei (Pearl, 2009). The error term Ei is not simply a regression residual but instead represents a combination of unobserved factors that only influence X~i, such as unobserved transcriptional regulators, certain genetic variants and specific environmental conditions. We thus also include the error terms E in the directed graph of Figure 2b. All root vertices are error terms and vice versa. The root causes of Y are the error terms that cause Y, or have a directed path into Y. We define the root causal strength (RCS) of X~i on Y as the following absolute difference (Figure 2c):(1)Φi=|E(Y|Pa(X~i),Ei)−E(Y|Pa(X~i))|=|E(Y|Pa(X~i),X~i)−E(Y|Pa(X~i))|.

We prove the last equality in the Materials and methods. As a result, RCS Φi directly measures the contribution of the gene X~i on Y according to its error term Ei without recovering the error term values. The algorithm does not impose distributional assumptions or functional restrictions such as additive noise to estimate the error term values as an intermediate step. Moreover Φi is patient-specific because the values of Pa(X~i) and X~i may differ between patients. We have Φi=0 when Ei is not a cause of Y, and we say that the gene X~i is a patient-specific root causal gene if Φi≫0, or its (conditional) root causal effect is large as depicted by the red ellipse in Figure 1b.

We propose an algorithm called Root Causal Strength using Perturbations (RCSP) that estimates Φ={Φ1,…,Φp} from genes measured in both bulk RNA-seq and Perturb-seq datasets derived from possibly independent studies but from the same tissue type. We rely on bulk RNA-seq instead of single-cell RNA-seq in order to obtain many samples of the label Y. We focus on statistical estimation rather than statistical inference because Φi>0 when Ei causes Y under mild conditions, so we reject the null hypothesis that Φi=0 for many genes if many gene expression levels cause Y. However, just like a machine typically breaks down due to only one or a few root causal problems, we hypothesize that only a few genes have large RCS scores Φi≫0 even in complex disease.

Estimating Φ requires access to the true gene expression levels X~ and the removal of the effects of confounding. We first control for batch effects representing unwanted sources of technical variation such as different sequencing platforms or protocols. We however can only obtain imperfect counts X from RNA sequencing even within each batch (Figure 2d). Measurement error introduces confounding as well because it prevents us from exactly controlling for the causal effects of the gene expression levels. Investigators usually mitigate measurement error by normalizing the gene expression levels by sequencing depth. We show in the Materials and methods that the Poisson distribution approximates the measurement error distribution induced by the sequencing process to high accuracy (Choudhary and Satija, 2022; Sarkar and Stephens, 2021). We leverage this fact to eliminate the need for normalization by sequencing depth using an asymptotic argument where the library size N approaches infinity. N takes on a value of at least ten million in bulk RNA-seq, but we also empirically verify that the theoretical results hold well in the Appendix. We thus eliminate the Poisson measurement error and batch effects by controlling for the batches B but not N in non-linear regression models.

We in particular show that Φi in Equation 1 is also equivalent to:(2)Φi=|E(Y|SP(X~i),Xi,B)−E(Y|SP(X~i),B)|,

where SP(X~i) refers to the surrogate parents of X~i, or the variables in X associated with Pa(X~i)⊆X~. RCSP can identify (an appropriate superset of) the surrogate parents of each variable using perturbation data because perturbing a gene changes the marginal distributions of its downstream effects – which the algorithm detects from data under mild assumptions (Figure 2e and f). The algorithm thus only transfers the binary presence or absence of causal relations from the single cell to bulk data – rather than the exact functional relationships – in order to remain robust against discrepancies between the two data types; we empirically verify the robustness in the Appendix. RCSP finally performs the two non-linear regressions needed to estimate E(Y|SP(X~i),Xi,B) and E(Y|SP(X~i),B) for each Φi. We will compare Φi against Statistical Dependence (SD), a measure of correlational strength defined as Ωi=|E(Y|Xi,B)−E(Y|B)| where we have removed the conditioning on SP(X~i).

We simulated 30 bulk RNA-seq and Perturb-seq datasets from random directed graphs summarizing causal relations between gene expression levels. We performed single gene knock-down perturbations over 2500 genes and 100 batches. We obtained 200 cell samples from each perturbation, and another 200 controls without perturbations. We therefore generated a total of 2501×200=500, 200 single cell samples for each Perturb-seq dataset. We simulated 200 bulk RNA-seq samples. We compared RCSP against the Additive Noise Model (ANM; Peters, 2014; Strobl and Lasko, 2023a), the Linear Non-Gaussian Acyclic Model (LiNGAM; Peters, 2014; Strobl and Lasko, 2022b), CausalCell (Wen et al., 2023), univariate regression residuals (Uni Reg), and multivariate regression residuals (Multi Reg). The first two algorithms are state-of-the-art approaches used for error term extraction and, in theory, root causal discovery. See Materials and methods for comprehensive descriptions of the simulation setup and comparator algorithms.

We summarize accuracy results in Figure 2g using the Root Mean Squared Error (RMSE) to the ground truth Φ values. All statements about pairwise differences hold true at a Bonferonni corrected threshold of 0.05/5 according to paired two-sided t-tests, since we compared RCSP against a total of five algorithms. RCSP estimated Φ most accurately by a large margin. ANM and LiNGAM are theoretically correct under their respective assumptions, but they struggle to outperform standard multivariate regression due to the presence of measurement error in RNA-seq (Appendix). Feature selection and causal discovery with CausalCell did not improve performance. Univariate regression performed the worst, since it does not consider the interactions between variables. RCSP achieved the lowest RMSE while completing in about the same amount of time as multivariate regression on average (Figure 2h). RCSP maintained the lowest RMSE even in the cyclic case, and the performance of the algorithm remained robust to differences between the directed graphs underlying the bulk RNA-seq and Perturb-seq data (Appendix). We conclude that RCSP both scalably and accurately estimates Φ.

We will cluster the RCS values in real data to find patient subgroups. We therefore also performed hierarchical clustering using Ward’s method (Ward, 1963) on the values of Φ estimated by RCSP with the synthetic data. We then computed the RMSEs and averaged them within each cluster. We found that RCSP maintained low average RMSE values regardless of the number of clusters considered (Figure 2i). We conclude that RCSP maintains accurate estimation of Φ across different numbers of clusters.

We ran RCSP on a bulk RNA-seq dataset of 513 individuals with age-related macular degeneration (AMD; GSE115828) and a Perturb-seq dataset of 247,914 cells generated from an immortalized retinal pigment epithelial (RPE) cell line (Ratnapriya et al., 2019; Replogle et al., 2022). The Perturb-seq dataset contains knockdown experiments of 2077 genes overlapping with the genes of the bulk dataset. We set the target Y to the Minnesota Grading System score, a measure of the severity of AMD based on stereoscopic color fundus photographs. We always included age and sex as a biological variable as covariates. We do not have access to the ground truth values of Φ in real data, so we evaluated RCSP using seven alternative techniques. See Materials and methods for a detailed rationale of the evaluation of real data. RCSP outperformed all other algorithms in this dataset (Appendix). We therefore only analyze the output of RCSP in detail here.

AMD is a neurodegenerative disease of the aging retina (Hadziahmetovic and Malek, 2020), so age is a known root cause of the disease. We therefore determined if RCSP identified age as a root cause. Note that RCSP does not need perturbation data of age to compute the RCS values of age, since age has no parents in the directed graph. The algorithm estimated a heavy tailed distribution of the RCS values indicating that most of the RCS values deviated away from zero (Figure 3a). The Deviation of the RCS (D-RCS), or the deviation from an RCS value of zero, measures the tailedness of the distribution while preserving the unit of measurement. The D-RCS of age corresponded to 0.46 – more than double that of the nearest gene (Figure 3d). We conclude that RCSP correctly detected age as a root cause of AMD.

Root causal genes typically affect many downstream genes before affecting Y. We therefore expect to identify few root causal genes but many genes that correlate with Y. To evaluate this hypothesis, we examined the distribution of D-RCS relative to the distribution of the Deviation of Statistical Dependence (D-SD), or the deviation from an SD value of zero, in Figure 3b. Notice that the histogram of D-RCS scores in Figure 3b mimics a folded distribution of Figure 1b. Thus, few D-RCS scores had large values implying the existence of only a few root causal genes. In contrast, most of the D-SD scores had relatively larger values concentrated around 0.10 implying the existence of many genes correlated with Y. We conclude that RCSP identified few root causal genes rather than many correlated genes for AMD.

The pathogenesis of AMD involves the loss of RPE cells. The RPE absorbs light in the back of the retina, but the combination of light and oxygen induces oxidative stress, and then a cascade of events such as immune cell activation, cellular senescence, drusen accumulation, neovascularization and ultimately fibrosis (Barouch and Miller, 2007). We therefore expect the root causal genes of AMD to include genes involved in oxidative stress during early pathogenesis. The gene MIPEP with the highest D-RCS score in Figure 3d indeed promotes the maturation of oxidative phosphorylation-related proteins (Shi et al., 2011). The second gene SLC7A5 is a solute carrier that activates mTORC1 whose hyperactivation increases oxidative stress via lipid peroxidation (Nachef et al., 2021; Go et al., 2020). The gene HEATR1 is involved in ribosome biogenesis that is downregulated by oxidative stress (Turi et al., 2018). The top genes discovered by RCSP thus identify pathways known to be involved in oxidative stress. We further verified that measurement error did not explain their large D-RCS scores in the Appendix.

We subsequently jointly analyzed the D-RCS values of all 2077 genes. We performed pathway enrichment analysis that yielded one pathway ‘amino acid transport across the plasma membrane’ that passed an FDR threshold of 5% (Figure 3c). The leading edge genes of the pathway included the solute carriers SLC7A5 and SLC1A5. These two genes function in conjunction to increase the efflux of essential amino acids out of the lysosome (Nicklin et al., 2009; Beaumatin et al., 2019). Some of these essential amino acids like L-leucine and L-arginine activate mTORC1 that in turn increases lipid peroxidation induced oxidative stress and the subsequent degeneration of the RPE (Nachef et al., 2021; Go et al., 2020). We conclude that pathway enrichment analysis correctly identified solute carrier genes involved in a known pathway promoting oxidative stress in AMD.

We next ran drug enrichment analysis with the D-RCS scores. The top compound arsenous acid inhibits RPE proliferation (Su et al., 2020), but the other three significant drugs have therapeutic potential (Figure 3e). Busulfan decreases the requirement for intravitreal anti-VEGF injections (Dalvin et al., 2022). Genistein is a protein kinase inhibitor that similarly attenuates neovascularization (Kinoshita et al., 2014) and blunts the effect of ischemia on the retina (Kamalden et al., 2011). Finally, a metabolite of the antiviral agent 3’-azido-3’-deoxythymidine inhibits neovascularization and mitigates RPE degeneration (Narendran et al., 2020). We conclude that the D-RCS scores identified promising drugs for the treatment of AMD.

Hierarchical clustering and UMAP dimensionality reduction on the patient-specific RCS values revealed four clear clusters of patients by the elbow method on the sum of squares plot (Figure 3f and g, respectively). The RCS scores of most of the top genes exhibited a clear gradation increasing only from the left to the right hand side of the UMAP embedding; we plot an example in Figure 3h. We found three exceptions to this rule among the top 30 genes (example in Figure 3i and see the Appendix). RCSP thus detected genes with large RCS scores primarily in the black cluster of Figure 3g. Pathway enrichment analysis within this cluster alone yielded supra-significant results on the same pathway detected in the global analysis (Figure 3j versus Figure 3c). Furthermore, drug enrichment analysis results by cluster confirmed that patients in the black cluster with many root causal genes are likely the hardest to treat (Appendix). We conclude that RCSP detected a subgroup of patients whose root causal genes have large RCS scores and involve known pathogenic pathways related to oxidative stress.

We next ran RCSP on 137 samples collected from CD4+ T cells of multiple sclerosis (MS; GSE137143) as well as Perturb-seq data of 1,989,578 K562 cells, which can be genetically engineered into artificial antigen-presenting cells for expanding T cells (Butler and Hirano, 2014; Replogle et al., 2022). We set the target Y to the Expanded Disability Status Scale score, a measure of MS severity. RCSP outperformed all other algorithms in this dataset as well (Appendix).

MS progresses over time, and RCSP correctly detected age as a root cause of MS severity with RCS values deviating away from zero (Figure 4a). The distribution of gene D-RCS scores concentrated around zero with a long tail, whereas the distribution of gene D-SD scores concentrated around a relatively larger value of 0.3 (Figure 4b). RCSP thus detected an omnigenic root causal model with a few root causal genes but many correlated genes.

MS is an inflammatory neurodegenerative disease that damages the myelin sheaths of nerve cells in the brain and spinal cord. T cells may mediate the inflammatory process by crossing a disrupted blood brain barrier and repeatedly attacking the myelin sheaths (Fletcher et al., 2010). Damage induced by the T cells also perturbs cellular homeostasis and leads to the accumulation of misfolded proteins (Andhavarapu et al., 2019). The root causal genes of MS thus likely include genes involved in T cell infiltration across the blood brain barrier.

Genes with the highest D-RCS scores included MNT, CERCAM, and HERPUD2 (Figure 4d). MNT is a MYC antagonist that modulates the proliferative and pro-survival signals of T cells after engagement of the T cell receptor (Gnanaprakasam and Wang, 2017). Similarly, CERCAM is an adhesion molecule expressed at high levels in microvessels of the brain that increases leukocyte transmigration across the blood brain barrier (Starzyk et al., 2000). HERPUD2 is involved in the endoplasmic-reticulum associated degradation of unfolded proteins (Kokame et al., 2000). Genes with the highest D-RCS scores thus serve key roles in known pathogenic pathways of MS.

We found multiple genes with high D-RCS scores in MS, in contrast to AMD where age dominated (Figure 4d versus Figure 3d). Measurement error did not account for the high scores (Appendix). We performed pathway enrichment analysis using the D-RCS scores of all genes and discovered two significant pathways at an FDR corrected threshold of 5%: ‘adenomatous polyposis coli (APC) truncation mutants have impaired AXIN binding’ and ‘EPH-ephrin signaling’ (Figure 4c). APC and AXIN are both members of the Wnt signaling pathway and regulate levels of beta-catenin (Spink et al., 2000). Furthermore, inhibition of Wnt/beta-catenin causes CD4+ T cell infiltration into the central nervous system via the blood brain barrier in MS (Lengfeld et al., 2017). Ephrins similarly regulate T cell migration into the central nervous system (Luo et al., 2016) and are overexpressed in MS lesions (Sobel, 2005). The APC-AXIN and EPH-ephrin pathways are thus consistent with the known pathophysiology of central nervous system T cell infiltration in MS.

We subsequently performed hierarchical clustering of the RCS scores. The within cluster sum of squares plot in Figure 4e revealed the presence of three clusters by the elbow method. We plot the three clusters in a UMAP embedding in Figure 4f. The clusters did not show a clear relationship with MS symptom severity (Appendix) or the levels of the top most genes of Figure 4d; we plot the MNT gene as an example in Figure 4g. However, further analyses with additional genes revealed that the distribution of many lower ranked genes governed the structure of the UMAP embedding (Appendix). The D-RCS scores of each cluster also implicated different mechanisms of T cell pathology including APC-AXIN in the green cluster, disturbed T cell homeostasis in the pink cluster and platelet enhanced T cell autoreactivity in the blue cluster (Appendix).

Global drug enrichment analysis did not yield any significant drugs even at a liberal FDR threshold of 10%. We thus ran drug enrichment analysis in each cluster of Figure 4f. The blue and pink clusters again did not yield significant drugs. However, the third green cluster identified the cysteine cathepsin inhibitors dipeptide-derived nitriles, phenylalinine derivatives, e-64, L-006235 and L-873724 (Figure 4h); statistical significance of the first three held even after correcting for multiple comparisons with the Bonferroni adjustment of 0.05/4 on the q-values. The leading edge genes of the significant drugs included the cathepsins CTSL, CTSS, and CTSB exclusively. These drug enrichment results corroborate multiple experimental findings highlighting the therapeutic efficacy of cathepsin inhibitors in a subgroup of MS patients responsive to interferon therapy (Haves-Zburof et al., 2011; Burster et al., 2007).

Prior research has also shown that EPH-ephrin signaling is more prevalent in relapsing-remitting multiple sclerosis than in other subtypes of the disease (Golan et al., 2021). EPH-ephrin signaling survived FDR correction in our analysis (Figure 4c). Furthermore, the pathway was more enriched in the pink cluster than in the other two (Figure 4i). The pink cluster indeed contained a higher proportion of patients with the relapsing-remitting subtype (Figure 4j). RCSP thus precisely identified the enrichment of EPH-ephrin signaling in the correct subtype of MS.

We denote a singleton variable like X~i with italics and sets of variables like X~ with bold italics. We can represent a causal process using a structural equation model (SEM) linking the p+1 variables in Z=X~∪Y using a series of deterministic functions:(3)Zi=fi(Pa(Zi),Ei),∀Zi∈Z

where fi is a function of the parents, or direct causes, of Zi and an error term Ei∈E. The error terms E are mutually independent. We will use the terms vertex and variable interchangeably. A root vertex corresponds to a vertex without any parents. On the other hand, a terminal or sink vertex is not a parent of any other vertex.

We can associate a directed graph to Z by drawing a directed edge from each member of Pa(Zi) to Zi for all Zi∈Z. A directed path from Zi to Zj corresponds to a sequence of adjacent directed edges from Zi to Zj. If such a path exists (or Zi=Zj), then Zi is an ancestor of Zj and Zj is a descendant of Zi. We collate all ancestors of Zi into the set Anc(Zi). A cycle occurs when there exists a directed path from Zi to Zj and the directed edge Zj→Zi. A directed acyclic graph (DAG) contains no cycles. We augment a directed graph by including additional vertices E and drawing a directed edge from each Ei∈E to Xi except when Xi=Ei is already a root vertex. We consider an augmented DAG G throughout the remainder of this manuscript.

The vertices Zi and Zj are d-connected given W⊆Z∖{Zi,Zj} in G if there exists a path between Zi and Zj such that every collider on the path is an ancestor of W and no non-collider is in W. The vertices are d-separated if they are not d-connected. Any DAG associated with the SEM in Equation 3 also obeys the global Markov property where Zi and Zj are conditionally independent given W if they are d-separated given W. The term d-separation faithfulness refers to the converse of the global Markov property where conditional independence implies d-separation. A distribution obeys unconditional d-separation faithfulness when we can only guarantee d-separation faithfulness when W=∅.

Performing causal discovery requires careful consideration of the underlying generative process. We therefore propose a causal model for RNA-seq. We differentiate between the biology and the RNA sequencing technology.

We represent a snapshot of a biological causal process using an SEM over X~∪Y obeying Equation 3. We assume that the phenotypic target Y is a terminal vertex so that gene expression causes phenotype but not vice versa. Each X~i∈X~ corresponds to the total number of RNA molecules of a unique gene in a single cell or bulk tissue sample. The error terms model root causes that are outside of gene expression, such as genetic variation or environmental factors. Moreover, the relation from gene expression to Y is stochastic because Y=fY(Pa(Y),EY), where EY introduces the stochasticity. Two individuals may therefore have the exact same error term values over X~ but different instantiations of Y.

We unfortunately cannot observe X~ in practice but instead measure a corrupted count X using single cell or bulk RNA-seq technology. We derive the measurement error distribution from first principles. We map an exceedingly small fraction of each X~i∈X~ within a sample at unequal coverage. Let πij denote the probability of mapping one molecule of X~i in batch j so that ∑i=1pπij is near zero. The law of rare events (Papoulis, 1984) implies that the Poisson distribution well-approximates the library size N so that N∼Pois(∑i=1pX~iπij).

We write the probability of mapping X~i in a given sample as:Pij=X~iπij∑i=1pX~iπij.

This proportion remains virtually unchanged when sampling without replacement because N≪∑i=1pX~i with small ∑i=1pπij. We can therefore approximate sampling without replacement by sampling with replacement using a multinomial: X∼MN(N;P1j,…,Ppj). This multinomial and the Poisson distribution over N together imply that the marginal distribution of each Xi∈X follows an independent Poisson distribution centered at (∑i=1pX~iπij)Pij=X~iπij, or:(4)Xi∼Pois(X~iπij).

We conclude that the measurement error distribution follows a Poisson distribution to high accuracy. Multiple experimental results already corroborate this theoretical conclusion (Grün et al., 2014; Sarkar and Stephens, 2021; Choudhary and Satija, 2022).

We can represent the biology and the RNA sequencing in a single DAG over X∪X~∪B∪Y, where B denotes the batch, and Y the target variable representing patient symptoms or diagnosis. We provide a toy example in Figure 6. We draw G over Z in black and make each X~i∈X~ a parent of Xi∈X in blue. We then include the root vertex B as a parent of all members of X in green. We augment this graph with the error terms of X~ in red and henceforth refer to the augmented DAG as G. Repeated draws from the represented causal process generates a dataset.

We provide an asymptotic argument that eliminates the need for normalization by sequencing depth when estimating conditional expectations using bulk RNA-seq. The argument applies to the conditional expectations as a whole rather than their individual parameters.

We want to recover the causal relations between X~ by removing batch B and depth N effects from the dataset because they correspond to the sequencing process rather than the underlying biology. We first consider removing sequencing depth by finding stably expressed housekeeping genes. Let A~ denote the set of housekeeping genes where X~i=x~i is a constant for each X~i∈A~; similarly A refers to the corresponding set with Poisson measurement error. Let N=n be large enough such that ∑Xi∈Axi>0 for each sample. Then dividing by L≜∑Xi∈AXi controls for sequencing depth in the following sense:limN→∞Xi∑Xi∈AXi=limN→∞Xi/N∑Xi∈AXi/N=Pij∑Xi∈APij=X~iπij/∑i=1pX~iπij∑X~i∈A~x~iπij/∑i=1pX~iπij=X~iπij∑X~i∈A~x~iπij,

where we have divided X~iπij by a constant in the last term. Thus, dividing by L removes measurement error within each batch as N→∞. We assume that N is so large that the approximation error is negligible. We only invoke the assumption in bulk RNA-seq, where the library size N is on the order of at least tens of millions.

We do not divide by L in practice because we may have L=0 with finite N. We instead always include L∪B in the predictor set of downstream regressions. Conditioning on L∪B ensures that all downstream regressions mitigate depth and batch effects with adequate sequencing depth, or that E(Y|U~,B)=E(Y|U,L,B) for any U~⊆X~ as N→∞. The equality holds almost surely under a mild smoothness condition:

Lemma 1. Assume Lipschitz continuity of the conditional expectation for all N≥n0:E|E(Y|U~)−E(Y|U,L,B)|≤ECN|U~−UdL|,

where d=πUB∑X~i∈A~x~iπiB, CN∈O(1) is a positive constant, and we have taken an outer expectation on both sides. Then E(Y|U~)=limN→∞E(Y|U,L,B) almost surely.

We delegate proofs to the Appendix unless proven here in the Methods. Note that limN→∞UdL=U~, so the Lipschitz assumption intuitively means that accurate estimation of U~ implies accurate estimation of E(Y|U~). Furthermore, conditioning on the library size N instead of L can introduce spurious dependencies because N depends on all of the genes rather than just the stably expressed ones.

We now eliminate the need to condition on L. Note that L is a sum of independent Poisson distributions given B per Expression (4). This implies Y⊥⊥L|(U,B) for any N, so that E(Y|U~)=limN→∞E(Y|U,L,B)=limN→∞E(Y|U,B) almost surely. We have proved:

Theorem 1. Consider the same assumption as Lemma 1. Then E(Y|U~)=limN→∞E(Y|U,B) almost surely, where we have eliminated the conditioning on L.

We emphasize again that these equalities hold for the conditional expectation but not for the regression parameters; the regression parameters do not converge in general unless we divide by L. We will only need to estimate conditional expectations in order to identify root causal genes.

We showed how to overcome Poisson measurement error without sequencing depth normalization in the previous section. We leverage this technique to define a measure for identifying the root causal genes of Y.

A root cause of Y corresponds to a root vertex that is an ancestor of Y in G. All root vertices are error terms in an augmented graph. We define the root causal effect of any Ei∈E on Y as Υi≜P(Y|Ei)−P(Y) (Strobl, 2024a; Strobl and Lasko, 2023c).

We can identify root causes using the following result:

Proposition 1. If Ei⊥̸⊥Y or Ei⊥̸⊥Y|Pa(X~i) (or both), then Ei is a root cause of Y.

We can also claim the backward direction under d-separation faithfulness. We however avoid making this additional assumption because real biological data may not arise from distributions obeying d-separation faithfulness in practice (Strobl, 2022a).

Proposition 1 implies that Ei is a root cause of Y when:Δi≜P(Y|Pa(X~i),Ei)−P(Y|Pa(X~i))≠0.

The above quantity corresponds to the conditional root causal effect but not the root causal effect Υi due to the extra conditioning on Pa(X~i). The two terms may also differ in direction; if Δi>0, then this does not imply that Υi>0, and similarly for negative values. The two variables thus represent different quantities but – in terms of priority – we would estimate Υi when we have nonzero Δi. Experimental results indicate that Υi and Δi take on similar values and agree in direction about 95% of the time in practice (Appendix).

We now encounter two challenges. First, the quantities Υi and Δi depend on the unknown error term Ei. We can however substitute Ei with X¯i in Δi due to the following result:

Proposition 2. We have P(Y|Ei,Pa(X~i))=P(Y|X~i,Pa(X~i)) under Equation 3.

We can thus compute the conditional root causal effect Δi without access to the error terms:Δi=P(Y|Pa(X~i),Ei)−P(Y|Pa(X~i))=P(Y|Pa(X~i),X~i)−P(Y|Pa(X~i)).

We can determine the root causal status of on when per Proposition 1. Nevertheless, the term ‘root cause’ in colloquial language refers to two concepts simultaneously: a root vertex that causes and has a large causal effect on . We thus say that is a root causal gene of if .

The second challenge involves computing the non-parametric probability distributions of which come at a high cost. We thus define the analogous expected version by:Γi≜∫y[p(y∣Pa(Xi~),Xi~)−p(y∣Pa(Xi~))]dy=E(Y∣Pa(Xi~),Xi~)−E(Y∣Pa(Xi~))=E(Y∣SP(Xi~),Xi,B)−E(Y∣SP(Xi~),B),Δi=ℙ⁢(Y|Pa⁢(X~i),Ei)-ℙ⁢(Y|Pa⁢(X~i))=ℙ⁢(Y|Pa⁢(X~i),X~i)-ℙ⁢(Y|Pa⁢(X~i)).

where p(Y) denotes the density of Y. Observe that if Δi=0, then Γi=0. The converse is not true but likely to hold in real data when a change in the probability distribution also changes its expectation. The set SP(X~i)⊆X denotes the surrogate parents of X~i corresponding to the variables in X associated with Pa(X~i)⊆X~. The last equality holds almost surely as N→∞ by Theorem 1.

We call Φi≜|Γi| the Root Causal Strength (RCS) of X~i on Y. The RCS obtains a unique value Φi=ϕij for each patient j. We say that X~i is a root causal gene of Y for patient j if ϕij≫0, since we posit a right skewed distribution of conditional root causal effects for each patient as in Figure 1(b). We combine the RCS scores across a set of n samples using the Deviation of the RCS (D-RCS) 1n∑j=1nϕij2, or the deviation of RCS from zero. We may compute D-RCS for each cluster or globally across all patients depending on the context. We thus likewise say that X~i is a root causal gene for a cluster of patients or all patients in a sample if its corresponding D-RCS score for the cluster or the sample is much lager than zero, respectively. Note that we do not specify a particular cutoff value for large (conditional) root causal effects, since the root causal effects likely lie on a continuous graduated scale as opposed to approximately two binary values. Nevertheless, visual inspection of the RCS or D-RCS histograms in disease should approximate a power law, where a large mass is concentrated around zero and a long tail extends to the right similar to folding Figure 1b.

We now design an algorithm called Root Causal Strength using Perturbations (RCSP) that recovers the RCS scores using Perturb-seq and bulk RNA-seq data.

Computing Φi for each X~i∈X~ requires access to the surrogate parents of each variable or, equivalently, the causal graph G. However, inferring G using causal discovery algorithms may lead to large statistical errors in the high dimensional setting (Colombo, 2014) and require restrictive assumptions such as d-separation faithfulness (Spirtes et al., 2000) or specific functional relations (Peters, 2014).

We instead directly utilize the interventional Perturb-seq data to recover a superset of the surrogate parents. We first leverage the global Markov property and equivalently write:(5)Φi=|E(Y|SA(X~i),Xi,B)−E(Y|SA(X~i),B)|,

where SA(X~i) denotes the surrogate ancestors of X~i, or the variables in X associated with the ancestors of X~i.

We discover the surrogate ancestors using unconditional independence tests. For any Xk∈X, we test Xk⊥⊥Pi by unpaired two-sided t-test, where Pi is an indicator function equal to one when we perturb Xi and zero in the control samples of Perturb-seq. Pi is thus a parent of Xi alone but not a child of B, so we do not need to condition on B. We use the two-sided t-test to assess for independence because the t-statistic averages over cells to mimic bulk RNA-seq. If we reject the null and conclude that Xk⊥̸⊥Pi then Xk must be a descendant of Pi by the global Markov property, so we include Xk into the set of surrogate descendants SD(X~i). Curating every Xj∈X such that Xi∈SD(X~j) into SA(X~i) yields the surrogate ancestors of X~i as desired.

We now introduce an algorithm called Root Causal Strength using Perturbations (RCSP) that discovers the surrogate ancestors of each variable X~ using Perturb-seq and then computes the RCS of each variable using bulk RNA-seq. We summarize RCSP in Algorithm 1.

RCSP takes Perturb-seq and bulk RNA-seq datasets as input. The algorithm first finds the surrogate descendants of each variable in X~ in Line 2 in order to identify the surrogate ancestors of each variable in Line 5. Access to the surrogate ancestors and the batches B allows RCSP to compute Φi for each Xi∈X from the bulk RNA-seq in Line 6. The algorithm thus outputs the RCS scores Φ as desired.

We certify RCSP as follows:

Theorem 2. (Fisher consistency) Consider the same assumption as Lemma 1. If unconditional d-separation faithfulness holds then RCSP recovers Φ almost surely as N→∞.

We engineered RCSP to only require unconditional d-separation faithfulness because real distributions may not obey full d-separation faithfulness (Strobl, 2022a).