A Bayesian model selection (BMS) framework for classifyingG×Tinteractions with molecular count phenotypes

Overview of the framework.

We focus on the effects of a treatment on the association between a genotype and molecular phenotype, such as gene expression and chromatin accessibility measured by RNA-sequencing (RNA-seq) [21] and assay of transposase-accessible chromatin sequencing (ATAC-seq) [22] techniques. In what follows, we refer to the unit being measured for a given molecular phenotype as a feature: for gene expression, the feature is a gene; for chromatin accessibility, the feature is an accessible chromatin region, often called a candidate cis-regulatory element (cCRE).

We aim to provide a principled approach to interpreting and prioritizing feature-SNP pairs that have been identified by standard methods, such as the “stratified”, “delta”, and “interaction” approaches (Introduction), for further experimental or bioinformatic interrogation. The types of feature-SNP pairs include expression QTLs (eQTLs) and chromatin accessibility QTLs (caQTLs). The framework takes the genotype and molecular phenotype data across subjects in control and treated conditions for a pre-selected set of feature-SNP pairs as input, and outputs posterior probabilities of eight possible models, representing whether the regression coefficients for the genotype, treatment, and G×T interaction terms are non-zero. We note that the well-recognized challenges of specifying an appropriate false discovery rate in detecting G×T or G×E interactions [12] are not directly relevant to our method, which aims to classify, rather than detect, G×T interactions for feature-SNP pairs that have already been selected.

We refer to the eight model categories using a vector of indicator variables , where m_g, m_t, and m_g×t are each 1 or 0 and denote the inclusion and exclusion of the genotype, treatment, and G×T interaction terms in the model, respectively (Fig 2). In the simplest form using linear regression, the model is

(1)

where i indexes samples, y_i is the log ⁡ -transformed molecular count data, g_i is the genotype coded as {0,1,2} or the imputation-based allelic dosage , t_i is an indicator variable for a treatment, denotes the regression coefficients, and is the residual error with error variance σ². Each of the eight models can be obtained by substituting elements of the corresponding m vector for m_g, m_t, and m_g×t in Eq (1). For example, m=(0,0,0) corresponds to the intercept only model,

m=(1,1,0) corresponds to

and m=(1,1,1) corresponds to

Thus, models corresponding to (0,0,1), (1,0,1), (0,1,1), and (1,1,1) include the G×T interaction term, whereas those corresponding to (0,0,0), (1,0,0), (0,1,0), and (1,1,0) do not. These model categories also apply to nonlinear regression that we propose in this work (Fig 2). It is useful to consider the biological meaning of the eight model categories through the taxonomy and corresponding nomenclature in Fig 2 and Table 1. Nonetheless, we will primarily refer to the models using the m vectors in the rest of the manuscript for simplicity.

We are primarily motivated by analyzing datasets generated in in-vitro cell systems with and without treatment. In this type of data, samples from both control and treatment conditions are available for the same donor; there may be replicate samples per combination of donor and condition; or replication may be consistently or sporadically absent. To account for a possible error correlation structure within a group of samples derived from the same donor, our framework can include effects of donors and/or genetic relatedness (kinship) between them as a random effect (SI Text).

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Fig 2. Illustration of types of G×T interactions.

Schematics of hypothetical linear regression where the phenotype is regressed on the genotype, coded as 0, 1, and 2, separately in the control (gray) and treated (brown) conditions. Shown on the top are the m vectors. Top: Linear models. Bottom: Nonlinear models. Shown below the schematics is the taxonomy of G×T interactions. (1,1,1). See text for details.

https://doi.org/10.1371/journal.pgen.1011561.g002

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Table 1. Types of G×T interactions.

https://doi.org/10.1371/journal.pgen.1011561.t001

Modeling the relationship between molecular phenotype and genotype:transformations and allelic additivity.

In the current practice of molecular QTL mapping, molecular count phenotypes are subject to a variance-stabilizing transformation, such as the log ⁡ transformation, and the effects of genotype are modeled linearly on the transformed scale. As described earlier, however, the allelic additivity assumption [18,19] posits that the effect of genotype is linear on the original count scale, such that it is preferable to model the log-transformed count y_i using a nonlinear model as

where μ_gi is defined for as

for a SNP with the major allele A and the minor allele B.

To illustrate the allelic additivity assumption, we consider the effect of a genetic variant in a cis-regulatory region on the expression of a target gene. A common assumption is that gene regulatory feedback mechanisms are rare and that gene expression from different alleles is independent. This leads to the assumption that, in a diploid cell, the total gene expression count is the sum of gene expression count from the two alleles and that the gene expression count is linear with respect to the genotype (Fig 3A). Fig 3B shows an example of previously published gene eQTL data in hNPCs in the original count scale, which is consistent with this idea. However, after variance-stabilizing transformation to achieve homoscedastic error, the mean expression values are no longer linear with respect to the genotype (Fig 3C). Similarly, RINT-transformation does not preserve the linear relationship unless there are approximately equal numbers of major and minor allele homozygous donors (Fig 3D). With the RINT transformation, moreover, adjustment for covariates, such as sex, cannot be handled properly, and large effects can be greatly underestimated. A similar argument also applies to molecular count data other than gene expression. Regardless, in practice, linearity is commonly assumed, which can lead to inaccurate inference. Although some previous studies have addressed this issue by accounting for the nonlinearity between transformed molecular count data and genotype [18,19] for eQTL mapping under a single condition, to our knowledge, this type of nonlinear model has not been used in response QTL mapping. As we shall see, failing to account for this nonlinear relationship can lead to reduced accuracy, and reduced separation of posterior probabilities of correct models from incorrect ones.

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Fig 3. The assumption of allelic additivity.

A. Barplots illustrating that allelic additivity results in the linear relationship between the phenotype and the genotype. The dark and pale gray boxes represent the hypothetical molecular count signals from the two alleles. B. An example of molecular count data on the original scale. Circles represent samples; dashed horizontal lines correspond to the mean expression values conditional on the genotype. C. The same as molecular count data in B but on the log ⁡ scale. Although the variance is more homogeneous than in B, the phenotype with respect to the genotype is no longer linear. This type of pattern is captured by log ⁡-NL in each condition. D. The same as the molecular count data in B but with RINT transformation.

https://doi.org/10.1371/journal.pgen.1011561.g003

To account for allelic additivity, we adapted the nonlinear model for single-condition molecular QTL mapping described above to G×T analysis (Eq (11)). The method, which we term log ⁡ -NL, explicitly models the nonlinear (NL) relationship between the genotype and the log ⁡-transformed molecular count phenotype. We call the commonly used methods, which assume a linear relationship between the genotype and the log ⁡- or RINT-transformed phenotype, as log ⁡-LM and RINT-LM, respectively, referring to the types of transformation. Throughout the manuscript, we consider comparisons between these three approaches.

Assessing the allelic additivity in hNPCs with growth stimulation

Previous studies have analyzed gene expression data from genetically diverse donors in the postmortem human tissues and found little evidence to suggest deviations from the allelic additivity [18,19]. To assess whether the conclusion also holds in in-vitro cell systems, we reanalyzed previously published eQTL data in hNPCs, a system in which numerous associations between regulatory features and SNPs have been identified [11,33–35]. In this analysis, we focused on 3073 feature-SNP pairs with significant G×T pairs (Methods) and fit the log ⁡-transformed data to the nonlinear and linear models by maximum likelihood estimation (MLE) for each condition separately (Methods). For comparison, we also fit a larger, more flexible model with a categorical genotype variable consisting of three levels, which can describe both nonlinear and linear relationships.

The analysis led to the following observations. First, across the feature-SNP pairs, the maximized likelihood values are higher for the nonlinear model than for the linear model or similar between the models (S1 Fig). Since the number of fitted parameters is identical, this suggests that the nonlinear allelic additivity model is more adequate than the linear model. Second, the values are similar between the nonlinear and larger models, suggesting that the nonlinear model sufficiently captures the complexity of the data with fewer parameters (S1 Fig). Third, the values are lower for the linear model than for the larger model, or they are similar between the models (S1 Fig). Overall, the results show little evidence for deviations from the allelic additivity, consistent with previous studies [18,19].

We next examined whether the allelic additivity holds for chromatin accessibility, focused on 83488 feature-SNP pairs with significant G×T pairs (Methods), and observed essentially the same trends as for the gene expression data (S1 Fig). To our knowledge, this is the first evidence to suggest that chromatin accessibility signals with respect to the genotype are linear in the original count scale in the majority of cases.

Simulation: classifying G×Tinteractions using BMS

We evaluated our G×T classification procedure using simulation. Since we observed little evidence of deviations from allelic additivity in experimental data, we assumed this mechanism to simulate data from the eight models specified in Eq (11) (Methods). The true parameter values were set according to the nonlinear regression results for the response eQTLs in hNPCs (Methods). The data comprised one observation per condition for 80 donors (total sample size n=160). We simulated four scenarios, by generating data with and without a donor random effect, and performing analyses with or without a donor random effect term in the model. Specifically, these correspond to a setting without random effect (scenario 1), that with donor random effect in model fitting but not in data generation (scenario 2), that with donor random effect in data generation but not in model fitting (scenario 3), and that with donor effect in both data generation and model fitting (scenario 4). We used MAP estimation followed by Laplace approximation to elicit the effect prior in a data-driven manner. For posterior inference, we additionally used MCMC followed by bridge sampling (Methods and SI Text). We consistently observed convergence of MCMC (S1 Table). In addition to assessing each of the eight models, we computed posterior probabilities of the aggregated model categories no-G×T, induced, and altered. Within the BMS framework, we compared the performance of the three modeling approaches, log ⁡-NL, log ⁡-LM, and RINT-LM as follows. First, we examined the distributions of posterior probabilities of the correct models (i.e., the data-generating model) as well as those of the incorrect models. Second, we stratified the data based on the correct model and the posterior mode model and inspected the distribution of posterior probabilities in each stratum. Third, we constructed receiver operating characteristics (ROC) curves assessing classification error at different posterior probability thresholds. Fourth, we assessed calibration of the procedure, that is, whether the posterior probabilities inferred for each model match the empirical frequencies with which those models were used to generate data [36]. Fifth, we investigated the quality of effect estimation by computing the root mean squared error (RMSE) across feature-SNP pairs.

The results were similar across the scenarios and led to the following points. Overall, the three modeling approaches show comparable performance as assessed by ROC curves (S2 Fig) as well as by calibration (S3 Fig) [37]. In the stratified analysis, the posterior mode models matched the correct models for a large fraction of instances across all methods (S4 Fig) [37]. Nonetheless, a superior performance of log ⁡-NL was evident in partial ROC curves for no-G×T and altered categories (Figs 4A and S5) [37]. There was also a consistent tendency for log ⁡-NL to outperform log ⁡-LM and RINT-LM, as assessed by the distribution of the posterior probability of the correct (i.e., data-generating) model and incorrect (i.e., not data-generating) models thereof (Figs 4B and S6) [37]. Specifically, for the aggregated model categories, no-G×T, induced, and altered, log ⁡-NL gave higher median posterior probabilities of the correct model as well as lower median posterior probabilities for the incorrect models than other approaches. This was also the case for the individual model categories except for (1,0,0) and (1,0,0), where the median posterior probability was higher for log ⁡ -LM than for log ⁡ -NL (S7 Fig) [37]. The posterior probability of the correct models may be low for a fraction of the simulation instances, due to the level of error variance chosen for the simulation; still, there exist clear distributional differences between the correct and incorrect models. The performance of BMS for selecting correct models can rather be evaluated by the calibration and ROC curves (Figs 4A, S3, and S2) [37]. Consistent with the superior performance of log ⁡ -NL, we observed that log ⁡ -LM gave strongly biased estimates for genotype and G×T interaction effects and that RINT-LM did so for all three effects (Table 2, S8 Fig) [37]. These trends did not significantly differ across computational strategies. However, with SNPs for which the minor allele homozygote was absent, we observed occasional, moderate inaccuracies as well as rare failures of MAP estimation followed by Laplace approximation (SI Text and S9 Fig) [37]. Hence, we propose to restrict the use of MAP estimation followed by Laplace approximation to hyperparameter optimization and to perform posterior inference using MCMC followed by bridge sampling in practice (SI Text and S10 Fig).

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Table 2. RMSE of effect estimates across methods.

https://doi.org/10.1371/journal.pgen.1011561.t002

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Fig4. BMS on simulated data.

A. Partial ROC curves comparing log ⁡ -NL, log ⁡ -LM, and RINT-LM for the no-G×T, induced, and altered model categories. B. Violin plots showing distributions of the posterior probability of the correct (top) and incorrect (bottom) models for log ⁡ -NL, log ⁡ -LM, and RINT-LM. Shown in the top of each panel is the model category. C. The same as A but for crossover interactions.

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Classifying G×Tinteractions for response eQTLs in hNPCs with growth stimulation

As an example application of our BMS framework, we computed posterior probabilities for the types of G×T interactions for a set of response eQTLs identified in hNPCs by Matoba et al. [11]. In particular, we focused on 98 response eQTLs on autosomes with the CHIR treatment (Methods). The response eQTLs represent pairs of genes and index SNPs for which the G×T interaction term was significantly non-zero based on hypothesis testing. CHIR, also known as CHIR99021, is an activator of the canonical Wnt pathway, which has been implicated in proliferation of hNPCs, cortical patterning, and complex brain traits [38–43]. As in the simulation experiments, we compared the three modeling approaches (log ⁡ -NL, log ⁡ -LM, RINT-LM), but using mixed effect models to account for the genetic relatedness between the donors (Methods). We used MAP estimation followed by Laplace approximation for hyperparameter optimization, and MCMC followed by bridge sampling for posterior inference (Methods and SI Text). We consistently observed convergence of MCMC (S2 Table).

Overall, the analysis led to the following two points. First, the BMS results were largely concordant with the previous response QTL mapping results at the level of calling significant interactions of genotype and treatment (S14 Fig). Specifically, of the 98 previously identified response eQTLs, G×T models achieved the highest posterior probabilities in 94 eQTL analyzed by log ⁡ -NL, with 95 for log ⁡ -LM, and 92 for RINT-LM. The small discrepancy is likely due to the difference in the data preprocessing (Methods). Second, the three approaches gave comparable but not identical classification and inference. Specifically, for the 90 out of 98 feature-SNP pairs, the posterior mode models were identical between approaches. Examples of feature-SNP pairs with varying degrees of concordance from the three modeling approaches are shown in Fig 5. For the long intergenic non-protein coding RNA LINC02073 and its neighboring SNP, rs7212610, the highest posterior probability was given to (1,1,1) by both log ⁡ -NL and log ⁡ -LM, but not by RINT-LM (Fig 5A and 5B). Furthermore, log ⁡ -NL captures the reduction of the genotype effect size upon treatment with higher certainty than log ⁡ -LM, likely due to its ability to account for the nonlinear relationship between the genotype and phenotype. The result is consistent with the idea that accuracy in effect size estimation can have a nonnegligible impact on the detection of G×T interactions. For the SLC35F3 gene encoding a thiamine transporter and its neighboring SNP, rs650866, all three modeling approaches assigned the highest and second highest posterior probabilities to (1,0,1) and (1,1,1), respectively. However, log ⁡ -NL gave larger certainty on the posterior mode model than log ⁡ -LM and RINT-LM did. A possible explanation for this observation is that the inadequate linear constraint of log ⁡ -LM as well as the nonparametric transformation in the RINT-LM approach led to an increased probability of an incorrect inference of the treatment effect for the major allele homozygote (Fig 5C and 5D). Indeed, the mean phenotype values for the major allele homozygote were -1.45 and -1.39 with the standard errors of 0.97 and 0.87 for the control and treated conditions, respectively, which did not differ significantly (P=0.68 from a paired t-test).

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Fig 5. Representative BMS results for the response eQTL in hNPCs.

The upper panels show fitted functional relationships between the genotype and the transformed molecular count phenotype based on the posterior mode model. The circles represent the data. The gray and brown colors represent the control and treated conditions, respectively. The arrows indicate deviations from regression lines (A, C, E, G). The lower panels show posterior probability of the different types of G×T interactions. The gray and blue colors represent the probability of the no-G×T and “interaction” categories, respectively. (B, D, F, H). Within each subfigure, the left, middle, and right panels show results from log ⁡ -NL, log ⁡ -LM, and RINT-LM, respectively.

https://doi.org/10.1371/journal.pgen.1011561.g005

We also computed aggregated probabilities of interest. For 16 and 75 feature-SNP pairs, the results from the three modeling approaches agreed in assigning the highest posterior probability to the induced and altered genotype effect patterns, respectively, among the three aggregated categories (no-G×T, induced, and altered). The TYR gene encoding a tyrosinase and its neighboring SNP, rs10830237, as well as the novel long non-coding RNA gene ENSG00000258922 and its neighboring SNP, rs4778055, provide examples of the induced category (Fig 5E, 5F, 5G and 5H).

Overall, the analysis illustrates the utility of our framework in that it provides interpretable posterior probabilities rather than hard classifications without uncertainty quantification. It also suggests potential advantages in employing the log ⁡-NL approach for response QTL. We note that similar results were obtained using mixed effect models to account for possible error correlation structures between samples derived from the same donor (donor random effect) rather than polygenic (kinship) effect (Methods and S15 Fig).

Identifying crossover interactions and accounting for signs of effects in the hNPCdata.

As discussed earlier, a crossover interaction, where the treatment has the opposite effect depending on the genotype, can imply a gene regulatory mechanism where a regulatory element mediates transcription activation and repression depending on the treatment condition (Introduction). To identify gene-SNP pairs with this type of association, we performed post-processing of the log ⁡-NL results to compute the posterior probability of crossover interaction for the previously identified 98 response eQTLs (Methods). Examples with high posterior probability of crossover interactions are shown in Fig 6. Interestingly, for the ZNHIT3 gene encoding the zinc finger HIT domain-containing protein 3, which is known to be defective in a severe encephalopathy, and a neighboring SNP, rs4796224, the posterior probability of crossover interaction was close to one (Fig 6A and 6B). In this case, the model category (1,1,1) solely received non-zero probability. By contrast, for the ENSG00000287315 gene encoding a novel antisense transcript and a neighboring SNP, rs10157612, the model categories (1,1,0) and (1,1,1) both received non-zero posterior probability (Fig 6C and 6D). For the TLCD4 gene encoding the TLC domain-containing protein 4 and a neighboring SNP, rs7556223, log ⁡ -NL, log ⁡ -LM, the posterior probability of a crossover interaction was lower than for other examples, representing the uncertainty (Fig 6E and 6F). We emphasize that this utility is uniquely provided by BMS and presents its additional advantage.

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Fig 6. Examples of response eQTLs with the crossover interaction in hNPCs.

The upper panels show fitted functional relationships between the genotype and the transformed molecular count phenotype based on the posterior mode model obtained by BMS with log ⁡ -NL. The circles represent the data. The gray and brown colors represent the control and treated conditions, respectively (A, C, E). The lower panels show posterior probability of the different types of G×T interactions obtained by BMS with log ⁡-NL. The gray color represents the probability of the no-G×T category. The pale and dark blue colors represent the probability of the “non-crossover” and crossover interactions, respectively (B, D, F).

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In practice, the distinction between positive and negative effects can be crucial for the interpretation and prioritization of response molecular QTLs. Post-processing of the output from our BMS framework can generate posterior probability of 27 categories that correspond to , accounting for the sign of the effect sizes. The results of such analysis are summarized as heatmaps in S14 Fig.

Classifying G×Tinteractions with respect to chromatin accessibility

To illustrate the usability of our framework on a molecular phenotype other than gene expression, we reanalyzed 1775 autosomal response caQTLs identified previously with the CHIR treatment [11]. Analogously to the analysis of the response eQTLs, this analysis led to the following observations. First, the BMS results are reasonably concordant with the response caQTL results at the level of calling significant interactions of genotype and treatment (S16 Fig). Specifically, for 1588, 1588, and 1523 of the 1775 feature-SNP pairs, the log ⁡ -NL, log ⁡ -LM, and RINT-LM approaches, respectively, assigned the highest posterior probability to one of the G×T interaction models. The three approaches gave comparable but not identical classification and inference. For the 1516 feature-SNP pars, the posterior mode models were identical between them. However, we observed a number of examples where the posterior probability patterns substantially differed between the approaches yet gave the same highest posterior probability model (S17 Fig). Among the three aggregated categories, no-G×T, induced, and altered, for 538 and 848 feature-SNP pairs, the results from the three modeling approaches agreed in assigning the highest posterior probability to the induced and altered effect patterns, respectively. An example of the induced effect pattern is shown in S17 Fig. We also observed a number of feature-SNP pairs with the crossover interaction. Examples are shown in S18 Fig. The classification results accounting for the sign of the effect size are summarized as heatmaps in S16 Fig.

Computational performance

Computational cost is a key issue in applying Bayesian methods, particularly those involving MCMC, to large genomics data. To systematically evaluate the computational performance of our framework, we measured the runtime and memory usage of BMS using MCMC followed by bridge sampling, or MAP estimation followed by Laplace approximation, on a Linux-based system. In this series of analyses, we focused on five randomly selected feature-SNP pairs from each set of simulated and experimental data and repeated measurements 10 times per feature-SNP pair. In all instances, the memory usage was less than 0.6 gigabytes (S3 and S4 Tables). The mean values of the 10 runtime measurements ranged from 107 to 306 seconds for MCMC followed by bridge sampling (S3 Table) and from 0.15 to 0.50 seconds for MAP estimation followed by Laplace approximation (S4 Table). Thus, MAP estimation followed by Laplace approximation was approximately one thousand times faster than MCMC followed by bridge sampling. Roughly put, even with MCMC, hundreds and thousands of feature-SNP pairs can be analyzed within a day if they are processed as parallel jobs on a standard computer cluster. For example, hyperparameter optimization for 1775 caQTLs over 1728 grid points using MAP estimation followed by Laplace approximation with all three modeling approaches required approximately 700 minutes; subsequent posterior inference for the same 1775 caQTLs using the hyperparameter-optimized MCMC followed by bridge sampling required approximately 100 minutes.

Datasets and preprocessing

Primary human neural progenitor cell (hNPC) genotypes, RNA-seq, and ATAC-seq data were obtained from a previous study [11] and were based on the GRCh38 genome assembly. The genotype data was coded as {0,1,2} to represent the number of minor (alternative) alleles and contained 78 and 72 donors for the RNA- and ATAC-seq data, respectively. Among the previously generated data with multiple treatments, we focused on the vehicle and CHIR treatments as control and treated conditions, respectively. The dataset contained one sample per combination of donor and treatment condition. That is, the donors were shared in both control and treated conditions, and there was no missingness.

For assessing the allelic additivity assumption, we considered a set of feature-SNP pairs for which a significant association between the genotype and the phenotype was identified at least in one of the control and treated conditions by the previous study [11]. We excluded feature-SNP pairs for which the number of donors at any genotype level is zero since such data is not suited to the larger model that assumes three genotype levels (Eq (7)). For eQTL and caQTL data, the filtering resulted in 3073 and 83488 feature-SNP pairs, respectively. For Bayesian model selection (BMS), we considered 98 response eQTLs as well as 1775 response caQTLs, identified by the previous study [11]. For both types of analysis, we only included SNPs on autosomes.

In the previous study, the count data was transformed using a variance stabilizing transformation other than the log ⁡ transformation [11]. To achieve compatibility with the nonlinear model that assumes the log ⁡ transformation (Eq (11)), we reprocessed the raw count data according to previous studies on the allelic additivity [18,19]. Briefly, we first scaled the original count data as

where i = 1 , … , n indexes the samples, g = 1 , … , G indexes the features, such as genes and ATAC-seq peak regions, d_i,g denote elements of the original count matrix, be the library size for sample i, and is the overall mean. We note that the experiments have similar sequencing depth and, thus, that the variance of the scaled counts is similar to that of the original counts. For a given gene g, we transformed c_i as

where y_i is the scaled, log ⁡ -transformed count data. Note that the subscript g is omitted for simplicity. The total numbers of genes and ATAC peak regions were 22354 and 172887, respectively. To aid comparisons between competing methods, we additionally transform count data using the rank inverse normal transformation (RINT),

where r_i is the rank order of y_i in y, and Φ is the cumulative density function of the standard Normal distribution. For response molecular QTL mapping and BMS, we applied the transformation to the phenotype data residualized with respect to confounding variables (see SI Text).

Assessing the allelic additivity assumption in a given condition

For assessing the allelic additivity assumption in a condition-specific manner, we computed the maximum likelihood estimates (MLE) of the model parameters based on the previously developed nonlinear model [18,19] (Eqs (s) and (3), and (4)) and compared the values with those obtained from a standard linear model for QTL mapping with a continuous genotype variable (Eq (5)) as well as those from a larger model with a categorical genotype variable (Eq (7)). Note that the model in Eq (7) subsumes those in Eqs (2) and (5). In our notation, for a given condition and the i-th donor (i = 1 , … , n), the nonlinear model representing the allelic additivity can be written as

(2)

(3)

(4)

where n is the number of donors, y_i is the log ⁡ -transformed molecular count data, g_i is the genotype coded as {0,1,2} or the imputation-based allelic dosage , ε_i is the residual error, and σ² is the residual error variance, and β₀ is the intercept. The parameter β_g represents the half the difference in the expected values of y_i for the major allele homozygote vs the minor allele homozygote. That is, the quantities can be defined as

but where for any non-zero β_g. Our definitions and interpretations of the coefficients differ from those in the previous studies [18,19] to make the scales comparable to those in the standard linear model for QTL mapping, which can be cast as

(5)

(6)

The larger model is cast as

(7)

(8)

where 1(⋅) is an indicator function, and parameters β₀, β₁, and β₂ are, respectively, the expected values of y_i for donors with the homozygous major allele, the heterozygote, and the homozygous minor allele. Note that, for a given condition, the hNPC data contained one measurement per donor and that, for the ease of exposition, confounding factors are omitted (for model formulation with confounding factors, see SI Text).

Extending the allelic additivity model forG×Tanalysis

The nonlinear allelic additivity model above can be modified for G×T analysis as follows. For the i-th sample (i = 1 , … , n), we model

(9)

(10)

(11)

where y_i, g_i, ε_i, σ², and β₀ are defined as earlier, t_i denotes an indicator variable for a treatment, β_g now represents the half the difference in the expected values of y_i for the major allele homozygote vs the minor allele homozygote under the control condition, β_t is the difference in the expected values of y_i in the control condition vs the treated condition for the major allele homozygote, and allows the effect of genotype to vary between treatment groups. That is, the quantities can be defined as

Our definitions and interpretations of the coefficients differ from those in the previous studies [18,19] to make the scales comparable to those in the standard linear model for G×T analysis, namely

(12)

(13)

Note that, for the ease of exposition, confounding factors are omitted (for model formulation with confounding factors, see SI Text).

Bayesian model selection

To classify the types of G×T interactions, we propose a Bayesian model selection approach considering eight models where the , , and parameters are either zero or non-zero in Eq (11). The approach consists of four steps as in Bayesian approaches more generally. First, we specify the likelihood and prior probabilities. Second, we compute the posterior probabilities. Third, we summarize the posterior probabilities of interest. Fourth, optionally, we make a decision based on a loss function.

Using a vector of indicators, , that specifies one of the eight candidate models in the set M, we cast the model as

(14)

(15)

(16)

where , , , , and are defined as earlier and denotes the regression coefficients. For concreteness, we write

where

Using j to index the models, we have

for j = 1 , … , 8. The conditional joint likelihood is

(17)

where denotes a vector of log ⁡ -transformed molecular count data, denotes a vector of genotypes, denotes a vector of treatment indicators, and denotes a vector of mean values in the untransformed scale. In an exact mathematical form, Eq (17) corresponds to

A BMS with linear regression can be formulated by replacing Eq (14) with

In BMS, it is necessary to specify a prior probability that each of the model is correct, , and prior distributions for the non-zero parameters in the models, , , and . We call the former and latter model and effect priors, respectively. For the model prior, by default, we place the uniform prior probability across the eight models, which corresponds to

for j = 1 , … , 8, where j indexes the models. This choice is reasonably justified since we apply BMS to a pre-selected set of feature-SNP pairs that are likely to have significant associations. For the effect prior, we use the Normal-Gamma prior, which is commonly used due to its conjugacy for linear models. That is, we set

(18)

(19)

(20)

where κ, ν, and are the prior hyperparameters. For the residual variance and the intercept , nearly non-informative priors are used by setting , . The hyperparameters control the effect sizes relative to the residual error standard deviation and need to be properly calibrated for the computation of the marginal likelihood. We employ an empirical Bayes approach where the hyperparameter values maximizing the sum of the marginal likelihood are sought by a grid search.

In our BMS procedure, the second step corresponds to computing the posterior probabilities of the model, , which is proportional to the product of the marginal likelihood conditional on the model, p ( y ∣ m ) , and the prior for the models, . The marginal likelihood is defined as

for j = 1 , … , 8. is a vector consisting of the non-zero coefficients. For example, and . The posterior probability of the jth model is computed as

This involves fitting each of the eight models separately. Since the posterior distributions of the parameters and the marginal likelihood, , are intractable, we obtain approximate values by either of two methods. In the first method, we fit the models using Markov Chain Monte Carlo (MCMC) as implemented in the R package “rstan” [51] and compute the marginal likelihood using bridge sampling [52]. In all analyses presented in this manuscript, for MCMC, we used the default setting of the sampling() function in the rstan package. That is, we set the number of chains to four. For each chain, the numbers of burn-in and post burn-in samples were set to 1000. In the second method, we obtain a maximum a posteriori (MAP) estimate using optimization and compute the marginal likelihood using Laplace’s method [48,53] (see SI Text for details). After computing the posterior probabilities of the eight models, in the third step of our BMS procedure, we summarize them in a way that is practically informative. We describe this step in detail in Refining posterior inference through post-processing. For the optional fourth step in BMS, we make a decision using a 0-1 loss function, which corresponds to selecting the model with the highest posterior probability (a maximum a posteriori, MAP, estimation).

In principle, this procedure is equivalent to fitting a Bayesian variable selection regression (BVSR) model where spike-and-slab priors are placed on the regression coefficients and all possible models are simultaneously considered in a sampling process [54]. For this type of model, sampling can be trapped in one model, resulting in poor mixing. By contrast, in our approach, all models are separately fit, which is possible since only a small number of models are considered.

Conditional and model averaged effect posteriors.

It is straightforward to obtain model-averaged posterior distributions of the effect sizes from the corresponding conditional distributions [55] as

Refining posterior inference through post-processing

In the third step of our BMS procedure, we summarize the posterior probability of interest.

Posterior probability of crossover interaction.

From MCMC samples, it is possible to compute posterior probability of the crossover interaction as follows. Let A be the event of a crossover interaction. Then, for j = 1 , … , 8, we have

where is defined as in Eq (16). Since the crossover interactions can appear only when m equals m₇ or m₈ and thus for j = 1 , … , 6, by the law of total probability, we have

For j = 7 , 8, we approximated the conditional posterior probability as

where S is the number of MCMC samples, B is the number of burn-in iterations, and denote samples at the s-th iteration. We set S=2000 and B=1000.

Subcategories of models based on the sign of effects.

To account for the sign of the effect sizes for a given model, we examined the sign of the posterior means or MAP estimates of non-zero coefficients and assigned the same amount of probability to the model with corresponding sign combination and zero probability to others. For example, if , we examined the sign of the estimates and given that the seventh model m₇ was correct. If and , we set

and

where is a vector specifying the 27 model categories.

Generating data for simulations

For simulation experiments, we generated 1000 sets of feature-SNP pairs for each of the eight models specified in Eq (16), summing to 8000 feature-SNP pairs in total. Each simulated feature-SNP pair comprised the genotypes for 80 individuals, with these drawn from a Binomial distribution, Binom ( n = 2 , p = π ) , where , corresponding to minor allele frequency (MAF) ranging from 0.05 to 0.5, and simulated normalized count data (i.e., the phenotype vector). In generating a simulated phenotype vector from a given model, we fixed the intercept and residual error standard deviation to and σ = 1, set the donor random effect variance to 0.2, and set the kernel matrix K to the identity. The regression coefficients were drawn from Normal distributions as in Eq (18), (19), and (20) with , with these values based on the results of hyperparameter calibration using the hNPC eQTL data. This generated 80 phenotype values for each of the control and treated conditions (160 in total) based on Eqs (14) and (16). For evaluating relative performance of log ⁡ -NL, log ⁡ -LM, and RINT-LM, we analyzed the entire set of 8000 simulated feature-SNP pairs using both MAP estimation followed by Laplace approximation to get posterior model probabilities, and MCMC followed by bridge sampling to compute the posterior probability of crossover interaction. In the other MCMC analyses comparing the three approaches, we used 100 feature-SNP pairs, summing to 800 in total, due to the computational cost. In our comparisons of estimating the marginal likelihoods via Laplace approximation vs bridge sampling, we used 10 feature-SNP pairs, summing to 80 in total, albeit for 125 combinations of hyperparameter values.

Additional methods

Additional details are in the Supplementary methods section in SI Text, specifically in the following subsections: The log-NL model of G×T analysis, Review of previously developed methods, Computing the marginal likelihood by Laplace approximation, Hyperparameter optimization using empirical Bayes, and Including covariates.