Author response: Sparse dimensionality reduction approaches in Mendelian randomisation with highly correlated exposures

Abstract

Full text Figures and data Side by side Abstract Editor's evaluation Introduction Results Discussion Materials and methods Appendix 1 Data availability References Decision letter Author response Article and author information Metrics Abstract Multivariable Mendelian randomisation (MVMR) is an instrumental variable technique that generalises the MR framework for multiple exposures. Framed as a regression problem, it is subject to the pitfall of multicollinearity. The bias and efficiency of MVMR estimates thus depends heavily on the correlation of exposures. Dimensionality reduction techniques such as principal component analysis (PCA) provide transformations of all the included variables that are effectively uncorrelated. We propose the use of sparse PCA (sPCA) algorithms that create principal components of subsets of the exposures with the aim of providing more interpretable and reliable MR estimates. The approach consists of three steps. We first apply a sparse dimension reduction method and transform the variant-exposure summary statistics to principal components. We then choose a subset of the principal components based on data-driven cutoffs, and estimate their strength as instruments with an adjusted F-statistic. Finally, we perform MR with these transformed exposures. This pipeline is demonstrated in a simulation study of highly correlated exposures and an applied example using summary data from a genome-wide association study of 97 highly correlated lipid metabolites. As a positive control, we tested the causal associations of the transformed exposures on coronary heart disease (CHD). Compared to the conventional inverse-variance weighted MVMR method and a weak instrument robust MVMR method (MR GRAPPLE), sparse component analysis achieved a superior balance of sparsity and biologically insightful grouping of the lipid traits. Editor's evaluation This paper investigated the identification of causal risk factors on health outcomes. It applies sparse dimension reduction methods on highly correlated traits in the Mendelian randomization framework. The implementation of this method helps to identify risk factors when given high dimensional traits data. https://doi.org/10.7554/eLife.80063.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Mendelian randomisation (MR) is an epidemiological study design that uses genetic variants as instrumental variables (IVs) to investigate the causal effect of a genetically predicted exposure on an outcome of interest (Smith and Ebrahim, 2003). In a randomised controlled trial (RCT) the act of randomly allocating patients to different treatment groups precludes the existence of systematic confounding between the treatment and outcome and therefore provides a strong basis for causal inference. Likewise, the alleles that determine a small proportion of variation of the exposure in MR are inherited randomly. We can therefore view the various genetically proxied levels of a lifelong modifiable exposure as a ‘natural’ RCT, avoiding the confounding that hinder traditional observational associations. Genetically predicted levels of an exposure are also less likely to be affected by reverse causation, as genetic variants are allocated before the onset of the outcomes of interest. When evidence suggests that multiple correlated phenotypes may contribute to a health outcome, multivariable MR (MVMR), an extension of the basic univariable approach can disentangle more complex causal mechanisms and shed light on mediating pathways. Following the analogy with RCTs, the MVMR design is equivalent to a factorial trial, in which patients are simultaneously randomised to different combinations of treatments (Burgess and Thompson, 2015). An example of this would be investigation into the effect of various lipid traits on coronary heart disease (CHD) risk (Burgess and Harshfield, 2016). While MVMR can model correlated exposures, it performs suboptimally when there are many highly correlated exposures due to multicollinearity in their genetically proxied values. This can be equivalently understood as a problem of conditionally weak instruments (Sanderson et al., 2019) that is only avoided if the genetic instruments are strongly associated with each exposure conditionally on all the other included exposures. An assessment of the extent to which this assumption is satisfied can be made using the conditional F-statistic, with a value of 10 for all exposures being considered sufficiently strong (Sanderson et al., 2019). In settings when multiple highly correlated exposures are analysed, a set of genetic instruments are much more likely to be conditionally weak instruments. In this event, causal estimates can be subject to extreme bias and are therefore unreliable. Estimation bias can be addressed to a degree by fitting weak instrument robust MVMR methods (Sanderson et al., 2020; Wang et al., 2021), but at the cost of a further reduction in precision. Furthermore, MVMR models investigate causal effects for each individual exposure, under the assumption that it is possible to intervene and change each one whilst holding the others fixed. In the high-dimensional, highly correlated exposure setting, this is potentially an unachievable intervention in practice. Our aim in this paper is instead to use dimensionality reduction approaches to concisely summarise a set of highly correlated genetically predicted exposures into a smaller set of independent principal components (PCs). We then perform MR directly on the PCs, thereby estimating their effect on health outcomes of interest. We additionally suggest employing sparsity methods to reduce the number of exposures that contribute to each PC, in order to improve their interpretability in the resulting factors. Using summary genetic data for multiple highly correlated lipid fractions and CHD (Kettunen et al., 2016; Nelson et al., 2017), we first illustrate the pitfalls encountered by the standard MVMR approach. We then apply a range of sparse principal component analysis (sPCA) methods within an MVMR framework to the data. Finally, we examine the comparative performance of the sPCA approaches in a detailed simulation study, in a bid to understand which ones perform best in this setting. Results Workflow overview Our proposed analysis strategy is presented in Figure 1. Using summary statistics for the single-nucleotide polymorphism (SNP)-exposure (γ^) and SNP-outcome (Γ^) association estimates, where γ^ (dimensionality 148 SNPs× 97 exposures) exhibits strong correlation, we initially perform a PCA on γ^. Additionally, we perform multiple sPCA modalities that aim to provide sparse loadings that are more interpretable (block 3, Figure 1). The choice of the number of PCs is guided by permutation testing or an eigenvalue threshold. Finally, the PCs are used in place of γ^ in an IVW MVMR meta-analysis to obtain an estimate of the causal effect of the PC on the outcome. Similar to PC regression and in line with unsupervised methods, the outcome (SNP-outcome associations (Γ^) and corresponding standard error (S⁢EΓ^)) is not transformed by PCA and is used in the second-step MVMR in the original scale. In the real data application and in the simulation study, the best balance of sparsity and statistical power was observed for the method of sparse component analysis (SCA) (Chen and Rohe, 2021). This favoured method and the related steps are coded in an R function and are available at GitHub (https://github.com/vaskarageorg/SCA_MR/, copy archived at Karageorgiou, 2023). Figure 1 Download asset Open asset Proposed workflow. Step 1: MVMR on a set of highly correlated exposures. Each genetic variant contributes to each exposure. The high correlation is visualised in the similarity of the single-nucleotide polymorphism (SNP)-exposure associations in the correlation heatmap (top right). Steps 2 and 3: PCA and sparse PCA on γ^. Step 4. MVMR analysis on a low dimensional set of principal components (PCs). X: exposures; Y: outcome; k: number of exposures; PCA: principal component analysis; MVMR: multivariable Mendelian randomisation. UVMR and MVMR A total of 66 traits were associated with CHD at or below the Bonferroni-corrected level (p=0.05/97, Table 1). Two genetically predicted lipid exposures (M.HDL.C, M.HDL.CE) were negatively associated with CHD and 64 were positively associated (Table 3). In an MVMR model including only the 66 Bonferroni-significant traits, fitted with the purpose of illustrating the instability of IVW-MVMR in conditions of severe collinearity, conditional F-statistic (CFS) (Materials and methods) was lower than 2.2 for all exposures (with a mean of 0.81), highlighting the severe weak instrument problem. In Appendix 1—figure 3, the MVMR estimates are plotted against the corresponding univariable MR (UVMR) estimates. We interpret the reduction in identified effects as a result of the drop in precision in the MVMR model (variance inflation). Only the independent causal estimate for ApoB reached our pre-defined significance threshold and was less precise (ORMVMR (95% CI): 1.031⁢(1.012,1.37), ORUVMR (95% CI): 1.013⁢(1.01,1.016) (Appendix 1—figure 4). We note that, for M.LDL.PL, the UVMR estimate (1.52⁢(1.35,1.71), p < 10-10)) had an opposite sign to the MVMR estimate (ORMVMR=0.905(0.818,1.001)). To see if the application of a weak instrument robust MVMR method could improve the analysis, we applied MR GRAPPLE (Wang et al., 2021). As the GRAPPLE pipeline suggests, the same three-sample MR design described above is employed. In the external selection GWAS study (GLGC), a total of 148 SNPs surpass the genome-wide significance level for the 97 exposures and were used as instruments. Although the method did not identify any of the exposures as significant at nominal or Bonferroni-adjusted significance level, the strongest association among all exposures is ApoB. Table 1 Univariable Mendelian randomisation (MR) results for the Kettunen dataset with coronary heart disease (CHD) as the outcome. Positive: positive causal effect on CHD risk; Negative: negative causal effect on CHD risk. PositiveNegativeVLDLAM.VLDL.C, M.VLDL.CE, M.VLDL.FC, M.VLDL.L,M.VLDL.P, M.VLDL.PL, M.VLDL.TG, XL.VLDL.L,XL.VLDL.PL, XL.VLDL.TG, XS.VLDL.L, XS.VLDL.P, XS.VLDL.PL,XS.VLDL.TG, XXL.VLDL.L, XXL.VLDL.PL,L.VLDL.C, L.VLDL.CE, L.VLDL.FC, L.VLDL.L, L.VLDL.P,L.VLDL.PL, L.VLDL.TG, SVLDL.C, S.VLDL.FC,S.VLDL.L, S.VLDL.P, S.VLDL.PL, S.VLDL.TGNoneLDLALDL.C, L.LDL.C, L.LDL.CE, L.LDL.FC, L.LDL.L, L.LDL.P, L.LDL.PL,M.LDL.C, M.LDL.CE, M.LDL.L, M.LDL.P,M.LDL.PL, S.LDL.C, S.LDL.L, S.LDL.PNoneHDLS.HDL.TG, XL.HDL.TGM.HDL.C, M.HDL.CE PCA Standard PCA with no sparsity constraints was used as a benchmark. PCA estimates a square loadings matrix of coefficients with dimension equal to the number of genetically proxied exposures K. The coefficients in the first column define the linear combination of exposures with the largest variability (PC1). Column 2 defines PC2, the linear combination of exposures with the largest variability that is also independent of PC1, and so on. This way, the resulting factors seek to reduce redundant information and project highly correlated SNP-exposure associations to the same PC. In PC1, very low-density lipoprotein (VLDL)- and low-density lipoprotein (LDL)-related traits were the major contributors (Figure 2a). ApoB received the 8th largest loading (0.1371, maximum was 0.1403 for cholesterol content in small VLDL) and LDL.C received the 48th largest (0.1147). In PC2, high-density lipoprotein (HDL)-related traits were predominant. The first 18 largest positive loadings are HDL-related and 12 describe either large or extra-large HDL traits. PC3 received its scores mainly from VLDL traits. Six components were deemed significant through the permutation-based approach (Figure 1, Materials and methods). Figure 2 Download asset Open asset Heatmaps for the loadings matrices in the Kettunen dataset for all methods (one with no sparsity constraints [a], four with sparsity constraints under different assumptions [b–e]). The number of the exposures plotted on the vertical axis is smaller than K=97 as the exposures that do not contribute to any of the sparse principal components (PCs) have been left out. Blue: positive loading; red: negative loading; yellow: zero. In the second-step IVW regression (step 4 in Figure 1), MVMR results are presented. A modest yet precise (OR = 1.002⁢(1.0015,1.0024), p<10−10) association of PC1 with CHD was observed. Conversely, PC3 was marginally significant for CHD at the 5% level (OR = 0.998 (0.998, 0.999), p=0.049). Since γ^ has been transformed with linear coefficients (visualised in loadings matrix, Figure 2), the underlying causal effects are also transformed and interpreting the magnitude of an effect estimate is not straightforward, since it reflects the effect of changing the PC by one unit on the outcome; however, significance and orientation of effects can be interpreted. When positive loadings are applied to exposures that are positively associated with the outcome, the MR estimate is positive; conversely, if negative loadings are applied, the MR estimate is negative. sPCA methods We next employed multiple sPCA methods (Table 2) that each shrink a proportion of loadings to zero. The way this is achieved differs in each method. Their underlying assumptions and details on differences in optimisation are presented in Table 2 and further described in Materials and methods. Table 2 Overview of sparse principal component analysis (sPCA) methods used. KSS: Karlis-Saporta-Spinaki criterion. Package: R package implementation; Features: short description of the method; Choice: method of selection of the number of informative components in real data; PCs: number of informative PCs. MethodPackageAuthorsFeaturesChoicePCsRSPCApcaPPCroux et al., 2013Robust sPCA (RSPCA), different measure of dispersion (Qn)Permutation KSS6SFPCACode in publication, Supplementary MaterialGuo et al., 2010Fused penalties for block correlationKSS6sPCAelasticnetZou et al., 2006Formulation of sPCA as a regression problemKSS6SCASCAChen and Rohe, 2021Rotation of eigen vectors for approximate sparsityPermutation KSS6 RSPCA (Croux et al., 2013) Optimisation and the KSS criterion pick six PCs to be informative (Karlis et al., 2003). The loadings in Figure 2 show a VLDL-, LDL-dominant PC1, with some small and medium HDL-related traits. LDL.C and ApoB received the 5th and 40th largest positive loadings. PCs 1 and 6 are positively associated with CHD and PCs 3 and 5 negatively so (Appendix 1—table 1). SFPCA (Guo et al., 2010) The KSS criterion retains six PCs. The loadings matrix (Figure 2) shows the ‘fused’ loadings with the identical colouring. In the two first PCs, all groups are represented. Both ApoB and LDL.C received the seventh and tenth largest loadings, together with other metabolites (Figure 2). PC1 (all groups represented) was positively associated with CHD and PC4 (negative loadings from large HDL traits) negatively so (Appendix 1—table 1). sPCA (Zou et al., 2006) The number of non-zero metabolites per PC was set at 14897∼16 (see Appendix 1—figure 6). Under this level of sparsity, the permutation-based approach suggested that six sPCs should be retained. Seventy exposures received a zero loading across all components. PC1 is constructed predominantly from LDL traits and is positively associated with CHD, but this does not retain statistical significance at the nominal level in MVMR analysis (Figure 3). Only PC4 that is comprised of small and medium HDL traits (Figure 2b) appears to exert a negative causal effect on CHD (OR (95% CI): 0.9975⁢(0.9955,0.9995)). The other PCs were not associated with CHD (all p values > 0.05, Appendix 1—table 1). Figure 3 Download asset Open asset Comparison of univariable Mendelian randomisation (UVMR) and multivariable MR (MVMR) estimates and presentation of the major group represented in each principal component (PC) per method. SCA (Chen and Rohe, 2021) Six components were retained after a permutation test. In the final model, five metabolites were regularised to zero in all PCs (CH2.DB.ratio, CH2.in.FA, FAw6, S.VLDL.C, S.VLDL.FC, Figure 2). Little overlap is noted among the metabolites. PC1 receives loadings from LDL and IDL, and PC2 from VLDL. The contribution of HDL to PCs is split in two, with large and extra-large HDL traits contributing to PC3 and small and medium ones to PC4. PC1 and PC2 were positively associated with CHD (Appendix 1—table 1, Figure 3). PC4 was negatively associated with CHD. Comparison with UVMR In principle, all PC methods derive independent components. This is strictly the case in standard PCA, where subsequent PCs are perfectly orthogonal, but is only approximately true in sparse implementations. We hypothesised that UVMR and MVMR could provide similar causal estimates of the associations of metabolite PCs with CHD. The results are presented in Figure 3 and concordance between UVMR and MVMR is quantified with the R2 from a linear regression. The largest agreement of the causal estimates is observed in PCA. In the sparse methods, SCA (Chen and Rohe, 2021) and sPCA (Zou et al., 2006) provide similarly consistent estimates, whereas some disagreement is observed in the estimate of PC6 for RSPCA (Croux et al., 2013) on CHD. A previous study implicated LDL.c and ApoB as causal for CHD (Zuber et al., 2020b). In Appendix 1—figure 7, we present the loadings for these two exposures across the PCs for the various methods. Ideally, we would like to see metabolites contributing to a small number of components for the sparse methods. Using a visualisation technique proposed by Kim and Kim, 2012, this is indeed observed (see Appendix 1—figure 7). In PCA, LDL.c and ApoB contribute to multiple PCs, whereas the sPCA methods limit this to one PC. Only in RSPCA do these exposures contribute to two PCs. In the second-step IVW meta-analysis, it appears that the PCs comprising of predominantly VLDL/LDL and HDL traits robustly associate with CHD, with differences among methods (Table 3). Table 3 Results for principal component analysis (PCA) approaches. Overlap: Percentage of metabolites receiving non-zero loadings in ≥1 component. Overlap in PC1, PC2: overlap as above but exclusively for the first two components which by definition explain the largest proportion of variance. Very low-density lipoprotein (VLDL), low-density lipoprotein (LDL), and high-density lipoprotein (HDL) significance: results of the IVW regression model with CHD as the outcome for the respective sPCs (the sPCs that mostly received loadings from these groups). The terms VLDL and LDL refer to the respective transformed blocks of correlated exposures; for instance, VLDL refers to the weighted sum of the correlated VLDL-related γ^ associations, such as VLDL phospholipid content and VLDL triglyceride content. †: RSPCA projected VLDL- and LDL-related traits to the same PC (sPC1). ‡: SCA discriminated HDL molecules in two sPCs, one for traits of small- and medium-sized molecules and one for large- and extra-large-sized. PCARSPCASFPCAsPCASCAOverlap10.93810.1870.196Overlap in PC1,PC210.43310.0100Sparse %00.4740.0820.8350.796VLDL significance in MR†YesNoYesNoYesLDL significance in MRNoYesNoNoYesHDL significance in MR‡YesYesYesNoNoSmall, medium HDL significance in MRYesNoYesYesYes Instrument strength Instrument strength for the chosen PCs was assessed via an F-statistic, calculated using a bespoke formula that accounts for the PC process (see Materials and methods and Appendix). The F-statistics for all transformed exposures cross the cutoff of 10. There was a trend for the first components being more strongly instrumented in all methods (see Appendix 1—figure 5), which is to be expected. In the MVMR analyses, the CFS for all exposures was less than three. Thus the move to PC-based analysis significantly improved instrument strength and mitigated against weak instrument bias. Simulation studies We consider the case of a data generating mechanism that reflects common scenarios found in real-world applications. Specifically, we consider a set of exposures X, which can be partitioned into blocks based on shared genetics. Certain groups of variants contribute exclusively to specific blocks of exposures, while having no effect on other blocks. This in turn leads to substantial correlation among the exposure blocks and a much reduced correlation of between exposure blocks, due only to shared confounding. This is visualised in Figure 4a. This data structure acts to reduce the instruments’ strength in jointly predicting all exposures. The dataset consists of n participants, k exposures, p SNPs (with both k and p consisting of b discrete, equally sized blocks) and a continuous outcome, Y. We split the simulation results into one illustrative example (for didactic purposes) and one high-dimensional example. Figure 4 Download asset Open asset Simulation Study Outline. (a) Data generating mechanism for the simulation study, illustrative scenario with six exposures and two blocks. In red boxes, the exposures that are correlated due to a shared genetic component are highlighted. (b) Simulation results for six exposures and three methods (sparse component analysis [SCA] [Chen and Rohe, 2021], principal component analysis [PCA], multivariable Mendelian randomisation [MVMR]). The exposures that contribute to Y (X1-3) are presented in shades of green colour and those that do not in shades of red (X4-6). In the third panel, each exposure is a line. In the first and second panels, the PCs that correspond to these exposures are presented as single lines in green and red. Monte Carlo SEs are visualised as error bars. Rejection rate: proportion of simulations where the null is rejected. Simple illustrative example We generate data under the mechanism presented in Figure 4a. That is, with six individual exposures X1,…,X6 split into two distinct blocks (X1-X3 and X4-X6). A continuous outcome Y is generated that is only causally affected by the exposures in block 1 (X1-X3). A range of sample sizes were used in the simulation in order to give a range of CFS values from approximately 2–80. We apply (a) MVMR with the six individual exposures separately, and (b) PCA and SCA. The aim of approach (b) is to demonstrate the impact of reducing the six-dimensional exposure into two PCs, so that the first PC has high loadings for block 1 (X1-X3) and the second PC has high loadings for block 2 (X4-X6). Although two PCs were chosen by both PCA methods using a KSS criterion in a large majority of cases, to simplify the simulation interpretation we fixed a priori the number of PCs at two across all simulations. Our primary focus was to assess the rejection rates of MVMR versus PCA rather than estimation, as the two approaches are not comparable in this regard. To do this we treat each method as a test, which obtains true positive (TP), true negative (TN), false positive (FP), and false negative (FN) results. In MVMR, a TP is an exposure that is causal in the underlying model and whose causal estimate is deemed statistically significant. In the PCA and sPCA methods, this classification is determined with respect to (a) which exposure(s) determine each PC and (b) if the causal estimate of this PC is statistically significant. Exposures are considered to be major contributors to a PC if (and only if) their individual PC loading is larger than the average loading. If the causal effect estimate of a PC in the analysis deemed statistically significant, major contributors that are causal and non-causal are counted as TPs and FPs, respectively. TNs and FNs are defined similarly. Type I error therefore corresponds to the FP rate and power corresponds to the TP rate. All statistical tests were conducted at the α/B = α/2 = 0.025 level. SCA, PCA, and MVMR type I error and power are shown in the three panels (left to right) in Figure 4b, respectively. These results suggest an improved power in identifying true causal associations both with PCA and SCA compared with MVMR when the CFS is weak, albeit at the cost of an inflated type I error rate. As sample size and CFS increase, MVMR performs better. For the PC of the second block’s null exposures, PCA seems to have a suboptimal type I error control (red in Figure 4b). In this low-dimensional setting, the benefit of PCA therefore appears to be limited. Complex high-dimensional example The aim of the high-dimensional simulation is to estimate the comparative performance of the methods in a wider setting that more closely resembles real data applications. We simulate genetic data and individual level exposure and outcome data for between K=30-60 exposures, arranged in B=4-6 blocks. The underlying data generating mechanism and the process of evaluating method performance is identical to the illustrative example, but the number of variants, exposures, and the blocks is increased. We amalgamate rejection rate results across all simulations, by calculating sensitivity (SNS) and specificity (SPC) as: (1) SNS=TPTP+FNSPC=TNTN+FP, and then compare all methods by their area under the estimated receiver-operating characteristic (ROC) curve (AUC) using the meta-analytical approach of Reitsma et al., 2005. Briefly, the Reitsma method performs a bivariate meta-analysis of multiple studies that report both sensitivity and specificity of a diagnostic test, in order to provide a summary ROC curve. A bivariate model is required because sensitivity and specificity estimates are correlated. In our setting the ‘studies’ represent the results of different simulation settings with distinct numbers of exposures and blocks. Youden’s index J (J=S⁢N⁢S+S⁢P⁢C-1) was also calculated, with high values being indicative of good performance. Two sPCA methods (SCA [Chen and Rohe, 2021], sPCA [Zou et al., 2006]) consistently achieve the highest AUC (Figure 5). This advantage is mainly driven by an increase in sensitivity for both these methods compared with MVMR. A closer look at the individual simulation results corroborates the discriminatory ability of these two methods, as they consistently achieve high sensitivities (Appendix 1—figure 10). Both standard and Bonferroni-corrected MVMR performed poorly in terms of AUC (AUC 0.712 and 0.660, respectively), due to poor sensitivity. PCA performed poorly, with almost equal TP and FP results (AUC 0.560). PCA and RSPCA did not accurately identify negative results (PCA and RSPCA median specificity 0 and 0.192, respectively). This extreme result can be understood by looking at the individual simulation results in Appendix 1—figure 10; both PCA and RSPCA cluster to the upper right end of the plot, suggesting a consistently low performance in identifying TN exposures. Specifically, the estimates with both these methods were very precise across simulations and this resulted in many FP results and low specificity. We note a differing performance among the top ranking methods (SCA, sPCA); while both methods are on average similar, the results of SCA are more variable in both sensitivity and specificity (Table 4). The Youden’s indexes for these methods are also the highest (Figure 5a). Varying the sample sizes (mean instrument strength in γ^ from F¯=221 to 1109 and mean conditional F-statistic C⁢F⁢S¯=0.34-12.81) (Appendix 1—figure 9) suggests a similar benefit for sparse methods. Figure 5 Download asset Open asset Extrapolated receiver-operating characteristic (ROC) curves for all methods. SCA: sparse component analysis (Chen and Rohe, 2021) sPCA: sparse PCA (Zou et al., 2006) RSPCA: robust sparse PCA (Croux et al., 2013); PCA: principal component analysis; MVMR: multivariable Mendelian randomisation; MVMR_B: MVMR with Bonferroni correction. Even with large sample sizes (F¯=1109.78, C⁢F⁢S¯=12.82), MVMR can still not discriminate between positive and negative exposures as robustly as the sPCA methods. A major determinant of the accuracy of these methods appears to be the number of truly causal exposures, as in a repeat simulation with only four of the exposures being causal, there was a drop in sensitivity and specificity across all methods. sPCA methods still outperformed other methods in this case, however (Appendix 1—table 2). Table 4 Sensitivity and specificity presented as median and interquartile range across all simulations. Presented as median sensitivity/specificity and interquartile range across all simulations; AUC: area under the receiver-operating characteristic (ROC) curve. PCASCAsPCARSPCAMVMR_BMVMRAUC0.560.9190.9410.6440.6600.712Sensitivity1,0.11,0.211, 0.0470.667, 0.2510.222, 0.20, 0.076Specificity0,0.020.925,0.7720.936, 0.0970.192, 0.1040.960, 0.0481,0Youden’s J00.5840.778–0.0610.1920.044 What determines PCA performance? In the hypothetical example of Figure 4 and indeed any other example, if two PCs are constructed, PCA cannot differentiate between causal and non-causal exposures. The only information used in this stage of the workflow (Steps 2 and 3 in Figure 1) is the SNP-X association matrix. Thus, the determinant of projection to common PCs is genetic correlation and correlation due to confounding, rather than how these blocks affect Y. Then, if only a few of the exposures truly influence Y, it is likely that, PCA will falsely identify the entire block as truly causal. This means the proportion of non-causal exposures within blocks of exposures that truly influence Y is a key determinant of specificity. To test this, we varied the proportion of non-causal exposures by varying the sparsity of the causal effect vector β vector