Log2021

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Last updated: 2021-09-02

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September 02

Keep cross mappable genes in the filtering step

1. Genes and modules containing cross mappable genes

All genes:	13634
With meta:	12585
Final use:	10461
All removed:	3173
including:	meta: 1049 auto: 1 prote&linc: 453 low_map: 1798 cross_map: 0
Number of modules:	158

Figure. Module v.s. #Genes. Bar height for module size. Red/blue bar for #cross mappable genes for each module. Left for genes cross mappable with any genes. Right for genes cross mappable with genes within the same module.

2. Run PCO with cross mappable genes

Figure. Illustration of PCO with cross mappable genes.

Past versions of 0902_PCO_keep_crossmap.jpeg

Version	Author	Date
786b185	liliw-w	2021-09-02

Problem: Computational infeasible

• In phase 1, for each SNP, filter out genes within the module that are cross mappable with any of its cis- genes. Not fast, as need to search pairs of (cis- gene, module gene) in the cross-mappability file with 23,727,385 cross-mappable gene pairs.

The code I use,

if_cross <- sapply(cis_Genes[1:10], function(x){
  colSums(matrix(outer(x, module_Geneid, FUN = "paste", sep=";") %in% cross_map$V4 |
                   t(outer(module_Geneid, x, FUN = "paste", sep=";")) %in% cross_map$V4, ncol = length(module_Geneid)) ) == 0
} )

Time elapsed,

Figure. Time used for a SNP with ~40 cis- genes and a module with 15 genes.

Past versions of 0902_time_cross_map.png

Version	Author	Date
786b185	liliw-w	2021-09-02

• In phase 2, each SNP could correspond to modules with different gene combinations, thus diff. \(\Sigma\) for with-in module correlation and \(\Sigma^0\), which is the estimated correlation of six tests involved in PCO using 2000 simulations. Thus, computing \(\Sigma^0\) for each SNP should take a long time.

July 08

How expression PCs affect the signals?

The following table gives the number of unique signals under various FDR levels for DGN regressing out expression PC as covariates (“DGN_wo_expPC”) and keeping expression PCs (“DGN_with_expPC”).

FDR level	DGN_wo_expPC	DGN_wo_expPC_maxp	DGN_with_expPC	DGN_with_expPC_maxp	overlap	#*
0.05	313	1.9e-08	169	4e-09	125	218
0.10	512	6.5e-08	225	2e-08	167	303

Correlation between meta-ed gene zscores from eQTLGen

Check document here.

July 06

How Sigma \(\hat{\Sigma}_{DGN_exp}\) and \(\hat{\Sigma}_{nullz}\) affect PCO pvalues for DGN?

I inspected three modules. Check here for the comparison.

For individual and clearer figures, check here, here, here, here, here, here, here, here, here.

June 29

“Try using zscores to build sigma, plot of number of SNPs with low zscores”

1. plot of number of SNPs with low zscores

In this section, I am going to check how the estimated Sigma \(\hat{\Sigma}_{K}\) (correlation matrix of zscores/residule expression across genes) can affect PCO’s pvalues. Use eQTLGen zscores. Consider two \(\hat{\Sigma}\)’s. (1) \(\hat{\Sigma}_{DGN}\), computed by \(cor(\tilde{Y}_{n \times K}^{DGN})\), where \(\tilde{Y}\) is the residual expression data from DGN, n is sample size, K is gene module size. (2) \(\hat{\Sigma}_{nullz}\), computed by \(cor(\tilde{Z}_{m \times K}^{eQTLGen})\), where z is genome-wide independent null zscores from eQTLGen, m is the number of null SNPs (m is in fact much less than genome-wide, but should be sufficient?).

Since for each null SNP, we should have its zscores on all genes (or at least all genes in the module we consider), but eQTLGen doesn’t provide zscores for all genome-wide SNPs (only disease-related variants for trans-eQTLs and close-to-gene variants for cis-eQTLs). Therefore, I will first extract the common SNPs for each module. Main steps include:

• Combine the zscores of all the significant/non-significant gene-SNP pairs in both eQTLGen cis-eQTL and trans-eQTL results.

• For each module, extract the common SNPs that have zscores available for all genes in the module.

• For each module, further extract the common SNPs that are “NULL SNPs”. A snp for a module is defined as a “null snp” if its z-scores for all genes in this module are below some z-score cutoff (i.e. all z-scores are small z’s").

• Among these common null SNPs, check how many of them are independent. (Use DGN genotype ‘plink –indep-pairwise 50 5 0.2’ ??? in this case, some SNPs not included in DGN are also considered as "independent. ref panel including all eQTLGen snps?)

Past versions of 0629_num_nullSNP.png

Version	Author	Date
1b79abb	liliw-w	2021-06-29

Here I explain the above figure. x-axis is for each module (75), y-axis is for number of SNPs under various scenarios that can be used for computing \(\hat{\Sigma}_{nullz}\). Color is for various z-scores cutoff for defining “NULL SNPs”. The cutoff is pvalue of z. Smaller cutoff, more insignificant z. Point shape is for unique SNPs and independent SNPs. The barplot in the lower panel gives the number of genes in each module.

Observations

1. For larger gene module, the common null SNPs (unique and independent) are fewer.

2. The average number of the unique null SNPs is ~8000 (under pvalue of z > 0.001). The independent null SNPs is ~4000.

3. Since we were particularly interested at eQTLGen’s module1, (as all eQTLGen trans-variants are significant for module1,) next I will focus on module1. I use the 2282 independent null SNPs (under pvalue of z > 0.001) for module1 to calculate \(\hat{\Sigma}_{nullz}(module1)\).

Build Sigma \(\hat{\Sigma}_{DGN}\) and \(\hat{\Sigma}_{nullz}\) and check PCO pvalues.

eQTLGen’s module1. Run PCO based on two Sigmas, \(\hat{\Sigma}_{DGN}\) and \(\hat{\Sigma}_{nullz}\) built on 2282 independent null SNPs. Calculate pvalues of 9918 SNPs across chromosomes.

Past versions of 0629_effect.of.Sigma.on.PCO.pvalue.module1.png

Version	Author	Date
1b79abb	liliw-w	2021-06-29

In the above figure, x-axis is for snp across chromosomes, y-axis is for log(p) for the module1, color is for different \(\hat{\Sigma}\), grey dashed line is the Bonferroni pvalue threshold (~1e-8).

Past versions of 0629_qqplot.effect.of.Sigma.on.PCO.pvalue.module1.png

Version	Author	Date
1b79abb	liliw-w	2021-06-29

The above figure gives QQ-plot for all pvalues across 22 chr’s of module1.

Observations

1. Using \(\hat{\Sigma}_{nullz}\) gives larger PCO pvalues, than using \(\hat{\Sigma}_{DGN}\). Number of signals decreases. Under Bonferroni pvalue threshold (~1e-8), when using \(\hat{\Sigma}_{nullz}\), there are 790 out of 9918 SNPs are significant, compared to 9794 significant SNPs using \(\hat{\Sigma}_{DGN}\).

June 25

Replication of eQTLGen results in trans-PCO of eQTLGen summary stats?

I first give some data statistics here.

Dataset	Method	#(gene, SNP)/(module, SNP)	#unique genes/modules	unique SNPs
eQTGen	FDR<0.05	59786	6298	3853
eQTGen	filter	20394	1857	2469
eQTLGen_trans-PCO	qvalue	406729	75	9918
eQTLGen_trans-PCO	Bonferroni	55634	74	9915

• The eQTGen’s original result here gives 59786 (gene, trans-eQTL) pairs under FDR \(<0.05\), corresponding to 6298 unique genes and 3853 unique SNPs.

• After some filtering, to make sure the eQTGen trans-eQTLs and genes are included in out trans-PCO result using eQTLGen summary stats. There are 20394 (gene, trans-eQTL) pairs left, corresponding to 1857 unique genes and 2469 unique SNPs.

• Orignal eQTLGen_trans-PCO result (under qvalue FDR correction): 406729 (module, trans-eQTL) pairs, corresponding to all 75 unique modules and 9918 unique SNPs.

• Under Bonferroni correction (p<6.721785e-08): 55634, 74, 9915.

Next, I will check how eQTGen signals are replicated in eQTLGen_trans-PCO results (see bold text above). The basic idea is, for an eQTGen trans-eQTL, find its trans-target genes, then look at how these genes are distributed in each gene modules we used for eQTLGen_trans-PCO, and whether the corresponding (module, trans-eQTL) are identified by eQTLGen_trans-PCO. Specifically, I will check two things.

1. For a trans-eQTL whose trans-target genes are included in one of our PCO module, is this (module, trans-eQTL) identified by eQTLGen_trans-PCO? (Power)

   Since in previous analysis, the intuition is PCO tends to pick modules where there are multiple trans-target genes for a SNP. Therefore, in the below figure, I picked out all the eQTGen trans-eQTLs that have >2 target genes and plotted them on x-axis. The target genes are distributed on the Y-axis for each PCO module. Number/color is for #target genes. These (module, SNP) pairs should be significant in eQTLGen_trans-PCO. I circled out (red) those (module, SNP) pairs that are missed by eQTLGen_trans-PCO. As a result, 11 (module, SNP) (11 unique SNPs) are missed.

Power

Past versions of 0625_power.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Check here for clearer version.

2. For a trans-eQTL whose trans-target genes are not included in any PCO modules (or only 1 or 2), is this (module, trans-eQTL) identified by eQTLGen_trans-PCO? (TIE)

   Similar plot, except Number/color is SNP with 0, 1, or 2 target genes in each module. These (module, SNP) pairs are less likely to be identified by eQTLGen_trans-PCO. Yet, I circled out (green) those (module, SNP) pairs that are still identified by eQTLGen_trans-PCO.

TIE

Past versions of 0625_TIE.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Check here for clearer version.

Observation

1. Almost all eQTGen trans-eQTLs with >2 target genes and their corresponding module are identified by eQTLGen_trans-PCO. This may suggest a good power of eQTLGen_trans-PCO.

2. However, the false positiva rate may be inflated, as a large amount of SNPs that has no target genes in the modules are also identified. (Maybe although none of the genes in the module are target genes for the SNPs, the genes may be cross-mappable with the SNP’s other traget genes that are filter out?)

How does zero values affect qvalue?

Short answer:

June 14

Why there are so many eQTLGen signals using PCO_\({\lambda<0.1}\)?

Check the QQ-plot of the observed pvalues from eQTLGen using PCO_\({\lambda<0.1}\) (9918 unique SNPs, all eQTLGen variants) and PCO_\({\lambda<1}\) (909 unique SNPs).

Past versions of 0614_qqplot.p_obs.eQTLGen.png

Version	Author	Date
9275134	liliw-w	2021-07-08

June 11

Compare with truncated ‘DGN_transeQTL_battle2014’

In this section, I removed trans-eGenes from Battle’s result that are not included in our analysis for being cross-mappable or poorly mapped. I do this to makes sure univariate and multivariate are based on the same set of genes. I go through the same analysis as below.

• Summary of the new version Battle trans-eQTL signals

After filter our some trans-eGenes, there are 157 (v.s. 736) significant (gene, SNP) pairs left, corresponding to 84 unique trans-eQTLs and 47 unique trans-eGenes.

The following is a a table of the number of trans-target genes of these 84 unique trans-eQTLs.

 1  2  3  4  5 15 16 29 
75  1  1  3  1  1  1  1 

We have identified 313 unique trans-eQTLs, among which 24 are also detected by Battle’s univariate method. The above table gives which Battle’s signals are replicated by trans-PCO.

Among 84 Battle’s trans-eQTLs, 24 (~28.57%) are also detected by trans-PCO. For 9 signals with more than one trans-target genes included in our analysis, all of them are detected by trans-PCO. Trans-PCO also detected other 15 signals with only one trans-target gene used in our analysis. For the other 60 univariate signals that are missed by trans-PCO, they have only one trans-target gene included in trans-PCO.

            sig N_transeGene if_transPCO num_included
 1:  3:56849749           29        TRUE           29
 2:  3:56858686           16        TRUE           16
 3:  3:56865776           15        TRUE           15
 4:  3:56880444            5        TRUE            5
 5:  3:56871432            4        TRUE            4
 6:  3:56874033            4        TRUE            4
 7:  7:50366637            4        TRUE            4
 8:  3:56851387            3        TRUE            3
 9:  7:50428445            2        TRUE            2
10: 7:106337250            1       FALSE            1
11: 12:54723028            1       FALSE            1
12: 12:54729872            1       FALSE            1
13: 4:103432974            1       FALSE            1
14: 4:103443569            1       FALSE            1
15: 4:103448582            1       FALSE            1
16: 4:103511114            1       FALSE            1
17: 1:118147675            1       FALSE            1
18: 1:118158079            1       FALSE            1
19: 19:37441365            1       FALSE            1
20: 19:37469098            1       FALSE            1
21: 19:37503899            1       FALSE            1
22: 19:37504596            1       FALSE            1
23: 19:37507729            1       FALSE            1
24: 19:37516220            1       FALSE            1
25: 19:37520927            1       FALSE            1
26: 19:37547219            1       FALSE            1
27: 19:37563211            1       FALSE            1
28: 19:37573074            1       FALSE            1
29: 19:37582471            1       FALSE            1
30: 19:37584650            1       FALSE            1
31: 19:37597395            1       FALSE            1
32: 19:37620248            1       FALSE            1
33: 19:37627113            1       FALSE            1
34: 19:37632877            1       FALSE            1
35: 19:37634291            1       FALSE            1
36: 19:37642165            1       FALSE            1
37: 19:37649577            1       FALSE            1
38: 19:37650929            1       FALSE            1
39: 19:37684966            1       FALSE            1
40: 17:33875262            1       FALSE            1
41: 1:248039451            1        TRUE            1
42: 1:248039713            1       FALSE            1
43: 1:248041182            1       FALSE            1
44: 1:248058265            1       FALSE            1
45: 19:44889660            1       FALSE            1
46: 19:44930738            1        TRUE            1
47: 19:44931995            1       FALSE            1
48: 19:44932972            1        TRUE            1
49: 19:44933947            1        TRUE            1
50: 19:44938980            1        TRUE            1
51: 19:44947050            1       FALSE            1
52: 19:44947184            1        TRUE            1
53: 19:44959866            1       FALSE            1
54: 19:44973969            1       FALSE            1
55: 19:44978298            1       FALSE            1
56: 19:44979443            1       FALSE            1
57: 19:44995990            1       FALSE            1
58: 19:45001346            1       FALSE            1
59: 19:45001619            1       FALSE            1
60: 14:35523266            1       FALSE            1
61: 14:35571651            1        TRUE            1
62: 14:35624831            1       FALSE            1
63: 21:44473062            1       FALSE            1
64: 21:44473425            1       FALSE            1
65: 21:44473446            1       FALSE            1
66:  12:8142798            1        TRUE            1
67:  12:8143483            1        TRUE            1
68:  12:8161503            1        TRUE            1
69:  12:8179891            1       FALSE            1
70:  12:8185853            1       FALSE            1
71:  12:8190206            1       FALSE            1
72: 6:159498130            1       FALSE            1
73: 15:80260554            1        TRUE            1
74: 15:80263217            1        TRUE            1
75: 15:80263345            1        TRUE            1
76: 15:80263406            1        TRUE            1
77:  7:50435777            1       FALSE            1
78: 6:169749100            1       FALSE            1
79: 14:20811332            1        TRUE            1
80: 17:33902635            1       FALSE            1
81: 17:34019488            1       FALSE            1
82: 17:34043021            1       FALSE            1
83: 17:34051176            1       FALSE            1
84:  9:99288951            1       FALSE            1
            sig N_transeGene if_transPCO num_included

Compare with ‘DGN_transeQTL_battle2014’

• Summary of Battle trans-eQTL signals

They gave 736 significant (gene, SNP) pairs, corresponding to 350 unique trans-eQTLs and 138 unique trans-eGenes.

The following is a a table of the number of trans-target genes of these 350 unique trans-eQTLs. We can see most of the trans-eQTLs have only 1 or 2 trans-target genes (\(283+34=317, \approx 91\%\)).

  1   2   3   4   5   6   7   8  11  13  14  22  23  24  29  33  59 
283  34   3   1  12   1   2   2   2   1   2   1   2   1   1   1   1 

Let’s take a look at the trans-eQTLs with more than 2 target genes. We can see these trans-eQTLs are mainly located around the regions 3:56849749, 3:101043565, 7:50366637, 17:33875262.

            sig N_transeGene if_transPCO num_included
 1:  3:56849749           59        TRUE           29
 2:  3:56858686           33        TRUE           16
 3:  3:56865776           29        TRUE           15
 4: 3:101242751           24       FALSE            0
 5: 3:101043565           23       FALSE            0
 6: 3:101106527           23       FALSE            0
 7: 3:101232736           22       FALSE            0
 8: 3:100963761           14       FALSE            0
 9: 3:100970240           14       FALSE            0
10: 3:100994497           13       FALSE            0
11:  3:56851387           11        TRUE            3
12:  3:56880444           11        TRUE            5
13: 3:100915962            8       FALSE            0
14:  7:50366637            8        TRUE            4
15:  3:56874033            7        TRUE            4
16: 3:101068620            7       FALSE            0
17:  3:56871432            6        TRUE            4
18: 3:101042594            5       FALSE            0
19: 3:101080694            5       FALSE            0
20: 3:101085413            5       FALSE            0
21: 3:101135210            5       FALSE            0
22: 3:101160339            5       FALSE            0
23: 3:101184134            5       FALSE            0
24: 3:101191663            5       FALSE            0
25: 3:101231921            5       FALSE            0
26: 3:101237307            5       FALSE            0
27: 3:101252514            5       FALSE            0
28: 3:101253091            5       FALSE            0
29: 3:101253605            5       FALSE            0
30:  7:50428445            4        TRUE            2
31:  7:50435777            3       FALSE            1
32: 17:33875262            3       FALSE            1
33: 17:33902635            3       FALSE            1
            sig N_transeGene if_transPCO num_included

• As for our results from trans-PCO

We have identified 313 unique trans-eQTLs, among which 24 are also detected by Battle’s univariate method. The above table gives which Battle’s signals are replicated by trans-PCO.

Table of the number of target genes for replicated trans-eQTLs.

 1  2  4  6  7  8 11 29 33 59 
12  3  1  1  1  1  2  1  1  1 

1. Signals detected by univariate but missed by trans-PCO.

   Since trans-PCO tends to identify SNPs that have effects on multiple genes, I will focus on the univariate trans-eQTLs that correspond to multiple genes (\(>2\)). For this type of signals, there are 33 trans-eQTLs in total, among which 9 are detected by trans-PCO. For the rest of 24 missed trans-eQTLs, there are basically three types: (1) 21 near the region “3:101043565”, (2) 1 at “7:50435777”, (3) 2 at “17:33875262” and “17:33875262”.

   1. Even though the 20 missed trans-eQTLs correspond to multiple genes, ranging from 5 ~ 24, they are still missed by trans-PCO. So I checked their trans-target genes, and it turns out that all of the target genes are filtered out at the beginning of our analysis for being cross-mappable or poorly mapped.

   2. As for the rest of three trans-eQTLs that are missed by trans-PCO, only one of their 3 trans-target genes are included in our analysis.

Take-aways

1. Trans-PCO tends to be more likely to detect trans-eQTLs with effects on multiple target genes.

2. Some signals could be missed at the step of filtering out genes.

May 26

Are trans-eQTLs cis-eQTLs?

Updated the way to define cis-eQTLs. Previously, cis-eQTL signals are gene-based (from Pheonix using fastQTL). In this section, I compare trans signals with SNP-based cis-eQTLs from two datasets, i.e. GTEx_v7 (whole blood) and DGN.

• GTEx_v7. Here provides all significant variant-gene associations.

• DGN. Two steps to obtain SNP-based significant associations.

  1. Gene-based associations. (eGenes using fastQTL, result from Pheonix)

  2. Find all cis-eQTLs for each gene.

     See “4.2 cis-eQTL mapping” described here. (p.s. I checked GTEx source code and turns out that the actual code GTEx used is a bit different. See below for key lines.)

pt = mean(c(max(eGene$bpval), min(eGene$bpval)))
eGene$pval_nominal_threshold = signif(qbeta(pt,
                                            eGene$shape1, eGene$shape2, ncp=0, lower.tail=TRUE, log.p=FALSE), 6)

Dataset	#signal	GTEx_v7_is_ciseQTL(SNP-based)	GTEx_v7_cis-eGene	DGN_is_ciseQTL(SNP-based)	DGN_cis-eGene	DGN_is_cissQTL(SNP-based)	DGN_cis-sIntro	DGN_cis-e/sQTL
DGN	313	51	16	156	39	101	71	182
eQTLGen_DGN	784	221	32	502	83	381	182	551

A diagram version:

Are trans-eQTLs cis-e/sQTLs?

Past versions of 0526-1.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Observations:

• For DGN, among 313 trans signals, 156 (~49.8%) are also cis-eQTLs, corresponding to 39 eGenes, including:

AC016753.7, BECN1, C17orf50, C2orf68, CFL2, CTD-2562J15.6, ERN1, FAM177A1, FIGNL1, FKRP, GGCX, HABP4, HSD17B1P1, ICAM2, KIAA0391, MAT2A, MLX, NEK6, NFE2, NFKBIA, OR2W3, PARP2, PEX12, PPP2R3C, PRKD2, PSMC3IP, RETSAT, RNF181, RP11-1094M14.8, RP11-69M1.4, RP11-85K15.2, SLC1A5, SLC2A3, SLFN12L, SYMPK, TMEM150A, TTC5, VAMP8, ZNF782

• For DGN, 101 (~32.3%) out of 313 trans signals are cis-sQTLs, corresponding to 71 introns. 26 trans-eQTLs are only sQTLs but not eQTLs. These signals are located on chromosome 15 and 19.

• Among 313 trans signals, 182 (~58.1%) are either cis-eQTLs or cis-sQTLs.

May 19

How many trans-eQTLs detected in DGN/GTEx whole blood are replicated in eQTLGen?

Replication of DGN signals in eQTLGen. Based on updated results, i.e. smaller and more gene modules & PCO\(_{\lambda>0.1}\).

Are trans-eQTLs cis-eQTLs?

1. Dataset: DGN

Dataset	#signal	is_ciseQTL(gene-based)	is_cissQTL(SNP-based)	is_cise(s)QTL(gene-based)
DGN	313	7	9	15
eQTLGen_DGN	784	17	30	45

Observations:

1.1 DGN

• Among 313 trans signals, 7 are also (gene-based) cis-eQTLs, corresponding to 5~7 independent SNPs. The cis-eGenes include SLC2A3, NFE2, KIAA0391, NFKBIA, ERN1, AC016753.7, MAT2A.

• Among 313 unique trans signals, 229 (~73%) have a neareset gene that are eGenes (with cis-eQTLs). Among 28 independent trans signals, 19 (~68%) have a neareset gene that are eGenes (with cis-eQTLs).

• Among 313 trans signals, 9 are also cis-sQTLs, corresponding to 6 independent SNPs.

1.2 eQTLGen_DGN

• Among 784 trans signals, 17 are also (gene-based) cis-eQTLs, corresponding to 9 independent SNPs. The cis-eGenes include AC016753.7, CPEB4, KIAA0391, MAT2A, MRFAP1, NFKBIA, PAQR6, PWP1, RNF181, RP11-539L10.2, RP11-69M1.4, S100P, SENP7, SLC25A44, SLC2A3, SMG5, TMEM79, ZNF782.

• Among 784 unique trans signals, 590 (~75%) have a neareset gene that are eGenes (with cis-eQTLs). Among 50 independent trans signals, 29 (~58%) have a neareset gene that are eGenes (with cis-eQTLs).

• Among 784 trans signals, 30 are also cis-sQTLs, corresponding to 10 independent SNPs.

2. Dataset: TCGA

3. eQTLGen?

March 8

Muscle

• Less modules, PCO\(_{\{\lambda: \lambda>0.1\}}\). 1 independent signal, large p.

• More modules, PCO\(_{\{\lambda: \lambda>0.1\}}\). 5 independent signal, small p.

• More modules, PCO\(_{\{\lambda: \lambda>1\}}\). 5 independent signal, relatively large p.

March 1

Real data analysis by PCO using more \(\lambda\)’s

Dataset: Muscle_Skeletal.

Fewer (independent) signals. \(PCO_{\{\lambda: \lambda>0.1\}}\) (1 signal) v.s. \(PCO_{\{\lambda: \lambda>1\}}\) (3 signals).

proteomics data

See the google doc here.

Other stuff

• RCC account storage

Feb 23

Updated simulation

1. Adjusted variance of effects

• tests

  • Oracle: \(\beta^T \Sigma^{-1}Z \approx \beta^T \hat\Sigma^{-1}Z = \beta^T (\sum_{\{\lambda_k: \lambda_k>0.1\}}\lambda_k u_k u_k^T) Z\)

  • PCO: Use \(\{\lambda_k: \lambda_k>0.1\}\).

• scenerios (\(K=100,K=105\))

  • N: var.b = 0.003; caus = 30%; N.seq = c(200, 400, 600, 800, 1000)

  • caus: var.b = 0.003; N = 500; caus.seq = c(1, 5, 10, 30, 50, 70, 100)

  • caus_fix: var.b.fix = 0.1; N = 500; caus.seq = c(1, 5, 10, 30, 50, 70, 100)

2. Run for other \(\Sigma\)’s. \(K=100,K=105\)

3. FDR level

The minimum null pvalue of PCO is \(\ge 10^{-9}\), similar as the significance threshold we use here \(10^{-9}\).

4. Results

power under Bonferroni corretion cutoff

Past versions of 0223_power.lambda0.1.K100.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Another Sigma (K=105)

Past versions of 0223_power.lambda0.1.K105.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Feb 16

simulation

• tests

  • Oracle: \(\beta^T \Sigma^{-1}Z \approx \beta^T \hat\Sigma^{-1}Z = \beta^T (\sum_{\{\lambda_k: \lambda_k>1\}}\lambda_k u_k u_k^T) Z\)

  • PC1;

  • minp = \(min\{p_1, \dots, p_K \} \times K\)

  • PCO: Use \(\{\lambda_k: \lambda_k>1\}\).

• Multiple testing

  The significance threshold is set based on the Bofferroni correction as the \(\frac{0.05}{50\times10^6} = 10^{-9}\) for univariate test and PC-based tests.

• scenerios (\(K=100\))

  • N: var.b = 0.01; caus = 100 (100%); N.seq = c(200, 400, 600, 800, 1000)

  • caus: var.b = 0.02; N = 500; caus.seq = c(1, 10, 30, 50, 70, 100)

  • caus_fix: var.b.fix = 0.2; N = 500; caus.seq = c(1, 10, 30, 50, 70, 100)

• results

power under Bonferroni corretion cutoff

Past versions of 0216_power.lambda1.png

Version	Author	Date
9275134	liliw-w	2021-07-08

• Try other tests

  • Oracle: \(\beta^T \Sigma^{-1}Z \approx \beta^T \hat\Sigma^{-1}Z = \beta^T (\sum_{\{\lambda_k: \lambda_k>0.1\}}\lambda_k u_k u_k^T) Z\)

  • PCO: Use \(\{\lambda_k: \lambda_k>0.1\}\).

power under Bonferroni corretion cutoff

Past versions of 0216_power.lambda0.1.png

Version	Author	Date
9275134	liliw-w	2021-07-08

• For talk

(This figure should be updated as for the case of ‘u1’, the threshold is falsely set as 0.05, which should be \(10^{-9}\).)

PC1 v.s. PCO

Past versions of 0216_talk.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Jan 26

simulation

• Methods: Oracle, PC1, PCO, MinP, minp.

• Models: \(z \sim N(\sqrt{N} \beta, \Sigma)\). \(\beta \sim N(0, \sigma_{b}^2 I_{K \times K})\).

1. various sample size \(N\). 200, 400, 600, 800, 1000.

2. various \(\sigma_{b}^2\). 0.005, 0.01, 0.05, 0.1, 0.2.

3. various causality percentage. ???

• simulation.

  Generate \(10^4\) zscores under each model. \(10^4\) simulations.

The following plot gives the boxplot, mean plot with standard deviation, and mean plot with standard error of the mean. Which one to use?

power under cutoff 0.05

Past versions of 0126_power_05.png

Version	Author	Date
9275134	liliw-w	2021-07-08

power comparison

Past versions of 0126_plot.error_bar.png

Version	Author	Date
9275134	liliw-w	2021-07-08

bar plot

Past versions of 0126_test.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Larger FDR level, #signals in large and small module?

Here I will look at how the number of signals changes in the original large and current small modules when I increase the FDR level.

The plot below is based on Muscle tissue from GTEx_v8. The x-axis is for various FDR levels, including 5%, 10%, 15%, and 20%. The y-axis is for the number of signals under these FDR levels. I plot three types of signals, i.e. module and SNP pairs (left subplot), unique SNPs (middle subplot), independent SNPs (right subplot). The two colors represent our previous WGCNA modules (red) and current smaller WGCNA modules (blue).

We can observe (1) When increasing FDR levels, number of signals only increase a little bit, say 3-5 more signals. This means that the corrected pvalues are bipolar, i.e. either very small or very large. (2) Signals based on smaller modules are similar to that of previous standard modules.

Muscle: #signals v.s.FDR

Past versions of 0126_signalsv.s.FDR.Muscle.png

Version	Author	Date
9275134	liliw-w	2021-07-08

DGN: #signals v.s.FDR

Past versions of 0126_signalsv.s.FDR.DGN.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Smaller gene modules, more signals?

I changed the parameter deepSplit in function cutreeDynamic from package WGCNA to get modules with smaller size. I run it on the Muscle tissue from GTEx (see GTEx_v8.Muscle_Skeletal.WGCNA in the GTEx results table). Compared to GTEx_v8.Muscle_Skeletal, though more modules (32 v.s. 18), they have same independent signals (3), located on chr5, chr10, and chr22.

The corresponding modules of these signals from small and large WGCNA share a large proportion of genes.

signal.Chr	module.small (#genes)	module.big (#genes)	#overlapped genes
chr5	module21 (89)	module15 (114)	88
chr10	module21 (89)	module15 (114)	88
chr22	module2 (297)	module4 (350)	257

The above table is for clusters obtained by method ‘WGCNA.min20_deep2’ (big) and ‘WGCNA.min20_deep4’ (small). Next, I give more info about the clusters by more methods, i.e. #genes in modules (A), #unclassfied genes (B), and #modules (C) by the six methods. I also give the pvalue distribution corresponding to these clusters below.

Generally, funcExplorer gives more modules, smaller module, and much more unclassified genes. Though the smaller modules, there aren’t many extreme pvalues. Say, the minimum p of all is about \(10^{-9}\), as compared to \(10^{-14}\) by other methods.

From the pvalue plot, it looks like ‘WGCNA.min10_deep4’ gives smaller clusters and some extreme pvalues. So next step I will run the whole pipleline for this setting.

Modules v.s. methods/parameters

Past versions of 0126_modules.png

Version	Author	Date
9275134	liliw-w	2021-07-08

p distribution of Muscle

Past versions of 0126_Muscle_Skeleta.p.funcExplorer.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Jan 05, Jan 19

Simulation

The simulations here aim to compare the power of PC-based tests including PC1 (using only the primary PC) and PCO (using combined PC’s), and non-PC based test MinP. The simulations consist of two parts: (1) verify that type 1 error is well controlled; (2) compare power of tests. At this time, I simply assume there is no LD among SNPs, i.e. tests are independent.

1. Verify that type 1 error is well controlled.

Since the tests are assumed to be independent, the null pvalues should be uniformly distributed. To verify that, I estimate the null pvalues by empirical p’s obtained from the following steps.

Step1. Generate \(z_0 \sim N(0, \Sigma) | H_0\).

Step2. Run tests including PC1, PCO, MinP for \(10^6\) times and obtain \(10^6\) p’s for each test.

Step3. Compute the empirical T1E at significance \(\alpha\) by \(\frac{I\{p<\alpha\}}{10^6}\). Draw qqplot of \(-log_{10}p\).

The correlation matrix \(\Sigma\) is the correlation of genes in module15 of GTEx dataset Muscle_Skeletal (Sigma.Muscle_Skeletal.module15.chr7.rds).

alpha	PC1	PCO	MinP
5e-02	0.050356	0.053584
1e-04	0.000093	0.000114

qqplot.simulation.null

Past versions of 0119_qqplot.simulation.null.png

Version	Author	Date
9275134	liliw-w	2021-07-08

2. Compare power of tests.

To compare the power of non-/PC-based tests, I will run the oracle test in addition to PC1, PCO, and MinP to benchmark the power. It’s mentioned in the PCO paper that given a fixed correlation relationship among the phenotypes, the power of a test depends on the relationship bewteen the true effects (\(\beta\)) and the phenotype correlation (\(\Sigma\)). Therefore, to see which tests perform well in what cases, I will consider different models, including (1) \(\beta=10 u_1\); (2) \(\beta=4 u_k\); (3) \(\beta=1.5 u_K\); (4) \(\beta=rnorm(K)\); (5) \(\beta=rnorm(0.7K)\); (6) \(\beta=rnorm(0.3K)\).

Step1. Generate \(z_0 \sim N(\beta, \Sigma) | H_1\).

Step2. Run tests including Oracle, PC1, PCO, MinP for \(10^6\) times and obtain \(10^6\) p’s for each test.

Step3. Compute the power at significance \(\alpha\) by \(\frac{I\{p<\alpha\}}{10^6}\).

Model	Oracle	PC1	PCO	MinP
u1	0.946341	0.902540	0.746301
uPCO	0.989835	0.050178	0.713434
uK	0.953727	0.049719	0.054140
100%	1.000000	0.058867	0.945918
70%	1.000000	0.058910	0.580368
30%	0.999995	0.050201	0.149904

GTEx

Results

The table below summarizes all results I have so far, followed by figures of the distributions of pvalues in various datasets.

Dataset	Nsample	All	Annotated	Filtered	Filtered_info	Final	Nmodule	FDR	Nperm	QTL_Module	unique_QTL	independent_QTL
DGN	913	13634	11979	8120	0;165;1663;8073	3859	21	combined chr+module	20	659	623	40
DGN_new	913	13634	12585	9939	1;453;1798;8705	3695	19	combined chr+module	10	331	275	26
TCGA	788	17656	15994	12694	336;3504;10697;598	4962
GTEx_v8.Whole_Blood	670	20315	20315	15245	619;4544;3677;13096	5070	22	combined chr+module	10	4	4	2
GTEx_v8.Muscle_Skeletal	706	21031	21031	15601	675;4447;3631;13490	5430	18	combined chr+module	10	38	38	3
GTEx_v8.Skin_Sun_Exposed_Lower_leg	605	25196	25196	18801	807;6458;4557;15650	6395	30	combined chr+module	10	1	1	1
GTEx_v8.Artery_Tibial	584	23304	23304	17390	795;5588;4202;14730	5914	20	combined chr+module	10	0	0	0
GTEx_v8.Muscle_Skeletal.cross	706	21031	21031	7141	675;4447;3631;0	13890	39	combined chr+module	10	43	43	4
GTEx_v8.Muscle_Skeletal.WGCNA	706	21031	21031	15601	675;4447;3631;13490	5430	32	combined chr+module	10	44	44	3
DGN_new.WGCNA	913	13634	12585	9939	1;453;1798;8705	3695	40	combined chr+module	10
TCGA_new

DGN

Past versions of 0119_DGN.p.png

Version	Author	Date
9275134	liliw-w	2021-07-08

DGN_new

Past versions of 0119_DGN_new.p.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Whole_Blood

Past versions of 0119_Whole_Blood.p.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Muscle_Skeletal

Past versions of 0119_Muscle_Skeletal.p.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Skin_Sun_Exposed_Lower_leg

Past versions of 0119_Skin_Sun_Exposed_Lower_leg.p.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Artery_Tibial

Past versions of 0119_Artery_Tibial.p.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Muscle_Skeletal.cross

Past versions of 0119_Muscle_Skeletal.cross.p.png

Version	Author	Date
9275134	liliw-w	2021-07-08

Remarks

1. The dataset “DGN_new” represents DGN through the standard filtering (see “Gene filter” on Dec 10).

2. GTEx datasets generally have few significant signals and relatively large pvalues (compared to DGN).

3. The dataset “GTEx_v8.Muscle_Skeletal.cross” uses Muscle_Skeletal samples from GTEx_v8 (similar as GTEx_v8.Muscle_Skeletal), but without removing cross-mapped genes before constructing gene modules. Therefore, there are 13890 genes (v.s. 5430 in GTEx_v8.Muscle_Skeletal) in total which result in 39 modules (v.s. 18 in GTEx_v8.Muscle_Skeletal). We do this step because we observed that there are relatively few signals using our original pipeline and we wonder if the reason to this observation is us filtering too many genes in the first step and leaving too few signals. To check on this, we put the “filtering” to the last step, i.e. including potentially cross-mapped genes into the analysis and generate significant variant-module pairs. We then exclude those where target eGene in the module is cross-mappable with any gene within 1Mb of the variant. Hopefully we could have more signals.

   However, though the increased genes and modules, the number of identified signals (43) is similar as that using the original pipeline (38). Next, I will look into these signals.

GTEx_v8.Muscle_Skeletal

variant-module	p	q
module15:10:48930105	8.63e-10	8.33e-03
module15:5:132440814	1.56e-09	2.16e-02
module15:5:132448891	3.36e-09	4.74e-02
module15:5:132450078	5.81e-12	0.00e+00
module15:5:132450726	5.96e-12	0.00e+00
module15:5:132450916	8.89e-12	0.00e+00
module15:5:132451361	5.97e-11	0.00e+00
module15:5:132451586	1.11e-11	0.00e+00
module15:5:132453865	1.05e-12	0.00e+00
module15:5:132454053	9.94e-13	0.00e+00
module15:5:132454171	1.03e-12	0.00e+00
module15:5:132454631	1.44e-12	0.00e+00
module15:5:132454724	1.08e-12	0.00e+00
module15:5:132455672	1.07e-12	0.00e+00
module15:5:132455979	1.04e-12	0.00e+00
module15:5:132456154	9.96e-13	0.00e+00
module15:5:132456710	6.04e-11	0.00e+00
module15:5:132458606	5.33e-11	0.00e+00
module15:5:132459905	4.84e-11	0.00e+00
module15:5:132459971	5.04e-11	0.00e+00
module15:5:132460190	5.05e-11	0.00e+00
module15:5:132460375	5.56e-11	0.00e+00
module15:5:132460917	5.03e-11	0.00e+00
module15:5:132461111	5.17e-11	0.00e+00
module15:5:132463834	5.47e-11	0.00e+00
module15:5:132464413	3.90e-11	0.00e+00
module15:5:132464907	1.01e-12	0.00e+00
module15:5:132466034	4.56e-11	0.00e+00
module15:5:132468333	1.32e-10	0.00e+00
module15:5:132468353	5.74e-11	0.00e+00
module15:5:132468564	8.53e-11	0.00e+00
module15:5:132469724	5.63e-11	0.00e+00
module15:5:132469899	6.96e-11	0.00e+00
module15:5:132470043	1.03e-10	0.00e+00
module15:5:132470796	5.42e-11	0.00e+00
module15:5:132471932	8.01e-11	0.00e+00
module15:5:132474927	2.71e-10	2.94e-03
module4:22:23508295	3.76e-10	5.71e-03

GTEx_v8.Muscle_Skeletal.cross

variant-module	p	q
module8:1:123955560	9.97e-12	0.00e+00
module8:1:123955561	1.06e-11	0.00e+00
module8:16:36353041	3.49e-11	6.67e-03
module8:16:36353056	8.29e-11	1.25e-02
module8:5:132453865	1.56e-09	3.12e-02
module8:5:132454053	1.59e-09	2.86e-02
module8:5:132454171	1.52e-09	3.33e-02
module8:5:132454631	1.43e-09	3.57e-02
module8:5:132454724	1.53e-09	3.23e-02
module8:5:132455672	1.50e-09	3.45e-02
module8:5:132455979	1.56e-09	3.03e-02
module8:5:132456154	1.78e-09	3.42e-02
module8:5:132456710	1.36e-09	3.33e-02
module8:5:132458606	3.37e-10	1.76e-02
module8:5:132459905	6.50e-10	1.74e-02
module8:5:132459971	3.92e-10	1.43e-02
module8:5:132460190	3.50e-10	1.58e-02
module8:5:132460375	3.37e-10	1.67e-02
module8:5:132460917	3.98e-10	1.36e-02
module8:5:132461111	3.57e-10	1.50e-02
module8:5:132463834	8.21e-10	2.40e-02
module8:5:132464413	9.93e-10	2.69e-02
module8:5:132466034	6.75e-10	1.67e-02
module8:5:132468333	1.86e-09	3.50e-02
module8:5:132468353	1.69e-09	3.06e-02
module8:5:132468564	2.43e-09	4.19e-02
module8:5:132469724	1.93e-09	3.66e-02
module8:5:132469899	1.73e-09	3.24e-02
module8:5:132470043	1.56e-09	2.94e-02
module8:5:132470796	1.85e-09	3.33e-02
module8:5:132471932	2.37e-09	4.29e-02
module8:5:46659000	9.86e-13	0.00e+00
module8:5:46906585	2.34e-11	7.14e-03
module8:5:47027928	0.00e+00	0.00e+00
module8:5:47068540	0.00e+00	0.00e+00
module8:5:47166316	3.33e-16	0.00e+00
module8:5:47210790	0.00e+00	0.00e+00
module8:5:47210990	0.00e+00	0.00e+00
module8:5:47258679	4.44e-16	0.00e+00
module8:5:47258917	3.33e-16	0.00e+00
module8:5:49809727	2.70e-13	0.00e+00
module8:5:49930686	0.00e+00	0.00e+00
module8:5:49936394	1.10e-11	0.00e+00

For GTEx_v8.Muscle_Skeletal, there are 38 variant-module pairs, corresponding to 2 module (module 15, module4) and 3 independent loci on (chr5, chr10, chr22). GTEx_v8.Muscle_Skeletal.cross has 43 variant-module pairs, corresponding to 1 module (module 8) and 4 independent loci on (chr1, chr5, chr16).

The signal on chr5 is significant for both module 15 (114 genes) and module 8 (394 genes) in two datasets, which have 82 shared genes. Take SNP [rs2706381][rs2706381 GTEx ref] (chr5:132474927) for example. It is ["in cis with IRF1 ($P \le 2\times10^{-10}$; Fig. 6c), a transcription factor that facilitates regulation of the interferon-induced immune response"][rs2706381 GTEx ref]. It is also ["associated in trans with PSME1 ($P \le 1.1\times10^{-11}$) and PARP10 ($P \le 7.8\times10^{-10}$)"][rs2706381 GTEx ref]. These two genes are included in module 15. The reference also gives additional results to "suggest that cis-regulatory loci affecting IRF1 are regulators of interferon-responsive inflammatory processes involving genes including PSME1 and PARP10, with implications for complex traits specific to muscle tissue".

I also looked at the enrichment of the genes in module 15. These genes are mainly enriched in immunity-related terms and tuberculosis. To reproduce, use the gene list [here](asset/0119_gene.GTEx_v8.Muscle_Skeletal.module15.txt).

SNP 10:48930105. module 15.

module4:22:23508295

+ Check if the signals' nearby genes are cross-mapped with genes in their corresponding module. The signal on chr5 corresponding to module8 has 20 genes +/- 500Kb away from its TSS. Many of them are crosspable with genes in the module 8. The number of of genes in the module crossmapped with each of the 20 genes are: `1, 144, 0, 62, 54, 0, 63, 4, 0, 0, 0, 59, 145, 0, 0, 128, 0, 23, 53`, where the nearest gene is crossmappable with 1 gene in the module.

• The other signals on chr1, chr16, and chr5:47210790 don’t have any nearby genes around its TSS.

• See the following plot for number of cross-mapped genes in each module.

Past versions of 0119_nCross.png

Version	Author	Date
9275134	liliw-w	2021-07-08

New TCGA by the standard filtering

eQTLGen

1. eQTLGen description

   This full dataset includes 19942 genes that showed expression in blood tested and 10317 SNPs that are trait-associated SNPs based on GWAS Catalog.

   After gene filter steps described in Dec 10, there are 4963 genes left. Applying the same filtering to 13634 DGN genes, there are 3695 genes left, among which 3642 are also included in eQTLGen. So, I will use these genes to do the downstream analysis, e.g. constructing co-expressed gene modules. These 3642 genes result in 19 gene modules.

2. Replication of DGN signals in eQTLGen

   The table below gives the signals found in eQTLGen and DGN.

   The first two rows give results based on eQTLGen zscores, with row 1 using qvalue for FDR correction (threshold \(0.05\)) and row 2 using \(\frac{0.05}{\#DGN signals}\) as significance threshold. The third row is based on the same gene modules and SNPs but tensorQTL zscores using DGN expression data. The FDR correction uses the empirical distribution of pvalues from the combined chr’s and modules (10 permutations).

Dataset	FDR	minp	(QTL, module)	unique QTL	independent QTL
eQTLGen	qvalue	5.82e-04	2195	909	348
eQTLGen	0.05/#DGN signals	1.18e-04	1707	762	286
eQTLGen_DGN	combined chr+module #10-perms	1.00e-07	420	374	28

Among 374 eQTLGen_DGN signals, 16 are replicated in 762 eQTLGen signals. These 16 replicated signals consists of 6 independent SNPs, including (based on GRCh37),

• rs12485738 : 3:56865776, intron variant of ARHGEF3.
• rs643381 : 6:139839423. (rs590856, 6:139844429).
• rs149007767 7:50370254: , intron variant of IKZF1.
• rs12718597 : 7:50428428, intron variant of IKZF1.
• rs35979828 : 12:54685880, 500B downstream variant of NFE2.
• rs7210990 : 17:16170764, intron variant of PIGL.

3. Replication of DGN signals in eQTLGen

DGN

Past versions of 0119_eQTLGen_DGN_venn.jpeg

Version	Author	Date
9275134	liliw-w	2021-07-08

Session information

R version 4.1.0 (2021-05-18)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Catalina 10.15.7

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] data.table_1.14.0 workflowr_1.6.2  

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.6        whisker_0.4       knitr_1.33        magrittr_2.0.1   
 [5] R6_2.5.0          rlang_0.4.11      fansi_0.5.0       highr_0.9        
 [9] stringr_1.4.0     tools_4.1.0       xfun_0.23         utf8_1.2.1       
[13] git2r_0.28.0      htmltools_0.5.1.1 ellipsis_0.3.2    rprojroot_2.0.2  
[17] yaml_2.2.1        digest_0.6.27     tibble_3.1.2      lifecycle_1.0.0  
[21] crayon_1.4.1      later_1.2.0       vctrs_0.3.8       promises_1.2.0.1 
[25] fs_1.5.0          glue_1.4.2        evaluate_0.14     rmarkdown_2.8    
[29] stringi_1.6.2     compiler_4.1.0    pillar_1.6.1      httpuv_1.6.1     
[33] pkgconfig_2.0.3