SumVg: Total Heritability Explained by All Variants in Genome-Wide Association Studies Based on Summary Statistics with Standard Error Estimates
Abstract
1. Introduction
2. Results
2.1. Overview of Methods
2.2. Simulation Results for SE Estimation
2.3. Performance of Different CI Construction Methods
2.4. Results on Immune Traits
2.5. R Package Implementation
3. Discussion
4. Materials and Methods
4.1. Estimation of the Total Heritability Explained (Vg)
4.1.1. Estimation of Total Vg Based on Tweedie’s Formula
4.1.2. Conversion of z-Statistics to Vg
4.1.3. Assumptions
4.1.4. An Alternative Conditional Estimator
4.2. Estimation of the Standard Error (SE) of Vg
4.2.1. Standard and Delete-d-Jackknife to Estimate SE
4.2.2. Parametric Bootstrap Approaches for Estimating SE
4.3. Construction of Confidence Intervals (CIs): An Exploratory Analysis
4.3.1. Normal Approximation (Standard Approach)
4.3.2. Percentile Approach
4.3.3. Union CI
- Normal approximation (standard approach), without bias correction (one estimator) or with bootstrap bias correction (3 estimators), then take the union of CIs;
- Percentile approach, without bias correction (3 estimators) and with bias correction (3 estimators), then take the union of CIs;
- Union of the final CI obtained from 1 and 2.
4.4. Simulation Studies
4.5. Application to Immune Traits
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Sum_of_Vg | Sample_Size | Mean_Est | TRUE_SE | SE | ||||
---|---|---|---|---|---|---|---|---|
jack_del_1 | jack_del_d | paraboot | fdrboot1 | fdrboot2 | ||||
0.295 | 5000 | 0.232 | 0.0482 | 0.0672 | 0.0524 | 0.0488 | 0.0519 | 0.0489 |
10,000 | 0.210 | 0.0265 | 0.0353 | 0.0295 | 0.0285 | 0.0312 | 0.0287 | |
20,000 | 0.244 | 0.0158 | 0.0208 | 0.0185 | 0.0165 | 0.0156 | 0.0168 | |
50,000 | 0.283 | 0.0076 | 0.0149 | 0.0167 | 0.0063 | 0.0081 | 0.0063 | |
0.312 | 0.0063 | 0.0143 | 0.0172 | 0.0051 | 0.0055 | 0.0054 | ||
0.321 | 0.0045 | 0.0134 | 0.0161 | 0.0036 | 0.0038 | 0.0041 | ||
0.191 | 5000 | 0.207 | 0.0491 | 0.0706 | 0.0523 | 0.0486 | 0.0500 | 0.0485 |
10,000 | 0.147 | 0.0242 | 0.0357 | 0.0274 | 0.0263 | 0.0285 | 0.0265 | |
20,000 | 0.158 | 0.0159 | 0.0208 | 0.0166 | 0.0156 | 0.0162 | 0.0160 | |
50,000 | 0.174 | 0.0064 | 0.0113 | 0.0113 | 0.0061 | 0.0070 | 0.0061 | |
0.195 | 0.0045 | 0.0110 | 0.0131 | 0.0040 | 0.0047 | 0.0041 | ||
0.207 | 0.0035 | 0.0103 | 0.0128 | 0.0031 | 0.0034 | 0.0035 | ||
0.101 | 5000 | 0.197 | 0.0521 | 0.0692 | 0.0524 | 0.0484 | 0.0496 | 0.0483 |
10,000 | 0.116 | 0.0260 | 0.0345 | 0.0265 | 0.0251 | 0.0257 | 0.0251 | |
20,000 | 0.098 | 0.0143 | 0.0202 | 0.0159 | 0.0150 | 0.0153 | 0.0154 | |
50,000 | 0.091 | 0.0058 | 0.0098 | 0.0078 | 0.0063 | 0.0057 | 0.0063 | |
0.094 | 0.0032 | 0.0069 | 0.0076 | 0.0027 | 0.0036 | 0.0027 | ||
0.107 | 0.0028 | 0.0072 | 0.0083 | 0.0023 | 0.0027 | 0.0025 |
Sum_Vg | N | Bias of the Estimator for SE | Variance of the Estimator for SE | RMSE of the Estimator for SE | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
jack_del_1 | jack_del_d | paraboot | fdrboot1 | fdrboot2 | jack_del_1 | jack_del_d | paraboot | fdrboot1 | fdrboot2 | jack_del_1 | jack_del_d | paraboot | fdrboot1 | fdrboot2 | ||
0.295 | 5000 | 1.91 | 4.26 | 5.92 × 10−4 | 3.73 | 7.16 | 1.77 | 5.14 | 1.26 | 1.15 | 9.30 × 10−6 | 2.32 | 8.34 | 3.59 | 5.04 | 3.13 × 10−3 |
10,000 | 8.81 | 2.98 | 2.00 × 10−3 | 4.66 | 2.25 | 7.34 | 1.21 | 3.87 | 4.99 | 3.62 × 10−6 | 1.23 | 4.58 | 2.80 × 10−3 | 5.17 | 2.94 | |
20,000 | 5.04 | 2.78 | 7.37 | −1.46 × 10−4 | 1.07 | 9.06 | 2.27 | 8.33 × 10−7 | 1.12 | 1.00 | 1.08 | 3.16 | 1.17 | 1.07 × 10−3 | 1.47 | |
50,000 | 7.25 | 9.03 | −1.32 | 4.68 × 10−4 | −1.30 | 1.29 | 1.45 | 1.36 | 1.93 | 1.30 × 10−7 | 1.35 | 9.11 | 1.37 | 6.42 × 10−4 | 1.35 | |
7.97 | 1.09 | −1.20 | −8.78 × 10−4 | −9.34 | 1.41 | 1.52 | 8.11 × 10−8 | 3.48 | 1.00 | 1.43 | 1.10 | 1.23 | 1.06 | 9.86 × 10−4 | ||
8.92 | 1.16 | −8.57 | −6.32 | −3.72 × 10−4 | 1.37 | 8.70 | 3.49 × 10−8 | 1.30 | 4.06 | 1.47 | 1.16 | 8.77 | 7.27 | 4.23 × 10−4 | ||
0.191 | 5000 | 2.16 | 3.21 | −5.02 × 10−4 | 9.69 | −5.23 | 5.53 | 5.41 | 1.07 | 1.02 | 8.58 × 10−6 | 3.19 | 8.03 | 3.31 | 3.34 | 2.98 × 10−3 |
10,000 | 1.15 | 3.16 | 2.04 × 10−3 | 4.22 | 2.29 | 2.80 | 1.43 | 4.88 | 2.98 × 10−6 | 5.03 | 2.03 | 4.93 | 3.01 × 10−3 | 4.56 | 3.20 | |
20,000 | 4.96 | 7.22 | −2.41 | 3.15 | 9.85 × 10−5 | 1.19 | 2.70 | 1.09 | 1.31 | 7.64 × 10−7 | 1.20 | 1.79 | 1.07 | 1.19 | 8.80 × 10−4 | |
50,000 | 4.90 | 4.83 | −2.92 × 10−4 | 5.76 | −2.97 | 1.10 | 8.17 | 1.66 | 1.50 | 1.24 × 10−7 | 1.16 | 4.91 | 5.01 | 6.94 | 4.60 × 10−4 | |
6.45 | 8.56 | −5.40 | 1.30 × 10−4 | −4.33 | 1.32 | 1.68 | 6.30 × 10−8 | 1.32 | 8.37 | 1.32 | 8.65 | 5.96 | 3.85 × 10−4 | 5.21 | ||
6.85 | 9.28 | −3.57 | −1.41 | −3.31 × 10−5 | 1.28 | 1.06 | 3.04 × 10−8 | 1.54 | 3.78 | 1.32 | 9.34 | 3.97 | 4.17 | 1.97 × 10−4 | ||
0.101 | 5000 | 1.71 | 3.38 × 10−4 | −3.72 | −2.54 | −3.81 | 1.81 | 6.26 | 1.01 | 1.07 | 7.70 × 10−6 | 2.17 | 7.92 | 4.90 | 4.14 × 10−3 | 4.71 |
10,000 | 8.45 | 4.62 | −8.81 | −2.84 × 10−4 | −9.03 | 1.66 | 1.45 | 3.49 | 1.66 × 10−6 | 2.46 | 1.54 | 3.83 | 2.07 | 1.32 × 10−3 | 1.81 | |
20,000 | 5.92 | 1.56 | 7.21 × 10−4 | 1.04 | 1.08 | 1.44 | 4.20 | 8.80 | 1.22 | 8.17 × 10−7 | 1.34 | 2.57 | 1.18 × 10−3 | 1.52 | 1.41 | |
50,000 | 4.03 | 2.04 | 4.91 | −1.02 × 10−4 | 5.85 | 6.00 | 4.31 | 1.61 | 1.08 × 10−7 | 1.26 | 8.73 | 2.15 | 6.34 | 3.44 × 10−4 | 6.85 | |
3.73 | 4.34 | −5.10 | 4.31 × 10−4 | −5.17 | 9.13 | 3.03 | 2.83 | 4.34 | 2.05 × 10−8 | 1.03 | 4.38 | 5.37 | 4.79 × 10−4 | 5.36 | ||
4.38 | 5.48 | −5.31 | −1.61 × 10−4 | −3.81 | 1.00 | 3.41 | 2.22 × 10−8 | 7.06 | 2.88 | 1.09 | 5.51 | 5.52 | 3.11 × 10−4 | 4.17 |
N | Union CI Type | Coverage (Vg = 0.295) | Coverage (Vg = 0.191) | Coverage (Vg = 0.101) |
---|---|---|---|---|
5000 | Standard | 0.75 | 0.97 | 0.77 |
Percentile | 1 | 1 | 0.78 | |
Standard + Percentile | 1 | 1 | 0.78 | |
10,000 | Standard | 0.6 | 0.67 | 0.94 |
Percentile | 0.99 | 1 | 1 | |
Standard + Percentile | 0.99 | 1 | 1 | |
20,000 | Standard | 0.89 | 0.84 | 0.96 |
Percentile | 0.91 | 1 | 1 | |
Standard + Percentile | 0.97 | 1 | 1 | |
50,000 | Standard | 1 | 1 | 0.9 |
Percentile | 1 | 1 | 1 | |
Standard + Percentile | 1 | 1 | 1 | |
Standard | 1 | 1 | 1 | |
Percentile | 1 | 1 | 1 | |
Standard + Percentile | 1 | 1 | 1 | |
Standard | 0.96 | 1 | 1 | |
Percentile | 0.13 | 0.66 | 1 | |
Standard+Percentile | 0.96 | 1 | 1 |
Trait | Abbreviation | GWAS ID | N | SNP_h2 (LDSC) | SNP_h2_se (LDSC) |
---|---|---|---|---|---|
Stem cell factor | SCF | ebi-a-GCST004429 | 8290 | −0.06 | 0.055 |
Interleukin-4 | IL4 | ebi-a-GCST004453 | 8124 | −0.0446 | 0.0595 |
Interleukin-17 | IL17 | ebi-a-GCST004442 | 7760 | −0.0407 | 0.0623 |
Hepatocyte growth factor | HGF | ebi-a-GCST004449 | 8292 | −0.0311 | 0.0579 |
Basic fibroblast growth factor | FGFBasic | ebi-a-GCST004459 | 7565 | −0.0159 | 0.0597 |
Stromal cell-derived factor-1 alpha (CXCL12) | SDF1a | ebi-a-GCST004427 | 5998 | −0.0116 | 0.0713 |
Interleukin-6 | IL6 | ebi-a-GCST004446 | 8189 | −0.0071 | 0.0568 |
Platelet derived growth factor BB | PDGFbb | ebi-a-GCST004432 | 8293 | −0.0043 | 0.0624 |
TNF-related apoptosis inducing ligand | TRAIL | ebi-a-GCST004424 | 8186 | 0.0125 | 0.0613 |
Interferon-gamma | IFNg | ebi-a-GCST004456 | 7701 | 0.0134 | 0.0624 |
Granulocyte colony-stimulating factor | GCSF | ebi-a-GCST004458 | 7904 | 0.0173 | 0.0601 |
Interleukin-10 | IL10 | ebi-a-GCST004444 | 7681 | 0.0186 | 0.0691 |
Trait | N | LDSC | SumVg | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
h2 | se | h2 | r2 | n_pruned_snp | se_jack1 | se_jack_del_d | se_paraboot | se_fdrboot1 | se_fdrboot2 | ||
SCF | 8290 | −0.06 | 0.055 | 0.333 | 0.1 | 428,593 | 0.0926 | 0.0822 | 0.0679 | 0.0443 | 0.0514 |
0.185 | 0.05 | 251,008 | 0.0526 | 0.0456 | 0.0467 | 0.0502 | 0.0517 | ||||
0.105 | 0.025 | 127,908 | 0.0307 | 0.0313 | 0.0272 | 0.0397 | 0.0335 | ||||
0.100 | 0.01 | 61,938 | 0.0310 | 0.0200 | 0.0220 | 0.0252 | 0.0265 | ||||
0.092 | 0.005 | 51,370 | 0.0229 | 0.0169 | 0.0201 | 0.0235 | 0.0230 | ||||
0.101 | 0.002 | 48,088 | 0.0319 | 0.0153 | 0.0220 | 0.0226 | 0.0198 | ||||
0.102 | 0.001 | 47,108 | 0.0316 | 0.0155 | 0.0223 | 0.0216 | 0.0188 | ||||
IL4 | 8124 | −0.0446 | 0.0595 | 0.503 | 0.1 | 427,005 | 0.1218 | 0.1133 | 0.0616 | 0.0563 | 0.0569 |
0.377 | 0.05 | 249,710 | 0.1000 | 0.0823 | 0.0484 | 0.0445 | 0.0453 | ||||
0.302 | 0.025 | 127,248 | 0.0650 | 0.0594 | 0.0318 | 0.0365 | 0.0336 | ||||
0.235 | 0.01 | 61,685 | 0.0529 | 0.0313 | 0.0247 | 0.0240 | 0.0236 | ||||
0.215 | 0.005 | 51,196 | 0.0472 | 0.0278 | 0.0227 | 0.0217 | 0.0221 | ||||
0.197 | 0.002 | 47,878 | 0.0571 | 0.0273 | 0.0228 | 0.0225 | 0.0253 | ||||
0.187 | 0.001 | 46,911 | 0.0482 | 0.0244 | 0.0198 | 0.0226 | 0.0242 | ||||
IL17 | 7760 | −0.0407 | 0.0623 | 0.352 | 0.1 | 427,226 | 0.1240 | 0.0946 | 0.0692 | 0.0625 | 0.0609 |
0.228 | 0.05 | 250,259 | 0.0683 | 0.0668 | 0.0499 | 0.0495 | 0.0495 | ||||
0.299 | 0.025 | 127,479 | 0.0877 | 0.0568 | 0.0360 | 0.0380 | 0.0323 | ||||
0.234 | 0.01 | 61,756 | 0.0485 | 0.0340 | 0.0239 | 0.0267 | 0.0256 | ||||
0.196 | 0.005 | 51,215 | 0.0475 | 0.0295 | 0.0237 | 0.0190 | 0.0249 | ||||
0.195 | 0.002 | 47,887 | 0.0634 | 0.0231 | 0.0231 | 0.0226 | 0.0210 | ||||
0.188 | 0.001 | 46,931 | 0.0568 | 0.0242 | 0.0183 | 0.0211 | 0.0215 | ||||
HGF | 8292 | −0.0311 | 0.0579 | 0.366 | 0.1 | 428,318 | 0.0917 | 0.0864 | 0.0569 | 0.0642 | 0.0593 |
0.242 | 0.05 | 250,843 | 0.0812 | 0.0722 | 0.0483 | 0.0492 | 0.0491 | ||||
0.205 | 0.025 | 127,850 | 0.0657 | 0.0488 | 0.0327 | 0.0326 | 0.0357 | ||||
0.098 | 0.01 | 61,906 | 0.0379 | 0.0224 | 0.0225 | 0.0260 | 0.0242 | ||||
0.115 | 0.005 | 51,301 | 0.0347 | 0.0199 | 0.0224 | 0.0230 | 0.0203 | ||||
0.111 | 0.002 | 47,878 | 0.0414 | 0.0162 | 0.0189 | 0.0211 | 0.0215 | ||||
0.108 | 0.001 | 46,934 | 0.0312 | 0.0171 | 0.0221 | 0.0208 | 0.0211 | ||||
FGFBasic | 7565 | −0.0159 | 0.0597 | 0.269 | 0.1 | 427,284 | 0.0835 | 0.0902 | 0.0656 | 0.0530 | 0.0577 |
0.217 | 0.05 | 249,930 | 0.0891 | 0.0604 | 0.0473 | 0.0504 | 0.0468 | ||||
0.117 | 0.025 | 127,587 | 0.0452 | 0.0431 | 0.0340 | 0.0363 | 0.0358 | ||||
0.133 | 0.01 | 61,911 | 0.0408 | 0.0301 | 0.0232 | 0.0239 | 0.0275 | ||||
0.135 | 0.005 | 51,259 | 0.0376 | 0.0243 | 0.0242 | 0.0267 | 0.0219 | ||||
0.143 | 0.002 | 47,874 | 0.0362 | 0.0218 | 0.0185 | 0.0233 | 0.0245 | ||||
0.126 | 0.001 | 46,914 | 0.0392 | 0.0206 | 0.0227 | 0.0214 | 0.0208 | ||||
SDF1a | 5998 | −0.0116 | 0.0713 | 0.395 | 0.1 | 425,165 | 0.1120 | 0.1068 | 0.0731 | 0.0757 | 0.0870 |
0.256 | 0.05 | 248,727 | 0.0872 | 0.0750 | 0.0580 | 0.0565 | 0.0631 | ||||
0.213 | 0.025 | 126,986 | 0.0707 | 0.0462 | 0.0431 | 0.0472 | 0.0468 | ||||
0.163 | 0.01 | 61,680 | 0.0497 | 0.0380 | 0.0359 | 0.0349 | 0.0324 | ||||
0.190 | 0.005 | 51,092 | 0.0708 | 0.0318 | 0.0250 | 0.0297 | 0.0270 | ||||
0.165 | 0.002 | 47,702 | 0.0447 | 0.0270 | 0.0294 | 0.0304 | 0.0301 | ||||
0.159 | 0.001 | 46,789 | 0.0512 | 0.0232 | 0.0279 | 0.0258 | 0.0308 | ||||
IL6 | 8189 | −0.0071 | 0.0568 | 0.422 | 0.1 | 427,566 | 0.0878 | 0.0896 | 0.0510 | 0.0575 | 0.0594 |
0.227 | 0.05 | 250,247 | 0.0620 | 0.0713 | 0.0402 | 0.0463 | 0.0468 | ||||
0.158 | 0.025 | 127,503 | 0.0672 | 0.0457 | 0.0372 | 0.0300 | 0.0360 | ||||
0.139 | 0.01 | 61,931 | 0.0606 | 0.0258 | 0.0220 | 0.0247 | 0.0220 | ||||
0.114 | 0.005 | 51,332 | 0.0288 | 0.0176 | 0.0196 | 0.0227 | 0.0236 | ||||
0.115 | 0.002 | 47,930 | 0.0302 | 0.0164 | 0.0191 | 0.0226 | 0.0202 | ||||
0.117 | 0.001 | 46,944 | 0.0319 | 0.0175 | 0.0227 | 0.0209 | 0.0211 | ||||
PDGFbb | 8293 | −0.0043 | 0.0624 | 0.432 | 0.1 | 427,743 | 0.0907 | 0.0993 | 0.0726 | 0.0653 | 0.0676 |
0.341 | 0.05 | 250,325 | 0.0670 | 0.0808 | 0.0600 | 0.0496 | 0.0576 | ||||
0.307 | 0.025 | 127,567 | 0.0735 | 0.0554 | 0.0370 | 0.0334 | 0.0326 | ||||
0.154 | 0.01 | 61,789 | 0.0372 | 0.0250 | 0.0213 | 0.0245 | 0.0243 | ||||
0.125 | 0.005 | 51,140 | 0.0310 | 0.0226 | 0.0234 | 0.0230 | 0.0221 | ||||
0.120 | 0.002 | 47,822 | 0.0258 | 0.0205 | 0.0214 | 0.0208 | 0.0233 | ||||
0.117 | 0.001 | 46,853 | 0.0392 | 0.0192 | 0.0201 | 0.0209 | 0.0226 | ||||
TRAIL | 8186 | 0.0125 | 0.0613 | 0.559 | 0.1 | 423,391 | 0.0613 | 0.1018 | 0.0785 | 0.0790 | 0.0750 |
0.304 | 0.05 | 247,717 | 0.0543 | 0.1190 | 0.0526 | 0.0503 | 0.0439 | ||||
0.242 | 0.025 | 126,350 | 0.0607 | 0.0647 | 0.0321 | 0.0362 | 0.0370 | ||||
0.128 | 0.01 | 61,114 | 0.0316 | 0.0251 | 0.0242 | 0.0229 | 0.0277 | ||||
0.127 | 0.005 | 50,633 | 0.0298 | 0.0231 | 0.0255 | 0.0239 | 0.0268 | ||||
0.128 | 0.002 | 47,359 | 0.0332 | 0.0216 | 0.0233 | 0.0215 | 0.0266 | ||||
0.121 | 0.001 | 46,415 | 0.0358 | 0.0195 | 0.0222 | 0.0229 | 0.0256 | ||||
IFNg | 7701 | 0.0134 | 0.0624 | 0.393 | 0.1 | 426,740 | 0.0946 | 0.0811 | 0.0528 | 0.0590 | 0.0594 |
0.241 | 0.05 | 249,818 | 0.0655 | 0.0628 | 0.0553 | 0.0520 | 0.0509 | ||||
0.244 | 0.025 | 127,514 | 0.0734 | 0.0582 | 0.0330 | 0.0406 | 0.0320 | ||||
0.138 | 0.01 | 61,890 | 0.0289 | 0.0303 | 0.0267 | 0.0239 | 0.0257 | ||||
0.138 | 0.005 | 51,314 | 0.0424 | 0.0201 | 0.0222 | 0.0248 | 0.0293 | ||||
0.141 | 0.002 | 47,918 | 0.0321 | 0.0204 | 0.0251 | 0.0248 | 0.0286 | ||||
0.137 | 0.001 | 46,934 | 0.0253 | 0.0183 | 0.0223 | 0.0246 | 0.0233 | ||||
GCSF | 7904 | 0.0173 | 0.0601 | 0.246 | 0.1 | 427,393 | 0.0707 | 0.0820 | 0.0620 | 0.0604 | 0.0580 |
0.198 | 0.05 | 250,222 | 0.0636 | 0.0607 | 0.0402 | 0.0436 | 0.0486 | ||||
0.164 | 0.025 | 127,583 | 0.0501 | 0.0415 | 0.0302 | 0.0360 | 0.0327 | ||||
0.142 | 0.01 | 61,846 | 0.0434 | 0.0257 | 0.0280 | 0.0239 | 0.0257 | ||||
0.122 | 0.005 | 51,266 | 0.0379 | 0.0196 | 0.0205 | 0.0247 | 0.0238 | ||||
0.120 | 0.002 | 47,919 | 0.0413 | 0.0183 | 0.0219 | 0.0236 | 0.0201 | ||||
0.112 | 0.001 | 46,939 | 0.0312 | 0.0159 | 0.0234 | 0.0207 | 0.0202 | ||||
IL10 | 7681 | 0.0186 | 0.0691 | 0.331 | 0.1 | 427,218 | 0.0621 | 0.1019 | 0.0584 | NA | NA |
0.310 | 0.05 | 250,109 | 0.0670 | 0.0858 | 0.0448 | NA | NA | ||||
0.198 | 0.025 | 127,543 | 0.0566 | 0.0463 | 0.0356 | 0.0382 | 0.0406 | ||||
0.130 | 0.01 | 61,944 | 0.0328 | 0.0225 | 0.0251 | 0.0268 | 0.0258 | ||||
0.141 | 0.005 | 51,257 | 0.0400 | 0.0220 | 0.0237 | 0.0282 | 0.0238 | ||||
0.148 | 0.002 | 47,880 | 0.0433 | 0.0183 | 0.0204 | 0.0271 | 0.0231 | ||||
0.142 | 0.001 | 46,898 | 0.0317 | 0.0194 | 0.0219 | 0.0261 | 0.0228 |
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So, H.-C.; Xue, X.; Ma, Z.; Sham, P.-C. SumVg: Total Heritability Explained by All Variants in Genome-Wide Association Studies Based on Summary Statistics with Standard Error Estimates. Int. J. Mol. Sci. 2024, 25, 1347. https://doi.org/10.3390/ijms25021347
So H-C, Xue X, Ma Z, Sham P-C. SumVg: Total Heritability Explained by All Variants in Genome-Wide Association Studies Based on Summary Statistics with Standard Error Estimates. International Journal of Molecular Sciences. 2024; 25(2):1347. https://doi.org/10.3390/ijms25021347
Chicago/Turabian StyleSo, Hon-Cheong, Xiao Xue, Zhijie Ma, and Pak-Chung Sham. 2024. "SumVg: Total Heritability Explained by All Variants in Genome-Wide Association Studies Based on Summary Statistics with Standard Error Estimates" International Journal of Molecular Sciences 25, no. 2: 1347. https://doi.org/10.3390/ijms25021347
APA StyleSo, H.-C., Xue, X., Ma, Z., & Sham, P.-C. (2024). SumVg: Total Heritability Explained by All Variants in Genome-Wide Association Studies Based on Summary Statistics with Standard Error Estimates. International Journal of Molecular Sciences, 25(2), 1347. https://doi.org/10.3390/ijms25021347