Table 1 Estimated contribution of GRS×E effects for 22 traits in the UK biobank.

From: Quantification of the overall contribution of gene-environment interaction for obesity-related traits

Trait

\(\widehat{{{\boldsymbol{\alpha }}}_{{\bf{1}}}}\)

\(\widehat{{{\boldsymbol{\alpha }}}_{{\bf{2}}}}\)

\(\widehat{{\boldsymbol{\beta }}}\)

\(\widehat{{\boldsymbol{\gamma }}}({\boldsymbol{P}})\)

\(\widehat{{{\boldsymbol{\gamma }}}_{{\boldsymbol{K}}}}\)

PΔ

\({{\hat{\boldsymbol{\beta }}}}_{{\boldsymbol{L}}}\)

\({{\hat{\boldsymbol{\gamma }}}}_{{\boldsymbol{L}}}\)

Body mass index (BMI)

0.232

0.013

0.65

0.139 (4.3e−78)

0.126

2.8e−01

0.65

0.14

Trunk fat-free mass

0.310

0.007

0.32

0.120 (9.0e−53)

0.193

1.3e−15

0.25

0.14

Whole-body fat-free mass

0.301

0.007

0.39

0.119 (5.4e−43)

0.191

3.7e−05

0.72

0.14

Whole-body fat mass

0.225

0.011

0.72

0.124 (3.5e−42)

0.117

6.3e−01

0.72

0.13

Leg fat mass

0.217

0.015

0.68

0.161 (3.6e−36)

0.149

5.0e−01

0.69

0.15

Whole-body water mass

0.301

0.007

0.39

0.125 (3.8e−31)

0.190

4.2e−02

0.36

0.14

Waist circumference

0.198

0.011

0.65

0.099 (1.7e−30)

0.111

3.9e−01

0.62

0.10

Hip circumference

0.221

0.010

0.69

0.119 (8.3e−29)

0.146

4.8e−02

0.69

0.11

Arm predicted mass

0.270

0.006

0.41

0.117 (9.3e−29)

0.203

3.0e−13

0.35

0.14

Weight

0.257

0.011

0.69

0.123 (2.2e−26)

0.122

9.3e−01

0.65

0.14

Arm fat mass

0.217

0.017

0.70

0.180 (3.6e−25)

0.140

6.3e−02

0.72

0.19

Arm fat percentage

0.204

0.006

0.26

0.108 (1.1e−24)

0.148

1.5e−03

0.28

0.11

Trunk predicted mass

0.309

0.007

0.32

0.120 (2.2e−24)

0.195

4.1e−09

0.24

0.14

Leg fat-free mass

0.276

0.008

0.54

0.126 (2.6e−23)

0.179

1.9e−04

0.62

0.14

Arm fat-free mass

0.271

0.006

0.41

0.120 (3.1e−23)

0.199

8.1e−03

0.35

0.14

Basal metabolic rate

0.287

0.009

0.48

0.136 (4.8e−23)

0.187

2.9e−02

0.48

0.14

Leg predicted mass

0.275

0.008

0.56

0.124 (4.2e−11)

0.181

2.7e−02

0.63

0.14

Impedance of leg

0.255

−0.001

0.08

0.070 (1.4e−07)

−0.162

1.7e−57

0.13

0.12

Leg fat percentage

0.203

0.005

0.45

0.064 (7.7e−07)

0.108

5.1e−03

0.33

0.08

FVC

0.235

0.008

0.56

0.071 (3.1e−05)

0.189

2.2e−10

0.29

0.06

Sitting height

0.357

0.001

0.09

0.059 (2.9e−03)

−0.258

1.7e−55

0.07

0.09

Impedance of whole body

0.264

0.003

0.29

0.048 (8.2e−03)

0.141

8.1e−07

0.25

0.11

  1. Only those 22 (of the 32) continuous traits are shown for which no maximum likelihood estimation convergence issues were detected. Column label abbreviations are as follows: α1: GRS effect, α2: GRS2 effect, β: environmental effect, γ: interaction effect, γK: interaction effect of fake GRS, PΔ: P-value for testing γ = γK, \({\widehat{\beta }}_{L}\): β estimate for the fake-Y approach, \({\widehat{\gamma }}_{L}\): γ estimate for the fake-Y approach. Note that significantly non-zero interactions are claimed only when the P-value of the estimate (\(\widehat{\gamma }\)) is below 0.05/32, i.e., Pγ < 1.5 × 10−3, which is based on the Bonferroni correction for multiple testing (ensuring family-wise error rate control at 5%). Pγ was calculated based on the Wald test (two-sided), and PΔ was derived from a two-sided Z test.
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