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Table 1 Scenarios used to generate the data a

From: A systematic comparison of statistical methods to detect interactions in exposome-health associations

Subscenario

Adjusted R 2

Pairwise corr. b

Interaction size (and sign)

Parameters c

Scenario 1. True model: F(E)=β 0+β 1 X 1+β 2 X 2+β 3 X 3+β 4 X 4+β 5 X 5 (Model size = 5; No interactions)

1a

0.10 (0.07, 0.16)

Mixed

  

σ=7.5

1b

0.30 (0.23, 0.39)

Mixed

  

σ=3.8

1c

0.11 (0.09, 0.12)

High

  

σ=13

1d

0.27 (0.25, 0.28)

High

  

σ=7.5

Scenario 2. True model: F(E)=β 0+β 1 X 1+β 2 X 2+β 3 X 3+β 4 X 4+β 5 X 5+γ 12 X 1 X 2 (Model size = 6; Only one 2-way interaction)

2a

0.09 (0.07, 0.14)

Mixed

Strong (+)

γ 12=1

σ=8.3

2b

0.09 (0.06, 0.15)

Mixed

Strong (−)

γ 12=−1

σ=8.3

2c

0.10 (0.06, 0.15)

Mixed

Moderate (+)

γ 12=0.5

σ=7.8

2d

0.10 (0.07, 0.14)

Mixed

Moderate (−)

γ 12=−0.5

σ=7.8

2e

0.13 (0.11, 0.14)

High

Strong (+)

γ 12=1

σ=12

2f

0.13 (0.11, 0.15)

High

Strong (−)

γ 12=−1

σ=12

2g

0.30 (0.28, 0.32)

High

Moderate (+)

γ 12=0.5

σ=7

2h

0.30 (0.28, 0.32)

High

Moderate (−)

γ 12=−0.5

σ=7

Scenario 3. True model: F(E) = β 0+β 1 X 1+β 2 X 2+β 3 X 3+β 4 X 4+β 5 X 5+γ 12 X 1 X 2+γ 13 X 1 X 3 (Model size = 7; X 1 involved in two 2-way interactions)

3a

0.11 (0.08, 0.15)

Mixed

Strong (+)

γ 12=γ 13=1

σ=8.3

3b

0.10 (0.08, 0.16)

Mixed

Strong (−)

γ 12=γ 13=−1

σ=8.3

3c

0.10 (0.06, 0.14)

Mixed

Moderate (+)

γ 12=γ 13=0.5

σ=7.8

3d

0.10 (0.07, 0.14)

Mixed

Moderate (−)

γ 12=γ 13=−0.5

σ=7.8

3e

0.29 (0.27, 0.32)

High

Strong (+)

γ 12=γ 13=1

σ=8

3f

0.29 (0.27, 0.31)

High

Strong (−)

γ 12=γ 13=−1

σ=8

3g

0.31 (0.29, 0.33)

High

Moderate (+)

γ 12=γ 13=0.5

σ=7

3h

0.31 (0.28, 0.33)

High

Moderate (−)

γ 12=γ 13=−0.5

σ=7

  1. aIn each of the three scenarios, the outcome Y was generated as Y=F(E)+ε, where F(E) is a function of the predictors X 1,…,X 5, and ε∼N(0,σ). In each scenario, subscenarios were considered according to the pairwise correlation of the predictors (“Mixed”, when selecting the predictors among the whole exposome, in which case the absolute pairwise correlation ranged from 0.0000 to 1.0000; or “High”, when selecting the predictors among the subset of the 13 variables in the exposome for which all absolute pairwise correlations were 0.62 or higher); the size of the interaction terms (“Strong”, corresponding to equal size than the main effects size; or “Moderate”, corresponding to size 1/2 of the “Strong”), and the sign of the interaction terms (+ or −). Values for the adjusted R 2 correspond to the mean and percentiles 2.5th and 97.5th as a result of fitting the model to 100 simulated datasets. bThe median of the mean pairwise correlation between the true predictors was 0.12 (percentiles 2.5th and 97.5th: (0.05, 0.25)) for “Mixed”, and 0.78 (percentiles 2.5th and 97.5th: (0.72, 0.87)) for “High”. The median of the mean pairwise correlation between the true predictors and the other exposures was 0.13 (percentiles 2.5th and 97.5th: (0.09, 0.16)) for “Mixed”, and 0.18 (percentiles 2.5th and 97.5th: (0.17, 0.19)) for “High”. cIn all scenarios, β 0=β 1=⋯=β 5=1
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Environmental Health

ISSN: 1476-069X