Table 1 Bias scenarios with covariate dependencies in the data generation and estimation functions. For each bias scenario, two experiment groups explore the effect of suitable negative controls (having the same potential confounding mechanism as the outcome of interest) and unsuitable negative controls. The third column shows the covariate dependencies of the data generating functions: f^⋆ for the outcome of interest and f⁻ for the negative controls. X is the set of measured confounders

Reference case

1

f^⋆ and f⁻ depend on X

Unmeasured confounder (U)

2.1

f^⋆ and f⁻ depend on X and U

2.2

Only f^⋆ depends on X and U

Model misspecification:

Quadratic term (\({X}_1^2\))

3.1

f^⋆ and f⁻ depend on X and \({\mathrm{X}}_1^2\)

3.2

Only f^⋆ depends on X and \({X}_1^2\)

Model misspecification:

Interaction between two confounders (X₁X₂)

4.1

f^⋆ and f⁻ depend on X and X₁X₂

4.2

Only f^⋆ depends on X and X₁X₂

Lack of positivity

Lack of overlap between treatment groups in terms of propensity score.

5.1

f^⋆ and f⁻ depend on X.

5.2

Only f^⋆ depends on X.

Measurement error in confounder

6

f^⋆ and f⁻ depend on X.

One of the confounders with large effect have random measurement error.