Use of recently vaccinated individuals to detect bias in test-negative case-control studies of COVID-19 vaccine effectiveness

Use of recently vaccinated individuals to detect bias in test-negative case-control studies of COVID-19 vaccine effectiveness

Supplementary Material

Hitchings et al

Supplementary Methods Derivation of bias-indicator

The odds ratio comparing vaccination status between cases and controls is given by 𝑂𝑅_𝑈^𝑅(𝑡) = 𝐶_𝑅+(𝑡)𝐶_𝑈−(𝑡)

𝐶𝑅−(𝑡)𝐶𝑈+(𝑡)

= 𝜋₋𝜆₋[𝜇_𝑈𝛼_𝑈−(1 − 𝑣)𝑡 +𝜇_𝑃𝛼_𝑃−𝑣𝑇_𝑉]𝑃

𝜋₊[𝜇_𝑈(1 − 𝑒^{−𝛼𝑈+𝜆+𝑡})(1 − 𝑣) +𝜇_𝑃(1 − 𝑒^{−𝛼𝑃+𝜆+𝑇𝑉})𝑣]𝑃

𝜋₊𝜇_𝑅(1 − 𝑒^{−𝛼𝑅+𝜆+𝑇𝑃})𝑣𝑃 𝜋₋𝜇_𝑅𝛼𝑅−𝜆₋𝑇𝑃𝑣𝑃

≈𝛼_𝑅+

𝛼_𝑅−

[𝜇_𝑈𝛼𝑈−(1 − 𝑣)𝑡 +𝜇_𝑃𝛼𝑃−𝑣𝑇𝑉] [𝜇_𝑈𝛼𝑈+(1 − 𝑣)𝑡 +𝜇_𝑃𝛼𝑃+𝑣𝑇𝑉], as 1 − 𝑒^−𝑥≈ 𝑥 when x is small.

Derivation of bias-corrected vaccine effectiveness estimate

The odds ratio comparing vaccination status between cases and controls is given by 1 − 𝑂𝑅_𝑅^𝐹(𝑡) = 1 − 𝐶_𝐹+(𝑡)𝐶_𝑅−(𝑡)

𝐶𝐹−(𝑡⁾𝐶𝑅+(𝑡⁾

= 1 − (𝜋+𝜇_𝑉[(1 − 𝜑)(1 − 𝑒^{−𝛼𝑉+𝜆+(𝑡−𝑇𝑃−𝑇𝑉)}) + 𝜑(1 − 𝑒^{−𝛼𝑉+𝜃𝜆+(𝑡−𝑇𝑃−𝑇𝑉)})]𝑣𝑃 𝜋₊𝜇_𝑉(1 − 𝑒^{−𝛼𝑉+𝜆+𝑇𝑃})𝑣𝑃 ⁾

𝜋−𝜇_𝑉𝛼𝑉−𝜆−𝑇𝑃𝑣𝑃 𝜋₋𝜇_𝑉𝛼_𝑉−𝜆₋(𝑡 − 𝑇_𝑃− 𝑇_𝑉)𝑣𝑃

≈ 1 − ([(1 − 𝜑)𝛼_𝑉+𝜆₊(𝑡 − 𝑇_𝑃− 𝑇_𝑉) + 𝜑𝛼_𝑉+𝜃𝜆₊(𝑡 − 𝑇_𝑃− 𝑇_𝑉)]

(𝛼_𝑉+𝜆₊𝑇_𝑃) ) 𝑇𝑃

(𝑡 − 𝑇_𝑃− 𝑇𝑉)

= 𝜑(1 − 𝜃).

Again, 1 − 𝑒^−𝑥≈ 𝑥 when x is small.

Derivation of bias-corrected vaccine effectiveness estimate with varying TV

If vaccination occurred on a set of days TVj, and a proportion pj of vaccinated individuals is vaccinated on each vaccination day, then

𝐶_𝐹+(𝑡) = ∑ 𝐶_𝐹+(𝑡|𝑇𝑉= 𝑇𝑉𝑗)𝑝_𝑗

𝑗

𝐶_𝐹−(𝑡) = ∑ 𝐶_𝐹−(𝑡|𝑇_𝑉= 𝑇_𝑉𝑗)𝑝_𝑗

𝑗

Expressions for 𝐶_𝑅+(𝑡) and 𝐶_𝑅−(𝑡) are independent of TV, and therefore remain as in the main text.

Thus,

1 − 𝑂𝑅_𝑅^𝐹(𝑡) = 1 − 𝐶𝐹+(𝑡⁾𝐶𝑅−(𝑡⁾ 𝐶_𝐹−(𝑡)𝐶_𝑅+(𝑡)

= 1 − (

∑_𝑗[(1 − 𝜑) (1 − 𝑒^{−𝛼𝑉+𝜆+(𝑡−𝑇𝑃−𝑇𝑉𝑗)}) + 𝜑 (1 − 𝑒^{−𝛼𝑉+𝜃𝜆+(𝑡−𝑇𝑃−𝑇𝑉𝑗)})]𝑝_𝑗

(1 − 𝑒^{−𝛼𝑉+𝜆+𝑇𝑃}) ) 𝑇𝑃

∑_𝑗(𝑡 − 𝑇𝑃− 𝑇𝑉𝑗)𝑝_𝑗

≈ 1 − (

𝛼_𝑉+𝜆₊[(1 − 𝜑) + 𝜑𝜃] ∑_𝑗(𝑡 − 𝑇_𝑃− 𝑇_𝑉𝑗)𝑝_𝑗 𝛼_𝑉+𝜆₊𝑇_𝑃 ⁾

𝑇𝑃

∑_𝑗(𝑡 − 𝑇𝑃− 𝑇𝑉𝑗)𝑝_𝑗

= 𝜑(1 − 𝜃).