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I’m trying to calculate the vaccine efficacy of a study in an effort to understand a bit more about the field but I can’t get to their result.

Here’s what I’m doing:

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This is a link to the full-test of the article whose results are presented in the post.

https://www.acpjournals.org/doi/full/10.7326/M21-1577

The study reported used a study design called a “test negative case-control” design. References 15-17 in the published explain the “test negative case-control design and the rationale for its use to evaluate vaccine effectiveness after the introduction into use. Only reference 17 is available full-text online for free.

https://academic.oup.com/aje/article/184/5/345/2389013

As described in the linked article, using this design, among people undergoing testing for the agent that is of interest (in this case, SARS-CoV-2 virus) in a health care setting (in this case, VA facilities):

“those testing positive for …….virus are defined as cases, and those testing negative form the comparison [control] group. Data on patients' vaccination histories and confounder profiles are also collected. Vaccine effectiveness is estimated from the odds ratio comparing the odds of testing positive for ……[the agent of interest] among vaccinated patients and unvaccinated patients, adjusting for confounders.”

In the study about the effectiveness of the Pfizer and Modern vaccines in preventing infection with the SARS-CoV-2 virus, to control for confounding, cases (test positive for SARS-CoV-2) and controls (test negative for SARS-CoV-2) were matched on potential confounders based on a propensity score.

“For each person who tested positive, we identified a propensity score–matched control participant who tested negative, matched by age, sex, race, body mass index, Charlson Comorbidity Index score, and geographic location.”

The actual derivation of the propensity score is not described in the paper.

After one-to-one matching on the propensity score, the analysis used conditional logistic regression to estimate the odd for being test-positive given being vaccinated and the odds for being test-positive given being non-vaccinated. Conditional logistic regression takes into account the matched nature of the study.

In the study, vaccine effectiveness was estimated as:

“1 − Odds(T+|Vaccinated) / Odds(T+|Nonvaccinated)”

This equation can also be expressed as:

1 - OR  (odds ratio from a 2 by 2 Table)

In the post, the calculations shown use the difference as a measure of vaccine effectiveness. While the risk difference CAN be used to measure vaccine effectiveness, the risk ratio (or odds ratio) is the most usual measure. See CDC.

https://www.cdc.gov/csels/dsepd/ss1978/lesson3/section6.html

The data shown in the past are from Table 2 of the published paper. The title of Table 2 indicates that the estimate of effectiveness is from a conditional logistic regression analysis. The estimated odds ratios based on a conditional logistic regression will be different from the odds ratio calculated from the “raw” data in Table 2. The conditional odds ratio estimates cannot be determined from the data shown in the Table.

As a quick check on how the use of a conditional analysis might have affected the estimates shown in the Table, the “raw” data in Table 2 can be used to estimate the odds ratios for being test-positive for SARS-CoV-2 (a case) >=7 days after 2 doses. This was the primary measure of effectiveness specified in the paper.

For the Moderna vaccine, using the “raw” data in Table 2, for vaccination >= 7 days after 2 doses, the odds ratio for being test positive is:

(45 x 36,538) / (1688 x 44,560) = 0.022

For the Moderna vaccine, the estimated effectiveness of vaccination >=7 days after 2 doses (compared with being unvaccinated) is:

1 – OR or 1 – 0.022 = 0.978 

which is 97.8% expressed as a percentage.

For the Pfizer vaccine, using the “raw” data in Table 2, for vaccination >= 7 days after 2 doses, the odds ratio for being test positive is:

(121 x 36,538) / (2245 x 44,560) = 0.044

For the Pfizer vaccine, the estimated effectiveness of vaccination >=7 days after 2 doses (compared with being unvaccinated) is:

1 – OR or 1 – 0.044 = 0.956

which is 95.6% expressed as a percentage.

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  • Thank you very much for the awesome answer.
    – Francois
    Jul 24 at 12:47

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