I think that the concern about time pertains to the attempt to estimate VE >120 days after full vaccination comparing the Moderna and Pfizer vaccines. The fully-adjusted estimates of VE at >120 days after full vaccination presented in the paper are 88% for Moderna and 77% for Pfizer.
If people who got the Pfizer vaccine got it earlier (in secular time—e.g., a higher proportion vaccinated in December, January, February, and March compared with Moderna), then the Pfizer vaccinated people would have accumulated more person-time at risk of having a breakthrough infection >120 days after full vaccination. The number of Pfizer vaccinated people hospitalized with a breakthrough infection >120 days after full vaccination would be higher simply because of a longer period at risk >120 days after full vaccination. The estimated odds ratio for COVID-19 hospitalization >120 days after full vaccination would be biased upward for the Pfizer vaccine (away from zero and toward 1.0) and the estimated VE for the Pfizer vaccine >120 days after full vaccination would be biased downward for the Pfizer vaccine (less effective when expressed as a percentage–(100 x (1 minus OR)).
The two vaccines had comparable EUA dates (December 11, 2020 for Pfizer and December 18, 2020 for Moderna) but “roll-out” of the two vaccines over time, by place, and between patient groups could have been differential.
Using the data in Table 2, it is possible to the calculate the percentage of all controls vaccinated with the Moderna vaccine who were >120 days after full vaccination. The estimate is 18.3% (77/422). The proportion of all controls vaccinated with the Pfizer vaccine who were >120 days after full vaccination is 18.9% (115/610). While reassuring, it is still possible that >120 days after full vaccination “hides” a fair number of people (perhaps really old people), who got the Pfizer vaccine very early, have been observed for a longer time than those who got the Moderna vaccine, and who are at greater risk of breakthrough infection (perhaps being older).
As more data accumulate, an analysis showing estimates of VE for the Moderna and Pfizer vaccines as a function of time after full vaccination would be of interest.
I hope others will weigh in on whether putting a variable—time since full vaccination—into the logistic model using the data at hand fixes the problem (if there is a problem).