Table 2 Equivalence of the adjusted RR assessed in a log-binomial regression model of the original cohort with N subjects and the adjusted OR assessed in a logistic regression model of the expanded cohort with N+N_.1 records, where \(N_{.1} = N_{e1} + N_{\bar {e}1}\) is the total number of cases in the original cohort

A) Cohort

Expected Y=1

Expected Y=0

X=e

N_e. exp{α+β+γZ}

N_e.(1−exp{α+β+γZ})

\(X = \bar {e}\)

\(N_{\bar {e}.} \exp {\{\alpha + \gamma Z\}}\)

\(N_{\bar {e}.} (1 - \exp {\{\alpha + \gamma Z\}})\)

B) Expanded cohort

Expected Y^∗=1

Expected Y^∗=0

Odds

X=e

N_e. exp{α+β+γZ}

N_e.

exp{α+β+γZ}

\(X = \bar {e}\)

\(N_{\bar {e}.} \exp {\{\alpha + \gamma Z\}}\)

\(N_{\bar {e}.}\)

exp{α+γZ}