Table 1 5 different methods of analysis to estimate the association between treatment and outcome

a

Univariable regression analysis

logit(P[Y + 1]) = β₀ + T ∗ β_T

b

Regression analysis with covariate adjustment

logit(P[Y + 1]) = C ∗ β_C + T ∗ β_T

c

Propensity score adjustment

PS = C ∗ β_C

logit(P[Y + 1]) = PS ∗ β_PS + T ∗ β_T

d

IV analysis

logit(P[Y + 1]) = Z ∗ β_Z

e

IV analysis with correction for hospital

logit(P[Y + 1]) = Z ∗ β_Z + H ∗ β_H

f

IV analysis with correction for hospital and all measured and unmeasured confounders

logit(P[Y + 1]) = Z ∗ β_Z + H ∗ β_H + C ∗ β_C + U ∗ β_U