How estimating nuisance parameters can reduce the variance (with consistent variance estimation)

We often estimate a parameter of interest psi when the identifying conditions involve a nuisance parameter theta. Examples from causal inference are Inverse Probability Weighting, Marginal Structural Models and Structural Nested Models. To estimate treatment effects from observational data, these methods posit a (pooled) logistic regression model for the treatment and/or censoring probabilities and estimate these first. These methods are all based on unbiased estimating equations. First, we provide a general formula for the variance of the parameter of interest psi when the nuisance parameter theta is estimated in a first step, using the Partition Inverse Formula. Then, we present 4 results for estimators psi-hat based on unbiased estimating equations including a nuisance parameter theta which is estimated by solving (partial) score equations, if psi does not depend on theta. This regularly happens in causal inference if theta describes the treatment probabilities, in settings with missing data where theta describes the missingness probabilities, and settings with measurement error where theta describes the measurement error distribution. 1. Counter-intuitively, the limiting variance of psi-hat is typically smaller when theta is estimated, compared to if a known theta were plugged in. 2. If estimating theta is ignored, the resulting sandwich estimator for the variance of psi-hat is conservative. 3. A consistent estimator for the variance of psi-hat can provide results fast: no bootstrap. 4. If psi-hat with the true theta plugged in is efficient, the limiting variance of psi-hat does not depend on whether theta is estimated. To illustrate we use observational data to estimate 1. the effect of cazavi versus colistin in patients with resistant bacterial infections and 2. how the effect of one year of antiretroviral treatment depends on its initiation time in HIV-infected patients.