  Approximate statistical inference via determination of the asymptotic
distribution of a statistic is routinely used for inference in applied medical
statistics (e.g. to estimate the standard error of the marginal or conditional
risk ratio). One method for variance estimation is the classical Delta-method
but there is a knowledge gap as this method is not routinely included in
training for applied medical statistics and its uses are not widely understood.
Given that a smooth function of an asymptotically normal estimator is also
asymptotically normally distributed, the Delta-method allows approximating the
large-sample variance of a function of an estimator with known large-sample
properties. In a more general setting, it is a technique for approximating the
variance of a functional (i.e., an estimand) that takes a function as an input
and applies another function to it (e.g. the expectation function).
Specifically, we may approximate the variance of the function using the
functional Delta-method based on the influence function (IF). The IF explores
how a functional $\phi(\theta)$ changes in response to small perturbations in
the sample distribution of the estimator and allows computing the empirical
standard error of the distribution of the functional. The ongoing development
of new methods and techniques may pose a challenge for applied statisticians
who are interested in mastering the application of these methods. In this
tutorial, we review the use of the classical and functional Delta-method and
their links to the IF from a practical perspective. We illustrate the methods
using a cancer epidemiology example and we provide reproducible and commented
code in R and Python using symbolic programming. The code can be accessed at
https://github.com/migariane/DeltaMethodInfluenceFunction
