Delta-method in Epidemiology

Abstract

Approximate statistical inference via determination of the asymptotic distribution of the statistics is a technique that epidemiologists and applied statisticians use routinely, we name it the classical delta-method, but there is a gap as it is not routinely taught and many applied researchers do not understand it neither its uses. The delta-method is a theorem which states that a smooth function of an asymptotically normal estimator is also asymptotically normally distributed. It can also be viewed as a technique for approximating the variance of a functional i.e., a nonlinear function of a random variable that can be approximated by averages. A fundamental problem in inferential statistics is to approximate the distribution of a statistic calculated from a functional derived from a transformation or the predictions from a regression model fitted parameters. Machine learning methods and causal inference problems applied to epidemiology requires to generate the point estimates from asymptotically linear parameters derived as functionals that are approximated by averages, but the standard error of these point estimates are not so easily calculated with analytical forms. Therefore, we need to approximate the distribution of the standard error for statistical inference using the functional delta-method based on the influence curve. The venue of new methods and techniques requires a constant update on valid statistical inference for applied epidemiologists and statisticians. In this tutorial we introduce the use of the classical and functional delta-method and its link to the Influence Curve in Epidemiology from a practical perspective. We provided a mostly applied overview of the classical and functional delta-method, the influence curve and the bootstrap including easy to understand R code and examples based on cancer epidemiology that may help to fill this gap.

Keywords: Statistical inference; Bootstrap; Delta-method; Epidemiology; Causal Inference; Tutorial.

Thank you

Thank you for reading the tutorial.

You can cite this repository as:
Luque-Fernandez MA, (2020) et al. Delta Method in Epidemiology: An Applied and Reproducible Tutorial. GitHub repository, http://migariane.github.io/DeltaMethodEpi.nb.html

If you have updates or changes that you would like to make, please send me a pull request. Alternatively, if you have any questions, please e-mail me.

Miguel Angel Luque Fernandez
E-mail: miguel-angel.luque at lshtm.ac.uk
Twitter @WATZILEI
Academic website: https://scholar.harvard.edu/malf/home
Projects website: https://maluque.netlify.app/