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Approximate statistical inference via determination of the asymptoticdistribution of a statistic is routinely used for inference in applied medicalstatistics (e.g. to estimate the standard error of the marginal or conditionalrisk ratio). One method for variance estimation is the classical Delta-methodbut there is a knowledge gap as this method is not routinely included intraining for applied medical statistics and its uses are not widely understood.Given that a smooth function of an asymptotically normal estimator is alsoasymptotically normally distributed, the Delta-method allows approximating thelarge-sample variance of a function of an estimator with known large-sampleproperties. In a more general setting, it is a technique for approximating thevariance of a functional (i.e., an estimand) that takes a function as an inputand applies another function to it (e.g. the expectation function).Specifically, we may approximate the variance of the function using thefunctional Delta-method based on the influence function (IF). The IF exploreshow a functional $\phi(\theta)$ changes in response to small perturbations inthe sample distribution of the estimator and allows computing the empiricalstandard error of the distribution of the functional. The ongoing developmentof new methods and techniques may pose a challenge for applied statisticianswho are interested in mastering the application of these methods. In thistutorial, we review the use of the classical and functional Delta-method andtheir links to the IF from a practical perspective. We illustrate the methodsusing a cancer epidemiology example and we provide reproducible and commentedcode in R and Python using symbolic programming. The code can be accessed athttps://github.com/migariane/DeltaMethodInfluenceFunction

The Delta-Method and Influence Function in Medical Statistics: a Reproducible Tutorial