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
翻译:通过确定统计的无症状分布而得出的近似统计推断,通常用于在应用医学统计中进行推断(例如,估算边际或有条件风险比率的标准误差)。差异估算的一种方法是古典三角洲方法,但存在一种知识差距,因为应用医学统计的培训中通常没有包括这种方法,而且其使用也没有被广泛理解。鉴于一个非正常的估测器的平稳功能通常也分布不均匀,德尔塔-方法允许在应用医学统计时,对一个具有已知大抽样属性的直径估算器函数的较大分布差异进行近似。在更一般的环境下,它是一种对应用功能(例如,估计值)进行匹配的方法,而该功能(例如,预测功能正常估测器)的顺利功能功能功能功能,也可以使用基于影响功能功能功能功能(IF)的Delta-method的功能变化进行近似似称。IFIF在应用的功能估算中,其功能模型的分布方法可以让正在使用的正位化方法(例如) 和直径的分布方法能够使正在使用的直观的流流化的流化方法在使用。