Analyzing small data sets using Bayesian estimation: the case of posttraumatic stress symptoms following mechanical ventilation in burn survivors

The analysis of small data sets in longitudinal studies can lead to power issues and often suffers from biased parameter values. These issues can be solved by using Bayesian estimation in conjunction with informative prior distributions. By means of a simulation study and an empirical example concerning posttraumatic stress symptoms (PTSS) following mechanical ventilation in burn survivors, we demonstrate the advantages and potential pitfalls of using Bayesian estimation. First, we show how to specify prior distributions and by means of a sensitivity analysis we demonstrate how to check the exact influence of the prior (mis-) specification. Thereafter, we show by means of a simulation the situations in which the Bayesian approach outperforms the default, maximum likelihood and approach. Finally, we re-analyze empirical data on burn survivors which provided preliminary evidence of an aversive influence of a period of mechanical ventilation on the course of PTSS following burns. Not suprisingly, maximum likelihood estimation showed insufficient coverage as well as power with very small samples. Only when Bayesian analysis, in conjunction with informative priors, was used power increased to acceptable levels. As expected, we showed that the smaller the sample size the more the results rely on the prior specification. We show that two issues often encountered during analysis of small samples, power and biased parameters, can be solved by including prior information into Bayesian analysis. We argue that the use of informative priors should always be reported together with a sensitivity analysis.

纵向研究中的小样本数据分析往往面临统计效力不足的问题，且常出现参数估计值有偏的缺陷。此类问题可通过结合信息先验分布的贝叶斯估计（Bayesian estimation）得到解决。本研究通过一项模拟研究，以及针对烧伤幸存者机械通气后创伤后应激症状（posttraumatic stress symptoms, PTSS）的实证案例，阐明了贝叶斯估计的应用优势与潜在陷阱。首先，我们演示了先验分布的设定流程，并通过敏感性分析展示了如何检验先验（误）设定的具体影响。其次，我们借助模拟实验明确了贝叶斯方法相较于默认极大似然估计法的优势场景。最后，我们对烧伤幸存者的实证数据进行了二次分析，该数据曾初步证实机械通气时长会对烧伤后创伤后应激症状的发展轨迹产生不利影响。不出所料，极小样本下的极大似然估计同样存在覆盖率不足与统计效力低下的问题。唯有结合信息先验的贝叶斯分析，可将统计效力提升至可接受水平。正如预期，样本量越小，分析结果对先验设定的依赖程度越高。本研究表明，小样本分析中常遇到的两大核心问题——统计效力不足与参数估计有偏——均可通过在贝叶斯分析中引入先验信息得以解决。我们主张，在使用信息先验时，必须同步报告敏感性分析结果。