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

Abstract

Background

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.

Methods

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.

Results

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.

Conclusion

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.

• Bayesian estimation
• maximum likelihood
• prior specification
• power
• repeated measures analyses
• small samples
• burn survivors
• mechanical ventilation
• PTSS

For the abstract or full text in other languages, please see Supplementary files under ‘Article Tools’

For the abstract or full text in other languages, please see Supplementary files under ‘Article Tools’

Conflict of interest and funding

Funding for this study was provided by a grant from The Netherlands Organization for Scientific Research: NWO-VENI-451-11-008. The original study was financially supported by the Dutch Burns Foundation.

Notes

For the abstract or full text in other languages, please see Supplementary files under ‘Article Tools’

¹We estimated the power using a repeated measure model with three outcome variables and regressing a dichotomous variable on the slope and parameters based on the empirical data. For more information and the syntax files go to the website of the first author.

² http://rensvandeschoot.com/433/additional-information-analyzing-small-data-sets-using-bayesian-estimation.

³ http://rensvandeschoot.com/433/additional-information-analyzing-small-data-sets-using-bayesian-estimation.

⁴ http://rensvandeschoot.com/433/additional-information-analyzing-small-data-sets-using-bayesian-estimation.

⁵Note that because we constrained the covariances between the latent variable to zero, an IG distribution is used in Mplus instead of the multivariate Inverse Wishart distribution, see Asparouhov and Muthén (2010).

⁶“The Bayesian counterpart of the frequentist confidence interval is the Posterior Probability Interval (PPI), also referred to as the credibility interval. The PPI is the 95% probability that in the population the parameter lies between the two values. Note, however, that the PPI and the confidence interval may numerically be similar and might serve related inferential goals, but they are not mathematical equivalent and conceptually quite different” (Van de Schoot et al., 2014, p. 849).

⁷ http://rensvandeschoot.com/433/additional-information-analyzing-small-data-sets-using-bayesian-estimation.