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.

Keywords: Bayesian estimation; PTSS; burn survivors; maximum likelihood; mechanical ventilation; power; prior specification; repeated measures analyses; small samples.

Fig. 1

PTSS scores over time of mechanical ventilation and non-mechanical ventilation group.

Fig. 2

Estimated model in Mplus.

Fig. 3

The effect of different estimators (ML vs. Bayes) and different values for ${\sigma }_0^2$, but with a fixed µ ₀ for β.

Fig. 4

Influence of miss-specification of prior mean for β (µ ₀) on the posterior mean for β (µ ₁) with different prior variances (${\sigma }_0^2$).

Fig. 5

Simulation results for different priors for IG(α₀, v₀).

Fig. 6

Trace plot of ${\mathrm{\sigma }}_{\mathrm{LS}}^{\mathrm{2}}$ with IG(−1,0)

Fig. 7

Trace plot of ${\mathrm{\sigma }}_{\mathrm{LS}}^{\mathrm{2}}$ with IG(0.001,0.001).

Fig. 8

Trace plot of ${\mathrm{\sigma }}_{\mathrm{LS}}^{\mathrm{2}}$ with IG(0.5, 0.5).

Fig. 9

The results of the simulation study in terms of power.

Fig. 10

The results of the simulation study in terms of coverage.