Bayesian analysis of fuel economy experiments

Statistical analysts can encounter difficulties in obtaining point and interval estimates for fixed effects when sample sizes are small and there are two or more error strata to consider. Standard methods can lead to certain variance components being estimated as zero which often seems contrary to engineering experience and judgement. Shell Global Solutions (UK) has encountered such challenges and is always looking for ways to make its statistical techniques as robust as possible. In this instance, the challenge was to estimate fuel effects and confidence limits from small-sample fuel economy experiments where both test-to-test and day-to-day variation had to be taken into account. Using likelihood-based methods, the experimenters estimated the day-to-day variance component to be zero which was unrealistic. The reason behind this zero estimate is that the data set is not large enough to estimate it reliably. The experimenters were also unsure about the fixed parameter estimates obtained by likelihood methods in linear mixed models. In this paper, we looked for an alternative to compare the likelihood estimates against and found the Bayesian platform to be appropriate. Bayesian methods assuming some non-informative and weakly informative priors enable us to compare the parameter estimates and the variance components. Profile likelihood and bootstrap based methods verified that the Bayesian point and interval estimates were not unreasonable. Also, simulation studies have assessed the quality of likelihood and Bayesian estimates in this study.