# Inference statistical analysis of continuous data based on confidence bands—Traditional and new approaches

## Abstract

In the analysis of continuous data, researchers are often faced with the problem that statistical methods developed for single-point data (e.g., *t* test, analysis of variance) are not always appropriate for their purposes. Either methodological adaptations of single-point methods will need to be made, or confidence bands are the method of choice. In this article, we compare three prominent techniques to analyze continuous data (single-point methods, Gaussian confidence bands, and function-based resampling methods to construct confidence bands) with regard to their testing principles, prerequisites, and outputs in the analysis of continuous data. In addition, we introduce a new technique that combines the advantages of the existing methods and can be applied to a wide range of data. Furthermore, we introduce a method enabling a priori and a posteriori power analyses for experiments with continuous data.

## Keywords

Bootstrap Hypothesis testing Power analysis Time series## Notes

### Author note

We thank Heiko Maurer for his methodological and mathematical advice. This research was funded by the Deutsche Forschungsgemeinschaft via the Collaborative Research Center on “Cardinal Mechanisms of Perception” (SFB-TRR 135) and the research projects MU 1374/3-1 and MU 1374 /5-1.

## Supplementary material

## References

- Efron, B. (1979). Bootstrap methods: Another look at the jackknife.
*Annals of Statistics*,*7*, 1–26. https://doi.org/10.1214/aos/1176344552 - Falkenstein, M., Hohnsbein, J., Hoormann, J., & Blanke, L. (1991). Effects of crossmodal divided attention on late ERP components: II. Error processing in choice reaction tasks.
*Electroencephalography and Clinical Neurophysiology*,*78*, 447–455. https://doi.org/10.1016/0013-4694(91)90062-9 CrossRefPubMedGoogle Scholar - Gehring, W. J., Goss, B., Coles, M. G. H., Meyer, D. E., & Donchin, E. (1993). A neural system for error detection and compensation.
*Psychological Science*,*4*, 385–390. https://doi.org/10.1111/j.1467-9280.1993.tb00586.x CrossRefGoogle Scholar - Joch, M., Hegele, M., Maurer, H., Müller, H., & Maurer, L. K. (2017). Brain negativity as an indicator of predictive error processing : The contribution of visual action effect monitoring.
*Journal of Neurophysiology*,*18*, 486–495. https://doi.org/10.1152/jn.00036.2017 CrossRefGoogle Scholar - Lenhoff, M. W., Santner, T. J., Otis, J. C., Peterson, M. G. E., Williams, B. J., & Backus, S. I. (1999). Bootstrap prediction and confidence bands: A superior statistical method for analysis of gait data.
*Gait and Posture*,*9*, 10–17. https://doi.org/10.1016/S0966-6362(98)00043-5 CrossRefPubMedGoogle Scholar - Makeig, S. (1993). Auditory event-related dynamics of the EEG spectrum and effects of exposure to tones.
*Electroencephalography and Clinical Neurophysiology*,*86*, 283–293. https://doi.org/10.1016/0013-4694(93)90110-H CrossRefPubMedGoogle Scholar - Makeig, S., Bell, A. J., Jung, T.-P., & Sejnowski, T. J. (1996). Independent component analysis of electroencephalographic data. In D. Touretzky, M. Moser, & M. Hasselmo (Eds.),
*Advances in neural information processing systems*(Vol. 8, pp. 145–151). Cambridge, MA: MIT Press. https://doi.org/10.1109/ICOSP.2002.1180091 Google Scholar - Makeig, S., Jung, T. P., Bell, A. J., Ghahremani, D., & Sejnowski, T. J. (1997). Blind separation of auditory event-related brain responses into independent components.
*Proceedings of the National Academy of Sciences*,*94*, 10979–10984.CrossRefGoogle Scholar - Maurer, L. K., Maurer, H., & Müller, H. (2015). Neural correlates of error prediction in a complex motor task.
*Frontiers in Behavioral Neuroscience*,*9*, 209:1–8. https://doi.org/10.3389/fnbeh.2015.00209 Google Scholar - Müller, H., & Sternad, D. (2004). Decomposition of variability in the execution of goal-oriented tasks: Three components of skill improvement.
*Journal of Experimental Psychology: Human Perception and Performance*,*30*, 212–233. https://doi.org/10.1037/0096-1523.30.1.212 PubMedGoogle Scholar - Tukey, J. W. (1958). Bias and confidence in not-quite large samples.
*Annals of Mathematical Statistics*,*29*, 614. https://doi.org/10.2307/2237363 CrossRefGoogle Scholar - Wilcox, R. R. (2012).
*Introduction to robust estimation and hypothesis testing*(3rd ed.). Amsterdam, The Netherlands: Academic Press. Retrieved from http://linkinghub.elsevier.com/retrieve/pii/B9780123869838000019 Google Scholar