Bootstrap

Least-squares fitting of compartmental model (or function) to measured data provides the model (function) parameters, and the sum-of-squares (SS) as an estimate of the goodness of the fit, but not the errors on the estimated model (function) parameters. With the means of classical statistics we could estimate the confidence intervals (CI), if the measurement errors of the data were normally distributed, with a known variance. Usually, this not the case, and most certainly not regarding the PET data. However, Monte Carlo simulation based bootstrap method (Efron and Tibshirani, 1986) allows estimation of the errors on the estimated model (function) parameters without knowing the distribution and variance of the errors in the data. When the assumptions of classical statistical methods can be reasonably satisfied, bootstrap methods tend to give similar results.

In the bootstrap method the model is fitted to the data as usual, but the residuals for the individual data points are stored, and randomly picked to generate synthetic datasets. The synthetic datasets are then fitted using the same method, each dataset fit providing us a set of bootstrapped parameter estimates. From these we can estimate standard deviation and confidence intervals for the model parameters. Generally, 200 synthetic datasets (replications) are needed for estimating SD, and 1000-2000 datasets for estimating the confidence intervals (Efron and Tibshirani, 1993; Davison and Hinkley, 1997).

Bootstrap method can be combined with AIC in model selection (Burnham & Anderson, 2002). Spectral analysis can be used in combination with bootstrapping (Turkheimer et al., 1998). Bootstrap method in model selection can incorporate plasma and metabolite data (Ogden et al., 2005).

PET image

Bootstrap approach has been validated for estimation of the statistical properties of PET and SPECT reconstructed images, including variance and noise correlation (Buvat, 2002; Dahlbom, 2002).

Patient movement during the PET scan can be detected using bootstrap method, if list-mode data is saved (Huang et al., 2011).

References

Burnham KP and Anderson DR. Model Selection and Multimodel Inference: A Practical Information-Theoretical Approach, 2nd ed., 2002, Springer. ISBN 0-387-95364-7. doi: 10.1007/b97636.

Chernick MR. Bootstrap Methods: A Guide for Practitioners and Researchers. 2nd ed. Wiley, 2008. ISBN 978-0-471-75621-7.

Davison AC, Hinkley DV. Bootstrap Methods and their Application. Cambridge University Press, 1997. ISBN 0-521-57471-4.

Efron B. Bootstrap methods: another look at the Jackknife. Ann Statist. 1979; 7(1): 1-26. doi: 10.1214/aos/1176344552.

Efron B, Tibshirani R. Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statist Sci. 1986; 1(1): 54-75. doi: 10.1214/ss/1177013815.

Efron B, Tibshirani RJ: An Introduction to the Bootstrap. Chapman and Hall, 1993. ISBN 978-0412042317.

Hughes IG, Hase TPA: Measurements and their Uncertainties - A Practical Guide to Modern Error Analysis. Oxford University Press, 2010. ISBN 978-0-19-956632-7.

Ikoma Y, Ito H, Yamaya T, Kitamura K, Takano A, Toyama H, Suhara T. Evaluation of error on parameter estimates in the quantitative analysis of receptor studies with positron emission tomography. IEEE Nuc Sci Sym Con. 2005: 2683-2685. doi: 10.1109/NSSMIC.2005.1596889.

Ikoma Y, Shidahara M, Ito H, Seki C, Suhara T, Kanno I. Evaluation of optimal scan time by bootstrap approach for quantitative analysis in PET receptor study. positron emission tomography. IEEE Nuc Sci Sym Con. 2006. doi: 10.1109/NSSMIC.2006.354335.

Ikoma Y, Ito H, Arakawa R, Okumura M, Seki C, Shidahara M, Takahashi H, Kimura Y, Kanno I, Suhara T. Error analysis for PET measurement of dopamine D2 receptor occupancy by antipsychotics with [11C]raclopride and [11C]FLB 457. Neuroimage 2008; 42(4): 1285-1294. doi: 10.1016/j.neuroimage.2008.05.056.

Manly BFJ. Randomization, Bootstrap and Monte Carlo Methods in Biology. 3rd ed., Chapman & Hall/CRC, 2007. ISBN 978-1-58488-541-2.

Wasserman L. All of Nonparametric Statistics. Springer, 2006. ISBN-13: 978-0387-25145-5. doi: 10.1007/0-387-30623-4.

Young GA. Bootstrap: more than a stab in the dark? Statist Sci. 1994; 9(3): 382-415. doi: 10.1214/ss/1177010383.

Updated at: 2019-07-08
Created at: 2014-10-08
Written by: Vesa Oikonen