# Bootstrap (draft)

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* allows estimation of
the errors on the estimated model (function) parameters without knowing the distribution and
variance of the errors in the data.

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 confidence intervals for the model parameters.

## See also:

## References

Efron B, Tibshirani RJ: *An Introduction to the Bootstrap*. Chapman and Hall, 1993.

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 D_{2} receptor occupancy
by antipsychotics with [^{11}C]raclopride and [^{11}C]FLB 457.
*Neuroimage* 2008; 42(4): 1285-1294.

Wasserman L. *All of Nonparametric Statistics.* Springer, 2006.
ISBN-13: 978-0387-25145-5

Created at: 2014-10-08

Updated at: 2017-10-22

Written by: Vesa Oikonen