The Annals of Statistics

Optimal experimental designs for fMRI via circulant biased weighing designs

Ching-Shui Cheng and Ming-Hung Kao

Full-text: Open access

Abstract

Functional magnetic resonance imaging (fMRI) technology is popularly used in many fields for studying how the brain reacts to mental stimuli. The identification of optimal fMRI experimental designs is crucial for rendering precise statistical inference on brain functions, but research on this topic is very lacking. We develop a general theory to guide the selection of fMRI designs for estimating a hemodynamic response function (HRF) that models the effect over time of the mental stimulus, and for studying the comparison of two HRFs. We provide a useful connection between fMRI designs and circulant biased weighing designs, establish the statistical optimality of some well-known fMRI designs and identify several new classes of fMRI designs. Construction methods of high-quality fMRI designs are also given.

Article information

Source
Ann. Statist., Volume 43, Number 6 (2015), 2565-2587.

Dates
Received: December 2014
Revised: June 2015
First available in Project Euclid: 7 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1444222085

Digital Object Identifier
doi:10.1214/15-AOS1352

Mathematical Reviews number (MathSciNet)
MR3405604

Zentralblatt MATH identifier
1331.62381

Subjects
Primary: 62K05: Optimal designs

Keywords
Circulant orthogonal array design efficiency Hadamard matrix hemodynamic response function $m$-sequence

Citation

Cheng, Ching-Shui; Kao, Ming-Hung. Optimal experimental designs for fMRI via circulant biased weighing designs. Ann. Statist. 43 (2015), no. 6, 2565--2587. doi:10.1214/15-AOS1352. https://projecteuclid.org/euclid.aos/1444222085


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References

  • Boynton, G. M., Engel, S. A., Glover, G. H. and Heeger, D. J. (1996). Linear systems analysis of functional magnetic resonance imaging in human V1. J. Neurosci. 16 4207–4221.
  • Buračas, G. T. and Boynton, G. M. (2002). Efficient design of event-related fMRI experiments using M-sequences. NeuroImage 16 801–813.
  • Cheng, C. S. (1978). Optimality of certain asymmetrical experimental designs. Ann. Statist. 6 1239–1261.
  • Cheng, C. S. (1980). Optimality of some weighing and $2^{n}$ fractional factorial designs. Ann. Statist. 8 436–446.
  • Cheng, C.-S. (1987). An optimization problem with applications to optimal design theory. Ann. Statist. 15 712–723.
  • Cheng, C.-S. (1992). On the optimality of $(\mathrm{M,S})$-optimal designs in large systems. Sankhyā Ser. A 54 117–125.
  • Cheng, C.-S. (2014). Optimal biased weighing designs and two-level main-effect plans. J. Stat. Theory Pract. 8 83–99.
  • Craigen, R., Faucher, G., Low, R. and Wares, T. (2013). Circulant partial Hadamard matrices. Linear Algebra Appl. 439 3307–3317.
  • Dale, A. M. (1999). Optimal experimental design for event-related fMRI. Human Brain Mapping 8 109–114.
  • Eccleston, J. A. and Hedayat, A. (1974). On the theory of connected designs: Characterization and optimality. Ann. Statist. 2 1238–1255.
  • Friston, K. J., Zarahn, E., Josephs, O., Henson, R. N. and Dale, A. M. (1999). Stochastic designs in event-related fMRI. NeuroImage 10 607–619.
  • Galil, Z. and Kiefer, J. (1980). $D$-optimum weighing designs. Ann. Statist. 8 1293–1306.
  • Golomb, S. W. and Gong, G. (2005). Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar. Cambridge Univ. Press, Cambridge.
  • Harville, D. A. (1997). Matrix Algebra from a Statistician’s Perspective. Springer, New York.
  • Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications. Springer, New York.
  • Horadam, K. J. (2007). Hadamard Matrices and Their Applications. Princeton Univ. Press, Princeton, NJ.
  • Kao, M.-H. (2013). On the optimality of extended maximal length linear feedback shift register sequences. Statist. Probab. Lett. 83 1479–1483.
  • Kao, M.-H. (2014). A new type of experimental designs for event-related fMRI via Hadamard matrices. Statist. Probab. Lett. 84 108–112.
  • Kao, M.-H. (2015). Universally optimal fMRI designs for comparing hemodynamic response functions. Statist. Sinica 25 499–506.
  • Kao, M.-H., Mandal, A. and Stufken, J. (2008). Optimal design for event-related functional magnetic resonance imaging considering both individual stimulus effects and pairwise contrasts. Stat. Appl. (N. S.) 6 225–241.
  • Kao, M.-H., Mandal, A., Lazar, N. and Stufken, J. (2009). Multi-objective optimal experimental designs for event-related fMRI studies. NeuroImage 44 849–856.
  • Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2 849–879.
  • Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In A Survey of Statistical Design and Linear Models (Proc. Internat. Sympos., Colorado State Univ., Ft. Collins, Colo., 1973) (J. N. Srivastava, ed.) 333–353. North-Holland, Amsterdam.
  • Kushner, H. B. (1997). Optimal repeated measurements designs: The linear optimality equations. Ann. Statist. 25 2328–2344.
  • Lazar, N. A. (2008). The Statistical Analysis of Functional MRI Data, Statistics for Biology and Health. Springer, New York.
  • Lin, Y.-L., Phoa, F. K. H. and Kao, M.-H. (2014). Partial Hadamard matrices: Construction via general difference sets and application to fMRI designs. Unpublished manuscript.
  • Liu, T. T. (2004). Efficiency, power, and entropy in event-related fMRI with multiple trial types. Part II: Design of experiments. NeuroImage 21 401–413.
  • Liu, T. T. and Frank, L. R. (2004). Efficiency, power, and entropy in event-related FMRI with multiple trial types. Part I: Theory. NeuroImage 21 387–400.
  • Low, R. M., Stamp, M., Craigen, R. and Faucher, G. (2005). Unpredictable binary strings. Congr. Numer. 177 65–75.
  • Masaro, J. C. (1988). On $A$-optimal block matrices and weighing designs when $N\equiv3\pmod{4}$. J. Statist. Plann. Inference 18 363–370.
  • Maus, B., van Breukelen, G. J. P., Goebel, R. and Berger, M. P. F. (2010). Robustness of optimal design of fMRI experiments with application of a genetic algorithm. NeuroImage 49 2433–2443.
  • Miezin, F. M., Maccotta, L., Ollinger, J. M., Petersen, S. E. and Buckner, R. L. (2000). Characterizing the hemodynamic response: Effects of presentation rate, sampling procedure, and the possibility of ordering brain activity based on relative timing. NeuroImage 11 735–759.
  • Paley, R. (1933). On orthogonal matrices. Journal of Mathematics and Physics 12 311–320.
  • Sathe, Y. S. and Shenoy, R. G. (1989). $A$-optimal weighing designs when $N=3$ $(\mathrm{mod}\ 4)$. Ann. Statist. 17 1906–1915.
  • Wager, T. D. and Nichols, T. E. (2003). Optimization of experimental design in fMRI: A general framework using a genetic algorithm. NeuroImage 18 293–309.
  • Worsley, K. J., Liao, C. H., Aston, J., Petre, V., Duncan, G. H., Morales, F. and Evans, A. C. (2002). A general statistical analysis for fMRI data. NeuroImage 15 1–15.