## The Annals of Statistics

### Optimal design of fMRI experiments using circulant (almost-)orthogonal arrays

#### Abstract

Functional magnetic resonance imaging (fMRI) is a pioneering technology for studying brain activity in response to mental stimuli. Although efficient designs on these fMRI experiments are important for rendering precise statistical inference on brain functions, they are not systematically constructed. Design with circulant property is crucial for estimating a hemodynamic response function (HRF) and discussing fMRI experimental optimality. In this paper, we develop a theory that not only successfully explains the structure of a circulant design, but also provides a method of constructing efficient fMRI designs systematically. We further provide a class of two-level circulant designs with good performance (statistically optimal), and they can be used to estimate the HRF of a stimulus type and study the comparison of two HRFs. Some efficient three- and four-levels circulant designs are also provided, and we proved the existence of a class of circulant orthogonal arrays.

#### Article information

Source
Ann. Statist., Volume 45, Number 6 (2017), 2483-2510.

Dates
Revised: November 2016
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.aos/1513328580

Digital Object Identifier
doi:10.1214/16-AOS1531

Mathematical Reviews number (MathSciNet)
MR3737899

Zentralblatt MATH identifier
06838140

#### Citation

Lin, Yuan-Lung; Phoa, Frederick Kin Hing; Kao, Ming-Hung. Optimal design of fMRI experiments using circulant (almost-)orthogonal arrays. Ann. Statist. 45 (2017), no. 6, 2483--2510. doi:10.1214/16-AOS1531. https://projecteuclid.org/euclid.aos/1513328580

#### References

• [1] Buračas, G. T. and Boynton, G. M. (2002). Efficient design of event-related fmri experiments using m-sequences. Neuroimage 16 801–813.
• [2] Chakravarti, I. M. (1956). Fractional replication in asymmetrical factorial designs and partially balanced arrays. Sankhyā 17 143–164.
• [3] Chakravarti, I. M. (1961). On some methods of construction of partially balanced arrays. Ann. Math. Stat. 32 1181–1185.
• [4] Chêng, C. S. (1978). Optimality of certain asymmetrical experimental designs. Ann. Statist. 6 1239–1261.
• [5] Chêng, C. S. (1980). Orthogonal arrays with variable numbers of symbols. Ann. Statist. 8 447–453.
• [6] Cheng, C.-S. (1995). Some projection properties of orthogonal arrays. Ann. Statist. 23 1223–1233.
• [7] Cheng, C.-S. (2014). Optimal biased weighing designs and two-level main-effect plans. J. Stat. Theory Pract. 8 83–99.
• [8] Cheng, C.-S. and Kao, M.-H. (2015). Optimal experimental designs for fMRI via circulant biased weighing designs. Ann. Statist. 43 2565–2587.
• [9] Cheng, C.-S., Kao, M.-H. and Phoa, F. K. H. (2017). Optimal and efficient designs for functional brain imaging experiments. J. Statist. Plann. Inference 181 71–80.
• [10] Craigen, R., Faucher, G., Low, R. and Wares, T. (2013). Circulant partial Hadamard matrices. Linear Algebra Appl. 439 3307–3317.
• [11] Dale, A. M. (1999). Optimal experimental design for event-related fmri. Human Brain Mapping 8 109–114.
• [12] Golomb, S. W. and Gong, G. (2005). Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar. Cambridge Univ. Press, Cambridge.
• [13] Hedayat, A. S., Sloane, N. J. A. and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications. Springer, New York.
• [14] Jansma, J. M., de Zwart, J. A., van Gelderen, P., Duyn, J. H., Drevets, W. C. and Furey, M. L. (2013). In vivo evaluation of the effect of stimulus distribution on fir statistical efficiency in event-related fmri. Journal of Neuroscience Methods 215 190–195.
• [15] Jones, B. and Nachtsheim, C. J. (2011). A class of three-level designs for definitive screening in the presence of second-order effects. Journal of Quality Technology 43 1–15.
• [16] Kao, M.-H. (2013). On the optimality of extended maximal length linear feedback shift register sequences. Statist. Probab. Lett. 83 1479–1483.
• [17] Kao, M.-H. (2014). A new type of experimental designs for event-related fMRI via Hadamard matrices. Statist. Probab. Lett. 84 108–112.
• [18] Kao, M.-H. (2015). Universally optimal fMRI designs for comparing hemodynamic response functions. Statist. Sinica 25 499–506.
• [19] 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.
• [20] Lazar, N. (2008). The Statistical Analysis of Functional MRI Data. Springer, New York.
• [21] Lin, Y.-L., Phoa, F. K. H. and Kao, M.-H. (2016+). Circulant partial Hadamard designs: Construction via general difference sets and its application to fmri experiments. Submitted.
• [22] Lin, Y.-L., Phoa, F. K. H. and Kao, M.-H. (2017). Supplement to “Optimal design of fMRI experiments using circulant (almost-)orthogonal arrays.” DOI:10.1214/16-AOS1531SUPP.
• [23] Lin, Y.-L. and Phoa, F. K. H. (2016). Constructing near-Hadamard designs with (almost) $D$-optimality by general supplementary difference sets. Statist. Sinica 26 413–427.
• [24] Lindquist, M. A. (2008). The statistical analysis of fMRI data. Statist. Sci. 23 439–464.
• [25] 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.
• [26] 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.
• [27] Low, R. M., Stamp, M., Craigen, R. and Faucher, G. (2005). Unpredictable binary strings. Congr. Numer. 177 65–75.
• [28] Maus, B., Van Breukelen, G. J., Goebel, R. and Berger, M. P. (2010). Robustness of optimal design of fmri experiments with application of a genetic algorithm. NeuroImage 49 2433–2443.
• [29] Paley, R. (1933). On orthogonal matrices. J. Math. Phys. 311–320.
• [30] Phoa, F. K. H. and Lin, D. K. J. (2015). A systematic approach for the construction of definitive screening designs. Statist. Sinica 25 853–861.
• [31] Radhakrishna Rao, C. (1945). Finite geometries and certain derived results in theory of numbers. Proc. Nat. Inst. Sci. India 11 136–149.
• [32] Radhakrishna Rao, C. (1946). Hypercubes of strength “$d$” leading to confounded designs in factorial experiments. Bull. Calcutta Math. Soc. 38 67–78.
• [33] Raktoe, B. L., Hedayat, A. and Federer, W. T. (1981). Factorial Designs. Wiley, New York.
• [34] Ryser, H. J. (1963). Combinatorial Mathematics. The Carus Mathematical Monographs 14. Wiley, New York.
• [35] Singer, J. (1938). A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 43 377–385.
• [36] Stanton, R. G. and Sprott, D. A. (1958). A family of difference sets. Canad. J. Math. 10 73–77.

#### Supplemental materials

• Supplement to “Optimal design of fMRI experiments using circulant (almost-)orthogonal arrays”. This supplementary material provides the generating vectors of $\mathit{COA}(n,K,2,2,0)$ when $8\leq n\leq 600$. These designs are obtained by Lemmas 3.8, 5.2 and Theorem 5.4 when $80\leq n\leq600$, and others are found by a computer search.