## The Annals of Statistics

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

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

Yuan-Lung Lin, Frederick Kin Hing Phoa, and Ming-Hung Kao

#### 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**

Received: January 2016

Revised: November 2016

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/16-AOS1531

**Mathematical Reviews number (MathSciNet)**

MR3737899

**Zentralblatt MATH identifier**

06838140

**Subjects**

Primary: 05B10: Difference sets (number-theoretic, group-theoretic, etc.) [See also 11B13] 05B15: Orthogonal arrays, Latin squares, Room squares

Secondary: 62K15: Factorial designs

**Keywords**

Circulant almost orthogonal arrays complete difference system design efficiency hemodynamic response function

#### 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

#### 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.Digital Object Identifier: doi:10.1214/16-AOS1531SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.