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Design Efficiency This article was originally published in Brain Mapping: An Encyclopedic Reference, published by Elsevier, and the attached copy is provided by Elsevier for the author's benefit and for the benefit of the author's institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues who you know, and providing a copy to your institution’s administrator. All other uses, reproduction and distribution, including without limitation commercial reprints, selling or licensing copies or access, or posting on open internet sites, your personal or institution’s website or repository, are prohibited. For exceptions, permission may be sought for such use through Elsevier's permissions site at: http://www.elsevier.com/locate/permissionusematerial Henson R.N. (2015) Design Efficiency. In: Arthur W. Toga, editor. Brain Mapping: An Encyclopedic Reference, vol. 1, pp. 489-494. Academic Press: Elsevier. Author's personal copy Design Efficiency RN Henson, MRC Cognition and Brain Sciences Unit, Cambridge, UK ã 2015 Elsevier Inc. All rights reserved. Abbreviations GLM General linear model AR(p) Autoregressive model of order p HRF Hemodynamic response function BOLD Blood oxygenation level-dependent (signal ReML Restricted maximum likelihood normally measured with fMRI) SOA Stimulus-onset asynchrony (SOAmin ¼minimal DCT Discrete cosine transform SOA) FIR Finite impulse response (basis set) TR Interscan interval (repetition time) fMRI Functional magnetic resonance imaging ÀÁ À1 Formal Definition of Efficiency e ¼ 1= cXTX cT [3] which can be seen as inversely related to the denominator of The general linear model (GLM) normally used for mass- the T-statistic in eqn [2]. Thus, if we increase e, we also increase univariate statistical analysis of functional magnetic resonance  imaging (fMRI) data can be written for a single voxel as T. (For multiple contrasts, where c is an M P matrix of M contrasts, such as an F-contrast, we can define the average ÀÁ T À1 T 2 efficiency as 1/trace{c(X X) c )}.) y ¼ Xb þ e, e N 0, s Ce [1] Note that the scaling of e is arbitrary (depending on the scaling of the contrast, scaling of regressors, and number of where y is an NÂ1 column vector of the data time series scans), so the precise relationship between e and T is best sampled every TR for N scans, X is an NÂP design matrix in assumed only to be monotonic. Note also that this statement which the P columns are regressors for the time series of pre- assumes that the estimate of the error variance (s^2) is indepen- dicted experimental effects, b is a PÂ1 column vector of dent of the design (X), which may not always be true (see parameters for each regressor in X (whose values are estimated later). Given these assumptions, and that the contrasts are when fitting the model to the data), and e is NÂ1 vector of specified a priori, then to maximize the efficiency of our residual errors. The second expression in eqn [1] denotes that design, we simply need to vary X. We now consider how X is the residuals come from a zero-mean, multivariate normal defined for fMRI. (Gaussian) distribution with covariance Ce. Normally, the residuals are assumed to be drawn independently from the same distribution (white residuals), or if not, then the data and model are filtered, or prewhitened, by an estimate of the 2 HRF Convolution error covariance (see later). This means that Ce ¼s I, corre- sponding to an NÂN identity matrix (I) scaled by a single We can start by assuming that stimuli elicit brief bursts of variance term s2. neural activity, or events, which are modeled by delta functions Assuming white residuals, the parameters can be estimated by every time a stimulus is presented. Then, for the jth of N event minimizing the sum of squares of the residuals, to give the so- j types (conditions), the neural activity over time, or neural time called ordinary least squares (OLS)estimates,b^. The planned com- course, u (t), can be expressed as parisons we want to test with our experiment are a linear combi- j  ¼ ðÞ nation of these parameter estimates, specified by a 1 Pcontrast i XNi j ÀÁ c ¼ ½À vector, . For example, c 1 1 would test whether the ujðÞ¼t d t À Tji parameter estimate for the first of two regressors is greater than i¼1 the second. Significance can be assessed by a T-statistic, defined by ¼ ... d where Tji is a vector of i 1 Ni(j) onset times and is the Dirac delta function. With fMRI, we do not measure neural cb^ TðÞ¼df qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÀÁ [2] activity directly, but rather the delayed and dispersed BOLD À1 cXTX cTs^2 impulse response, b(t), where t indexes poststimulus time (e.g., from 0 to 30 s). Given that b(t) may vary across voxels s^2 eTe T where is the error variance estimated by /df (where (and individuals), it can be modeled by linear combination of denotes the transpose of a matrix) and the degrees of freedom, t Nk hemodynamic response functions (HRFs), hk( ): df, are defined by NÀrank(X). The probability, p, of getting a ^ ¼ value of T or greater under the null hypothesis that cb ¼ 0, kXNk ðÞ¼t b ðÞt given the df, can then be calculated from Student’s T-distribu- bj kjhk ¼ tion, and the null hypothesis rejected if, for example, p<0.05. k 1 We are now in the position to define the efficiency of a where bkj are the parameters to be estimated for each HRF and contrast, e,as condition (and voxel). Brain Mapping: An Encyclopedic Reference http://dx.doi.org/10.1016/B978-0-12-397025-1.00321-3 489 Brain Mapping: An Encyclopedic Reference, (2015), vol. 1, pp. 489-494 Author's personal copy 490 INTRODUCTION TO METHODS AND MODELING | Design Efficiency Assuming that BOLD responses summate linearly (though of the BOLD impulse response, then a more general set such see later), the predicted BOLD time course over the experi- as an FIR is necessary. Using a canonical HRF would corre- ment, x(t), can then be expressed as the convolution of the spond to what Liu, Frank, Wong, and Buxton (2001) called neural time courses by the HRFs: detection power, while using an FIR would correspond to what they called estimation efficiency. This is important because ¼ ¼ ¼ ¼ ðÞ jXNj jXNj kXNk i XNi j ÀÁ the choice of HRF affects the optimal experimental design xtðÞ¼ ujðÞ t bðÞ¼t b hk t À Tji [4] kj (see later). j¼1 j¼1 k¼1 i¼1 resulting in a linearly separable equation that can be repre- ¼ sented by a design matrix X with P NjNk columns. At one extreme, we can assumed a fixed shape for the BOLD Filtering response by using a single canonical HRF (i.e., Nk ¼1). At the other extreme, we can make no assumptions about the shape So far, we have considered definition of the signal, x(t), but the of the BOLD response (up to a certain frequency limit) by other factor that affects the T-statistic in eqn [2] is the noise s^2 using a so-called finite impulse response (FIR) set (see Figure 1; variance, . fMRI is known to have a preponderance of low- for multiple basis functions, the contrasts become c INk , frequency noise, caused, for example, by scanner drift and by where c is a contrast across the Nj event types and INk is an biorhythms (e.g., pulse and respiration) that are aliased by Nk ÂNk identity matrix for the Nk basis functions). Normally, slower sample rates (1/TR). A common strategy therefore is to one is only interested in the magnitude of a BOLD response, high-pass filter the data. An example matrix, F, for implement- in which case a single canonical HRF is sufficient to estimate ing high-pass filtering within the GLM using a discrete cosine efficiency a priori (by assuming that a canonical HRF is a transform (DCT) set is shown in Figure 1. The reduction in sufficient approximation on average across voxels and individ- noise will improve sensitivity, as long as the filtering does not uals). If however one is interested in estimating the shape remove excessive signal too. Heuristics suggest that an Canonical HRF SOA t min hk( ) u1 (t) Amplitude (a.u.) and or 0 0 5 10 15 20 25 30 u (t) 2 Amplitude (a.u.) Post-stimulus time (s) Transition table 0 200 400 600 800 1000 FIR HRF set Time (s) t hk( ) x1 (t) X x2 (t) Amplitude (a.u.) 0 5 10 15 20 25 Post-stimulus time (s) Scans F c 1 0 e = 1/(c((KX)T (KX))-1 cT ) −1 Figure 1 Ingredients for efficiency: the minimal SOA, SOAmin, and stimulus transition table determine the neural time course, uj(t), which is convolved with HRFs of poststimulus time, hk(t), to create the design matrix, X. This, together with an a priori contrast, c, and any (high-pass) filter matrix, À1 K (here generated by K¼IN ÀFF ), then determines the efficiency, e. Brain Mapping: An Encyclopedic Reference, (2015), vol. 1, pp. 489-494 Author's personal copy INTRODUCTION TO METHODS AND MODELING | Design Efficiency 491 approximate inflection in the noise power spectrum typically baseline. For a canonical HRF, the efficiency of these two occurs at around 1/120 s, which is why it is inadvisable to have contrasts is plotted against SOA in Figure 2(a). As can be min designs with changes in signal slower than this (e.g., alternat- seen, the optimal SOA for the common effect is around 18 s, ing blocks of more than 60 s; see later).
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