^{*}

Edited by: Bertrand Thirion, Institut National de Recherche en Informatique et Automatique, France

Reviewed by: Matthew Brett, University of Cambridge, UK; Arnaud Delorme, Centre de Recherche Cerveau et Cognition, France

*Correspondence: Cyril R. Pernet, Brain Research Imaging Centre, Division of Clinical Neurosciences, Western General Hospital, Crewe Road, EH4 2XU, Edinburgh, UK e-mail:

This article was submitted to Brain Imaging Methods, a section of the journal Frontiers in Neuroscience.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

This tutorial presents several misconceptions related to the use the General Linear Model (GLM) in functional Magnetic Resonance Imaging (fMRI). The goal is not to present mathematical proofs but to educate using examples and computer code (in Matlab). In particular, I address issues related to (1) model parameterization (modeling baseline or null events) and scaling of the design matrix; (2) hemodynamic modeling using basis functions, and (3) computing percentage signal change. Using a simple controlled block design and an alternating block design, I first show why “baseline” should not be modeled (model over-parameterization), and how this affects effect sizes. I also show that, depending on what is tested; over-parameterization does not necessarily impact upon statistical results. Next, using a simple periodic vs. random event related design, I show how the hemodynamic model (hemodynamic function only or using derivatives) can affects parameter estimates, as well as detail the role of orthogonalization. I then relate the above results to the computation of percentage signal change. Finally, I discuss how these issues affect group analyses and give some recommendations.

A common way the analyze functional Magnetic Resonance Imaging (fMRI) time series is to use the General Linear Model (GLM—Friston et al., ^{T}X^{T}X

While the mathematical machinery behind mass univariate GLM analyses is described in many papers (see e.g., Monti,

All simulations were programmed in Matlab and the codes can be seen in annexes as well as available to download. The section on Model parameterization corresponds to the file

Let's consider first a simple controlled block design (one condition of interest—Figure ^{T}X^{T}X

^{2}. In addition, since the model degrees of freedom depend on the rank of the design matrix, all models have the same degrees of freedom giving the same

Despite different design matrices, all models provided the same fit, i.e., the same fitted data. This is explained by the fact that all design matrices can predict equally well the data. In model 1, the sum of the two first regressors is the constant term; and having this constant term in the model cannot thus change the fit, compared to model 2. The same sums of squares of the effect and the same residuals were therefore obtained, and the amount of variance explained was always the same (same ^{2}). Also, because degrees of freedom are defined by the rank (i.e., the number of independent regressors) of the design matrix, the over-parameterized model (model 1) had the same degrees of freedom as the other models, and therefore

Differences among models occurred when looking at the parameters of interest: the 1st model returned parameter estimate values different from the simulated data (_{1} = 3 for baseline, _{2} = 4 for activation), whilst model 2 returned parameter estimates that reflected directly the amount of change in the data (_{1} = 1 for activation, _{2} = 10 for baseline/constant). The reason why the estimated parameters in Model 1 do not reflect the simulations is because there is no unique solution, indeed there is an infinite number of possible solutions for the estimated parameter that can lead to the same error (the same sum of square of the error). Following Equations 1 and 2, the data are simply expressed as the sum of weighted regressors plus the error term. Model 2 (i.e., modeling activation only, plus the constant) thus follows Equation 5 and the constant term (the intercept) is given by Equation 6.

_{1} coding for activation and X_{2} coding for the constant term), _{1}, _{2}, are the parameter estimates and

It becomes apparent that the constant term (here _{2} * _{2}) represents the average across observations of the adjusted data, i.e., the estimated average of the data minus the effect of the activation regressors and the error. In this model, the constant term therefore models baseline, and _{1} reflects the signal change relative to it. In Model 1 (i.e., modeling activation and baseline, plus the constant), individual beta estimate values cannot be interpreted because they are not “estimable”. Since the design matrix is over-parameterized (i.e., ^{T}X^{T} = 0 (^{T}]. If we are using contrasts orthogonal to [1 1 –1] then our result is independent of the arbitrary constant

Another important aspect of the GLM is the scale of the design matrix. Since the design matrix is a model of the data, the parameters can be seen as values that simply scale the columns of _{1} = 1, _{2} = 10). In contrast, model 3, for which the activation regressor was scaled between 0 and 2, had a parameter estimate for activation of half the value of the signal change (_{1} = 0.5, _{2} = 10). In fMRI, after regressors are convolved by the hemodynamic response model, they are not always rescaled between 0 and 1 and this will matter when looking at the PSC because the parameter estimates do not then reflect directly changes in the signal. However, if we only focus on the statistics, and because

Consider now an

^{2}. Parameter estimates however differed. The fitted data for condition 1 are plotted in blue, for condition 2 in red and for the baseline (model 1 only) in black. In model 1, condition 1 and 2 are modeled as positive effects relative to the constant term (7.5 + 1.5 = 9 for condition 1, 7.5 + 3.5 = 11 for condition 2) whereas for model 2 and 3, they are modeled as a negative effect relative to constant for condition 1 (10 – 1 = 9 for model 2 or 10 – 0.5^{*} 2 = 9 for model 3) and a positive effect relative to constant for condition 2 (10 + 1 = 11 for model 2 and 10 + 0.5^{*} 2 = 11 for model 3). Despite those differences, contrasts

These examples illustrate the fundamental point that “contrast specification and the interpretation of results are entirely dependent on the model specification

Using a set of functions [here the hemodynamic response function (hrf) and its time derivative] rather than the hrf alone is usually considered desirable, because even minor miss-specification of the hemodynamic model can result in substantial bias and loss of power, possibly inflating the type I error rate (Lindquist et al., ^{2}—Figure

^{2}/

Data were analyzed using design matrices where events were convolved by the hrf (model 1) vs. the hrf and its 1st derivative. Three models were compared: adding the derivative without orthogonalization (model 2), adding the derivative orthogonalized onto the regressor convolved by the hrf [SPM (Friston et al., ^{2}, which is expected since more variance was explained compared with the hrf alone model (Figure _{1} = 6.96 (vs. 10 expected) and adding the temporal derivative, irrespective of orthogonalization, led to an increase of the hrf parameter estimates (7.44 for model 2, 7.39 for model 3, and 7.12 for model 4) thus giving a better estimate of the true hrf regressor. However, when applying the same simulation with the hemodynamic signal peaking earlier than the standard hrf model, adding the temporal derivative has the opposite effect, i.e., it gives lower estimates of the true hrf regressor (see annex

^{2}, Figure

Another important point to notice, it that despite orthogonalization, the parameter estimate for the regressor convolved by the hrf was different before and after adding the temporal derivative. It must be understood that this change in parameter estimate is function of (1) how the orthogonalization is performed (as exemplified above); (2) the correlation between the regressor of interest and the constant term (which itself depends on the inter-stimulus interval—see annex

Rather than using the raw parameter estimates to report or investigate local changes, it is often preferable to compute a more standard measure such as the Percentage Signal Change (PSC). The PSC is defined here as the ratio between the magnitude of the BOLD response and the overall mean of the adjusted time series. Because the parameter estimates from the GLM (Equation 2) are a scaled version of this magnitude, it is also mandatory to account for the value range in the design matrix (Poldrack et al., _{condition} _{constant} _{ss}

As explained below, the SF not only allows recovering the true signal change but also allows comparing results across different designs. Therefore, instead of using the maximum of a given trial in the experimental design matrix, we may choose a “typical” trial which does not have to be present in the actual design (Poldrack et al.,

To evaluate the impact of hemodynamic modeling on the computation of PSC and _{ss}

For both the periodic design and the fast event related designs, the data were created so that the mean activity was identical; with GLM parameters being different (Figure

To be comparable between designs, computation of the PSC has to account for differences in the height of the regressors because, as illustrated in Figure

In the simulations presented here, the maximum height in the periodic design was 0.105 vs. a maximum height in the fast event related design of 0.115. If one scales the PSC using those heights, we obtained values of 1.05 vs. 1.15%, even though the true signal changes are comparable. In contrast, if one uses the same SF for both designs, the estimates PSC become comparable. If one uses a SF of 0.105, we obtained 1.05% for the period design vs. 1.04% for the fast event related design. If one uses a SF 0.115, the PSC of the periodic design goes up to 1.157 vs. 1.1506% for the fast event related design. This simply illustrates that the PSC is a relative metric. To use the same analogy as Poldrack et al. (

Analysis of the down-sampled data showed that the models before/after down-sampling explained about the same amount of variance as with the original data.

To finish this tutorial, we considered how timing misspecification also impacts PSC computations. Analyses of the high resolution designs were replicated but using data with a temporal shift of +2 s, and a design matrix including the hrf and its time derivative. For these analyses, we compared the PSC computed using the parameter estimates of the hrf to the corrected parameter estimates, which are based on the combination of the hrf and its derivative (Steffener et al., _{1} the parameter estimates for the hrf, _{1} the regressor convolved by the hrf, _{2} the parameter estimates for the temporal derivative, _{2} the regressor convolved by the 1st derivative_{1} divided by its absolute value, allowing recovering the sign [as in Calhoun et al. (

Analysis and modeling of the periodic vs. fast event related designs with some temporal delay are displayed in Figure

The first misconception about the GLM has to do with modeling rest or null events and can be related to the understanding of (1) what the constant term is (2) what model (over) parameterization implies. Because the constant term is often referred to as the intercept, this is often interpreted as “baseline.” Physically, the constant term reflects the offset of the measured signal, which is not on average zero even without stimuli (Poline et al.,

Modeling rest periods or null events does not, however, necessarily impact statistical results (second misconception), as long as the right contrasts are used. For a simple block design (Figure

A third misconception is to think that adding temporal and/or dispersion derivatives never change the parameter estimate(s) of the hrf regressor(s), because of orthogonalization. In a linear system, fitting orthogonal regressors is indeed identical as fitting each regressor separately because orthogonal regressors are also uncorrelated (for further insight into independence vs. orthogonality vs. correlation, see Rodgers et al.,

The final and forth misconception related to the PSC. First, it is essential to define relative to what the PSC is computed. If using the GLM parameter estimates, the PSC is computed relative to the adjusted mean (which can be seen as the baseline in block designs or event related designs). In other cases, like the default in AFNI, this is relative to the temporal mean. This has to be reported because the actual values will differ between methods, even for the same data. Second, when using the GLM parameter estimates, it is also essential to define a reference trial at the resolution of the super-sampled design and report the scaling factor because of (1) the impact of the design matrix data range (scaling) on parameter estimates, (2) the impact of data resolution (i.e., TR) compared to the hemodynamic model, and (3) the differences in the way hemodynamic responses can summate. Unfortunately most software do not provide such information easily and one needs to regenerate the super-sampled model or recreate a “typical” trial. For group analyses, if the 2nd level was performed using the parameters of regressors convolved by hrf only, it makes sense to report the PSC computed using these same parameters (and use a scaling factor based on a reference trial sampled according to the design). If the analysis, however, uses derivatives and/or there are evidences of a temporal shift or narrowing/widening of the hfr, using a combination of parameters is more likely to reflect the true PSC as demonstrated in the simulation.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

I would like to thank Dr R. Henson (MRC CBU, University of Cambridge) for the numerous emails discussing issues of orthogonalization and his early look at the article. Also thank you to Dr T. Nichols (Dpt of statistics, Warwick University), and Dr Chen (NIMH, Bethesda) for discussing orthogonalization procedures in FSL and in AFNI. Thank you to my colleagues Dr M. Thrippleton and Prof. I. Marshall (Brain Research Imaging Centre, University of Edinburgh) for their comments on the early version of the article. Finally, thank you to Dr JB Poline (Neurospin, Paris) and Dr M. Brett (Brain Imaging Center, University of California Berkeley) for making numerous insightful comments during the review process.

The Supplementary Material for this article can be found online at:

The abstract of all articles published in the journal Neuroimage between January 2013 and June 2013 were examined, following which studies that used fMRI were retained (human and non-human alike). The method part of each of these articles was examined and only studies using the “standard” GLM procedure were included in the review, i.e., studies using in house methods, block averaging, FIR models or using multivariate pattern analyses were discarded. In total, 75 articles were included and listed below in an abbreviated reference format.

For each article, the following methodological criteria were recorded:

(i) Blocked or event related design,

(ii) Rest periods or null events presents,

(iii) Modeling of rest or null events if any,

(iv) Reporting of effect size estimates (beta/con or PSC),

(v) Were reported parameter estimates valid (i.e. using a contrast if rest was modeled),

(vi) If PSC reported, how was it computed.

Articles included in the review:

Agnew et al. (2013).

Andics et al. (2013).

Apps et al. (2013).

Archila-Suerte et al. (2013).

Baeck et al. (2013).

Bennett et al. (2013).

Bergstrom et al. (2013).

Blank and von Kriegstein (2013).

Bonath et al. (2013).

Bonner et al. (2013).

Bonzano et al. (2013).

Bradley et al. (2013).

Brown et al. (2013).

Callan et al. (2013).

Callan et al. (2013).

Campanella et al. (2013).

Causse et al. (2013).

Chen et al. (2013).

Cohen Kadosh et al. (2013).

Cservenka et al. (2013).

de Hass et al. (2013).

Demanet et al. (2013).

Di Dio et al. (2013).

Egidi and Caramazza (2013).

Ferri et al. (2013).

Fischer et al. (2013).

Freud et al. (2013).

Gilead et al. (2013).

Gillebert et al. (2013).

Gorgolewski et al. (2013).

Heinzel et al. (2013).

Helbing et al. (2013).

Henderson and Norris (2013).

Hermans et al. (2013).

Hove et al. (2013).

Indovina et al. (2013).

James and James (2013).

Kassuba et al. (2013).

Kau et al. (2013).

Killgore et al. (2013).

Kitayama et al. (2013).

Koritzky et al. (2013).

Kruger et al. (2013).

Kulakova et al. (2013).

Liang et al. (2013).

Liew et al. (2013).

Limongi et al. (2013).

Lutz et al. (2013).

Lutz et al. (2013).

Madlon-Kay et al. (2013).

Manginelli et al. (2013).

Moon et al. (2013).

Muller et al. (2013).

Pau et al. (2013).

Pomares et al. (2013).

Raabe et al. (2013).

Rothermich and Kotz (2013).

Simonyan et al. (2013).

Spreckelmeyer et al. (2013).

Stice et al. (2013).

Sun et al. (2013).

Sutherland et al. (2013).

Telzer et al. (2013).

Thornton and Conway (2013).

Turk-Browne et al. (2013).

Tusche et al. (2013).

Tyll et al. (2013).

van der Heiden et al. (2013).

van der Zwaag et al. (2013).

Witt and Stevens (2013).

Wu et al. (2013).

Yu-Chen et al. (2013).

Zeki and Stutters (2013).

Zhang and Hirch (2013).

Zhang et al. (2013).