General Linear Model.

The most common fMRI statistical analysis approach is by using voxel-wise GLM, to see how well observed BOLD response associates with expected BOLD response from stimuli [12]. Voxels that are activated by a task are identified through conducting statistical inference on a task-related regression parameter. For a given voxel, the GLM is: (1) where Y_t is a t × 1 vector of measured BOLD signal intensities of the voxel over time, with y(n) denoting observed BOLD response at time n, 1 ≤ n ≤ t. X is a t × (P + 1) design matrix that represents the expected BOLD signal change per task for P tasks, generated by convolving a hemodynamic response function (HRF) with variables that indicate the timing of when a respective task is administered. A commonly used canonical HRF is the double gamma HRF, which we will consider below in all analyses. Specifically, for a task p, x_p(n) represents the expected observed BOLD response at time n due to administration of task p. Additionally, a first column of 1’s is included for an intercept term. Thus, B equals [b₀ b₁···b_p···b_P]^T, a (P + 1) × 1 regression coefficients vector that includes an intercept b₀, as well as task-related parameters b_p, p = 1 … P. In this formulation, b_p respectively represents the magnitude of HRF activation associated with task p. The t×1 error vector, E_t, represents the error (or noise) component, and is assumed to be normally distributed with mean zero and variance σ²V_t. Note that σ² is the error variance and V_t, a t×t matrix, represents the temporal autocorrelation structure. Thus, Y_t is assumed to have a multivariate normal probability distribution, as follows: where |σ²V_t| is the determinant of σ²V_t. Major sources of noise in fMRI data that are represented by E_t include brain metabolism and physiology, and spontaneous fluctuations. These sources can involve temporal autocorrelation, so that a temporal autocorrelation structure is considered. Using generalized least square (GLS) estimation of regression parameters, one is able to make inferences about a voxel’s activation status through testing of hypotheses.

The quality of signal data for detecting activation for a specific task can be indicated by a voxel-wise calculation of time series SNR, which is defined here as follows (see also [13]):

SNR = (Beta value associated with task) / (within voxel standard deviation of error, σ)

Conventional fixed designs versus sequential designs.

In conventional analyses based on fixed designs, parameters are estimated only after acquisition of all images is completed. In contrast, in sequential testing, estimates of B are computed and updated after each scan volume is observed. Since the focus of experimentation is to obtain these estimates for inference, this updating process allows for real-time decisions to be made on how to adjust the experimentation dynamically. For instance, as we will describe next, based on the updated estimates and their statistical properties, decisions are made using SPRT for whether or not to continue administration of a task. The likelihood functions that comprise the SPRT rely on the same statistical models and assumptions as for the fixed design model, and hence the same statistical parameter estimation approaches. Inferentially, in the sequential designs to be proposed below, focus is given on hypotheses that assess whether the associated task parameter in the regression model is zero or greater than or equal to some predetermined value δ > 0. This differs from the standard alternative GLM hypotheses, in that the alternative assumes a minimum difference from zero. Ideally, this threshold represents a significant clinical difference. In the examples below, we select these values based on projected SNR associated with the alternative hypothesis threshold value (i.e. standard deviation units).

One-sided Voxel-wise SPRT.

We first consider SPRT for one-sided hypothesis tests on contrasts of voxel-specific regression parameters from GLM models. Wald’s SPRT methods can be implemented with either independent and identically distributed (i.i.d.) data or on non-i.i.d. data [14]. Two-sided analogues are similar, as will be seen later on. The general form of one-sided hypotheses is where c equals [c₁—c_p—c_P], a (P + 1) × 1 contrast vector. cB is a linear combination of corresponding coefficients. δ is considered with practical important difference from θ₀. For instance, the hypothesis test of the form H₀: b_p = 0 against H_a: b_p ≥ 1 represents the test of whether or not a voxel is activating in association with a task p, assuming that a magnitude of 1 is considered a practical difference from 0. Other comparisons of task activation magnitudes can be represented by linear combinations of regression parameters associated across several tasks, which we illustrate in the simulations.

, the least squares (or maximum likelihood) estimate, is assumed to be normally distributed with mean cB and variance. It takes the form: (2) The statistic that is the basis for a one-sided SPRT is a likelihood ratio of = θ₁ given observations collected up to a time point t divided by the likelihood of = θ₀ given the same observations. The formula of likelihood ratio [14,15] is: (3) However, Wald’s original formulation of SPRT doesn’t account for unknown “nuisance” parameters that are not the main focus of inference, in this case. Estimates of such values will consequently be assumed as true values in the calculation of the SPRT, following as in Cox [16]. These unknown parameters can vary between subjects. For example, the variance, , may differ from person to person and even change across various fMRI experimental designs. Therefore, according to Cox’s work [16], is replaced by corresponding maximum-likelihood estimations (MLEs) computed by the following equations: (4) where is the MLE for σ² up to t observations. For the scope of this investigation, the temporal autocorrelation structure, V_t, will be assumed known, for computational simplicity. Note that these variances are estimated at the voxel level.

Given this SPRT, stopping criteria at the single voxel level are as follows. Asymptotically, due to consistency in the estimators of unknown parameters used in Cox’s SPRT, the same horizontal stopping boundaries as for Wald’s SPRT can be employed [15]. Based on user specified values of Type I error α and Type II error β, the decision is made by following rejection/acceptance rules after collecting each scan image. Let T > 0 denote the upper bound. Supposing up to scan time t < T has been observed, decision rules for proceeding are defined as:

1. Continue sampling when B < Λ_t < A
2. Stop sampling and accept H₀ when Λ_t < B
3. Stop sampling and accept H_a when A < Λ_t

where stopping boundaries (A,B) = (log((1-β)/α), log(β/(1-α))). Note if t = T, then sampling is stopped, and H₁ is accepted if Λ_t > 0; otherwise, accept H₀. (5) Certainly, in practice, multiple voxels that comprise the ROIs will be analyzed simultaneously. The determination of whether to stop fMRI scanning for an experiment should involve jointly considering all these voxels. A global stopping rule will be proposed below that takes into account how many of the voxel-level analysis call for stopping, so these single voxel stopping rules are still pertinent to the overall goal of developing sequential methods for fMRI analyses.

Two-sided Voxel-wise SPRT.

It may also be of interest to test two-sided hypotheses about values of cB, such as of the form: (6) where a difference between cB and θ₀ greater than δ is considered as practically important.

As an example, suppose cB = b₁—b₂, where b₁ and b₂ respectively reflect activation levels for task 1 and 2. It may not be known a priori for which task activation may be higher for the voxel, and hence a two-sided hypothesis would be appropriate. Given normality of the error terms, the estimate is assumed normally distributed with mean of cB and. Let δ> 0 be as in (6). The likelihood given the alternative hypothesis is true is a weighted average of f(Y_t | = θ₀ - δ) and f(Y_t | = θ₀ + δ). An equal weight of 1/2 has been shown to be the optimum weighting amount [10,17]. The SPRT statistic for two-sided hypotheses for voxel-level analysis is: (7) Again, the SPRT statistic Λ_t, including the statistical estimates that comprise it, are computed sequentially after each new fMRI scan image is collected (in other words, in real-time). Following Cox, is replaced by the corresponding MLE computed as in (equation (4)). The decision is made by the same rejection/acceptance rules as for one-sided hypothesis tests, as in (5).

Multiple Comparisons Correction.

One fMRI three-dimensional whole brain image volume can contain hundreds of thousands of voxels. Conducting voxel-wise analysis thus can include a very large number of hypotheses if each voxel is statistically considered separately, as we consider here. This presents what is known as the “multiple comparisons” issue, and care is needed in determining decision rules for hypotheses, in order to preserve simultaneous Type I and Type II errors. Bonferroni correction is used in this study, as these corrections are easily reflected in the stopping boundary specifications. The Type I and Type II errors for a single voxel’s hypothesis are modified as α_n = α/N and β_n = β/N, where N is the total number of hypotheses (voxels) being considered at once [18], which often may involve focus on a specific ROI.

Global Stopping Rule.

Since we are possibly testing a large number of voxels at once, a “global” decision is needed on when to stop administration of a stimulus, based on aggregate performance of SPRTs across the voxel-wise tests. A well-known irony (or drawback) associated with SPRTs is that the largest number of observations required for stopping occurs precisely when we may be most indifferent to the actual parameter value being tested [17]. For instance, suppose that the true parameter value is half way between the null and alternative hypothesis values. This presents the most difficult problem in terms of number of scan volumes that will be required to stop in a classical SPRT. Yet, it also is the case where indifference to the classification results may be the greatest, as the value truly is “in-between” the null and alternative. For efficient stopping, we suggest a rule that stops when a pre-determined, user-defined percentage of voxel-level tests satisfy the stopping criteria specified as in (5). This allows for circumventing the waiting for voxel-level tests that are “stragglers” when the more clear-cut cases are already decisively decided upon as active or non-active. Therefore, a global stopping rule is proposed that allows for identification of the most distinctive voxel activation levels, but leaves some uncertainty for “borderline” cases that require relatively much more testing, and for which the consequences of misclassification are less. In return, as we will demonstrate, this allows for the chance to obtain substantial overall reductions in the number of scan volumes needed for activation determinations across an ROI. A key to success for this approach is to choose a pre-determined percentage that is reflective of the number of voxels with activation levels that don’t lie between the null and alternative hypotheses. The procedure for implementing a global stopping rule is:

1. Predetermine a targeted percentage level, denoted by G%, that is acceptable in terms of voxels that will have decisive classification
2. Stop fMRI scanning if at least G% of voxels satisfies (5) after multiple comparisons adjustment via Bonferroni correction. Otherwise, continue scanning.
3. At the stopping time point, T, the final decision on activation status for each of the voxels is made according to the following rules:
4. Accept H₀ when Λ_T ≤ 0
5. Accept H_a when Λ_T > 0

Summary of voxel-wise SPRT procedures.

In sum, the procedures of voxel-wise SPRT are recursively employed until a globally-determined stopping point for experimental administration is reached. The following steps are:

• Step 1: Collect one new fMRI image.
• Step 2: Across voxels, apply real time pre-processing procedures, such as spatial smoothing and normalized drift correction [19].
• Step 3: For each voxel-level GLM, compute MLEs of cB and based on equations (2) and (4).
• Step 4: For each voxel, compute an SPRT statistic Λ_t, based on (equation 3) or (7), depending on the form of the hypothesis test, and the corresponding MLEs from Step 3.
• Step 5: Determine if stopping would be invoked for each voxel-level test based on rejection/acceptance rules as in (5), incorporating Bonferroni correction as needed.
• Step 6: Assess the global stopping criterion for the pre-determined target G%, to determine if stopping of the administration of the stimulus should be invoked. If not, repeat from Step 1.
• Final Step: If the specified global stopping is rule satisfied, at each voxel all the fMRI signal data that has been collected up to stopping for that voxel is used to make a final, determination as to activation status. The likelihood ratio as in (3) for one-sided hypothesis or (7) for two-sided hypothesis will be computed, and rule for deciding between hypotheses is to select the associated hypothesized parameter value with the largest corresponding likelihood value.

1.1 Simulation model.

For one voxel with t scan volumes represented in time series data, the linear model in the simulation analysis, including intercept term and two task-related regressors. For the SPRT analysis, Type I error and Type II error are separately set as 0.01 and 0.1. In this analysis, suppose false positives are considered less desirable than false negatives. The stopping boundaries in the SPRT method are corrected for the multiple comparisons among voxels by the Bonferroni approach. In the fixed sample GLM analysis, the Bonferroni approach is too conservative to detect the active regions, as Type I error thresholds values are very small. Therefore, multiple comparison issue is corrected by controlling the false discovery rate (FDR) [23,24].

1.2 Simulation results.

2.1 Material and Methods.

2.2 Real fMRI studies results.

Real data is considered from a 20-year old female subject found to have successful (differential) activity localization from a prior GLM analysis involving the complete set of available scan data. Dynamic methods were applied to the existing data, and compared with the complete data findings of activation. The complete data was comprised of 200 scan volumes per session over the 3 sessions (600 in total, including 24 adult face stimuli and 24 house stimuli). Using voxel-wise SPRT, a dramatic saving in scan times was found with comparable localization findings.

We first considered a scenario where it is of interest to detect voxels that activate for the adult face stimulus. Within the ROI, only 160 scan volumes were required before a 60% global stop rule was satisfied. The complete data results determined that 328 voxels were active for the adult face stimuli, while the dynamic, SPRT-based approach found 399 such active voxels. There was an overlap of 303 voxels being found active with both approaches. See Fig. 8.

For differential activation detection, 209 scan volumes were required through our proposed sequential methods. Overall stopping of administration of stimuli was again determined when more than 60% of the voxel-level SPRTs called for stopping individually. A traditional GLM approach applied on all 600 scans identified 88 voxels that are active for the adult face but not house image and voxel-wise SPRT detected 123 active voxels. 68 voxels were classified as active by both methods. The results are presented in Fig. 9. For activity and differential activity detection, the reductions in the sample size that are required by the SPRT method respectively are around 73% and 65% from 600 scan volumes. Hence, large savings were found. Moreover, there was a high level of overlapping results between approaches, indicating that the findings are still similar. Initially, Bonferroni correction was applied for the complete sample GLM analysis. However, this led to conservative results, in terms of low numbers of detections of active voxels. The results based on a FDR–based approach are displayed here to compare with results from voxel-wise SPRT. The notation q = 0.05 follows as in [24].

Finally, to facilitate visualization and localization of the mask and activated voxels within the brain, we display these areas in relation to the subject’s T1-weighted MRI. The example given in Fig. 10 is for the SPRT case and the identification of differentially activated voxels within the mask.

Halving algorithm and SPRT for detecting neural compensation.

In detecting neural compensation for a given task, it may be that compensation is induced at different difficulty levels, depending on the individual. Such an instance is described in Stern 2012, where allowed time for response in a recall task was varied to adjust difficulty [28]. The goal of this example is to determine the minimum task difficulty level that will activate an ROI, such as one belonging to a neural compensation network. Again assume a block design for each level. Through dynamic adjustment of stimuli, it is not necessary to administer all difficulty levels, which can be time consuming. Instead, we propose a halving algorithm for dynamically selecting difficulty levels, that when used in conjunction with SPRT, can lead to substantial savings in scan volumes when identifying the minimum difficulty level for activation.

The approach of our proposed halving algorithm is to start from a mid-range difficulty level. Then, depending on whether or not activation is determined, based on BOLD responses observed in real time, a lower or higher difficulty level is administered. The assumption is that if a subject is not activating at a higher level, then that subject won’t be active at a lower one either. There would be no need, therefore, to test at relatively lower levels if activation is not observed. On the other hand, if activation is seen at the selected level, a lower level is administered next. Certainly, there is the possibility for error in the activation determinations, but we will see in a simulation study that classification performance can be quite good.

A halving algorithm is shown in a tree representation in Fig. 11 for a task with five difficulty levels, to represent adaptive selection sequences for difficulty level administration. As shown in Fig. 11, the implemented stimulus starts from a task with difficulty level 3, where the difficulty levels are ranked 1 through 5, from the easiest one to the hardest one. A harder or easier task is administered depending on the estimated task activation status for difficulty level 3. Applying the halving algorithm reduces the number of difficulty levels that must be administered from all five levels in a conventional fixed design to at most three. In conjunction, applying SPRT to determine activation status at each task difficulty level reduces the number of blocks that need to be administered even further.

3.1 Simulation study.

We generated activation response curves over the five task difficulty levels according to hypothesized activation curves from Stern’s investigations [36,38]. The corresponding activation magnitudes over task demand levels, as reflected by regression parameter values, are shown in Table 5. They serve as the basis for simulating subject responses, as shown in Fig. 12. The R package “nruRosim” [20] was used to generate simulated fMRI BOLD signals, and again, the simulated datasets were analyzed within the Matlab environment. For simplicity and illustrative purposes, in this example we consider time series consisting of one BOLD value per scan to represent an ROI, such as the mean of the voxel BOLD values in the ROI [39,40,41]. Although not studied here, these methods can be extended to SPRTs conducted on single voxel analyses across an ROI. For instance, an ROI can be considered activated if a certain percentage of voxels are declared activated, and global stop rules can be applied at each difficulty level.

In order to model realistic BOLD signals, a nonlinear Balloon model was used [42]. The simulated BOLD signals were produced by combining time series associated with activation activity and noise, as in (1). The corresponding activation amplitudes were defined as the regression coefficient values associated with the experimental design in GLM, as shown in Table 5. The thermal noise modeled by normal distribution was also added in the simulated BOLD signal. The SNR was kept to at most to 0.5 or below to fit clinical fMRI signal observation. One hundred individual signals were generated for each difficulty level being the minimum one.

In this simulation, at each difficulty level, the task was administered in a block design. Experimental blocks start with an 8-second rest period followed by a 3-second task stimulus. This cycle was repeated 30 times, ending with one 8-second rest period, for a total of 338 seconds or 169 scan volumes with a 2-second TR. For a design of this length, and given the simulated signal characteristics, accuracy in determining the true minimum activation length is seen to be fairly high.

3.2 Simulation results.

Again, the goal is to identify the difficulty level at which an individual starts to activate a specific ROI. Given a task with five difficulty levels, a traditional fixed design in this setting would require a total of 150 blocks. At each difficulty level, a GLM-based, one-sided hypothesis test of H₀: b = 0 versus H_a: b > 0 is conducted once all blocks have been administered, to make a decision on activation status. Consider the fixed design results, as displayed in Table 6. Each column represents the true minimum activity difficulty level, from 1 to 5, plus the inactive at any level case. Each row represents the classification decision made after observing the BOLD signals from 150 alternating cycles administered, using a GLM framework (t-test) at significance level α = 0.05. The last row shows the percentage of minimum difficulty levels of activation that were correctly classified from the 100 simulations per level. The accuracies are around 70% to 80%.

For the sequential design involving the halving algorithm and SPRT, the decision rule can be denoted by the terminal node of the “path” taken in the tree representation shown in Fig. 11. At each non-terminal node, the path direction of the next stage is determined by the result of hypothesis test of H₀: b = 0 versus H_a: b ≥ δ = 0.38. This alternative hypothesis value is considered, due to small to the moderate SNR value associated with this value (σ² = 1). The same signals as in the fixed design analysis were analyzed by SPRT. The results are displayed in Table 7. Columns represent the true difficulty level associated with initial activation, from 1 to 5, plus the inactive case. The average total number of blocks administered per individual is given in adjacent columns. Each row represents the classification decision made after dynamically observing the BOLD signal using the halving algorithm and SPRT, which are set at significance level of 0.05 and with Type II error of 0.10. The last row shows the percentage of correctly classified minimum difficulty levels among 100 simulations per actual difficulty level. The accuracies are around 70% to 90%. Note that our simulations show that sequential design can result in a savings of 72–83% in scan time over fixed designs, depending on a subject’s minimum level, while maintaining comparable accuracy.