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Sign-Consistency Based Variable Importance for Machine Learning in Brain Imaging

  • Vanessa Gómez-Verdejo
  • Emilio Parrado-Hernández
  • Jussi TohkaEmail author
  • Alzheimer’s Disease Neuroimaging Initiative
Open Access
Original Article

Abstract

An important problem that hinders the use of supervised classification algorithms for brain imaging is that the number of variables per single subject far exceeds the number of training subjects available. Deriving multivariate measures of variable importance becomes a challenge in such scenarios. This paper proposes a new measure of variable importance termed sign-consistency bagging (SCB). The SCB captures variable importance by analyzing the sign consistency of the corresponding weights in an ensemble of linear support vector machine (SVM) classifiers. Further, the SCB variable importances are enhanced by means of transductive conformal analysis. This extra step is important when the data can be assumed to be heterogeneous. Finally, the proposal of these SCB variable importance measures is completed with the derivation of a parametric hypothesis test of variable importance. The new importance measures were compared with a t-test based univariate and an SVM-based multivariate variable importances using anatomical and functional magnetic resonance imaging data. The obtained results demonstrated that the new SCB based importance measures were superior to the compared methods in terms of reproducibility and classification accuracy.

Keywords

Bagging Support Vector Machines Variable importance MRI Alzheimer’s Disease Schizophrenia 

Introduction

Machine Learning (ML) is a powerful tool to characterize disease related alterations in brain structure and function. Given a training set of brain images and the associated class information, here a diagnosis of the subject, supervised ML algorithms learn a voxel-wise model that captures the class information from the brain images. This has direct applications to the design of imaging biomarkers, and the inferred models can additionally be considered as multivariate, discriminative representations of the effect of the disease to brain images. This representation is fundamentally different from conventional brain maps that are constructed based on a voxel-by-voxel comparison of two groups of subjects (patients and controls) and the patterns of important voxels in these two types of analyses provide complementary information (Kerr et al. 2014; Haufe et al. 2014; Tohka et al. 2016).

A key problem in using voxel-based supervised classification algorithms for brain imaging applications is that the dimensionality of data (the number of voxels in the images of a single subject, i.e., the number of variables1) far exceeds the number of training subjects available. This has led to a number of works studying variable selection within brain imaging; see Mwangi et al. (2014) for a review. However, in addition to selecting a set of relevant variables, it is interesting to rank and study their importance to the classification. This problem, termed variable importance determination, has received significantly less attention and is the topic of this paper.

The simplest approach to assess the importance of a variable is to measure its correlation with the class labels, for example, via a t-test. This is exactly what massively univariate analysis does. However, it considers variables independently of others and, therefore, may miss interactions between variables. Indeed, a variable can be meaningful for the classification despite not presenting any linear relationship with the class label (Haufe et al. 2014). Further, there is evidence that this importance measure does not perform well for variable selection in discrimination tasks (Chu et al. 2012; Tohka et al. 2016) and, therefore, multivariate importance measures might be more appropriate.

The use of ML opens a door for more sophisticated methods to assess the importance of variables, able to capture some of the interactions among variables. A straightforward approach is to base each variable importance on the difference between the performances observed in an ML method when the variable is present and when it is absent from the training instances. A significant decrease in performance after removing a variable indicates its relevance. While this methodology is straight-forward, it is not suitable for brain imaging applications as it would fail to recognize the relevance of important but mutually redundant variables. A few ML methods favor a richer and more detailed analysis as they enable to assess the role that each individual variable plays in the composition of the architecture of the model. Two clear examples of this class are linear models and Random Forests2 (RFs).

If the variables have been properly standardized, the weights of a linear classifier can be considered as measures of variable importance (Caragea et al. 2001) (see, e.g, Cohen et al. 2010; Khundrakpam et al. 2015 for neuroimaging examples). However, when the variables are redundant as often in neuroimaging, the weights may change erratically in response to small changes in the model or the data. This problem is known as multicollinearity in statistics. Linear regressors can be endowed with Lasso and Elastic Net regularizations (Friedman et al. 2008; Zou and Hastie 2005), in order to deal with problems with a high number of input variables. These regularizations force sparsity and remove variables of reduced relevance from the linear model, enhancing the contribution of the remaining variables. More elaborated methods take a further step in the exploitation of the relationship between the weight of each variable in a linear classifier/regressors and its relevance (Guyon et al. 2002). The starplots method of Bi et al. (2003) learns an ensemble of linear Support Vector Regressors (SVR) endowed with a Lasso type regularization in the primal space. The regularization filters out the non-relevant variables from each regressor, while the starplots determine the relevance of the non-filtered variables by looking for smooth and consistent patterns in the weights they achieve across all the regressors in the ensemble. These linear methods with Lasso regularization present two significant drawbacks in very high dimensional scenarios. First, the computational burden of the resulting optimization in the primal space is high. Second, they enforce an aggressive sparsity, usually reducing the number of non-filtered variables to a final quantity comparable to the number of training instances. This last feature is specially harmful in neuroimaging problems, where the typical situation is that a disease affects a set of focal brain areas. This reflects in groups of clustered variables being important with a strong correlation. A Lasso regularization would filter out most of the important (but correlated) variables, what complicates the interpretation of the discrimination pattern. To combat this problem, for example, Grosenick et al. (2013) and Michel et al. (2011) have introduced brain imaging specific regularizers which take into the account the spatial structure of the data. The application of these methods is complicated by a challenging parameter selection (Tohka et al. 2016) and deriving a variable-wise importance measure is complicated by the joint regularization of weights of the different variables.

Also RFs (Breiman 2001) facilitate the assessment of variable importance. RFs are ensembles of decision trees where each tree is trained with a subset of the available training subjects, and with a subset of the available variables. RFs offer two main avenues for assessing the variable importance: Gini importance and permutation importance based on the analysis of out-of-bag samples (Archer and Kimes 2008). Both measures have found applications in brain imaging: Langs et al. (2011) studied voxel selection based on Gini importance, Moradi et al. (2015) ranked the different types of variables (imaging, psychological test scores) for MCI-to-AD conversion prediction based on the out-of-bag variable importance and Greenstein et al. (2012) ranked the importance of cortical ROI volumes to schizophrenia classification. However, these applications have considered at most tens of variables while our focus is on a voxel-wise analyses of whole brain scans, where we have tens of thousands of variables. Indeed, the usability of RFs to capture variable importance in a multivariate fashion is questionable in high dimensional scenarios. This is partially due to the use of decision trees as base classifiers. Each decision tree in an RF comes out of a training set that includes a sample of the observations and of the variables. In a data set in which the number of variables is far larger than the number of observations each tree definition will rely on a very reduced set of variables (notice that a set of 100 samples is split completely by a tree with 100 nodes, each implementing a threshold test in one of the variables). This means that in order to get a chance to assess the importance of all the variables (namely, that each variable shows a large enough number of times across the trees in the ensemble), the forest must contain an extraordinarily large number of trees, and this makes the method computationally less attractive than the use of an ensemble of linear classifiers. In an ensemble of linear classifiers every variable can get a weight in every classifier, while in a random forest typically only a reduced fraction of variables will be used as splits in every tree. In addition, the weight of a variable in a linear classifier results from a global optimization process that takes into account the joint contributions of all variables simultaneously, while the usage of a variable in a split within a decision tree is strongly dependent of the particular subset of splits that lead to the branch in which this split is used (Strobl et al. 2008). Note that this criticism pertains only to the variable importance scores from RFs, and not to the accuracy of predictions derived from RFs.

To overcome the limitations of the regularized linear models and RFs, we introduce and study a new variable importance measure based on the sign consistency of the weights in an ensemble of linear Support Vector Machines (SVMs). Briefly, we train an ensemble of SVMs using only a part of the subjects available for each SVM in the ensemble. The main idea is to define the importance of a variable using its sign consistency, i.e., the fraction of members of the ensemble in which its weight is positive (or negative). We thereafter prune the variable importances using the ideas from transductive conformal analysis. We derive a paramatric hypothesis test of the variable importance measures and show that the new importance measures are an improvement over p-value estimation for SVM weights of Gaonkar and Davatzikos (2013) and Gaonkar et al. (2015).

The results presented in this paper build on our previous work on variable selection (Parrado-Hernández et al. 2014) and thoroughly extend a preliminary conference paper focused on assessing variable relevance (Gomez-Verdejo et al. 2016). In particular, the main novel contributions of this paper over (Parrado-Hernández et al. 2014) and Gomez-Verdejo et al. (2016) are:
  • We explain the variable importance measure and its properties in more detail than before and provide algorithms for its computation.

  • We derive a parametric hypothesis test for variable importance. The variable selection is carried out with a hypothesis test that replaces the cross validation of the previous works. The hypothesis test improves the stability and reduces the processing time by an order of magnitude.

  • We perform new large-scale experiments to assess the accuracy and stability of our novel importance measure in comparison to related approaches (Gaonkar and Davatzikos 2013; Gaonkar et al. 2015) and show that this new measure is an improvement over them.

Methods

Variable Importance with Ensembles of Linear SVMs

We start by introducing the basic variable importance computation, termed Sign Consistency Bagging (SCB). Thereafter, in the next subsection, we enhance the basic algorithm by introducing a conformal variant (SCBconf) of the basic algorithm that outperforms the basic algorithm for certain problems.

Consider a binary classification task to differentiate between two groups of subjects. We refer to these two groups as patients and controls. The training data is formed by N pairs (xi,yi), i = 1,…,N where \(\textbf {x}^{i} = [{x_{1}^{i}},\dots ,{x_{P}^{i}}]^{T}\) is a vector with the variables corresponding to the brain scan of the i-th subject and yi = 1 if the i-th subject is a patient and yi = − 1 otherwise. We assume \({x_{j}^{i}}\ge 0\). This is often a natural requirement and, in any case, it can be fulfilled by adding a suitable constant to all \({x_{j}^{i}}\).

We use linear SVM classifiers as base learners. The predicted label \(\hat y\) for a test sample x is given by
$$ \hat y = \text{sign}(w_{0} + \mathbf{w}^{T}\textbf{x}) \doteq g(\textbf{x}) $$
and the classifier is defined in terms of its parameters w0 (bias) and w = [w1,…,wP]T (weight vector).

We train S linear SVMs, each with a different subset of γN training instances selected at random without replacement; γ ∈ [0,1] is the subsampling rate, usually selected as γ = 0.5. The s-th SVM is therefore described by the bias term \({w_{0}^{s}}\) and the weight vector \(\mathbf { w}^{s} = [{w_{1}^{s}}, {w_{2}^{s}}, \dots , {w_{P}^{s}}]\). Once the learning of the ensemble is finished, each variable in the input data xj is related with a set of S weights \({w_{j}^{1}},\dots ,{w_{j}^{S}}\). Since \({x_{j}^{i}} \ge 0\) for all i,j, there is a straightforward qualitative interpretation of the sign consistency across all the \({w_{j}^{s}}\) in the set. If \({w_{j}^{s}}\ge 0\) for all s, we have that the j-th variable pushes the classifier towards the positive class while a result of \({w_{j}^{s}}<0\) for all s means that j-th variable pushes the classifier towards the negative class. Since the linear SVM uses a L2 norm regularization that does not enforce sparsity in the primal space it is very unlikely that \({w_{j}^{s}}= 0\), for all j,s. It is rare that all \({w_{j}^{s}}\) would share the same sign and we introduce an importance scoring Ij, j = 1,…,P, quantifying the sign consistency:

$$ I_{j} = 2 \max(\hat{p}_{j},1 - \hat{p}_{j}) - 1 = 2 \vert \hat{p}_{j} - 0.5 \vert $$
(1)
with
$$ \hat{p}_{j} = \frac{1}{S}\sum\limits_{s = 1}^{S} \mathcal{I}({w_{j}^{s}}>0), $$
(2)
where \(\mathcal {I}(x)\) is the indicator function, \(\mathcal {I}(x)= 1\) if x is true and \(\mathcal {I}(x)= 0\) otherwise. If all weights corresponding to a same variable \({w_{j}^{s}}\), s = 1,…,S receive the same sign, variable j would get an importance scoring Ij = 1, meaning it is very relevant for the classification. As the proportion of \({w_{j}^{s}}\) with opposite sign gets balanced, the interpretation of the variable j for the classification gets blurred, since in some members of the ensemble a high value of xj would indicate member of one of the classes while in some other members it would indicate membership of the other class. In this case, as the interpretation varies depending on the sub-sample, the variable is not useful (or is even harmful) for the generalization performance and should have a low importance score.

Transductive Refinement of Variable Importance

In this sub-section, we introduce a variant of SCB variable importance, SCBconf. This algorithm builds on the results of SCB and enhances the identification of the relevant variables by borrowing ideas from transductive learning and conformal analysis.

Transduction refers to learning scenarios in which one has access to the observations, but not the labels, of the test set3. Conformal analysis goes a step further and proposes to learn a set of models, one per each training set that arises from adding the test instance with each possible label (in binary classification one would learn two models). Then, the test instance is classified with the label that led to the model that better conformed to the data.

Conformal analysis is used to enhance variable importance scores as follows: Let \({\mathbf {u}^{r}_{1}},\ldots ,{\mathbf {u}^{r}_{M}}\) be a subset of M testing data selected randomly in the r-th conformal iteration with r = 1,…,R. Now, M independent labellings \({a_{1}^{r}},\ldots , {a_{M}^{r}}\) are generated at random. Label \({a_{i}^{r}}\) is the one generated for sample \({\mathbf {u}_{i}^{r}}\) in the r-th iteration. The correct labels of these test samples are never used along this procedure because they are not accessible. For each of these iterations, we compute the importance measures Ij(r), j = 1,…,P, based on the training data x1,…,xN, the test samples u1,…,uM and the labels \(y_{1}, \ldots , y_{N},{a_{1}^{r}},\ldots , {a_{M}^{r}}\). After running R iterations, we set

$$ I_{j}^{\text{conf}} = \min_{r} I_{j}(r). $$
(3)

For a variable to be important, \(I_{j}^{\text {conf}}\) requires that it is important in all of the R labellings. The underlying intuition is that the importance of variables that yield a high Ij(r) in a few the subsets, but not in all of them, strongly depends on particular labellings. Therefore these variables should not be selected as their importances are not aligned with the labeling that leads to the disease discrimination, but labellings that stress other partitions not relevant for the characterization of the disease.

Hypothesis Test for Selecting Important Variables

The previous subsections have introduced two scorings, Ij and \(I_{j}^{\text {conf}}\), able to assess the relevance of the variables. This subsection presents a hypothesis test to fix qualitative thresholds so that variables with scorings above the threshold can be considered as relevant for the classification and variables with scorings below the threshold can be safely discarded since their importance is reduced. We first present the test for importances Ij and thereafter generalize this test for \(I_{j}^{conf}\). We adopt a probabilistic framework in which the sign of the weight of variable j in the SVM of bagging iteration s, \(\text {sign}({w_{j}^{s}})\), follows a Bernoulli distribution with the unknown parameter pj ∈ (0,1); this indicates that wj > 0 with probability pj. In this framework, an irrelevant variable j is expected to yield positive and negative values in wj with the same probability, thus one would declare variable j as irrelevant if pj = 0.5 with high probability. We formulate this scenario via the following hypothesis test:

$$ \left\{ \begin{array}{ccl} H_{0}: & p_{j}= 0.5, & j\text{ is not relevant} \\ H_{1}: & p_{j} \neq 0.5, & j\text{ is relevant,} \end{array}\right. $$
(4)
which we use to detect relevant variables by rejecting the null hypothesis. We solve the test (4) with a statistic zj:
$$ z_{j} = \frac{\hat{p}_{j}-0.5}{\sqrt{\mathbb{V}\text{ar} \left\lbrace \hat{p}_{j} \right\rbrace}}. $$
(5)
where estimate \(\hat {p}_{j}\) of pj is computed as the sample mean of the observed signs of \({w_{j}^{s}}\): \(\hat {p}_{j} = \frac {1}{S}{\sum }_{s = 1}^{S} \mathcal {I}({w_{j}^{s}} > 0).\) The parameter \(\hat {p}_{j}\) can be considered to follow a binomial distribution rescaled by the factor S and, thus, its variance can be estimated as \(\displaystyle \frac {1}{S}\hat {p}_{j}(1-\hat {p}_{j})\) from the observations. As the observations come from a bagging process, they are correlated and independence cannot be assumed. Therefore, we resource to the following estimator of \(\mathbb {V}\text {ar} \left \lbrace \hat {p}_{j} \right \rbrace \) (Nadeau and Bengio 2003):
$$ \frac{\mathbb{V}\text{ar} \left\lbrace \hat{p}_{j} \right\rbrace} {S}= \left( \frac{1}{S} + \frac{\rho}{1-\rho}\right) \hat{\sigma}_{j}^{2},\qquad \rho<1 $$
where ρ represents the correlation among samples and \(\hat {\sigma }_{j}^{2}\) denotes to the estimator variance if independence could be assumed. Moreover, according to Nadeau and Bengio (2003), since the bagging corresponds to a scenario in which, at each iteration, n1 samples are used for training the SVM and n2 = Nn1 are left out, ρ can be estimated as n1/(n1 + n2). Since the proposed bagging scheme uses n1 = γN training samples in each iteration, we can approximate ρ with γ and, noticing also that S ≫ 1, we get that
$$ \mathbb{V}\text{ar} \left\lbrace \hat{p}_{j} \right\rbrace = S \left( \frac{1}{S} + \frac{\gamma}{1 - \gamma}\right)\hat{\sigma}_{j}^{2} \simeq S \frac{\gamma}{1 - \gamma} \hat{\sigma}_{j}^{2}. $$
With these approximations, the statistic zj of Eq. 5 becomes
$$ z_{j} = \frac{\hat{p}_{j}-0.5}{\sqrt{\frac{\gamma}{1 - \gamma} \hat{p}_{j}(1-\hat{p}_{j})}}. $$
(6)
The statistic zj of Eq. 6 follows a t-student distribution with S − 1 degrees of freedom (Nadeau and Bengio 2003). When S is large enough, as in our case, one can safely approximate the distribution by the standard Gaussian distribution with zero mean and unit variance.

We now generalize the hypothesis test to the importances \(I_{j}^{conf}\) of “Transductive Refinement of Variable Importance”. The selection of \(I_{j}^{\text {conf}}\) as the minimum of the R scorings Ij(r) is equivalent to select as \(\hat {p}_{j}^{\text {conf}}\) the \(\hat {p}_{j,r}\) that lies closest to 0.5. The \(z_{j}^{\text {conf}}\) can be then computed using Eq. 6 and substituting \(\hat {p}_{j}\) by \(\hat {p}_{j}^{\text {conf}}\). An equivalent definition would be to select zj with the smallest absolute value among the R candidates.

Implementation

Algorithms 1 and 2 sketch the implementation of the method to assess variable importance and its version with transductive refinement. In both cases, the ensemble contains a total of S = 10.000 SVMs and each SVM is trained with half of the available training data (γ = 0.5). The SVM regularization parameter C was fixed to 100 as it was observed to be large enough to solve properly these linearly separable problems. A supplement further demonstrates the use of the SCB method with a small toy example.

If any of the training sets presents unbalanced class proportions, the subsampling process at each bagging iteration is used to correct for it. If the transductive refinement is applied, the number of conformal iterations is set to R = 20. For each of these iterations, the number of selected test data, M, has been fixed in such a way that no more than two test data samples is used per each 100 training samples. The hypothesis test described in “Hypothesis Test for Selecting Important Variables” to identify the subset of important variables is applied with a significance level of α = 0.05.

Finally, the overall goodness of the proposed variable importance measure is evaluated by checking the discriminative capabilities of a linear SVM trained using only the important variables. This SVM is also trained with C = 100, since in most cases there are still more variables than samples. However, unlike in the bagging iterations, in this final classifier the class imbalance is solved by using a re-weighting the regularization parameter of the samples of the minority class in the training of the SVM. This way the contribution of the samples of both minority and majority class to the SVM loss function is equalized. This is a standard procedure within SVMs, contained in most SVM implementations (Chang and Lin 2011).

The software implementation of all the methods has been developed in Python4. The SVM training relies on the Scikit-learn package (Pedregosa et al. 2011) which is based on the LIBSVM (Chang and Lin 2011).

Materials

Simulated Data

We generated 10 simulated data sets to evaluate the methods against the known ground-truth and to demonstrate their characteristics with a relatively simple classification task. The datasets contained simulated images of 100 controls and 100 patients and the images had 29852 voxels, similarly to ADNI magnetic resonance imaging (MRI) data in the next subsection.

The simulations were based on the AAL atlas (Tzourio-Mazoyer et al. 2002), downsampled to 4mm3 voxel-size. We selected six regions as important modeling dementia related changes in structural MRI. The voxels of these regions are given in sets Q1,…,Q6 which are left and right Hippocampus (Q1,Q2), Thalamus (Q3,Q4), and Superior Frontal Gyrus (Q5,Q6). We generated the brain scan corresponding to the each subject in a way that each relevant region (Q1,…,Q6) presents correlated voxels (within a class), to make the task of finding them difficult for multivariate variable selection/importance methods. For every subject i, i = 1,…,N, we draw a subject bias bi from a Gaussian distribution with zero mean and variance 0.01. Define Q as the union of the six relevant regions Qk. If i models a patient, its voxel values are generated with the rule:
$$ x_{ij} = \left\{ \begin{array}{ll} \displaystyle\frac{1}{|Q_{k}|} \sum\limits_{j \in Q_{k}} (1 + b_{i} + e_{ij}) + v_{ij}, & ~~\text{if \(j \in Q\)} \\ e_{ij}, & \text{otherwise} \end{array}\right. $$
If i models a control subject, its voxel values are generated with:
$$ x_{ij} = \left\{ \begin{array}{ll} \displaystyle\frac{1}{|Q_{k}|} \sum\limits_{j \in Q_{k}} (b_{i} + e_{ij}) + v_{ij}, & ~~\text{if \(j \in Q\)} \\ e_{ij}, & \text{otherwise} \end{array}\right. $$
In all cases eij and vij were drawn from zero-mean Gaussian distributions with variances 1 and 0.01, respectively. Thereafter, to the relevant voxels, we added white noise with the variance \(\sqrt {2}\) projected to the Bayes-optimal decision hyperplane. This operation maintains the Bayes error rate, but it makes the task of finding important voxels more difficult. Finally, we filtered the images with an isotropic 4-mm FWHM Gaussian kernel to model the smoothness in brain images. The Bayes error for this data was 2.2 %. To evaluate the classification accuracy of the methods, we simulated a large test set with the same parameters as the training set.

ADNI Data

A part of the data used in the preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). For up-to-date information, see http://www.adni-info.org.

We studied the classification between MCI (Mild Cognitive Impairment) and healthy subjects (NCs) using structural MRIs from ADNI. This problem is more challenging than NC vs. Alzheimer’s Disease (AD) classification (Tohka et al. 2016) and therefore offers better insight into the capabilities of different variable importance methods. We did not consider stable vs. progressive MCI classification as the number of MCI subjects is not large enough for the reproducibility analysis performed with this data (see Tohka et al. 2016 for a more detailed discussion).

We used MRIs from 404 MCI subjects and 231 NCs (T1-weighted MP-RAGE sequence at 1.5 Tesla, typically 256 x 256 x 170 voxels with the voxel size of 1 mm x 1 mm x 1.2 mm). The MRIs were preprocessed into gray matter tissue images in the stereotactic space, as described in Gaser et al. (2013) and Moradi et al. (2015), and thereafter they were smoothed with the 8-mm FWHM Gaussian kernel, resampled to 4 mm spatial resolution and masked into 29852 voxels. We age-corrected the data by regressing out the age of the subject on a voxel-by-voxel basis (Moradi et al. 2015). This has been observed to improve the classification accuracy in dementia related tasks (Tohka et al. 2016; Dukart et al. 2011) due to overlapping effects of normal aging and dementia on the brain.

With these data, we studied the reproducibility of the variable importance using split-half resampling (aka 2-fold cross-validation) akin to the analysis in Tohka et al. (2016). We sampled without replacement 100 subjects from each of the two classes, NC and MCI, so that N = 200. This procedure was repeated L = 100 times. We denote the two subject samples (split halves train and test) by Al and Bl for the iteration l = 1,…,L. The sampling was without replacement so that the split-half sets Al and Bl were always disjoint and therefore can be considered as independent train and test sets. The algorithms were trained on the split Al and tested on the split Bl and, vice versa, trained on Bl and tested on Al. All the training operations, including the estimation of regression coefficients for age removal, were done in the training half. The test half was used only for the evaluation of the algorithms. We used 2-fold CV instead of more typical 5 or 10-fold CV as we needed to ensure that the training sets between different folds are independent in order not to overestimate the reproducibility of variable importance scores.

COBRE Data

To demonstrate the applicability of the method for the resting state fMRI analysis, we used the pre-processed version5 of the COBRE sample (Bellec et al. 2015). The dataset, which is a derivative of the COBRE sample found in International Neuroimaging Data-sharing Initiative (INDI)6 includes preprocessed resting-state fMRI for 72 patients diagnosed with schizophrenia (58 males, age range = 18-65 yrs) and 74 healthy controls (51 males, age range = 18-65 yrs). The fMRI dataset features 150 EPI blood-oxygenation level dependent (BOLD) volumes (TR = 2 s, TE = 29 ms, FA = 75 degrees, 32 slices, voxel size = 3x3x4 mm3 , matrix size = 64x64) for each subject.

We processed the data to display voxel-wise estimates of the long range functional connectivity (Guo et al. 2015). It is well documented that a disruption of intrinsic functional connectivity is common in schizophrenia patients, and this disruption depends on connection distance (Wang et al. 2014; Guo et al. 2015). First, the fMRIs were preprocessed using the NeuroImaging Analysis Kit (NIAK7) version 0.12.14 as described at 4. The preprocessing included slice timing correction and motion correction using a rigid-body transform. Thereafter, the median volume of fMRI of each subject was coregistered with the T1-weighted scan of the subject using the Minctracc tool (Collins and Evans 1997). The T1-weighted scan was itself non-linearly transformed to the Montreal Neurological Institute (MNI) template (symmetric ICBM152 template with 40 iterations of non-linear coregistration Fonov et al. 2011). The rigid-body transform, fMRI-to-T1 transform and T1-to-stereotaxic transform were all combined, and the functional volumes were resampled in the MNI space at a 3 mm isotropic resolution. The ”scrubbing” method (Power et al. 2012) was used to remove the volumes with excessive motion (frame displacement greater than 0.5 mm). A minimum number of 60 unscrubbed volumes per run, corresponding to 180 s of acquisition, was required for further analysis. For this reason, 16 controls and 29 schizophrenia patients were rejected from the subsequent analyses, yielding 43 patients and 58 healthy controls to be used in the experiment. The following nuisance parameters were regressed out from the time series at each voxel: slow time drifts (basis of discrete cosines with a 0.01 Hz high-pass cut-off), average signals in conservative masks of the white matter, and the lateral ventricles as well as the first principal components of the six rigid-body motion parameters and their squares (Giove et al. 2009). Finally, the fMRI volumes were spatially smoothed with a 6 mm isotropic Gaussian blurring kernel and the gray matter (GM) voxels were extracted based on a probabilistic atlas (0.5 was used as the GM probability threshold).

Following this preprocessing, we computed the correlations between the time series of GM voxels which were at least 75 mm apart from each other. We use \(N_{j}^{75}\) to denote the set of voxels at least 75 mm apart from the voxel j and \(z(r)_{jj^{\prime }}\) to denote the Fisher transformed correlation coefficient between the voxels j and j. Then, two features \(x_{j}^{-},x_{j}^{+}\) are defined per voxel:

$$ x_{j}^{-} = \sum\limits_{\underset{z(r)_{jj^{\prime}} < 0}{j^{\prime} \in N_{j}^{75}}} - z(r)_{jj^{\prime}}; \quad \quad \quad x_{j}^{+} = \sum\limits_{{j^{\prime} \in N_{j}^{75}; z(r)_{jj^{\prime}} > 0}} z(r)_{jj^{\prime}}. $$
(7)
The long-range connection threshold of 75 mm is rather arbitrary, but it has been previously used to define short and long range connections (e.g., by Guo et al. 2015; Wang et al. 2014). We separated the positive and negative connections as Guo et al. (2015). This preprocessing yielded altogether 81404 variables, corresponding to two times 40702 GM voxels.

Compared Methods

SVM with Permutation Test

The closest approach to SCBs is training a linear SVM and studying the importance of the weights of the primal variables in the SVM by means of a permutation test (Mouro-Miranda et al. 2005; Wang et al. 2007). Here, we use two analytic implementations of this approach. The first one involves a null hypothesis test on a statistic over the j-th SVM primal weight (Gaonkar and Davatzikos 2013),wj. The second one involves a test on its contribution to the SVM margin (Gaonkar et al. 2015). As the number of primal variables (voxels) greatly exceeds the number of dual variables (subjects), we can consider that all the training samples would eventually become support vectors. Therefore, we approximate the SVM primal weight vector by the LS-SVM as Suykens and Vandewalle (1999):

$$ \textbf{w} = B\textbf{y} = X^{T} \left[ \hat{X}^{-1} + \hat{X}^{-1} J \left( -J^{T} \hat{X}^{-1} J \right)^{-1} J^{T} \hat{X}^{-1} \right] \textbf{y}, $$
(8)
where X is the N × P (number of subjects × number of variables) training data matrix, \(\hat {X} = XX^{T}\), y = [y1,…,yN]T is the associated class label vector, and J is a vector of ones. Considering that the permutation test randomly generates different label values with probabilities P {yi = 1} = p1,P {yi = − 1} = 1 − p1 where p1 is the percentage of patient data, we can define the expected value and variance of the labels during permutations as: \( \mathbb {E} \left \lbrace y_{i} \right \rbrace = 2 p_{1} -1\), \( \mathbb {V}\text {ar} \left \lbrace y_{i} \right \rbrace = 4 p_{1} -4 {p_{1}^{2}}\). Now, using Eq. 8, we define the distribution of the null hypothesis under the first statistic (denoted as SVM+perm) as a Gaussian distribution with mean and variance given by:
$$ \mathbb{E} \left\lbrace w_{j} \right\rbrace = \left( 2 p_{1} -1 \right) \sum\limits_{i = 1}^{N} B_{ij} $$
(9)
$$ \mathbb{V}\text{ar} \left\lbrace w_{j} \right\rbrace = \left( 4 p_{1} -4 {p_{1}^{2}} \right) \sum\limits_{i = 1}^{N} B_{ij}^{2} $$
(10)
where B is as in Eq. 8. The second approach, called SVMmar+perm, defines a statistic over the contribution of the weight of the j-th voxel to the SVM margin as:
$$ s_{j} = \frac{\delta w_{j}} {2 \Vert\textbf w\Vert} $$
where δ is the SVM margin. In this case, its null distribution is approximated by a Gaussian distribution with zero mean and variance
$$ \mathbb{V}\text{ar} \left\lbrace s_{j} \right\rbrace = \frac{\mathbb{V}\text{ar} \left\lbrace w_{j} \right\rbrace}{\left[{\sum}_{j^{\prime}}\mathbb{V}\text{ar} \left\lbrace w_{j^{\prime}} \right\rbrace\right]^{2}} $$
(11)

Thus, the test SVM+perm (or SVMmar+perm) will claim that a variable is relevant with a confidence level of α, if the probability that a Gaussian distribution, with mean (9) and variance (10) (or zero mean and variance (11)), generates the value wj (or sj) in the interval \([\frac {\alpha }{2}, 1- \frac {\alpha }{2}]\). For the experiments we will set the confidence level α to 0.05.

T-test and Gaussian Naive Bayes (T-test+NGB)

Although the central part of the discussion is focused on the advantages of SCB over the SVM+perm methodology of the previous sub-section, we demonstrate the advantages of SCB over a typical univariate filter-based variable selection/importance. The most widely used massively univariate approach to variable importance is to apply the t-test to each variable separately. Once these tests are applied, the selection of the variables that will be used during the classification can be performed by determining a suitable α-threshold on the outcome of the tests, and selecting as important variables those that exceed the corresponding threshold. We use the Gaussian Naive Bayes classifier (John and Langley 1995) as the classifier that consumes the variables selected with the t-test filters. As with the other approaches, we set the α-threshold to 0.05, two-sided.

Results

Synthetic data

Table 1 lists the results achieved on the synthetic data. We evaluated:
  • the classification accuracy (ACC) computed using a separate and large test sample;

  • the sensitivity (SEN) of the variable selection defined as the ratio between the number of correctly selected important variables and the number of important variables;

  • the specificity (SPE) of the variable selection defined as the ratio between the number of correctly identified noisy variables and the number of noisy variables;

  • the mean absolute error (MAE) defined as:

    $$ \text{MAE }= \sum\limits_{j \in \mathcal{I}} \hat{\rho}_{j}/|\mathcal{I}| + \sum\limits_{j \in \mathcal{N}} (1 - \hat{\rho}_{j})/|\mathcal{N}|, $$
    (12)
    where \(\hat {\rho }_{j}\) is the estimated p-value for the variable j to be important (the lower the p-value the more important the variable), and \(\mathcal {I}\), \(\mathcal {N}\) are the sets of the important and noise variables, respectively. For the SCB methods, \(\hat {\rho }_{j}\) values were computed based on Eq. 6.
Table 1

Quantitative results with synthetic data

Method

ACC

SEN

SPE

MAE

SCB

0.916 ± 0.004

0.369 ± 0.013

0.889 ± 0.002

0.392 ± 0.004

SCBconf

0.879 ± 0.008

0.208 ± 0.011

0.957 ± 0.002

0.380 ± 0.006

SVM+perm

0.797 ± 0.005

0.076 ± 0.004

0.992 ± 0.001

0.411 ± 0.004

SVMmar+perm

0.891 ± 0.007

0.233 ± 0.011

0.945 ± 0.001

0.411 ± 0.004

t-test+NGB

0.818 ± 0.010

0.259 ± 0.013

0.949 ± 0.002

0.396 ± 0.004

The values shown are averages and standard deviations over 10 different training sets. ACC is the classification accuracy evaluated using a large test set, SEN is the sensitivity of the variable selection, SPE is the specificity of the variable selection, and MAE is the mean absolute error. See the text for details. Variables are selected using the α-threshold of 0.05

The result of the best performing method is highlighted in bold-face

ACC, SEN, SPE measures depend on a categorization of variables into important ones and noise. The categorization, since all the studied methods provide p-values for the variable importance, was determined by a (two-sided) α-threshold of 0.05.

Table 1 shows that the accuracies of SCB methods and SVMmar+perm were better than those of the other methods. Indeed, a hypothesis test comparing the accuracies of the methods (t-test, not to be confused with the hypothesis tests about variable importance) over the 10 different training sets indicated a p-value < 0.001 in every case. The MAEs achieved by the SCB methods compared very favorably to all baseline approaches (the statistical significance evaluated with t-tests in the 10 data partitions provided a p-value < 0.05). Notice that the MAE is independent of the thresholds used to categorize variables as important or not. The specificity (or 1 - SPE) values of the methods are interesting as they can be compared to the nominal α-threshold of 0.05; SCB without conformal analysis was too lenient compared to the nominal threshold while the SCBconf well attained the nominal threshold. SVM+perm was overly conservative while the t-test filter and SVMmar+perm attained well the nominal level. The examples in Fig. 1 visualize the same conclusions. Interestingly, as visible in Fig. 1, there was a tendency for all methods to give a high importance to the same variables. This was as expected with a relatively simple simulation.
Fig. 1

Variable importance Z-scores (absolute values) on a plane cutting through Thalami and Superior Frontal Gyri with synthetic data. The voxels in the areas surrounded by red color were important in the ground-truth and the voxels outside those areas were not

ADNI

With ADNI data, we performed a split-half resampling (2-fold cross-validation) type analysis akin to Tohka et al. (2016). This analysis informs us, in addition to the average performance of the methods, about the variability of variable importances due to different subject samples in the same classification problem.

The quantitative results are listed in Table 2. We recorded the test accuracy (ACC) of each algorithm (the fraction of the correctly classified subjects in the test half) averaged across L = 100 re-sampling iterations. Moreover, we computed the average absolute difference in ACC between the two split-halves, i.e.,

$$ {\Delta} ACC = \frac{1}{L}\sum\limits_{i = 1}^{L} \left|ACC(A_{i},B_{i}) - ACC(B_{i},A_{i})\right|, $$
(13)
where ACC(Ai,Bi) means accuracy when the training set is Ai and the test set is Bi. SCBconf and SCB performed similarly in terms of the classification accuracy and ΔACC. SCB methods were significantly more accurate than t-test+NGB (p-value < 0.05) according to a conservative corrected repeated 2-fold CV t-test (Bouckaert and Frank 2004; Nadeau and Bengio 2003) of the classification accuracy. However, this conservative test did not indicate a significant difference between the accuracy of the SCB methods and SVM+perm and SVMmar+perm. ΔACC was markedly smaller with the SCB based methods than with the three other methods.
Table 2

Quantitative results with the ADNI split-half experiment

 

SCBconf

SCB

SVM+perm

SVMmar+perm

T-test+NGB

ACC

0.769

0.766

0.713

0.745

0.704

ΔACC

0.030

0.029

0.047

0.041

0.045

Nsel

2067 ± 255

4420 ± 420

1884 ± 2286

14686 ± 8838

10253 ± 2278

mHD

1.536 ± 0.105

1.174 ± 0.049

2.952 ± 0.843

1.648 ± 2.736

0.669 ± 0.144

mHDsta

1.536 ± 0.105

1.546 ± 0.111

2.938 ± 3.590

2.719 ± 2.774

1.707 ± 0.705

MAE

0.194 ± 0.006

0.278 ± 0.007

0.267 ± 0.064

0.445 ± 0.051

0.197 ± 0.020

The values listed are the averaged values over 100 resampling runs followed, where reasonable, by their standard deviations. mHD and mHDsta are computed in voxels. ACC is the classification accuracy, ΔACC is the variability of the ACC (13), Nsel is the number of selected voxels, mHD is the modified Hausdorff distance (14), mHDsta is the modified Hausdorff distance when all methods are forced to select the same number of variables, and MAE is the mean absolute error between the variable importance p-values obtained using independent training sets

The result of the best performing method is highlighted in bold-face

The average number of selected voxels (with α-threshold of 0.05) was the smallest with SCBconf and SVM+perm. SCB selected roughly twice as many voxels as SCBconf. The t-test and SVMmar+perm were the most liberal selection methods. When evaluating the standard deviations in the numbers of selected voxels, SCB and SCBconf were the most stable methods. Especially, the number of voxels selected by SVM+perm and SVMmar+perm varied considerably as demonstrated in Fig. 2. We interpret this as a handicap of SVM+perm and SVMmar+perm as the α-threshold was always the same and the variability was not expected. The numbers of selected voxels were more variable with T-test+NGB than with SCB-based methods. According to the MAE measure, SCBconf and t-test were the most reproducible (see Table 2).
Fig. 2

The difference in the numbers of selected voxels between two independent training sets within each split-half resampling run. The SCB methods were more stable with respect to the number of selected voxels than the other methods. Especially, SVM+perm and SVMmar+perm suffered from an excess variability

We quantified the similarity of two voxel sets selected on the split-halves Ai and Bi using modified Hausdorff distance (mHD) (Dubuisson and Jain 1994). This has the advantage of taking into account spatial locations of the voxels. Let each of the voxels a be denoted by its 3-D coordinates (ax,ay,az). Then, the mHD is defined as

$$ H(V_{A},V_{B}) = \max (d(V_{A},V_{B}),d(V_{B},V_{A})), $$
(14)
where
$$ d(V_{A},V_{B}) = \sum\limits_{\mathbf{ a} \in V_{A}} \min_{\mathbf{ b} \in V_{B}} ||\mathbf{ a} - \mathbf{ b}||. $$

Tohka et al. (2016) showed that reproducibility measures of the voxel selection are correlated with the number of selected voxels. To overcome this limitation and make the comparison fair, we here studied standardized sets of voxels by forcing each algorithm to select the same number of voxels as SCBconf in the split half Ai. For each algorithm, we then selected the voxels in the Bi according to the α-threshold obtained for the split-half Ai. The mHD computed using this standardization is denoted by mHDsta in Table 2. As shown in Table 2, the t-test filter was the most reproducible according to the uncorrected mHD. However, this was an artifact of the over-liberality of the test. When standardized with the respect to the number of selected voxels (the row mHDsta), the SCB based methods were most reproducible; however, the difference to the t-test filter was not statistically significant. The SVM+perm and SVMmar+perm were significantly less reproducible than any of the other methods.

Figure 3 shows examples of visualized variable importance maps. All methods displayed, for example, Hippocampus and Amygdala as important as would be expected. An interesting difference can be observed in middle frontal gyrus, where there was a cluster of highly important voxels according to the SCB methods. However, the t-test filter did not consider these voxels as important. Both SCB methods identified several clusters of important voxels, with SCBconf being more conservative. SVM+perm importance appeared to be more scattered and the t-test was the most liberal selecting many more voxels than the other methods.
Fig. 3

Variable importance Z-scores from a randomly selected example run of the ADNI split-half experiment. The Z-scores are thresholded at |Z| > 1.96, corresponding to two-sided alpha threshold of 0.05. Positive Z values indicate positive weights. Axial slices at the z-coordinate of the MNI stereotactic space of 0mm, -10mm -20mm, and -30mm are shown

COBRE

The classification accuracies and numbers of selected voxels with the COBRE data are listed in Table 3. In this experiment, SCBconf was significantly more accurate than the other methods (p-value always < 0.01, according to the corrected resampled t-test Bouckaert and Frank 2004; Nadeau and Bengio 2003). The other methods performed similarly in terms of the cross-validated classification accuracy. This indicates that the conformal analysis was an essential addition to SCB, probably because the COBRE dataset can be assumed to be more heterogeneous than the ADNI dataset. The heterogeneity of COBRE data probably stems from multiple sources. For example, schizophrenia has often been characterized as a heterogeneous disorder (Seaton et al. 2001), the subjects suffering from schizophrenia were receiving various medications at the time of scanning (Kim et al. 2016), the age range of the subjects in the dataset was large, and resting state fMRI is more prone to noise due to, for example, subject motion than anatomical T1-weighted MRI. It is particularly in these kinds of applications where we expect the conformal analysis to be most useful. The classification accuracy achieved with SCBconf appeared to outperform recent published analyses of the same data (Chyzhyk et al. 2015; Kim et al. 2016). However, note that the direct comparison of the classification performance with these works is not fair since it is subject to the differences in variable extraction (different variables were used), data processing (different subjects were excluded) and evaluation (different cross-validation folds were used).
Table 3

Average accuracy and number of selected voxels with the 10-fold CV with the COBRE experiment

 

SCBconf

SCB

SVM+perm

SVMmar+perm

T-test+NGB

ACC

0.952 ± 0.069

0.695 ± 0.154

0.731 ± 0.136

0.692 ± 0.129

0.709 ± 0.170

Nsel

4251 ± 598

11085 ± 588

2433 ± 216

6975 ± 442

26757 ± 3397

The values after ± refer to the standard deviations over 10 CV-folds

The SCBconf selected, on average, 4251 variables and was more conservative than the plain SCB, which selected 11085. SVM+perm selected only 2433 variables on average whereas SVMmar+perm selected 6975. The numbers of variables selected by SVM+perm and SVMmar+perm were less variable than in the ADNI experiment where this variation was clearly a problem for these methods. The t-test was overly liberal. Interestingly, the t-test selected many more variables corresponding to the negative correlation strength (on average 24283) than to the positive correlation strength (on average 2474). Instead, SCB and SVM+perm methods selected similar numbers of variables corresponding to the positive and negative correlation strength. This is also visible in Fig. 4, where the median magnitudes of the variable importances are visualized (medians of absolute value of z-scores, see Eq. 6, over 10 CV runs). Concentrating on the SCBconf, widely distributed and partially overlapping areas were found to be important for both negative and positive correlation strength. Particularly, the most important variables (with medians of absolute z-scores exceeding 15 or equivalent p-values smaller than 10− 51) were found in left cerebellum, left inferior temporal gyrus, left and right thalamus, left inferior parietal gyrus, right inferior frontal gyrus, left medial frontal gyrus, and left middle frontal gyrus for negative correlation strength. For positive correlation strength, median absolute z-scores exceeding 15 were found in left and right cerebellum, left inferior frontal gyrus, left caudate, right lingual gyrus, right middle temporal gyrus and left medial frontal gyrus. We note that a high z value of 15 was selected as threshold in this discussion to concentrate only to the most important variables. We have made the complete maps of variable importance available at NeuroVault service (Gorgolewski et al. 2015) at http://neurovault.org/collections/MOYIOPDI/.
Fig. 4

Median magnitudes of variable importance Z scores among 10 CV runs with COBRE data. The Z-scores are thresholded at |Z| > 1.96. Note that if a variable lights up then it was selected during at least half of the CV runs. ’Pos’ and ’Neg’ quantifiers refer to the strength of the positive and negative connectedness that were separated in the analysis. We do not visualize whether the classifier weights are negative or positive to avoid clutter. Axial slices at the z-coordinate of the MNI stereotactic space of 15mm, 0mm -15mm, and -30mm are shown. Complete maps are available in the NeuroVault service http://neurovault.org/collections/MOYIOPDI/

With the COBRE data, we studied the effect of multiple comparisons correction to the classification accuracy and to the number of selected variables. For multiple comparisons correction, we used variable-wise false discovery rate (FDR) correction with Benjamini-Hochberg procedure (assuming independence) (Benjamini and Hochberg 1995). The classification accuracies and the numbers of selected variables, with and without FDR correction, are shown in box-plots of Fig. 5. SVM+perm and SVMmar+perm were excluded from this experiment as the multiple comparisons problem is different with them (Gaonkar and Davatzikos 2013) and it was found to produce an empty set of variables in some cases. As is shown in Fig. 5, including multiple comparisons correction had no influence to the classification performance with any of the methods.
Fig. 5

The classification accuracy and the number of selected variables across 10 CV folds with COBRE data with and without FDR based multiple comparisons correction. Whether FDR correction is included or not made no difference to the classification performance of the methods

Computation Time

The experiments were run in a computer with Intel Xeon 2.40Ghz processor with 20 cores and 128 Gb of RAM. The training of several SVMs that took place in the bagging stages of both SCB and SCBconf was parallelized across all the cores of the computer. All the other computations (the weight aggregation that leads to the final measure of variable importance, the hypothesis testing and the evaluation of the final SVM) were done using a single core. The baseline methods, SVM+perm, SVMmar+perm, and t-test+ NGB, were run using a single core.

The baseline methods SVM+perm, SVMmar+perm, and t-test+ NGB required between 1 and 5 seconds depending on the size of the dataset (number of samples and dimensionality) and on the number of selected important variables, as this last quantity determines the training time of the final classifier. The computation time of the SCB was in the range 5 to 6 minutes due to the bagging. The computation time was up to 2 hours in the case of the SCBconf, as each conformal analysis iteration involves a complete bagging and we carried out R = 20 of these iterations. Since bagging can be run in parallel, these times could be substantially reduced by further parallelization.

Discussion

We have introduced and evaluated new variable importance measures, termed SCB and SCBconf, based on sign consistency of classifier ensembles. The measures are specially designed for very high dimensional scenarios with far more variables than samples such as in neuroimaging, where many widely used multivariate variable importance measures fail. The SCB variable importance measures extend and generalize ideas for the voxel selection we have introduced earlier (Parrado-Hernández et al. 2014). We have derived a novel parametric hypothesis test that can be used to assign a p-value to the importance of the variable for a classification. We have shown that the variable selection using SCB and SCBconf importance measures leads to a more accurate classification than the variable selections based on a standard massively univariate hypothesis testing, or on two statistical tests based on a parametric SVM permutation test (Gaonkar and Davatzikos 2013; Gaonkar et al. 2015). These three methods were compared to the SCB methods because 1) they are applicable to high-dimensional data and 2) they come with a parametric hypothesis test to assign p-values to variable importance. We have also demonstrated that the proposed SCB and SCBconf measures were robust and that they can lead to better classification accuracies than the state of art in schizophrenia classification based on resting state fMRI.

The basic idea behind the SCB methods is to train several thousand linear SVMs, each with a different subsample of data, and then study the sign-consistency of weights assigned to each variable. These weights having the same sign is a strong indication of the stability of the interpretation of the variable with respect to random subsampling of the data and, thus, a strong indication of the importance of the variable. Therefore, we can quantify the importance of the variable by studying the frequency of sign of the classifier weights assigned to it. While the ideas of random subsampling and random relabeling are widely used for variable importance and selection, for example, in the out-of-bag variable importances of RFs (Breiman 2001), the idea of sign consistency is much less exploited and novel in brain imaging. The reason to study signs of the weights of linear SVMs rather magnitudes of those weights is that the signs of the weights are much less sensitive to the redundancy of the data (multicolinearity). In addition, the signs of the weights have a simple interpretation in terms of the classification task while the weight magnitude is much more difficult to interpret in high dimensions. A positive weight means that the associated variable pushes the classifier towards the positive class while a negative weight means that the associated variable pushes the classifier towards negative class. The conformal variant, SCBconf, refines SCB variable importance by utilizing test data by assigning random labels to it. This is essentially relabeling in the transductive setting and it is especially useful in situations where the data is heterogeneous as we demonstrated using the COBRE resting-state fMRI sample. The reason why the refinement is needed is that most variables in a brain scan contribute to separating that one brain from the other brains in the training set. Expressed in terms of linear classification, the number of linearly independent components among the training input variables exceeds the number of training instances so that any set of binary labels leads to a linearly separable problem, and randomizing a part of the labeling helps in separating truly important variables for the characterization of the disease from the variables that just separate one brain from the rest.

We note that the SCB variable importances are probabilistic, i.e., if the signs of the weights of classifiers from the bagging process are consistent with a high probability, then the associated variable is deemed as important for the classification. The nature of the quantifier ’with a high probability’ is made exact by the associated p-value resulting from the hypothesis test against the null hypothesis of that the sign is inconsistent. We further note that even for the most important variables, two subsets of training data leading to opposite signs for weight of that variable probably exist. However, for the important variable, the signs are the same with high probability.

We have used uncorrected p-values to threshold the variable importance scores. There are two reasons for this. First, the variable importance scores might be interesting also for variables that do not pass stringent multiple comparisons corrected threshold. Second, retaining also variables that are borderline important could improve the generalization performance of the classifier. With the COBRE fMRI dataset, we have shown that ultimately this is a matter of preference and whether using corrected or uncorrected thresholds makes no difference to the generalization performance of the classifier. We also experimented this with synthetic data and observed a slight drop in the classification performance when using the FDR corrected thresholds. As Gaonkar and Davatzikos (2013) noted the classifier weights of an SVM are not independent and thus FDR based multiple comparisons correction probably over-corrects. In a data-rich situation, cross-validation based estimate of the generalization error might be used to select the optimal α-threshold, however, one should keep in mind that cross-validation based error estimates have large variances (Dougherty et al. 2011) and this might offset the potential gains of not setting the importance threshold a-priori (Tohka et al. 2016; Huttunen and Tohka 2015; Varoquaux et al. 2017).

The SCB method has two parameters: the number of resampling iterations S and the subsampling rate γ. In our target applications, where the number of variables is larger than the number of samples, the parameter C for the SVMs can always be selected to be large enough (here C = 100) to ensure full separation. For the parameter S, the larger value is always better and we have found that S = 10.000 has been sufficient. We have selected the subsampling rate to be 0.5 and previously we have found that the method is not sensitive to this parameter; in fact, these parameter settings agree with those previously used by Parrado-Hernández et al. (2014). SCBconf has one extra parameter R (the number of random labelings of the test samples). We have here selected R = 20 and we do not expect gains by increasing this value.

It is important to compare our approach to RFs (Breiman 2001), which also use bagging to derive variable importances from several classifiers. An important difference is the choice of the base classifier used to construct an ensemble. In our case, this is a linear classifier, where each variable receives a different weight in each classifier of the ensemble. These different weights received by a same variable across all the classifiers in the ensemble can be combined in a score that decides the importance of the variable. RFs, instead, use decision trees as the base classifiers. As we have argued in the introduction, the use of linear classifiers is advantageous when the number of variables is in the order of tens of thousands as the number of trees in a forest would have to be extraordinarily large to decide importance for each variable. We also remark that when the number of samples is smaller than the number of variables, the classification tasks are necessarily linearly separable and thus solvable by linear classifiers (Duda et al. 2012). Techniques such as oblique RFs (oRFs) (Menze et al. 2011), offer an interesting half way between RFs based on univariate splits and linear classifiers as in each node the decision is based on a linear classifier with a randomly selected subset of variables (Menze et al. 2011 suggest to use subsets of the size \(\sqrt {P}\)). In the cases where subsets of variables used for training a classifier is larger than the number of training subjects, each tree in an oRF reduces to a single linear classifier because the classification problem in the first node is already linearly separable and thus the subsequent nodes would face data subsets where all the instances belong to the same class. Variable importance in oRFs depend on the choice of an additional tuning parameter to decide if a variable is important in an individual split or not.

Placing a p-value to variable importances in RFs has turned out to be a difficult problem. The early proposal by Breiman and Cutler (2007) suffers from substantial problems identified by Strobl and Zeileis (2008). Later proposals (Hapfelmeier and Ulm 2013; Altmann et al. 2010) rely on permutation approaches requiring training hundreds of RFs even when the number of variables is in the order of tens (Hapfelmeier and Ulm 2013). Consequently, variable selection approaches using RFs rely on different heuristics or cross-validation (Díaz-Uriarte and De Andres 2006; Genuer et al. 2010).

Information Sharing Statement

Only open datasets were used in the paper. The Alzheimer’s Disease Neuroimaging Initiative (ADNI; RRID:SCR_003007) is available at http://www.adni-info.org. The pre-processed form of the COBRE (RRID:SCR_010482) dataset is available at https://figshare.com/articles/COBRE_preprocessed_with_NIAK_0_12_4/1160600. The original COBRE dataset can be found in International Neuroimaging Data-sharing Initiative (INDI)8. The software implementation of all the methods has been developed in Python and a Python notebook containing the implementations is https://github.com/vgverdejo/ResearchActivities/blob/master/Neuroimage/Sign-consistency.ipynb. Statistical maps are available at http://neurovault.org/collections/MOYIOPDI/ (NeuroVault, RRID:SCR_003806).

Footnotes

  1. 1.

    In most scenarios relevant to this work, a single variable corresponds to a single voxel, but this does not have to be the case. We opt to use the general term variable here.

  2. 2.

    We refer to those RFs in which each tree split is defined in terms of a single variable.

  3. 3.

    Transductive algorithms can be contrasted with inductive ones, for example, the k-nearest neighbor algorithms are transductive algorithms (Gammerman et al. 1998). Because the label information of the test data is not used in the learning process, transduction does not lead to upward biased classifier performance estimates.

  4. 4.
  5. 5.
  6. 6.
  7. 7.
  8. 8.

Notes

Acknowledgments

Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics.

J. Tohka’s work was supported by the Academy of Finland (grant no. 316258) and V. Gómez-Verdejo’s work has been partly funded by the Spanish MINECO projects TEC2014-52289R and TEC2016-81900-REDT/AEI.

Compliance with Ethical Standards

Conflict of interests

No conflicts of interest exist for any of the named authors in this study.

Supplementary material

12021_2019_9415_MOESM1_ESM.pdf (1001 kb)
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Authors and Affiliations

  1. 1.Department of Signal Processing and CommunicationsUniversidad Carlos III de MadridLeganésSpain
  2. 2.A.I. Virtanen Institute for Molecular SciencesUniversity of Eastern FinlandKuopioFinland

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