fMRI and SNP data collection

A total of 208 subjects, after signing informed consent, were recruited in the study, including 96 SCZ cases (age: 34 ± 11, 74 males) and 112 healthy controls (age: 32 ± 11, 68 males). Both SNP and fMRI data were collected from each of those 208 subjects. The healthy controls have no history of psychiatric disorders and were free of any medical. SCZ cases met the DSM-IV diagnostic criteria for schizophrenia. After pre-processing, 153594 fMRI voxels and 759075 SNP loci were obtained for the following biomarker selections. Please refer to [22] for detailed description of data collection and pre-processing.

Generalized sparse model

where $y\in R^{n\times 1}$ is the phenotype vector of the subjects; matrix $X_1\in R^{n\times p_1}$ and $X_2\in R^{n\times p_2}$ represent data sets of different modalities having normalized column vectors (e.g., ${\left|\right|*\left|\right|}_2=1$); $X=\left[{\alpha }_1{X}_1,{\alpha }_2{X}_2\right]\in R^{n\times p}$; ${\alpha }_1+{\alpha }_2=1$, and ${\alpha }_1,{\alpha }_2>0$ are the weight factors for $X_1$ and $X_2$ respectively. The measurement error $\epsilon \in R^{n\times 1}$. We aim to reconstruct the unknown sparse vector $\delta =\left[\begin{array}{c}\hfill {\delta }_1\hfill \\ \hfill {\delta }_2\hfill \end{array}\right]\in R^{p\times 1}$ based on $y$ and $X$, where ${\delta }_1\in R^{p_1\times 1}$, ${\delta }_2\in R^{p_2\times 1}$, and $p_1+p_2=p$.

It can be proven that when $p>35n$, the matrix $X\in R^{n\times \mathsf{\text{p}}}$ has the difficulty to satisfy the restricted isometry property (RIP) condition [24] for signal recovery. In this work, $p=759075+153594=912669$ and $n=208$. Thus $p\gg 35n=7280$. To overcome this problem, we propose the SRVS algorithm described as follows.

SRVS algorithm

where ${\left|\right|*\left|\right|}_2$ represents $L_p$ norm; $p\in \left[0,1\right]$. The SRVS Algorithm given below is used to solve the $L_p$ minimization problem and select the phenotype relevant column vectors out of $X$.

Sparsity control using $k$

In Step 2 of the SRVS Algorithm, we exploit Fisher-Yates Shuffling algorithm [31] with a window of length $k$ to select $X_l\in R^{n\times k}$ from $X\in R^{n\times \mathsf{\text{p}}}$. The length $k$ determines the resolution of the SRVS algorithm. When $k=p$, the number of variables selected will be generally equal to the sample number $n$[23]. The smaller the $k$, the more the variables selected, and those variables generally include the variables selected with bigger $k$, as shown in Figure 1. This multi-resolution property enables us to select different number of variables at different significance levels.

Further sparsity control using $\epsilon$

The parameter $\epsilon$ given in Eq. (2) can be used for further sparsity control. The magnitudes of entries of $\delta$ reflect the significance of the corresponding columns of $X$[21]. Thus, a threshold can be selected for $\delta$ using cross-validation [32]. Another way to determine a threshold is using the error term $\epsilon$ (as shown in Figure 2), which reflects the residual of $\left|\right|y-X\delta \left|\right|$[20]. When $\epsilon =0$, noises may be involved in the columns selected [20]. In this study, we set $\epsilon =\tau {\left|\right|y\left|\right|}_2$. From Figure 2, we show that if the first 400 variables with amplitudes larger than 0.002 are selected (i.e. points (400, 0.002) on 'Regression coefficients' curve), it corresponds to the point (400, 0.4) on the 'Error term coefficient' curve; it indicates that with these 400 variables, the error term $\epsilon =0.4{\left|\right|y\left|\right|}_2$.

Validation

To validate the variable selected using our proposed SRVS algorithm, we compared our selected SNPs and fMRI voxels with that of previous studies. In addition, we used the selected SNPs and fMRI voxels to identify SCZ patients from healthy controls with the sparse representation based classifier (SRC) [15][18]. Then a leave one out (LOO) cross-validation approach was carried out to evaluate the identification accuracy. We compared the classification results with that of Li et al.'s method [21]. Furthermore, we compared the results of using variables selected from one type of data and that of both types of data. We also studied the influences of selecting different number of variables.

Variable selection with different weight factors

Sparse model given by Eq. (1) with different weight factors were solved by our proposed SRVS method and by Li et al.'s method, respectively, as shown in Figure 3. It can be seen that at the two ends (${\alpha }_1=0.3\mathsf{\text{or}}0.6$), the variables were selected form one type of data.

In each of the 16 trials given by Figure 3, we selected the top 200 biomarkers by our proposed SRVS method and by Li et al.'s method [21]. As shown in Figure 3, the weight factor has similar effects on the variable selection of the two methods. It was interesting to see that even though the number of SNPs was much larger than that of fMRI voxels (759075 vs. 153594), similar number of variables were selected from both data sets when weight factor ${\alpha }_1$ for SNP data set was around 0.5 (0.46 for SRVS method with $L_0$ norms, and 0.47 for Li et al.'s method). This suggests that the two data sets may contain similar information for the SCZ case/control study.

Comparison with Li et al.'s method

We selected 200 variables (SNPs and fMRI voxels) in each trial by our proposed SRVS method and by Li's et al.'s method respectively, as shown in Figure 3. However, further study showed that the variables selected by the two methods were significantly different (overlap <10%) (see Figure 4). Thus it was necessary to validate and compare those different groups of variables selected. We first compared the selected SNPs and the corresponding genes with the publicly reported SCZ genes for both methods. Then we compared the brain regions identified using those two methods. In addition, we compared the classification accuracies using the variables selected by our proposed SRVS method and Li et al.'s method.

To further compare the two methods at different sparsity level, we studied more top variables in each of the 16 trials. To reach this purpose, we set $\epsilon =0.3y_2$ and $k=0.05$ for SRVS method. For Li's method [21], the number of subjects selected in each run was one tenth of total number of subjects; and we set the threshold $\theta$ = 0.01(please refer to [21] for the meaning of $\theta$). As a consequence, 500 to 800 variables (SNPs and fMRI voxels) were selected in each trial. In this case, our proposed method selected 20 reported genes. For Li et al.'s method, 14 reported genes were located, and 11 of the top 45 genes were identified by both methods [22]. However, the genes identified by the two methods have <10% overlaps. For the top 50 genes selected by the two methods, there was only one gene, CSMD1, was identified by both methods. We listed the top 50 genes and the corresponding SNPs chosen by the two methods in Additional file 2.

When comparing the fMRI voxels selected (follow the approach shown in Figure 3), we showed that the SRVS method were capable of selecting fMRI voxels that were clustered in specific regions, as shown in Figure 5 (a). Those voxels located within a same region will have high correlations with each other. Therefore the results indicate the capability of our proposed SRVS method in selecting significant biomarkers that are highly correlated. Further study showed that the brains regions selected by our proposed SRVS method were mostly reported being associated with SCZ [33–35], including temporal lobe, lateral frontal lobe, occipital lobe, and motor cortex (see Table 2). However, Li et al.'s method tended to select voxels that were scattered over different brain regions (see Figure 5 (b)). Besides, the brain regions selected by those two methods were largely different from each other. Thus we used multivariate classification approach to evaluate the effectiveness of the variables selected by two methods.

Multivariate classification

In this study, a LOO cross validation was carried out to evaluate the classification accuracy. In each run of the LOO validation, one sample was used for testing while the rest ones were used for variable selection. Results were presented in Figure 6. We showed that our proposed SRVS algorithm provided significantly higher classification ratios (CRs) ($p-\mathsf{\text{value}}<{\mathsf{\text{1}}}^{\mathsf{\text{e}}-\mathsf{\text{11}}}$) for both the 200-selected-variable case and the 800-selected-variable case. However, using different number of top selected variables showed no significant differences for neither of the two methods (p-value > 0.1).

From Figure 6 (a) we showed that the highest classification accuracy was achieved at the weight factor ${\alpha }_1=0.5$, where around equal sized SNPs and fMRI voxels were selected by the SRVS method. At the two ends (${\alpha }_1=0.3\mathsf{\text{or}}0.6$), the classification accuracies were relatively lower. This suggests that using biomarkers from both types of data may lead to better identification accuracy.